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International Journal of Pure and Applied Mathematics
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Volume 63 No. 2 2010, 183-195
MODELING OF TWO-FLUID FLOW AND
HEAT TRANSFER WITH SOLIDIFICATION IN
CONTINUOUS STEEL CASTING PROCESS
UNDER ELECTROMAGNETIC FORCE
T. Mookum1, B. Wiwatanapataphee2§, Y.H. Wu3
1,2Department of Mathematics
Faculty of Science
Mahidol University
272, Rama 6 Road, Rajthevee, Bangkok, 10400, THAILAND
2e-mail: scbww@mahidol.ac.th
3Department of Mathematics and Statistics
Curtin University of Technology
Perth, WA 6845, AUSTRALIA
e-mail: yhwu@maths.curtin.edu.au
Abstract: This paper is concerned with the two-fluid flow and heat trans-
fer in the continuous steel casting process under electromagnetic (EM) force.
The governing equations consist of the Navier-Stokes equations, the continuity
equation, and the energy equation. The influence of the EM field on the flow
pattern, the meniscus shape, and temperature distribution in the EM caster is
modeled by the addition of the EM force and the surface tension force in the
Navier-Stokes equations. The EM force is defined by the cross product of cur-
rent density and magnetic flux density obtained from the Maxwell’s equations.
A surface tension force is a function of the level set function which can be solved
from the level set equation. A complete set of governing equations is solved by
the level set finite element method. The numerical results demonstrate that
the EM field applied to the system has significant effect on the two-fluid flow,
meniscus profile, and temperature distribution.
AMS Subject Classification: 35Q60, 74A50, 74F15, 74S05, 83C50
Key Words: electromagnetic stirring, continuous steel casting process, two-
fluid flow, heat transfer, level set finite element method
Received: June 23, 2010 c
2010 Academic Publications
§Correspondence author
184 T. Mookum, B. Wiwatanapataphee, Y.H. Wu
1. Introduction
In the continuous steel casting process, it has been recognized that most of sur-
face defects occur around the meniscus region. Understanding of the complex
phenomena around this region is important. These phenomena include two-
fluid flow and heat transfer with solidification. Over the last decade, extensive
studies have been carried out worldwide to model the fluid flow and steel so-
lidification using the EM stirring, see [16, 19, 21]. By ignoring the meniscus
behavior, the results obtained from those studies give basic understanding of the
fluid flow and heat transfer. Recently, meniscus behavior has been studied by
both experimental models and numerical models. In experimental models, [8]
including water model, [6], water/air model [9], the results indicate that there
are many parameters affecting the meniscus shape. The meniscus amplitude in-
creases significantly when the inlet velocity increases, the port angle increases,
or the immersion depth of submerge entry nozzle (SEN) decreases. By apply-
ing a magnetic field in the continuous casting, the meniscus always keeps the
parabolic shape near the mould wall, see [8]. In computational models with
EM effect, extensive studies have focused on the model of the heat transfer and
single fluid flow. Very little attempt has been made to couple the heat transfer
and two-fluid flow in EM continuous casting process. The meniscus may be
regarded as the gradient of the magnetic pressure (see [10, 25]) and may be
calculated by the action of Lorentz force, see [19, 11]. The results show that
the meniscus height increases when the magnetic flux density and the frequency
of magnetic field increase. In the computational models with no EM effect, the
meniscus shape is calculated by the volume-tracking method, see [2, 18] and
the Bikerman equation, see [14, 17]. The Bikerman equation can capture the
meniscus shape only near the mould wall whereas the volume-tracking method
can determine the meniscus shape in the mould. Another method that has
been used widely to determine the interface between two fluids is the level set
method. The level set method originally introduced by Osher and Sethian [13]
is the most popular technique for solving two-fluid flow problems. This method
is applied in several problems such as dam breaking (see [7, 23]), rising bubble
(see [12, 22]), and droplet (see [20, 24]).
