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The Effect of Turbulence on Two-Fluid Flow and Heat Transfer in
Continuous Steel Casting Process
THEERADECH MOOKUM
Mae Fah Luang University
School of Science
333 Moo 1, Thasud, Muang, Chiang Rai 57100
THAILAND
t.mookum@sci.mfu.ac.th
BENCHAWAN WIWATANAPATAPHEE
Mahidol University
Department of Mathematics, Faculty of Science
272 Rama 6 Road, Rajthevee, Bangkok 10400
THAILAND
scbww@mahidol.ac.th
YONG HONG WU
Curtin University of Technology
Department of Mathematics and Statistics
Perth, WA 6845
AUSTRALIA
yhwu@maths.curtin.edu.au
Abstract: In this paper, we develop a mathematical model to study the coupled turbulent two-fluid flow and heat
transfer process in continuous steel casting. The complete set of field equations are established. The turbulence
effect on a flow pattern of molten steel and lubricant oil, meniscus shape and temperature field as well as solidifi-
cation are presented in the paper.
Key–Words: Continuous steel casting process, turbulent flow, heat transfer, two-fluid flow, level set method
1 Introduction
Continuous casting (CC) is the process whereby hot
steel is solidified into a “semifinished” billet, bloom,
or slab for subsequent rolling in the finishing mills.
Hot steel is drained from the tundish through a
submerged entry nozzle (SEN) into a water-cooled
mould, where intense cooling causes a thin shell of
steel next to the mould walls solidifies before the mid-
dle section, called as a strand. The strand is continu-
ously extracted from the base of the mold at a con-
stant speed into a spray-chamber. The strand is im-
mediately supported by closely-spaced, water cooled
rollers after leaving the mould. During the process,
the mould itself oscillates vertically to facilitate the
process and to prevent the steel sticking to the mould
walls. A lubricant can also be added to the steel in the
mould to prevent sticking, and to trap any slag par-
ticles, including oxide particles or scale. Finally, the
strand is cut into predetermined lengths by mechanical
shears. In industrial practice, many problems still fre-
quently occur. These problems mainly include molten
steel breakouts, formation of surface defects and seg-
regation of solute elements. As industry development
is moving toward casting thin steel plates, these prob-
lems are expected to become more and more critical
to the success of the process. Thus, further study of
the CC process, development of robust mathematical
models for simulating the complex phenomena occur-
ring in the process become more and more important
for the design of new systems and optimization of the
process. Over the last three decades, extensive studies
have been carried out to model many aspects of the
CC process [1, 4, 5, 6, 7, 8, 10]. Little work has been
done to solve the coupled turbulent two-fluid flow and
heat transfer with solidification [11, 12, 13]. These
studies have resulted in a basic understanding of the
physics of the CC process and provided some basic
guidelines for the design of the process. However,
many phenomena such as the heat transfer process,
the formation of oscillation marks and the meniscus
behavior, have not been fully understood nor well con-
trolled.
In this study, we propose a mathematical model
for the problem of turbulent two-fluid flow and heat
transfer with solidification in the continuous steel
casting process. The effect of turbulence on the veloc-
ity field, temperature distribution and meniscus sur-
face is investigated. The rest of the paper is orga-
nized as follows. In section two, a complete set of
field equations is given. Section three represents a nu-
merical study to demonstrate the effect of turbulence
on the flow pattern of two fluids, the heat transfer with
solidification and the meniscus shape.
Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements
ISBN: 978-960-474-298-1
53
2 Mathematical Model
In the CC process, there are two fluids, including hot
steel and liquid lubricant. Materials are assumed to
be incompressible Newtonian fluids. Three different
computation regions of hot steel are the solidified re-
gion, the mushy region and the molten region. The
solidified region presents near the edge of the casting
in which every point moves along the casting direction
at the constant casting speed. The molten region is in
the center, and the mushy region is in between. For
velocity field in the other both regions, we assumed
that the fluid flow in the mushy region obeys Darcy’s
law for porous media and the influence of turbulence
on the transport momentum and energy is modeled by
the addition of the turbulent viscosity to the laminar
viscosity and turbulent conductivity to the molecular
conductivity, yielding the effective viscosity µeff and
the effective thermal conductivity kef f given by
µeff =µ+µt, keff =k+cµt
σt
,(1)
where µ,kand cdenote respectively the laminar vis-
cosity, thermal conductivity and heat capacity of two
fluids; σtis the turbulence Prandtl number assume as
0.9 [1]. The turbulent viscosity is determined by
µt=ρCµ
K2
ε,(2)
where ρis the density of two fluids, the coefficient Cµ
is suggested to be 0.09 [7], Kis the turbulent kinetic
energy and εis the turbulent dissipation rate. Thus,
the governing equations are the continuity equation
(3) and the Navier-Stokes equations (4) and the en-
ergy equation (5), namely
∇ · u= 0,(3)
ρ(∂u
∂t +u· ∇u)−∇·(−pI+µef f (∇u+ (∇u)T))
=F(u,x, t) + ρg+fst,(4)
ρc (∂T
∂t +u· ∇T)=∇ · (kef f ∇T) + QT,(5)
where uis the velocity of the fluids, pis the fluid
pressure, g= (0,0, g)with grepresenting the grav-
itational acceleration, Fis the forcing function, fst
is the surface tension force, Tis the temperature, and
QTis the heat source.
The influence of turbulence on the transport mo-
mentum and energy is modeled by the addition of
the turbulent viscosity to the laminar viscosity and
turbulent conductivity to the molecular conductiv-
ity. Various models have been proposed for cal-
culating these turbulent parameters, namely simple
mixing-length type models, one-equation models,
two-equation models. Based on a critical review, the
mixing-length type models is recommended for most
boundary-layer type flows in the absence of recircu-
lation, the one-equation model is suitable for simple
recirculation flows, while the two-equation model can
be used to model more complex flows [Launder &
Spalding (1972) [7] and Ferziger (1987) [4]]. In the
CC mould, the flow field is complex with circulation.
Thus, we use the two-equation (K−ε)model for cal-
culating µt.
ρ(∂K
∂t +u· ∇K)=∇ · [(µ+µt
σK)∇K]
−µt
σt
βg· ∇T+µtP(u)−ρε (6)
ρ(∂ε
∂t +u· ∇ε)=∇ · [(µ+µt
σε)∇ε]
+C1(1 −C3)εµt
Kσt
βg· ∇T+C1εµt
KP(u)−C2ρε2
K
(7)
where P(u) = ∇u: (∇u+(∇u)T),βis the thermal
expansion of steel, the coefficients C1= 1.44, C2=
1.92, σK= 1 and σε= 1.3[7].
To determine the movement of the interface, the
level set function is obtained by solving the following
equation [9]:
∂ϕ
∂t +u· ∇ϕ=γ∇ · (ϵ∇ϕ−ϕ(1 −ϕ)ˆn),(8)
where ϕis the level set function defined by:
ϕ(x, t) =
0 if x∈Ωs
0.5 if x∈Γint
1 if x∈Ωo.
(9)
The quantities γand ϵare reinitialization parameter
and thickness of the interface, and ˆnis the unit normal
vector at the interface. The reinitialization of equation
(8) is given by
∂ϕ
∂t =γ∇ · (ϵ∇ϕ−ϕ(1 −ϕ)ˆn).(10)
The equation (10) is solved for obtaining the initial
condition for the level set equation (8). The physi-
cal properties of fluids are represented in terms of the
Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements
ISBN: 978-960-474-298-1
54
level set function as
ρ=ρs+ (ρo−ρs)ϕ, (11)
µ=µs+ (µo−µs)ϕ, (12)
k=ks+ (ko−ks)ϕ, (13)
c=cs+ (co−cs)ϕ, (14)
where the subscripts sand odenote respectively the
molten steel and lubricant oil. The heat source QT
in equation (5) occurring only in the steel region and
representing the rate of change of the volumetric latent
heat is given by
QT=−ρs(∂HL
∂t +u· ∇HL)(1 −ϕ),(15)
where HL=Lf(T)is the latent heat in which Lrep-
resenting the latent heat of liquid steel. The liquid
fraction f(T)is given by
f(T) =
0 if T≤TS,
T−TS
TL−TS
if TS< T < TL,
1 if T≥TL,
(16)
where TSand TLare the solidification temperature
and melting temperature of the steel, respectively.
