ISIJ International, Vol. 48 (2008), No. 2, pp. 161–169
A New Semi-analytical Model for Prediction of the Strand
Surface Temperature in the Continuous Casting of Steel in the
Mold Region
Mostafa ALIZADEH,1) Ahmad Jenabali JAHROMI1) and Omid ABOUALI2)
1) Department of Materials Science, Shiraz University, Shiraz, Iran. E-mail: [email protected], [email protected]
2) Department of Mechanical Engineering, Shiraz University, Shiraz, Iran. E-mail: [email protected]
(Received on August 27, 2007; accepted on December 17, 2007 )
In this research, a new semi-analytical model is presented for the strand surface temperature in the continuous casting of steel. Firstly with a dimensional analysis approach a general relation between the strand
surface temperature and the other effective variables such as conductivity of the steel, pouring temperature, casting velocity, distance from the meniscus, volume rate of cooling water, solidus temperature and
heat flux density at the meniscus is deduced. The constants appeared as coefficients or powers of variables
in presented relation were computed by a numerical simulation of continuous casting for a breakout bloom.
The resulted semi-analytical model for strand surface temperature was extended to predict the solidified
shell thickness. The resulted semi-analytical model was validated with comparison to experimental, analytical and numerical results of slab, billet and bloom and good agreement was seen. The new presented
model at this research is in versus of controllable parameters and specifications of the mold and it can be
used for design of a process control system for continuous casting of slab, billet and bloom.
KEY WORDS: continuous casting; analytical modeling; numerical modeling; strand surface temperature;
shell thickness.
1.
used to predict the shell thickness and time relationship.13)
Introduction
dshellKt N ...................................(1)
There are some analytical methods which are still being
used as the standard references to validate numerical models for continuous casting.1–3) For example Meng et al.2) and
Park et al.3) presented an numerical approach and compared
their results for the solidified shell thickness of slab and billet with analytical models. Moreover in real time modeling
of the continuous casting an analytical model is more applicable.4) Detecting and monitoring of the continuous casting
mold process certainly plays an important role to the technology of producing non-defect strand. The visualization of
the continuous casting mold process, which should include
visualization of temperature, heat flux, liquid steel solidification, strand stress etc., is the basis of an advanced control.5–7) Having a reliable analytical model can simplify this
advanced control.8) It seems the surface temperature of the
strand in the mold plays an important role in the analytical
modeling.
Many investigators calculated the heat transfer across the
interfacial gap as a function of the surface temperatures of
the steel shell and the hot face of the mold wall.9–11) Some
researchers applied the surface temperature of the strand in
solidification cracking studies.12)
All of the measurement methods of the solidified shell
thickness (breakout measurement, sensing method, radioactive tracer and chemical tracer) show that the Eq. (1) can be
The exponent N varies between 0.5–0.9 for low-carbon
steels. Table 1 lists some of the results for Eq. (1).1,13,14)
Using the heat flow theory, Neumann found that N0.5 and
K factor is in the range of 0.09–0.23 for low carbon steels.
Neumann’s method and the heat balance integral method
are the main analytical models for prediction of the solidified shell thickness.1) Equations (2), (3) and (4) are results
of the Neumann’s solution.1) Vogt and Wuennenberg preTable 1. Some measurements and results about shell thickness.
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ISIJ International, Vol. 48 (2008), No. 2
sented Eq. (5) for solidified shell thickness based on the
heat balance integral method.13)
d shell K
π ∆H f
d shell ( z ) z
..............................(2)
VC
k st
.............................(3)
ρstCst
K 2φ
0
Cst (Tsol Tsur
)
mathematical formulation of the heat transfer is based on
the fundamental equation of heat conduction. Because of
the high heat extraction rate at the strand surface, the axial
conduction (z) of the heat is negligible compared to the heat
transferred by the bulk motion of the strand.15) Therefore,
heat conduction equation reduces to the following form16):
ρ(T )C (T )
φ exp(φ 2 )erf (φ ) ...............(4)
∂T
∂
∂t
∂x

∂T 
∂ 
∂T 
 k (T ) ∂x  ∂y  k (T ) ∂y  q˙




...........................................(6)
The latent heat source (q˙) is expressed by the following
equation:
k st ∆ H f
Cst q( z )
q˙ ρ(T )∆ H f
2
 k ∆ Hf 
2k st
 st

