Simulation of continuous casting of steel by a meshless technique

Simulation of continuous casting of steel by a
meshless technique
R. Vertnik* 1 and B. Sˇarler 2
A recently developed local radial basis function collocation method is used for the solution of the
transient convective–diffusive heat transport in continuous casting of steel. The solution of the
thermal field with moving boundaries due to phase-change and the growing computational
domain is based on the mixture continuum formulation. The growth of the domain and the
movement of the starting block are described by activation of additional nodes and by the
movement of the boundary nodes through the computational domain, respectively. Time-stepping
is performed in an explicit way by a simple characteristic procedure. A two-dimensional transient
test case solution is shown at different times and its accuracy is verified by comparison with the
reference finite volume method results. The method is very attractive in the present context due to
its trivial implementation of curved geometry for two and three dimensions, accuracy and stability
of the results.
Keywords: Steel, Continuous casting, Simulation, Meshless, Radial basis functions, Simple characteristic procedure
Introduction
Continuous casting 1 is currently the most common
process for production of steel. During casting, the
process parameters are changing with time. The
fluctuations of the process parameters, i.e. casting
temperature or temperature of the cooling water, can
significantly influence the quality of the strand. Surface
and subsurface defects usually appear at places where
transient phenomena occur. To keep a uniform quality
of the strand and to prevent defects during the casting
process, an accurate online control must be obtained.
The accuracy and flexibility of the online control can be
achieved by appropriate preliminary studies on the basis
of off-line computer models. Off-line computer models
can greatly improve the understanding of transients. On
the basis of off-line models, simplified on-line models are
developed, which are used to control and automate the
casting process by compensating for the transient
changes.
Many different numerical methods have been used in
the past to solve the related continuous casting models.
In this paper, the recently developed local radial basis
function collocation method (LRBFCM) is used to
calculate transient temperature fields in the curved
strand. The method was already developed for diffusive
problems, 2 convective–diffusive problems with phasechange
3 and direct-chill (DC) casting problems for
aluminium alloys with growing domains. 4 The method
was extended to cope with the curved geometry of a
casting machine and with highly convection dominated
1 Sˇ tore Steel d.o.o., Zˇ elezarska cesta 3, SI-3220 Sˇ tore, Slovenia
2 University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia
*Corresponding author, email robert.vertnik@gmail.com
problems, 5 specific for continuous casting of steel. The
adaptive upwind technique (AUT) was used to stabilise
the numerical approximation due to the convection
dominated situation. Here, a simple characteristic
procedure (SCP) 6 is proposed as an alternative to the
AUT, because of its trivial numerical implementation.
Governing equations
Consider a connected growing domain V with boundary
c occupied by a phase change material described with
the density r b, specific heat at constant pressure c b of the
phase b, effective thermal conductivity k, and the
specific latent heat of the solid–liquid phase change hm.
The mixture continuum formulation of the enthalpy
conservation for the assumed system is
L
Lt rh ð Þz+: ðr~vhÞ~+ : ðk+TÞ (1)
The mixture density is defined as r~f V S rSzf V L rL, the
mixture velocity is defined as r~v~f V S rS ~vSzf V L rL ~vL, and
the mixture enthalpy is defined as h~f V S hSzf V L hL, with
subscripts S and L denoting the solid and the liquid
phase, respectively. The constitutive mixture temperature
– mixture enthalpy relationships are
hS~
ð T
Tref
cSdT, hL~hS(T)z
ð T
TS
(cL{cS)dTzhm (2)
with Tref and TS standing for the reference temperature
and solidus temperature, respectively. All material
properties can arbitrarily depend on the temperature.
The liquid volume fraction f V L is assumed to linearly
vary from 0 to 1 between solidus TS and liquidus
temperature TL. By assuming the initial temperature,
velocity field, and boundary conditions at time t0, the
ß 2009 W. S. Maney & Son Ltd.
Received 17 June 2008; accepted 12 September 2008
DOI 10.1179/136404609X368064 International Journal of Cast Metals Research 2009 VOL 22 NO 1–4 311

