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DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2011.16.xx
DYNAMICAL SYSTEMS SERIES B
Volume 16, Number 4, November 2011 pp. X–XX
NUMERICAL SIMULATION OF TWO-FLUID FLOW AND
MENISCUS INTERFACE MOVEMENT IN THE
ELECTROMAGNETIC CONTINUOUS STEEL CASTING
PROCESS
B. Wiwatanapataphee and T. Mookum
Department of Mathematics, Faculty of Science
Mahidol University
Bangkok, 10400, THAILAND
Y. H. Wu
Department of Mathematics and Statistics
Curtin University of Technology
Perth, WA, 6845, AUSTRALIA
(Communicated by the associate editor name)
Abstract. This paper presents a mathematical model and numerical tech-
nique for simulating the two-fluid flow and the meniscus interface movement in
the electromagnetic continuous steel casting process. The governing equations
include the continuity equation, the momentum equations, the energy equa-
tion, the level set equation and two transport equations for the electromagnetic
field derived from the Maxwell’s equations. The level set finite element method
is applied to trace the movement of the interface between different fluids. In an
attempt to optimize the casting process, the technique is then applied to study
the influences of the imposed electromagnetic field and the mould oscillation
pattern on the fluid flow, the meniscus shape and temperature distribution.
1. Introduction. Electromagnetic (EM) continuous casting is an industrial heat
extraction process for converting molten steel to steel products. In the EM casting
process, molten steel is continuously poured into a water-cooled mould where inten-
sive cooling causes a thin solidified steel shell to form around the edge of the steel,
and the solidified steel is continuously withdrawn from the bottom of the mould.
To prevent the molten steel from oxidation and sticking to the mould wall, mould
powder or lubricant fluid is added on the top of the mould and the mould oscillates
vertically for the lubricant fluid to form a layer on the top of the molten steel and
between the mould wall and the steel. Electromagnetic field is also imposed on
the system to control the flow pattern of fluids in the mould and the solidification
process particularly in the meniscus region, as it has been recognized that the sur-
face quality of the casting products is mainly associated with the phenomena in the
meniscus region.
2000 Mathematics Subject Classification. Primary: 74A50; Secondary: 74S05.
Key words and phrases. Electromagnetic caster, continuous steel casting process, meniscus
shape, two-fluid flow, level set method, finite element method.
The authors is supported by the OHEC and the TRF through the RGJ Ph.D. Program grant
PHD/0212/2549.
1
2 B. WIWATANAPATAPHEE, T. MOOKUM AND Y. H. WU
To achieve high production rate and ensure high quality of the casting products,
intensive studies have been carried out worldwide over the last few decades to model
various aspects of the continuous casting process, in particular the heat transfer
- solidification process [10,27], the flow phenomena [22,25], the electromagnetic
stirring [12,14,15,21] and formation of oscillation marks [11]. However, as analyzed
by Thomas [23], the continuous casting process involves a staggering complexity of
at least 18 interacting phenomena at the mechanic level, and due to the complexity,
previous work mainly focus on each individual phenomena or interaction of two or
three phenomena only. On the other hand, to have optimal control of the casting
process, it is essential to fully understand the interaction of the main physical
phenomena occurring in the casting process. Hence, development of sophisticated
models capable of simulating the staggering complexity of the major interacting
phenomena including the heat transfer, phase change, electromagnetic stirring, fluid
flow and evolution of the interface between different fluids is still a challenge.
In this work, we further develop our previous works [25,29] in various aspects.
In [25], an enthalpy finite element method was developed to model the interacting
phenomena of heat transfer, solidification and turbulence in the casting process.
The work of [29] extend the model of [25] by incorporating the effect of the electro-
magnetic field in the coupled heat transfer - turbulence flow model. In this paper,
we generalize the previous models to include the mould oscillation effect and the in-
teraction between the lubricant fluid and the molten steel. With this generalization,
it becomes possible to simulate and quantitatively analyze the complex interacting
phenomena occurring in the meniscus region of the casting mould, particularly the
meniscus interface movement and the influence of the mould oscillation pattern.
The rest of the paper is organized as follows. In section two, the complete set of
equations for the model is established. In section three, the method of solution is
described briefly. In section four, a numerical study is presented.