ρstCstVC q( z )
 Cst q( z ) 
∫
Z
∂f s
...........................(7)
∂t
Equation (6) can be simplified with substitution of following formulation:
q( z )dz
0
∂ρ(T ) H (T )
∂T
∂f
ρ(T )C (T )
ρ(T )∆ H f s .....(8)
∂t
∂t
∂t
...........................................(5)
These analytical models have some limitations which reduce their application. For example Eq. (2) is based on con0
stant surface temperature (T sur
) and the effect of the super
heat in the melt was not considered. Also determination of
the K factor needs a complex mathematic method, or experimental correlation such as listed equations in Table 1.
Moreover the variable parameters in the listed equations in
the Table 1 are only the carbon content and a special section of the strand (slab or billet). Equation (5) is derived for
the pure iron solidifying with a planar solidification front;
consequently it is applicable only for low carbon steels.
In this work a semi-analytical model without above limitations is presented. The resulted semi-analytical model for
strand surface temperature is extended for prediction of solidified shell thickness. Also the liquid flow with superheat
is treated by the heat conduction method. This method simply increases the conductivity of the liquid to approximate
the effect of fluid flow.
Which the enthalpy of the material is computed as the sum
of the sensible enthalpy, h, and the latent heat, DHf :

H (T )  href 


C (T )dT  f s ∆ H f ...........(9)

T ref

∫
T
The solid fraction is computed by the linear model17):

