5th International & 26th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12th–14th, 2014,
IIT Guwahati, Assam, India
MATHEMATICAL MODELLING OF SOLIDIFICATION IN A CURVED
STRAND DURING CONTINUOUS CASTING OF STEEL
Ambrish Maurya1*, Pradeep Kumar Jha2
1
*MIED, Indian Institute of Technology, Roorkee, 247667, [email protected]
2
MIED, Indian Institute of Technology, Roorkee, 247667, [email protected]
Abstract
A two dimensional fluid flow, heat transfer and solidification model has been developed for a curved shape
continuous steel slab caster. The strand has been divided in various sections depending upon cooling conditions
in the mold and secondary cooling zone. Themodel was validated against the experimental results for solid shell
thickness in the moldas reported by Hakaru et al.(1984)CFD software Ansys Fluenthas been used for solving
the differential equations to predict the temperature distribution,solid shell thickness by finding the liquid
fraction of steelwithin the domain. Process parameters, casting speed and cooling rate has been varied to analyse
their effects on metallurgical length and solid shell thickness at the mold exit.
Keywords: Solidification, CFD, Mathematical Modelling, Metallurgical Length.
1 Introduction
Continuous casting is a process whereby molten
metal get solidified into semi-finished products.
During continuous casting of steel, solidification
starts as molten metal comes from tundish to mold
(water cooled copper mold) and forms a solid shell at
the inner surface of the mold. This solid shell should
have adequate thickness(Li & Thomas 2004)at the
exit of mold to withstand the ferrostatic
pressure(Brimacombe et al. 1977)of liquid metal
present inside it. Due to insufficient shell thickness at
the mold exit causes breakout that will not only
decrease the productivity and quality but also cause
fatal accident. About two-thirdto half of the superheat
given to the liquid steel is removed in the
mold(Huang et al. 1992, Zhao et al. 2005).
Solidification in the moldmay promote the defects and
cracks formation. Further, solidification and
transformation of strand from vertical to
horizontal(W. I. Irving 1993) takes place in secondary
cooling zone (SCZ. Water sprays are used for cooling
and rollers are used for pulling the solid shell from
mold and guiding it to move in a predefined
curvature.Quality of the cast slab is directly linked
with the cooling intensity and thermal gradient of the
slab during solidification in SCZ, as most of the
surface and internal defects are found due to improper
spray cooling of strand in SCZ. It was found that, in
continuous casting productivity and quality are
coupled with each other, as high casting speed is
required to increase productivity,while cooling
conditionshas to be alteredaccordingly to have
continuous defect free casting.However, chemical
compositions of the steel and other casting parameters
also havemuch significance in defining the cooling
rate of the strand.
1.1 Casting Speed
High productivity is foremost requirement from a
continuous casting machine, which means high
casting speed of the caster. As casting speed increases
the solid shell thickness gets reduced if the cooling
rate is not increased accordingly(Liu et al. 2012).
However, casting speed can’t be beyond a certain
value, because cooling capacities are limited in
practice. Breakout and bulging (causing centreline
segregation) phenomenon are most commonly
observed at high speed casting(Binder 1992, Ha et al.
2001)because of ferrostatic pressure of the liquid
metal inside solid shell and the pressure due to rolls.
High casting speed may leads to increase the
metallurgical length beyond the cut-off point. It is to
be noticed that liquid core should exist at the
straightening(W. I. Irving 1993) of the strand to
reduce the strain and stress development.
1.2 Cooling Rate
Surface temperature of the strand is most
significant parameter during continuous casting which
directly affect the surface quality of cast. So,
secondary cooling rate is required to be reviewed
carefully in order to achieve the desired surface
temperatures at different sections of the strand. At the
same time metallurgical length of the strand should
also be controlled not to exceed the cut-off
point(Sadat et al. 2011) and required cut-off
temperature(Binder 1992). In a real caster machine,
the secondary cooling zone is subdivided into many
sections which are engaged by various water nozzles
to provide desire cooling and surface temperature.