In this study, we propose a mathematical model for the problem of cou-
pled two-fluid flow and heat transfer with solidification in the EM continuous
steel casting process. A finite element based level set method is developed for
simulating the coupled two-fluid flow, heat transfer and steel solidification phe-
nomena. The effect of source current density on the meniscus shape, the flow
of lubricant oil and molten steel, and solidification of steel is investigated. The
MODELING OF TWO-FLUID FLOW AND... 185
rest of the paper is organized as follows. In Section 2, a complete set of field
equations is given. In Section 3, the method of solution is presented. Section
4 represents a numerical study to demonstrate the influence of source current
density on the flow of two fluids, the meniscus shape, and the heat transfer with
solidification.
2. Mathematical Model
In this work, we study the motion of the molten steel and lubricant oil, the
meniscus shape, and the heat transfer with solidification in the EM continuous
steel casting process. The molten steel and lubricant oil are assumed to be
incompressible Newtonian fluid. The lubricant oil acting as thermal insulator
is filled up on the top of the molten steel to protect the steel from oxidation. The
governing equations describing the coupled two-fluid flow and heat transfer with
solidification in the EM caster are the continuity equation, the Navier-Stokes
equations, the energy equation, and the level set equation:
∇ · u= 0,(1)
ρ∂u
∂t +u· ∇u− ∇ · −pI+µ(∇u+ (∇u)T)=ρg+Fem +Fst,(2)
ρc ∂T
∂t +u· ∇T=∇ · (k∇T) + QT,(3)
and
∂φ
∂t +u· ∇φ= 0,(4)
where u, p, T and φdenote respectively the velocity field, pressure, temperature
field, and level set function; ρand µare the density and viscosity of fluid; g
is the gravitational acceleration; cand kare the heat capacity and thermal
conductivity of the fluid.
The level set function in equation (4) is defined to be negative in the molten
steel Ωs, positive in the lubricant oil Ωo, and zero level on the interface Γint:
φ(x, t)
<0 if x∈Ωs,
= 0 if x∈Γint,
>0 if x∈Ωo,
(5)
Since the physical properties of the two fluids are different, they are discon-
tinuous on the interface. To smooth the discontinuities, the density, viscosity,
heat capacity, and thermal conductivity are represented in terms of the smooth
186 T. Mookum, B. Wiwatanapataphee, Y.H. Wu
Heaviside function H(φ) as
ρ(φ) = ρs+ (ρo−ρs)H(φ),(6)
µ(φ) = µs+ (µo−µs)H(φ),(7)
c(φ) = cs+ (co−cs)H(φ),(8)
k(φ) = ks+ (ko−ks)H(φ),(9)
where the subscripts sand odenote respectively the molten steel and lubricant
oil and H(φ) is defined by
H(φ) =
0 if φ < −ε,
1
21 + φ
ε+1
πsin πφ
ε if |φ| ≤ ε,
1 if φ > ε,
(10)
for the small thickness of the interface ε, see [4, 15].
The EM force, Fem, in equation (2) is determined by
Fem =J×(∇ × A),(11)
where Aand Jare the magnetic vector potential and current density, respec-
tively. Based on our previous work in [1], Aand Jcan be determined by the
following two equations:
∇ × (1
ν∇ × A) = J,(12)
∇ · (−η∇ϕ+Js) = 0,(13)
where J=−η∇ϕ+Js,νand ηdenote respectively the magnetic permeability
and electroconductivity, Jsis the source current density, ϕis a scalar potential
function. The surface tension force in equation (2) acting only at the interface
can be expressed as
Fst =σκ(φ)δ(φ)ˆn, (14)
where σis the surface tension coefficient, ˆn=∇φ
|∇φ|is the unit normal on the
interface, κ(φ) = −∇ · ˆnis the interfacial curvature, δ(φ) is the delta function
[3, 4]. In this study, the delta function for the simulation is chosen to be
δ(φ) = 6|∇φ||φ2−α2|for α=3
2ε.