1

 T Tliq
fs 
 Tsol Tliq

0

T Tsol
Tsol T Tliq ..............(10)
Tliq T
For metallic alloys at the range of temperatures where solidification occurs, the physical properties will be evaluated
using following equations18):
k(kSkl) · fskl ...........................(11)
2.
Numerical Simulation of Solidification
C(CSCl) · fsCl ..........................(12)
The solidification of steel in the mold region in the continuous casting process is controlled by three types of parameters:
1. Steel parameters such as chemical composition, physical properties (conductivity, density and specific heat),
solidus temperature and liquidus temperature.
2. Machine parameters such as mold dimensions, mold
taper and mold material.
3. Operational parameters such as mold level, volume
flow rate of cooling water, type of the mold powder,
casting speed, pouring temperature and mold oscillation conditions.
Determination of the effects of these parameters on the
strand surface temperature by the experimental method is
very difficult and expensive. Therefore in this work, for this
aim, a numerical simulation for solidification of steel in the
mold region is applied.
In this study, the numerical modeling of the strand has
been developed to track a transverse slice of a steel bloom
as it moves down through the mold with casting speed. The
© 2008 ISIJ
r (r Sr l) · fsr l ..........................(13)
Usually to consider of the liquid flow effect on the thermal
field, an effective thermal conductivity is employed in the
liquid zone.12,17–19) the effective thermal conductivity depend on the some operational parameters such as electromagnetic mold stirrer conditions, geometrical shape and
size of submerged entry nozzle and the amount of superheat. In the present semi-analytical model the simple suggestion of Louhenkilpi et al.19) was used which is an increase of two times for the conductivity of the liquid for
bloom with electromagnetic stirrer.
• Boundary Conditions
Figure 1 shows the Schematic diagram of the computational domain. Two-dimensional heat transfer phenomenon
was considered, with heat flux being admitted to be negligible along the vertical direction z, i.e. ∂T/∂Z)0. The bloom
symmetry permits that only one quarter of the transverse
slice would be modeled for a full thermal evolution characterization. It is supposed that the transverse slice moves in
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ISIJ International, Vol. 48 (2008), No. 2
Fig. 1. The Schematic diagram of coordinate and calculation domain for solidification simulation of bloom with listed
condition in Table 2.
the direction of z from the meniscus (z0) to the end of the
mold (zHme) with velocity of VC, therefore the local heat
flux density is necessary as boundary condition. Equations
(14), (15), (16) and (17) depict the boundary conditions of
the cross-section shown in Fig. 1.18)
at y 0 and 0 x xs k (T )
∂T
∂y
Fig. 2. Schematically a vertical section of the mold showing the
mold length, meniscus and direction of z.
0 ......(14)
∂T
0 ......(15)
∂x
∂T
k (T )
q .....(16)
∂y
at x 0 and 0 y ys k (T )
at y ys and 0 x xs
∂T
q .....(17)
∂x
TTpouring ........................(18)
at x xs and 0 y ys k (T )
at
t0
The local heat flux density in Eqs. (16) and (17) is calculated by the Schwerdtfeger’s law20) as follows:
Fig. 3. The vertical breakout shell section. This breakout has
happened in the fix sector area. This figure indicates the
meniscus and the shell thickness along of the mold and
below of the mold.
q(z)qaea z ...............................(19)
The exponent a is the slope of the straight lines in q–z halflogarithmic plot and its value is almost 1.5 m1.20) qa is the
apparent local heat flux density at the meniscus. In this
work, qa is evaluated using the heat balance formulation.
Figure 2 shows schematically a vertical section of the mold
in which the meniscus is in z0, the end of the mold is in
zHme and the top of the mold is in zHmeHm. The cooling water enters to the water jacket from the bottom of the
mold at the volume rate of Q and at the temperature of T 0w.
The cooling water absorbs the heat flux from the mold and
exits from the top of the mold at temperature of T 0wDT Tw.
Therefore it can be written:
at
at
zHme
z0
temperature differential at the mold length as follows:
dT qa Pm
eα z dz .....................(23)
ρ wC w Q
Integration of Eq. (23) from zHme to z0 (see Eqs. (20)
and (21)), leads to the local heat flux density at the meniscus, as follows:
qa α∆TwT ρ wC w Q
1
.............(24)
Pm
1 eα H me
Substituting Eq. (24) into Eq. (19) leads to the equation of
the local heat flux density as follows:
TwT 0w .....................(20)
TwT 0wDT Tw..................(21)
q( z )
The local heat flux density absorbed by the cooling water in
the differential control surface is as follows:
eα z
α∆TwT ρ wC w Q
............(25)
Pm
1 eα H me
This work has focused on the solidification modeling of a
bloom with breakout problem. Figure 3 depicts the vertical
breakout section of this bloom. Table 2 lists the practical
conditions and the mold characteristics of solidification of
the bloom in a steel making factory. The measured thickness of the solidified shell in the breakout section would be
ρ C QdT
q( z ) w w
.........................(22)
dz Pm
Where dzPm is the partial surface area of the total mold
surface area. Equality of Eqs. (19) and (22) leads to the
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ISIJ International, Vol. 48 (2008), No. 2
Table 2. The practical conditions for numerical and analytical simulation of bloom, slab and billet.
compared with the calculated thickness.
3.
Strand Surface Temperature Modeling by Dimensional Analysis Method
The dimensional analysis is a method for grouping of the
variables which are effective on a physical phenomenon
such as solidification. Using this method, the number of
variables is reduced to some dimensionless variables. The
number of variables is an important factor in experimental
evolution of a physical phenomenon. By reducing the number of variables, the numbers of experimental and numerical tries to evaluate the physical phenomenon will reduce.21)
With using this method in this research the effective variables on the strand surface temperature are grouped in
some non-dimensional numbers.
The effective parameters on the strand surface temperature are: conductivity of the solid steel (kst), pouring temperature (Tpour), casting velocity (VC), distance from the
meniscus (Z), volume flow rate of cooling water (Q),
solidus temperature (Tsol) and local heat flux density at the
meniscus (qa). So the following relation can be written:
Tsurf(Tpour, Tsol, VC, Z, Q, kst, qa) ...............(26)
The number of the variables in the Eq. (26) is equal to
eight including Tsur. Figures 4(a) and 4(b) show the results
of the numerical simulation for variation of surface temperature with pouring and solidus temperature. According
to curve fitted equations resulted by numerical simulation,
to a good approximation, the strand surface temperature
0.65
is proportional to T 1.0
(the exact value of expour and Tsol
ponents are 0.918 and 0.645 respectively). Therefore the
number of the parameters can be reduced to six
(Tsur /(TpourTsol0.65), VC, Z, Q, qa, kst) in Eq. (26) and it can be
summarized as the Eq. (27):
© 2008 ISIJ
Fig. 4. Effect of the pouring and solidus temperatures on the
strand surface temperature obtained from the present numerical prediction for bloom.
Tsur
0.65
TpourTsol
164
f (VC , Z , Q , qa , k st ) ...............(27)
ISIJ International, Vol. 48 (2008), No. 2
Fig. 6. Comparison of predicted strand surface temperature by
present analytical model (Eq. (32)) and numerical prediction (Ref. 2)) for slab.
Fig. 5. Variation of the p 1 with p 2, where p 1 varies with z and
Tsur , p 2 varies with z.
With a dimensional analysis approach21) it can be shown
that all six variables can be grouped in two non-dimensional numbers as following:
π1 Tsur
0.65
TpourTsol
π2
 Zqa 
 k 
 st 
This new semi-analytical correlation (Eq. (32)) is controlled by the steel parameters such as conductivity and
solidus temperature which both are representative of chemical composition. Also this model is controlled by the machine parameters such as mold dimension (in qa) and also it
is controlled by the casting parameters such as the casting
velocity and the pouring temperature and the mold level.
Moreover the local heat flux density at the meniscus (qa) is
depended to DT Tw which is affected by chemical composition and the mold powder. The main limitation of the present semi-analytical model is the simple modeling of the
liquid flow effect on thermal filed by a heat conduction
method.
0.65
....................(28)
VC Z 2
...............................(29)
Q
To identify the strand surface temperature formula, it is
necessary determine the function of Eq. (30):
p 1f(p 2) .................................(30)