The surface temperature of strand during secondary
cooling should be controlled between 825oC and
1225oC (depending on steel composition) in order to
avoid the surface cracks caused by fluctuation of
temperature. In order to avoid transverse surface
cracking the strand surface temperature should be
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MATHEMATICAL MODELLING OF SOLIDIFICATION IN A CURVED STRAND DURING CONTINUOUS CASTING OF STEEL
either more than the high-temperature limit or lesser
than the lower-temperature limit of the ductility
trough. Longitudinal cracks at the bending and
unbending regions are more usual in steels with
carbon contents of about 0.08– 0.14%. To avoid that,
strand surface temperature should in between the
upper and lower limit temperature of elasticity at
bending and unbending zone(Kandeil 1991). In the
present work strand surface temperature is kept
between the limiting temperatures ranges discussed
above by controlling the cooling rate.
solid shell formation to estimate the optimal condition
for continuous casting process.Various process
parameters have been varied by taking into account of
metallurgical requirements to have defect free casting.
There is no universal method to optimize the process
however; many researchers have tried to give
optimum modelwithin the constraints and variables
selected. The objective of the study is to find the
relation between casting speed and cooling rate by
fixing the metallurgical length and solid shell
thickness at mold exit.
1.3 Superheat
Superheat has an adverse effect on the surface
quality. Increase in superheat temperature slows down
solid shell growth.The hotter and thinner shell is more
susceptible to deformation, bulging and crack
formation(Zhao et al. 2005). On the other hand, by
lowering the superheat, the temperature of the liquid
steel during the first stage of solidification may be too
low, which may cause freezing of the meniscus and
excessive solidification of a thick slag rim. This could
lead to transverse cracks in the cast. Hence, superheat
has an important influence on the microstructure and
internal quality.In continuous casting process, casting
speed is restricted according to superheat of the liquid
metal to minimize the chance of breakout.
The measurement of the temperature variations
during continuous casting process is very difficult, as
it is operated above the melting pointtemperature of
steel. So, mathematical modelling of continuous
castingbecomes a successful way to analyse and
optimize the process. Over the years many
mathematical models have been developed to find the
solidification behaviour in mold and secondary
cooling zone. Most of the models developed are
limited to mold region or for vertical stranddue
togeometrical
and
computational
simplicity.Louhenkilpi et al. (1993) have predicted
the temperature distribution and solidifying shell
thickness in a vertical slab caster.However, it’s very
difficult to developmodel for surface and internal
cracks prediction, because the temperature, strain and
stress fields have to be calculated over the process by
considering the industrial cooling conditions. Huang
et al. (1992) investigated the solidification process by
varying superheat and casting speed (Kandeil et al.
1991) within and below the mold and reported that
casting speed and superheat temperature directly
raises the rate of superheat extraction. Hong et al.
(2002) have reported that due to bend type mold,
recirculating flow appears in the upper-outer region,
resulting in less solid shell thickness at the inner
region and at the lower part of the mold.Ideal taper of
the mold has been reported by Meng and Thomas
(2003)by predicting the interfacial gap between mold
wall and solidified shell.
In the present work two dimensional heat transfer
solidification model has been developed for a curved
slab caster. The model predicts the surface
temperature, temperature profile in the strand and
2 Governing Equations
2.1 Heat transfer
Energy conservation equation for solidification is
given as:
ߩ
߲‫ܪ‬
+ ߩ∇. ሺ‫ݑ‬ത ‫ܪ‬ሻ = ∇൫݇௘௙௙ ∇ܶ൯ + ܳ௅
߲‫ݐ‬
Where, H is the enthalpy of the material and can be
computed as the sum of sensible heat (h) and latent
heat content (∆H),
‫ = ܪ‬ℎ + ∆‫ܪ‬
Sensible enthalpy and latent heat content are
defined as,
ℎ = ℎ௥௘௙ + න
்
்ೝ೐೑
∆‫ߚܮ = ܪ‬
ܿ௣ ݀ܶ
Where, L = latent heat of material,β = liquid fraction,
href= reference enthalpy, Tref = reference temperature,
keff = effective conductivity, cp = specific heat, ρ =
density,
‫ݑ‬ത = velocity, QL = source term.
The source term QL, has two terms in it; explicit
latent heat term and convective term. In a single phase
solidification model,QLcan be expressed as
߲݂௦
ܳ௅ = ߩ‫ܮ‬
+ ߩ‫ݑܮ‬ത௣௨௟௟ . ∇݂௦
߲‫ݐ‬
Latent heat of material is released from the
mushy zone. The solidified shell formed is pulled out
at a constant casting velocity “‫ݑ‬ത௣௨௟௟ ”. That means the
region having solid fraction “fs” equals to one, will
move along the casting direction with the casting
speed. It is to be noted that sum of liquid fraction “β”
and solid fraction “fs” is always 1(Hong et al. 2002).
