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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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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.
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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.
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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.
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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
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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
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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
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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
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List of symbols
Vm instantaneous mold velocity, mm/min;
^average 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;
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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;
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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;
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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.
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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]
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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],
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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]
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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
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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]:
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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].
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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
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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.
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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
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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.
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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).
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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
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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].
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2 7
time
meniscus
Figure 1.8 Takeuchi and Brimamacombe model o f OM
formation with subsurface hook [22]
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meniscus
Figure 1.9 Takeuchi and Brimacombe model o f OM
formation without subsurface hook [22]
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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
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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.
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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)
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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
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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
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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].
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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]
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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]
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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.
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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).
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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.
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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)
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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
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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.
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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
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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.
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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.
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229696895952492690978251144225999^8356^6452693918057514928615642299195
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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,
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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):
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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;
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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.
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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;
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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)
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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:
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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
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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
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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
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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)
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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).
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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:
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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 -sin^o)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:
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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 )
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61
3.99
8.092 0 2 4 6 -2 0 2 4 *
b ) d )
- 0.1
I.W3.99
8.096
e) f)
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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.
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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.
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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
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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)
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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
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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)
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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:
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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)
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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
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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)
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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),
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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:
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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:
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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
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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;
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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):
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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:
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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;
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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)
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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)
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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)
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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
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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
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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.
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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
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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.
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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.
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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
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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.
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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 ;
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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.
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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
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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)
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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
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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
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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.
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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.
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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.
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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.
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101
Bibliography
1. Irving, W.R. ’’Continuous casting o f steel”, The Institute o f Materials, London 1993
2. J. Savage, W.H. Pritchard, “The problem o f Rupture o f the Billet in the Continuous
Casting O f Steel”, the Journal o f the Iron and Steel Institute, Vol. 178, November 1954,
pp.269-277.
3. J. Elfsberg, “Oscillation M ark Formation in Continuous Casting Process”, Casting o f
Metals, Royal Sweden Institute o f Technology, 2003.
4. B.G. Thomas, “Continuous Casting: Modeling,” The Encyclopedia o f Advanced
Materials, (J.Dantzig, A. Greenwell, J. Michalczyk, eds.) Pergamon Elsevier Science
Ltd., Oxford, UK, Vol.2, 2001, 8p., (Revision 3, Oct. 12, 1999).
5. McGraw-Hill Dictionary o f Physics, Sybil P. Parker (Editor in Chief), New-York,
1985.
6. Riboud P. V., Larrecq M., Steelmaking Conference Proceedings, Vol. 74, Washington
D.C., USA, 1991, pp. 78-82.
7. Mills K. C., Steel Technology International, 1994.
8. Abratis H., Hofer F., Jtinemann M., Sardemann J., Stoffel H., Stahl und Eisen 116
(1996), Nr. 4, pp. 85-91.
9. Pinheiro C. A., Samarasekara I.V., Brimacombe J.K., Iron and Steelmaker, October
1994, pp.55-56
10. Branion R.V., Mold Powders for Continuous Casting and Bottom Pour Teeming, Iron
and Steel Society, AIME, pp. 3-14.
11. Me Cauley W. L., Apelian D., Iron and Steelmaker, August 1983.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
12. Gray R., Marston H., Steelmaking Conference Proceedings, Vol. 74, W ashington
D.C., USA, 14-17 Apr 1991, pp. 93-102.
13. Szekeres E. S., “Overview o f mold oscillation in continuous casting”, Iron and steel
Engineer, July 1996, pp. 29-37.
14. W olf M. M., M old Powders for Continuous Casting and Bottom Pour Teeming, Iron
and Steel Society, AIME, pp. 33-44.
15. N.A. McPherson and R.E. Mercer. Ironmaking and Steelmaking. 1980, vol. 67,
pp. 167-79.
16. T. Okazaki, H. Tomono, K. Ozaki, and Y. Akabane: Tetsu-to-Hagane, 1982, vol. 68,
p.S929.
17. H. Oka, Y. Eda, T. Koshikawa, H. Nakato, T. Nozaki, and Y. Habu: Tetsu-to-Hagane,
1983, vol. 69, p. S I032.
18. T. Kuwano, N. Shigematsu, F. Hoshi, and H. Ogiwara: Ironmaking and Steelmaking,
1983, vol.10, pp.75-81.
19. T. Emi, H. Nakato, Y. Iida, K. Emoto, R. Tachibana, T. Imai, and H. Bada: Proc. 61
st NOH-BOSC, 1978, pp.350-61.
20. Kawakami, T. Kitagawa, H. Mizukami, H. Uchibori, S.Miyahara, M. Suzuki, and Y.
Shiratani: Tetsu-to- Hagane, 1981, vol.67, pp.l 190-99.
21. N.A. McPherson, A.W. Hardie, and G. Patric: ISS Transactions, 1983, vol.3, pp.21-
36.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
103
22. Takeuchi E., Brimacombe J.K., “The Formation o f Oscillation Marks in the
Continuous Casting o f Steel Slabs”, Metallurgical Transactions B, Vol. 15 B, September
1984, pp. 493-509.
23. Waters, B.H.C., “Continuous casting o f Non-Ferrous Metals, Parts V and VI”, Metal
Treatment & Drop Forging, Vol.20, 1953, p.79 and 103.