This was done in this research by a numerical simulation.
Figure 5 depicts the variation of the p 1 with p 2. The regression investigation of the results (Fig. 5) shows that the
best equation fitted on the curves has a form via Eq. (31):
4.
4.1. Strand Surface Temperature
The predicted strand surface temperature by the present
semi-analytical model (Eq. (32)) was validated by comparing with two numerical results presented by Meng and
Thomas2) for slab and present numerical simulation for
bloom. Meng and Thomas2) simulated solidification of a
slab with numerical solution of the one-dimensional heat
transfer equation at a steady state condition. The conditions
of their work are listed in Table 2. They used the heat transfer resistor model between the mold wall and the shell surface (mold flux layer and gas gap) as a boundary condition.
Figure 6 compares the numerically simulated results of
Meng and Thomas,2) with the results of the present semianalytical model (Eq. (32)). The necessary parameters for
solution of Eq. (32) are set similar to Meng and Thomas’s2)
work (Table 2). Figure 6 shows that both predictions are
reasonably in agreement. However the existence of some
approximations in both models and also approximation of
heat flux by liquid flow in the present semi-analytical model
caused a small difference between them. In the present
semi-analytical model the thermal resistances considered
by Meng and Thomas2) is hidden in the term of DT Tw (in qa)
in this comparison.
In Figs. 7(a)–7(b) the strand surface temperature of the
bloom predicted by the numerical simulation, is compared
with the results of the present semi-analytical model (Eq.
p 19.01p 20.275 .............................(31)
Using of Eqs. (28), (29) and (31) lead to the final equation
of the strand surface temperature via Eq. (32):
k T 
Tsur 9 Tpour  st sol 
 Zqa 
0.65
 V Z2 
 C 
 Q 
0.275
.....(32)
Whereas the selection of the power form for Eq. (30) minimizes the error of the regression work, leads to a problem
in the final result (Eq. (32)). In Eq. (32), when the term of z
tends to zero, the strand surface temperature (Tsur) tends to
infinite value. To resolve this problem a minimum value for
z is defined. To define the zmin it is assume that the temperature of the steel which is in contact to the mold wall at the
meniscus is equal to the solidus temperature. Therefore replacing Tsur in Eq. (32) with Tsol and solving of Eq. (32) for
z leads to zmin as follows:
z min 0.35 1010 (Tpour )10 (Tsol )3.5
k 
 st 
 qa 
6.5
V 
 C 
 Q
Model Validation
2.75
.........(33)
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ISIJ International, Vol. 48 (2008), No. 2
models caused a small difference between them.
There are two points that should be noted here:
1. The used boundary conditions in the present numerical
simulation of the bloom, is the uniform local heat flux
density at the surface of the bloom, whereas for the
slab (by Meng and Thomas2)) the boundary condition
is thermal resistance between the mold wall and the
shell surface.
2. The slab was simulated with one-dimensional assumption by Meng and Thomas whereas the bloom in present work was simulated with two-dimensional conditions.
These points show that the present semi- analytical
model can be used for slab and bloom and its application is independent of the boundary conditions.
4.2. Solidified Shell Thickness
Another way for validation of the presented semi-analytical model is the using of the Eq. (32) in calculation of the
solidified shell thickness. In this work it is assumed that the
conductivity of the solidified shell is independent of the
temperature. The one-dimensional heat transfer among the
solidified shell is used to determine the solidified shell
thickness. Therefore the solidified shell thickness is defined
by:
q( z ) k st
dT
k st
→ dn dT .............(34)
dn
q( z )
Where n is the direction normal to solidified shell. The
boundary conditions of this equation are:
n0
at
TTsol ..........................(35)
ndshell at
TTsur ........................(36)
Integration of Eq. (34) using above boundary conditions resulted in:
d shell k st
(Tsol Tsur ) .....................(37)
q( z )
Where Tsur and Tsol are the sides temperatures of the solidified shell. In this equation Tsur is calculated using Eq. (32)
and q is calculated using Eq. (25). Substituting Eq. (32)
into Eq. (37) is conduced the equation of the solidified shell
thickness as follows:
d shell
0.275  
0.65