The liquid fraction can be calculated by determining
the temperature as
0
‫ۓ‬
ۖ ܶ − ܶ௦௢௟௜ௗ௨௦
ߚ=
‫ܶ۔‬௟௜௤௨௜ௗ௨௦ − ܶ௦௢௟௜ௗ௨௦
ۖ
1
‫ە‬
݂݅ ܶ < ܶ௦௢௟௜ௗ௨௦
݂݅ ܶ௦௢௟௜ௗ௨௦ < ܶ < ܶ௟௜௤௜ௗ௨௦ ݂݅ ܶ > ܶ௟௜௤௜ௗ௨௦
2.2 Fluid flow
The continuity equation can be expressed as
߲
ሺߩ‫ݑ‬௜ ሻ = 0
߲‫ݔ‬௜
While, transient Navier-Stokes equation
momentum conservation can be expressed as
for
߲
ሺߩ‫ݑ‬ሻ + ߩ∇ሺ‫ݑݑ‬ሻ = −∇ܲ + ∇൛ߤ௘௙௙ ሺ∇. ‫ݑ‬ሻൟ + ߩ݃ + ܵ
߲‫ݐ‬
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5th International & 26th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12th–14th, 2014,
IIT Guwahati, Assam, India
Where, µeff = µl + µt is the effective viscosity due to
turbulence realizablek-ε model is used for present
study, µl is the dynamic viscosity and µt is the
turbulent viscosity. In the enthalpy-porosity technique
mushy zone is treated as porous medium and the
porosity of every cell is set equal to the liquid fraction
in that cell. The porosity equals to zero if the zone is
fully solidified, which extinguishes the velocity in that
zone. Thus the momentum sink term “S” was added to
the right hand side of the Navier-Stokes equation. The
presence of this term allows the newly solidified
material to move downward at constant pull velocity.
The momentum sink can be expressed as
ܵ=
ሺ1 − ߚሻଶ
‫ܣ‬
൫‫ݑ‬ത − ‫ݑ‬ത௣௨௟௟ ൯
ሺߚ ଷ − ߦሻ ௠௨௦௛
Where, ξ is a small number (0.001) just to prevent
zero in denominator, Amush is mushy zone constant.
The mushy zone constant measures the amplitude of
the damping; the higher this value, the steeper the
transition of the velocity of the material to zero as it
solidifies. Very large value may cause the solution to
oscillate. In the momentum sink term, the relative
velocity between the molten liquid and the solid is
used rather than the absolute velocity of the liquid.
For simulating turbulence, the realizable k-ε
turbulence model was used, which is found to be
mostsuitable(Shih et al. 1995).
The turbulence
viscosity is given by
ߤ௧ = ߩ‫ܥ‬ఓ
݇ଶ
ߝ
The two partial differential equations for
turbulent kinetic energy (k) and dissipation rate (ε) (9)
are given by
߲݇
+ ߩሺ∇݇‫ݑ‬ሻ = ∇൫ߙ௞ ߤ௘௙௙ ∇݇൯ + ‫ܩ‬௞ + ߩߝ + ܵ௞
߲‫ݐ‬
߲ߝ
ߝ
ߝଶ
∗
ߩ + ߩሺ∇ߝ‫ݑ‬ሻ = ∇൫ߙఌ ߤ௘௙௙ ∇ߝ൯ + ‫ܥ‬ଵఌ ‫ܩ‬௞ − ‫ܥ‬ଶఌ
ߩ + ܵ௞
߲‫ݐ‬
݇
݇
ߩ
Sinks are added to all of the turbulence equations
in the mushy and solidified zones to account for the
presence of solid matter. The sink term(Ansys®,
academic research)is very similar to the momentum
sink term.
ܵ௞ =
ሺ1 − ߚ ଶ ሻ
‫ܣ‬
߮
ሺߚ ଷ − ߝሻ ௠௨௦௛
Where φ represents the turbulence quantity being
solved (k, ε, ω, etc.), and the mushy zone
constant, Amush, is the same as the one used in equation
above. αk and αε are the inverse effective Prantdl
∗
numbers, C1ε and ‫ܥ‬ଶఌ
are the model parameters, Gk is
the generation of the turbulence kinetic energy due to
mean velocity gradients.