24. Thornton, D. R., “ An Investigation on the Function o f Ingot Mold Dressings”,
Journal o f the Iron and Steel Institute, vol. 7, July 1956, p.300-315.
25. Savage J., “A New Reciprocation M old Cycle to Improve Surface Quality o f
Continuously Cast Steel”, Iron and Coal Trades Review, Vol. 182, 4, April 14, 1961,
p.787-795.
26. Sato R.,” Powder Fluxes for Ingot Making or Continuous Casting”, Bull. Jap.
Institute o f Metals, Vol. 12, 1973 (No. 6), p. 391.
27. Sato R., “Powder Fluxes for Ingot Making and Continuous Casting”, 62nd National
Open Hearth and Basic Oxygen Steel Conference proceedings, Vol. 62, Detroit,
Michigan, 25-28 Mar, American Institute o f Mining, Metallurgical, and Petroleum
Engineers, Vol.62, 1979, pp. 48-67.
28. Davies I.G., and Sharp R.M., “The Formation o f Reciprocation Marks During
Continuous Casting o f Slabs”, British Steel Corporation Technical Note No.
PDN/433/77, 1977.
29. Saucedo I.G., Beech J., Davies G. J., “The Development o f Surface in Steel Ingots
and Strands”, Proceedings o f 6th International Vacuum Metallurgy Conference on Special
Melting: San Diego, California, April 23-27, 1979, pp. 885-904.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
104
30. Saucedo I. G., “Early Solidification During the Continuous Casting o f Steel”, 74th
Steelmaking Conference Proceedings, Vol. 74, Washington D.C., USA, 14-17 Apr 1991,
pp. 79-89.
31. Tomono H., Ackermann P., Kurz W., Heinemann W., “Surface mark formation in
continuous casting o f steel”, Proceedings o f a Symposium: Continuous Casting o f Small
Cross Sections, Pittsburgh, Pa., 8 Oct 1980, Metallurgical Society o f AIME, pp. 55-73.
32. Takeuchi E., Brimacombe J.K.,’’Effect o f Oscillation-Mark Formation on the Surface
Quality o f Continuously Cast Steel Slabs”, Metallurgical Transactions B, Vol. 16B,
September 1985, pp. 605-625
33. Samarasekera I.V., Brimacombe J.K., Bommaraju R., “Mold Behaviour and
Solidification in the Continuous Casting o f Steel Billets”, ISS Transactions, Vol. 5, 1984,
pp.79-94.
34. Suzuki T., Miyata Y., Kunieda T., J. Japan Inst. Metals, Vol. 50, No.2 (1986),
pp.208-214
35. Delhalle A., Larrecq M., Petegnief J., Radot J.P., La Revue de Metallurgie -C IT ,
June 1989, pp.483-489.
36. Lainez E., Busturia J. C., “The E.L.V. Solidification Model in Continuous Casting
Billet Molds Using Casting Powder”, 1st European Conference on Continuous Casting,
Florence, Italy September23-25, 1991, pp. 1.621-1.631.
37. J. Sengupta, H.J. Shin, B.G Thomas, S.H. Kim, “M icrograph evidence o f meniscus
solidification and sub-surface microstructure evolution in continuous cast ultralow-
carbon steels”, Acta Materialia, 54(2006),pp. 1165-1173.
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105
38. M ikloukhine S, Tsukerman V., “The mechanism o f the slag renewing during
continuous casting with low extraction speed”, proceeding o f a 9th Conference: Problems
o f ingots casting, Volgograd, USSR, 1990.
39. Joonpyo Park, Heetae Jeong, Hoyoung Kim and Jongkeun Kim, “Laboratory Scale
Continuous Casting o f Steel Billet with High Frequency Magnetic Field”, ISIJ
International, Vol. 42 (2002), No. 4, pp. 385-391.
40. Rajil Saraswat, A. B. Fox, K. C. Mills, P. D. Lee and B. Deo, “The factors affecting
powder consumption o f mold fluxes”, Scandinavian Journal o f M etallurgy Vol. 33, 2004,
pp. 85-91
41. Hermann Schlighting, Boundary layer theory. New York, McGrow-Hill Book
Company, 1968, p.271.
42. Samoylovich Y.A., Krulevitskiy S.A., Goriainov V.A., Kabakov Z.K. Heat processes
in continuous casting o f steel. Moscow, Metallurgy, 1982, p. 152.
43. D. Hudson, Statistic for physics. Moscow, World, 1970.
44. W. T. Eadie, D. Dryard, F.E. James, M. Roos, B. Sadoulet, “Statistical methods in
experimental physics” . Nort-Holland Publishing Company - Amsterdam-London, 1971
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106
Appendix A
Thermophysical model of meniscus for 1st stage formation
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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
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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
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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 “
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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
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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
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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.)
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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
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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
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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;
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1500
UU0 i!480
1460
0 0.1 0.2
xLj
1500
1460
8.5 9 9.5 10
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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) + Psl^i 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
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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 . := u^O, 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
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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
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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 ^u := 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
u^O .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
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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 » ^u ,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
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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
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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 )
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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
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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 ^u ,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
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126
u(Lm,t) = Tm
u := Pdesolve 0 W o ^u,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
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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
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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
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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 )
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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
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131
Appendix B
Adequacy test
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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 )
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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
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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)
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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.
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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
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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
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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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