2. Mathematical Model. The EM casting process involves various complex phys-
ical phenomena including heat transfer with phase change, electromagnetic stirring,
fluid flow and evolution of the interface between the lubricant fluid and the molten
steel. These phenomena interact each other. For the lubricant fluid - molten steel
flow problem, the key modeling element is the tracking of the evolution of the inter-
face between the two fluids in the dynamic casting process. Many techniques may
be used for solving this kind of problems such as the Arbitrary Lagrangian-Eulerian
method [5,7,17] and the level set method [16]. In this work, we develop an algo-
rithm based on the conservative level set method in which the interface is defined
as a φ0level of a level set function, namely
τint ={(x, y, z)∈R3:φ(x, y, x, t) = φ0}
The level set function is set to zero in the molten steel, φ0= 0.5 on the interface
and one in the lubricant fluids. When the fluids flow with velocity v, the level set
function evolves with time according to the following convection-diffusion equation
∂φ
∂t +v· ∇φ=γ∇ · [ε∇φ−φ(1 −φ)n] (1)
where γis a reinitialization parameter and εis the thickness of the interface, and
n=∇φ
|∇φ|is the unit normal vector on the interface [18].
TWO-FLUID FLOW AND INTERFACE MOVEMENT IN CC PROCESS 3
To construct the governing equations for the flow of both fluids in a unified form,
let
ρ=ρs+ (ρo−ρs)φ, (2)
µ=µs+ (µo−µs)φ, (3)
c=cs+ (co−cs)φ, (4)
k=ks+ (ko−ks)φ, (5)
where the subscripts sand odenote respectively molten steel and lubricant fluid.
Thus, in terms of the ρ, µ, c and kdefined above, the flow of both the lubricant
fluid and the molten steel can be expressed in vector form as follows.
∇ · v= 0 (6)
∂v
∂t − ∇ · 1
Re
∇v−vv −1
ρpI=g+1
ρFem +1
ρFst,(7)
where pis pressure, gis the gravitational acceleration, vdenotes the fluid velocity,
Re>0 denotes the Reynolds number. The forcing function Fem is the EM force
which, from our previous work in [1], can be calculated by
Fem =J×(∇ × A),(8)
where Aand Jdenote respectively the magnetic vector potential and current den-
sity. In [1], it has been demonstrated that from the Maxwell’s equations, by ne-
glecting the influence of the displacement current and the flow induced current on
the magnetic field, Aand Jare governed by the following trnasport equations
∇ × (1
ν∇ × A) = J,(9)
∇ · J= 0,(10)
where J=Js−η∇ϕ,νand ηdenote respectively the magnetic permeability and
electroconductivity, Jsis the source current density, and ϕis a scalar potential
function.
The forcing function Fst is the surface tension force acting only on the interface
and is given by Brackbill et al. [3] and Chang et al. [4] by
Fst =σδ(φ)κ(φ)n,(11)
where σand δ(φ) denote respectively the surface tension coefficient and the delta
function, κ(φ) = −∇ · nis the interfacial curvature. The smooth delta function for
the simulation here is chosen to be
δ(φ) = 6|∇φ||φ(1 −φ)|.(12)
For the problem of heat transfer with phase change occurring in the casting process,
we utilize the single domain enthalpy method developed in our previous work [28]
as one of the elements toward tacking of the coupled problems in question. In this
method, the temperature field is governed by the following energy equation
∂T
∂t +∇ · (vT−α∇T) = ST,(13)
where Tis temperature, α=k/ρ is the thermal diffusivity, and STis the heat
source due to phase change and is zero everywhere except in the mushy region of
4 B. WIWATANAPATAPHEE, T. MOOKUM AND Y. H. WU
Figure 1. Computation domain (a=0.1 m.) and boundary conditions.
steel where
ST=−1
c∂HL
∂t +v· ∇HL(1 −φ).(14)
Here HLdenotes the enthalpy function which can be approximated by the linearly
distributed function of temperature namely HL=Lf(T) in which Lis the latent
heat of liquid steel and f(T) is liquid fraction. The liquid fraction is zero in the
solidified region, one in the liquid region and can be approximated as a linear
function of temperature in the mushy region namely
f(T) =
0 if T≤TS,
T−TS
TL−TS
if TS< T < TL,
1 if T≥TL,
(15)
where TSand TLare respectively the solidification temperature and melting tem-
perature of the steel.
Equations (1) - (16) constitute a system of ten partial differential equations in
terms of six coordinates and time-dependent unknown functions vx, vy, vz, p, T and
φand four time-independent unknown functions A= (Ax, Ay, Az) and ϕ. To
completely define the problem, we specify the boundary conditions for the velocity
field, temperature field, level set function and EM field as shown in Figure 1.