The forcing function F(u,x, t)in equation (4) is
proportional to the velocity of the liquid relative to the
porous media (mushy media) and is given by
F(u,x, t) = Cµt(1 −f(T))2
f(T)3(u−Ucast),(17)
where Ucast = (0,0, Ucast)with Ucast representing
the constant downward casting speed and Cis the
morphology constant. The surface tension force in
equation (4) acting only at the interface can be ex-
pressed as
fst =σκδ ˆn,(18)
where σis the surface tension coefficient, ˆn=∇ϕ
|∇ϕ|
is the unit normal on the interface, κ=−∇ · ˆnis the
interfacial curvature, δis the delta function [3]. The
smooth delta function used in the surface tension force
is chosen to be [2]
δ= 6|∇ϕ||ϕ(1 −ϕ)|.(19)
3 Numerical Results
To investigate the effect of turbulence on velocity
field, temperature distribution and meniscus surface,
Figure 1: Isosurface plot of turbulent kinetic energy
K.
Figure 2: Isosurface plot of turbulent dissipation rate
ε.
we consider the case where the port angle is 12◦
downward, the inlet velocity of molten steel is 0.12
m/s, the molten steel has 5◦C of super-heat, the de-
livery turbulent kinetic energy and its dissipation rate
are respectively 0.0502 m2/s2and 0.457 m2/s2. The
values of other parameters are given in Table 1.
The models with turbulence effect and with no
turbulence effect have been used in computation to in-
vestigate its impact on the two-fluid flow, heat transfer
and meniscus shape in the casting region. Distribu-
tion of the turbulence quantities Kand εare shown
in Figures 1 and 2. The quantities of turbulence vari-
ables Kand εare very high near the nozzle opening.
Close to the solid boundary, the level of turbulence ap-
proaches zero. Values of turbulent kinetic energy and
its dissipation rate rapidly decrease in the circulation
region and then reach the smallest level in the solid-
Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements
ISBN: 978-960-474-298-1
55
Table 1: Parameters used in numerical simulation
Parameters Value
Ucast -0.000575 m/s
ρs7800 kg/m3
ρo2728 kg/m3
µs0.001 P a ·s
µo0.0214 P a ·s
σ1.6 N/m
γ0.01 m/s
ϵ0.001 m
g-9.8 m/s2
β3×10−5◦C−1
TL1465 ◦C
TS1525 ◦C
cs465 J/kg◦C
co1000 J/kg◦C
ks35 W/m◦C
ko1W/m◦C
L2.72 ×105J/kg
C1.8×106m−2
(a) (b)
Figure 3: Vector plot of velocity vectors in xz−plane
near symmetry; (a) With turbulence effect; (b) With
no turbulence effect.
ified steel shell. The results of the computed veloc-
ity field, turbulent kinetic energy, dissipation rate of
turbulent kinetic energy, temperature distribution and
meniscus shape are compared in Figures 3−7. The
turbulence is found to have considerable effect on the
flow pattern, heat transfer in the casting process and
meniscus shape.
Figure 3 shows the velocity vectors in xz−plane
(a) (b)
Figure 4: Vector plot of velocity vectors on the hor-
izontal plane near the meniscus; (a) With turbulence
effect; (b) With no turbulence effect.
near symmetry, Figure 4 shows the velocity vectors on
the horizontal plane near the meniscus, and Figure 5
shows the streamline plot of the velocity fields. In the
model with turbulence effect, the depth of the lower
recirculation zone decreases, and the upper recircula-
tion zone occurs near the nozzle. The velocity from
the model with no turbulence effect is stronger than
that with turbulence effect. Figure 6 shows compar-
ison of the temperature profiles in the mould region.
It indicates that the temperature field in both models
with and with no turbulence effect drops very fast near
the strand surface. The average temperature from the
model with turbulence effect is lower than that with
no turbulence effect in the region.
Figures 7(a) and 7(b) compare the meniscus shape
obtained respectively by models with and with no tur-
bulence effect. The results indicate that when turbu-
lence is taken into account, the depth of the menis-
cus shape near the edge of casting is predicted to be
deeper.