 VC Z 2 
 k stTsol 
k st 


Tsol 9 Tpour 
  Q 
q( z ) 


 Zqa 





.........................................(38)
Validation of the presented model for the solidified shell
thickness was done for slab, billet and bloom. For this purpose the results of the experimental measuring of the solidified shell thickness reported by Meng and Thomas2) (for
slab), Park and Thomas3) (for billet) and present authors
(for bloom) were used. Meng and Thomas2) used a breakout
shell of slab for measuring of the solidified shell thickness.
Also in the present work a breakout shell of bloom (see Fig.
3) was used which occurred in the listed conditions in Table
Fig. 7. Comparison of the present semi-analytical model (Eq.
(32)) and numerical simulation result for bloom.
(32)). This comparison is done for all parameters of Eq.
(32). This figure shows that semi-analytical model is in
agreement with the results of the numerical simulation.
This agreement validates the present dimensional analysis.
However the existence of some approximations in both
© 2008 ISIJ
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ISIJ International, Vol. 48 (2008), No. 2
Fig. 8. Comparison of measured shell thickness with the results of the present semi-analytical model (Eq. (38)), Neumann’s analytical model (Eqs. (2)–(4)), Vogt and Wuennenberg’s analytical model (Eq. (5)) and the numerical results for bloom (a), slab (b) and billet (c).
2. Park and Thomas3) have used FeS tracer to investigate
solid shell growth. FeS tracer was suddenly added into the
liquid pool during steady state casting. Because FeS cannot
penetrate the solid shell, the position of the solid shell front
at that instant can be clearly recognized after casting using
a sulphur print.
The solidified shell thickness of slab, billet and bloom
measured by above techniques are compared with the results of the present semi-analytical model (Eq. (38)), Neumann’s analytical model (Eqs. (2)–(4)), Vogt and Wuennenberg’s analytical model (Eq. (5)) and numerical results
(Meng and Thomas2) for slab, Park and Thomas3) for billet
and present work for bloom). The results of this comparison
are shown in Fig. 8(a) for bloom, Fig. 8(b) for slab and Fig.
8(c) for billet for the listed conditions in Table 2.
Figures 8(a)–8(c) show the results of the present semianalytical model, Neumann analytical model and the numerical results are in a reasonable agreement with the
measured data, but the results of Vogt and Wuennenberg’s
analytical model are not in agreement with the measured
data. In derivation of Eq. (5) it has been assumed that the
decrease of temperature at the position z is half of the tem-
perature decrease between the solidification front and the
shell surface temperature. Also the temperature of the solidification front is equal to melting temperature of the pure
iron. This assumption may have caused the deviation from
the measured data. The consistency or no consistency of the
results of the Neumann’s analytical model with the measured data depends on the selection of the constant surface
0
). In Fig. 8(b) the selected constant surface
temperature (T sur
temperature by Meng and Thomas3) was equal to 1 300°C
and this assumption might have been the reason of poor accuracy for this model and this is one of the major limitations of Neumann’s analytical model. The results of present
semi-analytical model and numerical solution are in good
agreement with measured data but the numerical results are
closer to the measured data compared with the present
semi-analytical model (in Figs. 8(b) and 8(c)) because in
semi-analytical model the role of liquid flow has some approximations.
Figure 9 shows the high dependency of the Neumann’s
0
. This figure was plotted using billet
analytical model to T sur
data in Table 2. The results show that the solidified shell
thickness at mold exit at the temperature of 1 000°C is
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ISIJ International, Vol. 48 (2008), No. 2
ume flow rate of cooling water, pouring temperature and
mold level. Additionally the term of total increase in the
cooling water temperature (DT Tw) in presented model, includes the effect of the some indirect effective parameters
such as the type of the powder and steel composition. However the effect of the steel composition also is appeared in
the solidus temperature and steel conductivity. The results
of present semi-analytical model are in good agreement
with the measured data and the results of numerical simulations reported in earlier works in literature and also numerical simulation of present work. Also this new model does
not have limitation of earlier analytical models such as Vogt
and Wuennenberg’s (Eq. (5)) and Neumann’s models (Eqs.
(2)–(4)).
Acknowledgements
The authors would like to acknowledge the authorities of
the Iran Alloy Steel Co. for their efforts in conducting the
plant trials.
Fig. 9. The solidified shell thicknesses predicted by the Neumann’s analytical model (Eqs. (2)–(4)) for listed conditions of billet in Table 2 at various strand surface temperature.
Nomenclature
Cst : Specific heat of solidified shell (J kg1 K1)
Cw : Heat capacity of water at average temperature
of water in the mold length (J kg1)
C(T) : Temperature dependent specific heat of steel
(J kg1 K1)
dshell : Solidified shell thickness (m)
fs : Solid fraction
H(T) : Enthalpy of the material (J kg1)
Hm : Mold length (m)
Hme : Mold length in contact to the melt (mold level
mold length) (m)
h : Sensible enthalpy (J kg1)
K : Coefficient in Eq. (1) (ms0.5)
k(T) : Temperature dependent conductivity of steel
(W m1 K1)
kst : Conductivity of solidified shell (W m1 K1)
N : Exponent of time in Eq. (1)
Pm : Perimeter of the tube mold (billet and bloom) or
the width of the slab (m)
Q : Volume flow rate of the cooling water (m3 s1)
q : Local heat flux density (W m2)
qa : Apparent heat flux density at the meniscus
(W m2)
q̇ : Latent heat source (W/m3)
T : Temperature (°C)
0
T sur
: Constant surface température (°C)
t : time (s)
VC : Casting velocity (m s1)
x, y : Rectangular coordinates (m)
a : experimental exponent in Eq. 6 (m1)
DT Tw : Total increase of the cooling water temperature
(°C)
DHf : Heat of fusion (J kg1)
f : Non-constant and dimensionless parameter which
is calculated by Eq. (4)
r st : Density of solidified shell (kg m3)
r w : Density of water at average temperature of water
in the mold length (kg m3)
r (T) : Temperature dependent density of steel (kg m3)
Fig. 10. Predicted shell thicknesses by the present semi-analytical model for the listed slab conditions in Table 2 at various casting velocity.
12.35 mm while at the 1 200°C it is 9.8 mm therefore the
selection of a true surface temperature is important in the
Neumann’s analytical model.
Figure 10 compares the predicted shell thicknesses by
the present semi-analytical model for the listed conditions
for slab in the Table 2 at various casting velocity. The shell
thickness decreases with casting velocity as it was expected.
5.
Conclusion
In this work a simple analytical model for the strand surface temperature in continues casting of steel was presented
with a dimensional analysis approach. The model was completed with the numerical simulation results of a bloom and
a semi-analytical model was deduced for the strand surface
temperature and the solidified shell thickness. The obtained
results in this work show that the present semi-analytical
models (Eqs. (32) and (38)) are applicable for the prediction of the strand surface temperature and the solidified
shell thickness for slab, billet and bloom. An advantage of
the present semi-analytical model is their controllability
with manageable parameters such as casting velocity, vol© 2008 ISIJ
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