3. Method of Solution. The electromagnetic field problem as shown in section
two can be uncoupled from the two-fluid flow and heat transfer problem. Thus, the
electromagnetic field problem governed by (9) and (10) is solved first for Aand ϕ
TWO-FLUID FLOW AND INTERFACE MOVEMENT IN CC PROCESS 5
by using the standard Galerkin finite element method to yield the electromagnetic
force for the subsequent two-fluid flow and heat transfer analysis.
We thus have the closed system of six partial differential equations in terms of
six coordinates and time dependent unknown functions (vx, vy, vz, p, T , and φ). A
numerical algorithm based on the finite element method is then developed to solve
the coupled problem. The fractures of the algorithm include: using the penalty
function method to weaken the continuity requirement for the fluid flow problem;
using a single domain enthalpy method to track the interface between the lubricant
fluid and the molten steel; using the Galerkin finite element method for discretiza-
tion in space. To keep detail of the paper to a minimum, the lengthy formulation is
omitted here. The resultant equations after discretization in space can be expressed
by
M˙
q+Kq=f,(16)
where q= (U,P,T,Φ) represents the values of the corresponding unknown at the
nodes of the finite element mesh. The matrix Mcorresponds to the transient terms
in the governing partial differential equations. The matrix Kcorresponds to the
advection and diffusion terms, and the vector fdepends nonlinearly on Uand Φ.
The numerical solutions to the nonlinear discretization system are then obtained
by using an iterative scheme developed based on Euler’s backward scheme. The
following convergent condition was used in the simulation
kRm+1
i−Rm
ik ≤ T ol, (17)
where the subscript m+ 1 and mdenote iterative computation steps, Ridenotes the
solution vector of the ith variable on the finite element nodes, k · k is the Euclidean
norm and T ol is a small positive constant.
4. Numerical Investigation and Discussion. We study here the three-dimensional
two-fluid flow and heat transfer process in the continuous steel casting. The example
under investigation is a square caster with size 0.2×0.2×0.4m3, and a submerged
entry nozzle with port outlet angle of 15odownward. The computation region, as
shown in Figure 1, represents just one quadrant of the casting steel system con-
sisting of the strand region occupied by the steel and lubricant oil on the top, the
mould region surrounding by mounted coil and the environment region. The finite
element mesh with finer grid around the meniscus region, used in this study, con-
sists of 16,478 tetrahedral elements with a total of 133,437 degrees of freedom. The
values of model parameters used in this simulation are listed in Table 1.
With the EM field, Aand ϕ, determined, we can determine the current density
vector J, the magnetic flux density vector Band the EM force Fem. Figure 2shows
the J,Band Fem fields. The results show that the current density circulates in a
clockwise direction parallel to the horizontal plane and the magnetic flux density
flows downward and parallel to the vertical plane, while the EM forces act on the
molten steel basically in the horizontal direction toward the central line.
Figure 3shows the pattern of molten steel flow on a symmetry plane. It is noted
that molten steel leaving the nozzle as a strong hot jet hits the top part of the mould
wall and then splits into two parts respectively in upward and downward directions.
This leads to two recirculation zones including an upper small recirculation zone
below the meniscus and a lower big recirculation zone below the nozzle port. The
comparison indicates that the EM force is an important factor dominating the
velocity field and the meniscus shape on the top region of the mould wall. The
6 B. WIWATANAPATAPHEE, T. MOOKUM AND Y. H. WU
Table 1. Parameters used in numerical simulation
Parameters Value Unit
Delivery velocity of molten steel uin 0.12 m/s
Density of molten steel ρs7800 kg/m3
Density of lubricant oil ρo2680 kg/m3
Effective viscosity of molten steel µs0.001 P a ·s
Effective viscosity of lubricant oil µo0.321 P a ·s
Surface tension coefficient σ1.6 m/s2
Thickness of the interface ε0.001 m
Gravitational acceleration g-9.8 m/s2
Pouring temperature Tin 1535 oC
Molten steel temperature TL1525 oC
Solidified steel temperature TS1465 oC
Shell surface temperature T∞150 oC
Environment temperature Te100 oC
Mould wall temperature Tm1400 oC
Heat capacity of molten steel cs465 J/kgoC
Heat capacity of lubricant oil co1000 J/kgoC
Thermal conductivity of molten steel ks35 W/moC
Thermal conductivity of lubricant oil ko1W/moC
Latent heat of liquid steel L2.72 ×105J/kg
Heat transfer coefficient of molten steel h∞1079 W/m2oC
Amplitude of mould oscillation A0.02 m
Angular frequency of mould oscillation ω150(2π/60) rad/s
Emissivity of solid steel $0.4
Stefan-Boltzmann constant ς5.66 ×10−8W/m2K4
Magnetic permeability of vacuum ν4π×10−7Henry/m
Electric conductivity of steel ηs4.032 ×106
Electric conductivity of coil ηc1.163 ×107
Electric conductivity of air ηa10−39
Source current density |Js|2×105A/m2
velocity field around the meniscus region as shown in Figure 4indicates that the
EM force leads to the reduction of flow speed especially in the top region near the
mould wall. The EM force thus contributes to preventing molten steel from sticking
to the mould wall and smoothing the steel casting surface.