The heat source QTin equation (3) occurring only in the steel region rep-
resents the phase change and is zero everywhere except in the mushy region
where
QT=−ρs∂HL
∂t +u· ∇HL(1 −H(φ)),(15)
MODELING OF TWO-FLUID FLOW AND... 187
Figure 1: Boundary conditions
where HLis the latent heat given by
HL=Lf(T) =
0 if T≤TS,
LT−TS
TL−TS
if TS< T < TL,
Lif T≥TL,
(16)
in which Lis the latent heat of liquid steel, f(T) is the liquid fraction, TS
and TLare the solidification temperature and melting temperature of the steel,
respectively.
For two-dimensional problems, equations (1)-(16) with the boundary con-
ditions as shown in Figure 1 can be manipulated to yield two closed systems of
partial differential equations. One is a stable system for determining the EM
field and another is an unstable system of five coordinate and time dependent
unknown functions (u1, u2, p, T, and φ).
188 T. Mookum, B. Wiwatanapataphee, Y.H. Wu
3. Method of Solution
In this study, we are concerned with two-dimensional cases. The EM field in
the system of equations (12) and (13) is solved first to yield the EM force in
equation (11) for subsequent two-fluid flow and heat transfer analysis (see more
detail in paper [21]). The governing partial differential equations of unsteady
unknown functions (u1, u2, p, T, and φ) are then discretized in space by the level
set finite element method to yield the following system of nonlinear ordinary
differential equations
M˙
q+Kq=f,(17)
where qrepresents the values of the corresponding unknown U,P,T,and Φ
at the nodes of the finite element mesh. The matrix Mcorresponds to the
transient terms in the governing partial differential equations, the matrix K
corresponds to the advection and diffusion terms depends nonlinearly on U
and Φ, and the vector fdepends nonlinearly on U,T,and Φ.
The numerical solutions to the nonlinear discretization system with ap-
propriate boundary conditions are then obtained by using an iterative scheme
developed based on the Euler’s backward scheme. The following convergent
condition was used in the simulation
kRm+1
i−Rm
ik ≤ T ol, (18)
where the subscripts m+1 and mdenote iterative computation steps, Ridenotes
the solution vector of the i-th 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
The influence of the EM field on the coupled two-fluid flow and heat transfer
solidification process is investigated in the present study. The example under
investigation is a rectangular caster which has a width of 0.1 m and a depth
0.4 m in the x−zplane. The computation domain as shown in Figure 2 was
discretized using 14,927 triangular elements with a total of 128,941 degrees
of freedom. The finer grid was used in the lubricant oil region and near the
interface. The values of model parameters used in this simulation are as listed
in Table 1.
Figure 3 shows the EM force Fem corresponding to different source current
density Je
s. The results show that the EM force acts on the molten steel in
MODELING OF TWO-FLUID FLOW AND... 189
Figure 2: Computation domain Ω
Figure 3: Influence of external current density Je
son the EM force (a)
Je
s= 104A/m2; (b) Je
s= 5 ×104A/m2; (c) Je
s= 105A/m2
the horizontal direction toward the center line. This force will prevent the steel
from sticking to the mould wall. The results also show that the magnitude of
190 T. Mookum, B. Wiwatanapataphee, Y.H. Wu
Parameters Value Unit
Delivery velocity of molten steel Uin 0.012 m/s
Density ρ
-molten steel 7800 kg/m3
-lubricant oil 2728 kg/m3
Effective viscosity µ
-molten steel 0.001 P a ·s
-lubricant oil 0.0214 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
Ambient temperature T∞100 oC
External temperature Text 100 oC
Mould wall temperature Tm1000 oC
Heat capacity c
-molten steel 465 J/kgoC
-lubricant oil 1000 J/kgoC
Thermal conductivity k
-molten steel 35 W/moC
-lubricant oil 1 W/moC
Latent heat L2.72 ×105J/kg
Heat transfer coefficient of molten steel h∞1079 W/m2oC
Emissivity of solid steel 0.4
Stefan-Boltzmann constant ς5.66 ×10−8W/m2K4
Magnetic permeability of vacuum ν4π×10−7Henry/m
Electric conductivity η