Vertnik and Sˇ arler Simulation of continuous casting of steel by a meshless technique
1 Geometry of billet caster with typical node arrangement and 5 point influence domain schematics
mixture temperature at the time t 0zDt can be found,
where t 0 is the starting time of the casting process and Dt
is the time step.
Solution procedure
The governing equation is discretised in 2D real curved
geometry (see Fig. 1) with Cartesian coordinates p x, p y
with base vectors i x, i y by using the SCP in its explicit
form. In this approach, the equation (1) is written in
Lagrange formulation as:
L(rh)
(p(t),t)~+
Lt
: (k0+T0); p(t)
~(px(t0)zDx) : ixz(py(t0)zDy) : iy (3)
where Dx and Dy are the distances travelled in the x- and
y-direction described as:
Dx~-vxDt, Dy~-vyDt : (4)
where -vx and -vy are average velocity values of v x and v y
along the characteristic
-vx~v0x 1{Dt Lv0x
Lx , -vy~v0y 1{Dt Lv0y
Ly , (5)
where v0x and v0y are known velocity values at the initial
time. The characteristic is a line that connects the
material point at the beginning of the time step with the
same material point at the end of the time step. In
equation (3) the convective term disappears and
diffusive and source terms are averaged quantities along
the characteristic. The explicit time discretisation of
equation (3) along the characteristic is
1
r0 Dt h{h0 ðx{Dx,y{DyÞ ~+ : ðk0+T0Þ ðx{Dx,y{DyÞ (6)
It is known that the solution of the above equation in
moving coordinates leads to mesh updating and presents
difficulties. An alternative way is to expand the terms
into a Taylor expansion
h0 (x{Dx,y{Dy)&h0{Dx Lh0
Lx
(x,y){Dy Lh0
Ly (x,y), (7)
312 International Journal of Cast Metals Research 2009 VOL 22 NO 1–4
+ : ðk0+T0Þ (x{Dx,y{Dy)&+ : (k0+T0)h0
{Dx L
Lx +: ½ (k0+T0) Š (x,y){Dy L
Ly +: ½ (k0+T0) Š (x,y) (8)
In the LRBFCM the domain and the boundary are
covered by overlapping influence domains. Each of the
influence domains includes N neighboring nodes. The
arbitrary function W is represented over a set of N nodes
on each of the influence domains in the following way
W(p)& XN
n~1
y n(p)an; y n(p)~ r 2 n (p)zc2 ;
r 2 n (p)~(p{p n) : (p{p n), (9)
where yn stands for the shape functions, an for the
coefficients of the shape functions, r n for the radial
distance between two collocation points in a subdomain,
and c for the shape parameter, respectively. The
coefficients an are calculated by collocation with the
multiquadric radial basis function 7,8 as the shape
function.
The growth of the domain is performed by moving the
bottom boundary nodes according to the casting
velocity and time step length in casting direction
Dc l5v 0l?Dt. The unknown values of the moving boundary
nodes G(t 0zDt) are set from equation (9). When the
distance between the moving boundary nodes and the
fixed domain nodes exceeds two times the typical grid
distance of the node arrangement, new inner nodes are
inserted between the moving boundary nodes and
the fixed inner nodes. Their values are obtained by
equation (9).
Numerical examples
The transient simulation of the Sˇtore–Steel billet caster 9
with the simplified boundary conditions and material
properties presented. The boundary conditions are:
(i) the inner and outer surface – Robin boundary
condition with the following heat transfer
coefficients: mould 2000 W m 22 K 21 , radiation
and sprays 600 W m 22 K 21 , and rolls

2 Calculated temperature on centreline (upper curves)
and at outer side (lower curves). t5340 s
1200 W m 22 K 21 , with the reference temperature
set to 293?15 K
(ii) the top surface – Dirichlet boundary condition
with the temperature 1763?5 K and
(iii) the bottom surface – Neumann boundary condition
with zero heat flux
with the following simplified material properties:
r S~r L~7800 kg m {3 , kS~35 W m {1 K {1 ,
kL~52 W m {1 K {1 , cS~700 J kg 1 K {1 ,
cL~800 J kg 1 K {1 and hm~227 kJ kg {1
The results are compared with the results obtained from
the fluid dynamics software Fluent, which is based on
the finite volume method (FVM). In Fluent, the growth
of the domain was realised by the dynamic mesh mode.
Figure 1 represents the geometry of the caster with the
node arrangement during the simulation. Due to the
high temperature gradients at the boundary, the node
arrangement is denser near the boundary.
Figure 2 shows the calculated temperature along the
billet on the outer side and the centreline obtained with
the LRBFCM and the FVM.
Conclusions
This paper represents a solution of the transient
temperature field in the two-dimensional curved geometry
of the steel billet caster by a meshless technique.
The governing equation is solved in its strong form on a
moving domain, where both the material and the
inter-phase boundary are simultaneously moving. The
growing of the computational domain is performed by
moving the bottom boundary nodes and by inserting
Vertnik and Sˇ arler Simulation of continuous casting of steel by a meshless technique
new inner nodes. A minimal discretisation man-effort
was put into the transition of the discretisation from the
flat to the real curved geometry. The extension of the
method to cope with higher dimensions is straightforward.
The stability problem of the dominated convection
is for the first time solved in the LRBFCM context
by using the explicit SCP. The accuracy of the new
method is tested by comparison of the results with the
Fluent software package. A very good agreement
between both methods is observed. The method was
chosen as the core in the simulation system of the Sˇtore
Steel billet caster. 9,10 Our ongoing research is focused on
the extension of the method to calculate the turbulent
fluid flow of the continuous casting processes and
to couple the macroscopic with the microscopic
simulations.
Acknowledgements
The first author would like to acknowledge the Public
Agency for Technology of the Republic of Slovenia for
support in the framework of the PhD programme. The
second author would like to thank the Slovenian
Research Agency for support in the framework of the
project L2–7204 Modelling and Optimisation of
Continuous Casting, which was part-financed by the
European Union, European Social Fund.
References
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