C1 = 1.44, C2 = 1.92, Cµ = 0.09, σk = 1.0, σε =
1.3(Versteegand Malalasekra 2012)
3 Assumptions and Boundary Conditions
The following assumptions were made during
formulation of the solidification model to simplify the
governing equations:
• Liquid steel as Newtonian incompressible fluid.
• Only two dimensional heat transfers (lateral
direction) is considered.
• Convective boundary condition has been taken for
extraction of heat from mold and strand.
• Density and specific heat of steel are invariant.
• Moldoscillation, effect of segregation, etc. has been
ignored.
• Perfect contact between the shell and mold is
considered as shrinkage due to solidification is
ignored.
• No slip boundary condition prevails at the walls.
Based on the above assumptions, the material
properties of steel anddimensions of the model are
listed in Table 1(Huang et al. 1992) and Table 2
respectively.
Table1Material properties of steel being used
Material Properties
Density (Kg.m-3)
Solidus temperature (K)
Liquidus temperature (K)
Degree of superheat (K)
Specific heat (J.Kg-1K-1)
Thermal conductivity (W.m-1K-1)
Viscosity (Kg.m-1s-1)
Latent heat of solidification (J.Kg-1)
Steel emissivity
Value
7020
1783
1795
15
750
41
0.0067
272000
0.8
Table2 Dimensions of the model
Dimensions(m)
Mold length
SCZ-1
SCZ-2
SCZ-3
SCZ-4
SCZ-5
SCZ-6
SCZ-7
SCZ-8
Radius of curve
Cut-off length
Value
1.0
1.2
0.5
2.4
3.0
4.5
5.4
6.5
9.5
9.74
34
3.1 Inlet
The mold is fed from inlet port with velocity inlet
of the molten steel. The velocity component at the
inlet is only in z-direction (casting direction) while,
inlet velocity was obtained by balancing the inlet flow
rate with the casting speed. However, inlet
temperature (Tinlet) of the molten steel was fixed
according to the superheat (∆T) supplied to the steel
above the liquidus temperature (TL). The inlet
temperature can be expressed as:
Tinlet=TL+∆T
3.2 Wall
To avoid the computational difficulties associated
with the heat extraction from steel through cooling
water flowing within mold, heat transfer by
convection(Li &Thomas 2004, Meng & Thomas
2003) has been considered for the mold walls. The
solidified shell was set to move with casting velocity
in the casting direction. In secondary cooling zone,
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MATHEMATICAL MODELLING OF SOLIDIFICATION IN A CURVED STRAND DURING CONTINUOUS CASTING OF STEEL
different heat transfer rates werequantifiedfor
eightsub-sections
sections modelled with consideration of
different cooling rateswhile moving down the strand.
3.3 Outlet
To maintain the constant casting speed,
speed velocity
outlet boundary condition has been applied at the exit
(cut-off point).. The velocity at the outlet is equal to
the casting speed in casting direction.
Computational domain has been broadly divided in
into
two parts: mold and secondary cooling zone (SCZ).
Further, SCZ is subdivided into eight sections, as
cooling rate varies while moving down the strand.
strand
The geometry of the model is shown in Fig.1. The
geometrical length is measured from the inlet and
outlet along the axis of the strand.
Ansys 14.0.Second
Second order implicit scheme and
realizable k – ε turbulence model were used to solve
the fluid flow equations by finite volume method. For
solidification, instead of tracking liquid
liquid-solid front
explicitly, enthalpy-porosity
porosity technique has been used
as explained in previous section.. The solution was
executed in transient state with time step of 0.01
second. The solution convergence has been achieved
with momentum residuals ˂ 10-4 and energy residuals
˂ 10-7. Liquid
iquid fraction of the steel and temperature
distribution in the strand has been calculated. The
liquid fraction in the complete domain and velocity
vector of the steel flow in the mold at casting speed of
1.6 m/min is shown in Fig.3. A discontinuity in the
solid shell formation can be found at the top portion
of the mold due to circulation of the liquid
steel;however a continuous solid shell formed later.
model
Figure 1 Schematic diagram of the model.