Figures 5and 6compare the interfaces (φ= 0.5) obtained from the model with
EM force and with no EM force. The results show that the EM force gives an
expanded and weakened upper recirculation zone near the mould wall, and deeper
interface around the billet corner.
Figures 7-9show temperature distributions in the continuous steel casting. The
results indicate that the temperature field is more uniform with the EM forces and
that the electromagenetic field can be used to control the solidification process.
To investigate the effect of the mould oscillation, we study the dynamic phenom-
ena occurring in a complete mould oscillation cycle. Figure 10 gives the pattern of
the mould oscillation and various instants of time for which the dynamic phenom-
ena is to be presented. Figure 11 plots the velocity field on the symmetry plane at
various instants of time during an oscillation cycle of the mould. It is indicated that
TWO-FLUID FLOW AND INTERFACE MOVEMENT IN CC PROCESS 7
-x
6
y
1 xQ
Qk
y6
z
J(A/m2)B=∇ × A(tesla) Fem (N/m3)
(a) Top view
(b) Side view
Figure 2. Simulated solutions of: the current density J(A/m2) in
the coil, mould and molten steel pool (first column); the magnetic
flux density B(tesla) in the molten steel pool (second column);
and the EM force Fem (N/m3) in the molten steel pool (third
column).
-x
6
z
(a) (b)
Figure 3. Velocity field on a vertical symmetric plane obtained
from two different models: (a) with no EM force; (b) with EM
force.
the lubricant fluid is pushed into the gap along the mould wall during the downward
period of the mould wall, and flows out of the gap during the upward period. Figure
12 presents the variation of the meniscus level at various instants of time during
8 B. WIWATANAPATAPHEE, T. MOOKUM AND Y. H. WU
(a) (b)
Figure 4. velocity field around the meniscus region under two
different cases: (a) with no EM force; (b) with EM force.
(a) (b)
Figure 5. Meniscus surface under two different cases: (a) with no
EM force; (b) with EM force.
Figure 6. Meniscus profile along the mould wall from the billet
corner to the symmetry plane.
an oscillation cycle of the mould. It shows that the interface moves away from the
TWO-FLUID FLOW AND INTERFACE MOVEMENT IN CC PROCESS 9
Figure 7. Temperature distribution under two different cases: (a)
with no EM force; (b) with EM force.
Figure 8. Temperature distribution on the meniscus surface un-
der two different cases: (a) with no EM force; (b) with EM force.
mould wall during the downward period of the mould wall, and moves toward the
mould wall during the upward period of the mould wall.
Therefore, we can conclude that the EM force and the mould oscillation have
significant effect on the solidification process and the movement of the interface
between the molten steel and the lubricant fluid. The electromagnetic force can be
used to control the velocity field in the mould region to achieve more uniform melt
flow in the mould.
10 B. WIWATANAPATAPHEE, T. MOOKUM AND Y. H. WU
Figure 9. Temperature distribution on a horizontal plane (z=
0.96) munder two different cases: (a) with no EM force; (b) with
EM force.
Figure 10. Mould oscillation velocity V=Acos(ωt), and instants
of time for showing the velocity field and meniscus profiles.
Acknowledgments. The first and the second authors gratefully acknowledge the
support of the Office of the Higher Education Commission and the Thailand Re-
search Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0212/2549).
The third author acknowledges the support of an Australia Research Council Dis-
covery project grant.
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Received xxxx 20xx; revised xxxx 20xx.
E-mail address:scbww@mahidol.ac.th
E-mail address:t.mookum@gmail.com
E-mail address:yhwu@maths.curtin.edu.au