-molten steel 4.032 ×106
-coil 1.163 ×106
-air 10−39
Table 1: Parameters used in numerical simulation
the force increases as the current density increases. Figure 4 shows the influence
of source current density on the velocity and temperature fields. The results
indicate that the magnitude of source current density has a considerable effect
on the flow and temperature fields. It is noted that a higher source current
MODELING OF TWO-FLUID FLOW AND... 191
Figure 4: Influence of external current density Je
son two-fluid flow
(first column), heat transfer and meniscus shape (second column), (a)
Je
s= 104A/m2; (b) Je
s= 5 ×104A/m2; (c) Je
s= 105A/m2
generates a lower speed of molten steel near the mould wall and this leads to
a reduction of temperature and hence the solidified steel shell is thicker. The
results also show the influence of EM on the formation of oscillation marks and
the meniscus shape. Figure 4(a) shows a subsurface hook on the steel surface
next to the mould wall occurring at 0.08 mbelow the free surface and the depth
of 0.005 mobtained from the mould with Je
s= 105A/m2. Higher Je
sreduces
the depth of the hook and the deformation of the meniscus and the height of
the meniscus. With increasing Je
sfrom 104to 105A/m2, the meniscus changes
192 T. Mookum, B. Wiwatanapataphee, Y.H. Wu
Figure 4: Continuation: Influence of external current density Je
son
two-fluid flow (first column), heat transfer and meniscus shape (second
column), (a) Je
s= 104A/m2; (b) Je
s= 5 ×104A/m2; (c) Je
s= 105A/m2
Figure 4: Influence of external current density on the temperature on
the meniscus
from the parabolic shape to flat shape and the interface is more smooth as
shown in Figure 4(a) −4(c). In Figure 4(c), the meniscus may freeze speedily
MODELING OF TWO-FLUID FLOW AND... 193
to form a thicker solidified shell.
Figure 4 shows the effect of varying the source current density on the tem-
perature profiles in the meniscus. It can be clearly seen that, with the increase
of source current density, the temperature on the meniscus decreases signifi-
cantly. In the case with Je
s= 105A/m2, the upper flow is too weak and this
causes the temperature on the upper region drops faster.
We can conclude that the EM field applied to the system has significant
effect on the flow, the meniscus shape, and the temperature distribution. Using
high current density, the velocity reduces and consequently the meniscus shape
becomes flat and the temperature decreases. We should also emphasize that to
improve the accuracy of results, the effect of mould movement must be included.
Therefore, further research will be carried out to include the effect of mould
movement.
Acknowledgments
The authors gratefully acknowledge the support of the Office of the Higher
Education Commission and the Thailand Research Fund through the Royal
Golden Jubilee Ph.D. Program (Grant No. PHD/0212/2549), and an Australia
Research Council Discovery project grant.
The paper entitled “Numerical Simulation of Three-Dimensional Fluid Flow
and Heat Transfer in Electromagnetic Steel Casting” published in 2009; volume
52: page 373-390 is also supported by the above sponsors.
References
[1] J. Archapitak, B. Wiwatanapataphee, Y.H. Wu, A finite element scheme
for the determination of electromagnetic force in continuous steel casting,
Int. J. Comp. Numer. Anal. App.,5(2004), 81-95.
[2] J. Anagnostopoulos, G. Bergeles, Three-dimensional modeling of the flow
and the interace surface in a continuous casting mold model, Metall. Mater.
Trans. B,30B (1998), 1095-1105.
[3] J.U. Brackbill, D. Kothe, C. Zemach, A continuum method for modeling
surface tension, J. Comp. Phy.,100 (1992), 335-353.
194 T. Mookum, B. Wiwatanapataphee, Y.H. Wu
[4] Y.C. Chang, T.Y. Hou, B. Merriman, S. Osher, A level set formulation
of Eulerian interface capturing methods for incompressible fluid flows, J.
Comp. Phy.,124 (1996), 449-464.
[5] K. Cukierski, B.G. Thomas, Flow control with local electromagnetic brak-
ing in continuous casting of steel slabs, Metall. Mater. Trans. B,39B
(2007), 94-107.