Figure 3 Liquid fraction and velocity vector (mold)
4 Validation
In order to validate the computational model,
model a
three dimensional model of mold has been prepared
and validated against the experimental resultsfor solid
shell thickness at the narrow wall as reported
byHakaru et al.(1984).. The comparison of both the
results is shown in Fig.2. Form the figure,
figure a good
agreement between the two can be observed.
observed
Figure 2 Validation of the model
5 Results and Discussion
The two dimensional model has been prepared
and discretised equations were solved in CFD tool
5.1 Effect of casting speed
The variation of surface temperature at the inner
wall of the strand with different casting speed is
shown in Fig. 4. The rise and fall in the curve is due
to the change in cooling rate in different cooling zone.
The cooling rate for all cooling zone were kept
constant. It is found
ound that with the increase in casting
speed, surface temperature of the strand increases
significantly. Onn the other hand, due to fluctuation in
temperature, high surface temperature may le
lead to
surface cracks. In Fig. 5temperature
temperature distribution at
cut-off point (outlet) is shown.It
.It was found that
temperature curve at the outlet become steeper with
the increase in casting speed and maximum centre
temperature was predicted forcasting
casting speed 2.0
m/min, as solidification was not completed till cut-off
point.
Solid shell thickness at the mold exit for slab
caster is to be 10-14
14 mm to avoid breakout and
bulging defect. For finding the shell thickness, Liquid
Fraction has been calculated and liquid fraction of 0.7
is considered as a solid zone (mushy zone having 30%
solid is assumed to have strength
ngth and can withstand
the load). From Fig. 6, it is clear that to avoid these
defects, casting speed should be below 1.8 m/min. On
the other hand, casting speed below 1.4m/min will
solidify more than the minimum requirement, as slag
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5th International & 26th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12th–14th, 2014,
IIT Guwahati, Assam, India
solidification may occur in mold and also the process
may be too slow.
Figure 7 shows the difference between the cut-off
length and the metallurgical length. It was found that
at casting speed of 1.2 m/min slab get completely
solidified before the straightening zone, due to which
large stress will develop during straightening and
surface defects likely to be generated. At the speed of
2.0 m/min metallurgical length is more than the cutoff length, while very little difference as for 1.8
m/min is also not preferable because any fluctuation
in the process parameter may stop the process.
Figure 6 Shell thicknesses at mold exit
Figure 4 Surface temperatures at inner wall
Figure 7Difference between cut-off length and
metallurgical length
5.2 Effect of cooling rate
The cooling rate of previous section has been
considered as a standard cooling rate and casting
speed was fixed at 1.6 m/min. The standard cooling
rate has been reduced to 75 and 85 percent and
increased to 115 and 125 percent in secondary cooling
zone, while heat loss from mold was kept constant to
maintain the shell thickness at the mold exit.
Figure 5 Temperature distributions at cut-off point
Figure 8 Surface temperatures at inner wall
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MATHEMATICAL MODELLING OF SOLIDIFICATION IN A CURVED STRAND DURING CONTINUOUS CASTING OF STEEL
It was found that with the increase in cooling
rate, surface temperature decreases, shown in Fig. 8.
The surface temperature for 75 percent is high
enoughand may cause surface defects in present
model. Centre temperature at the cut-off point is
maximum when cooling rate is lowered to 75 percent
also lowering to 85 percent is very close to it.
Conversely surface temperature for both cases has
significant difference, shown in Fig. 9. By finding the
metallurgical length, it was found that solidification
completed before cut-off point for all the cases. It can
be seen from Fig. 10 that difference between
metallurgical length and cut-off length is low for 75
and 85 percent case which may cause critical problem
during any fluctuation in cooling rate as well as
casting speed. At the same time metallurgical length
for 125 percent is much more than requiredfor the
process.
Figure 9 Temperature distributions at cut-off point
Figure 10 Difference between cut-off length and
metallurgical length
6 Conclusions
A two dimensional numerical model has been
developed to study the solidification of continuous
casting of steel slab. The temperature distribution and
solid shell thickness has been calculated to study the
effect of casting speed and cooling rate. It was
observed that casting speed can be increased up to a
certain limit to increase the productivity, after which
incomplete solidification can be found at the cut-off
point or insufficient shell thickness at the mold exit.
The increase in casting speed has more pronounced
effect on the chance of formation of defects such as
bulging, cracks etc. depending on shell thickness and
surface temperature.Conversely, cooling rate can be
lowered in the SCZ at a fixed casting speed by
maintaining the metallurgical length and the surface
temperature within the specified range.
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IIT Guwahati, Assam, India
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