[6] D. Gupta, A.K. Lahiri, Water-modeling study of surface distributes in
continuous slab caster, Metall. Mater. Trans. B,25B (1993), 227-233.
[7] J.H. Jeong, D.Y. Yang, Finite element analysis of transient fluid flow with
free surface using VOF (volume-of-fluid) method and adaptive grid, Int.
J. Numer. Meth. Fluids,26 (1998), 1127-1154.
[8] T. Li, S. Nagaya, K. Sassa, S. Asai, Study of meniscus behavior and surface
properties during casting in high-frequency magnetic field, Metall. Mater.
Trans. B,26B (1994), 353-358.
[9] R. Miranda, M.A. Barron, J. Barreto, L. Hoyos, J. Gonzales, Experimen-
tal and numerical analysis of the free surface in a water model of a slab
continuous casting mold, ISIJ Int.,45 (2005), 1626-1635.
[10] H. Nakata, J. Etay, Meniscus shape of molten steel under alternating mag-
netic field, ISIJ Int.,32 (1992), 521-528.
[11] F. Negrini, M. Fabbri, M. Zuccarini, E. Takeuchi, M. Tani, Electromanetic
control of the meniscus shape during casting in a high frequency magnetic
field, Ener. Conv. Manag.,41 (2000), 1687-1701.
[12] H. Oka, K. Ishii, Numerical analysis on the motion of gas bubbles using
level set method, J. Phy. Soc. Japan,68 (1999), 823-832.
[13] S. Osher, J.A. Sethian, Fronts propagating with curvature-dependent
speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phy.,
79 (1988), 12-49.
[14] K. Schwerdtfeger, H. Sha, Depth of oscillation marks forming in continuous
casting of steel, Metall. Mater. Trans. B,31B (2000), 813-826.
[15] M. Sussman, P. Smereka, S. Osher, A level set approach for computing
solutions to incompressible two-phase flow, J. Comp. Phy.,114 (1994),
146-159.
MODELING OF TWO-FLUID FLOW AND... 195
[16] K. Takatani, Effects of electromaghetic brake and meniscus electromanetic
stirrer on transient molten steel flow at meniscus in a continuous casting
mold, ISIJ Int.,43 (2003), 915-922.
[17] E. Takeuchi, J.K. Brimacombe, The formation of oscillation marks in the
continuous casting of steel slabs, Metall. Mater. Trans. B,15B (1984),
493-509.
[18] A. Theodorakakos, G. Bergeles, Numerical investigation of the interface
in a continuous steel casting mold water model, Metall. Mater. Trans. B,
29B (1997), 1321-1327.
[19] T. Toh, E. Takeuchi, M. Hojo, H. Kawai, S. Matsumura, Electromagnetic
control of initial solidification in continuous casting of steel by low fre-
quency alternating magnetic field, ISIJ Int.,37 (1997), 1112-1119.
[20] Y. Watanabe, A. Saruwatari, D.M. Ingram, Free-surface flows under im-
pacting droplets, J. Comp. Phy.,227 (2008), 2344-2365.
[21] Y.H. Wu, B. Wiwatanapataphee, Modelling of turbulent flow and multi-
phase heat transfer under electromagnetic force, Disc. Cont. Dyn. Sys. B,
8(2007), 695-706.
[22] Z. Yu, L.S. Fan, Direct simulation of the buoyant rise of bubbles in infinite
liquid using level set method, The Canadian J. Chem. Eng.,86 (2008),
267-275.
[23] Y. Zhang, Q. Zou, D. Greaves, Numerical simulation of free-surface flow
using the level-set method with global mass correction, Int. J. Numer.
Meth. Fluids,29 (2009), 657-684.
[24] H.K. Zhao, B. Merriman, S. Osher, L. Wang, Capturing the behavior of
bubbles and drops using the variational level set approach, J. Comp. Phy.,
29 (2009), 657-684.
[25] X.R. Zhu, R.A. Harding, J. Campbell, Calculation of the free surface shape
in the electromagnetic processing of liquid metals, Appl. Math. Model.,21
(1997), 207-214.
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