4 Concluding Remarks
A sophistication mathematical model for simulating
the coupled turbulent two-fluid flow-heat transfer-
solidification process in the continuous casting has
been constructed utilizing a modified K−εturbu-
lence model and level set equation. Based on the
highly nonlinear model established, an efficient level
set Bubnov-Galerkin finite element method has been
developed and applied to study the turbulent two-fluid
flow and temperature distribution in the casting re-
gion. The results clearly show that the effect of tur-
bulence on solidification is big. Thus, in most case,
we can use the model with turbulence effect taking
into account the two-fluid flow to obtain a reasonable
approximation.
Acknowledgements: The authors gratefully ac-
Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements
ISBN: 978-960-474-298-1
56
(a) (b)
Figure 5: Streamline plot of velocity fields; (a) With
turbulence effect; (b) With no turbulence effect.
(a) (b)
Figure 6: Temperature distribution at 1465, 1500,
1525, and 1530◦C; (a) With turbulence effect; (b)
With no turbulence effect.
knowledge the support of the Office of the Higher
Education Commission and the Thailand Research
Fund through the Royal Golden Jubilee Ph.D. Pro-
gram (Grant No. PHD/0212/2549), and an Australia
Research Council Discovery project grant.
(a)
(b)
Figure 7: Meniscus surface pattern; (a) With turbu-
lence effect; (b) With no turbulence effect.
References:
[1] M.R. Aboutalebi, M. Hasan and R.I.I. Guthrie,
Coupled turbulent flow, heat and solute trans-
port in continuous casting processes, Met-
all. Mater. Trans. 26B, 1995, pp. 731–744.
[2] J.P.A. Bastos and N. Sadowski, Comsol mul-
tiphysics modeling guide version 3.4, COM-
SOLAB 2007.
[3] J.U. Brackbill, D. Kothe and C. Zemach, A Con-
tinuum method for modeling surface tension,
J. Comput. Phys. 100, 1992, pp. 335–353.
[4] J.H. Ferziger, Simulation of incompressible tur-
bulecnt flow, Comput. Phys. 69, 1987, pp. 1-48.
[5] K.G. Kang, H.S. Ryou and N.K. Hur, Coupled
turbulent flow, heat, and solute transport in con-
tinuous casting process with an electromagnetic
brake, Numer. Heat Trans. A. 48, 2005, pp. 461–
481.
[6] D.S. Kim, W.S. Kim and K.H. Cho, Numeri-
cal simulation of the coupled turbulent flow and
macroscopic solidification in continuous casting
with electromagnetic brake, ISIJ. Inter. 40, 2000,
pp. 670–676.
Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements
ISBN: 978-960-474-298-1
57
[7] B.E. Launder and D.B. Spalding, The numerical
computation of turbulent flows, Comput. Meth-
ods Appl. Mech. 3, 1974, pp. 269–289.
[8] P.R. Lopez, R.D. Morales, R.S. Perez and
L.G. Demedices, Structure of turbulent flow in
slab mold, Metall. Meter. Trans. B. 36B, 2005,
pp. 787–800.
[9] E. Olssen, G. Kreiss and S. Zahedi, A conser-
vative level set method for two phase flow II,
J. Comput. Phys. 225, 2007, pp. 785-807.
[10] R. S´
anchez-Perez, R.D. Morales, M. D´
ıaz-Cruz,
O. Olivares-Xometl and J. Palafox-Ramos, A
physical model for the two-phase flow in a
nontinuous casting mold, ISIJ. Inter. 32, 1992,
pp. 521-528.
[11] Y.H. Wu and B. Wiwatanapataphee, Mod-
elling of turbulent flow and multi-phase
heat transfer under electromagnetic force,
Disc. Cont. Dyn. Sys. B 8, 2007, pp. 695–706.
[12] Y.H. Wu and B. Wiwatanapataphee, An entralpy
control volume method for transient mass and
heat transport with solidification, Int. J. Com-
put. Flu. Dyn 18, 2004, pp. 577–584.
[13] B. Wiwatanapataphee, Y.H. Wu, J. Archapi-
tak, P.F. Siew and B. Unyong, A numerical
study of the turbulent flow of molten steel
in a domain with a phase-change boundary,
J. Comp. App. Math. 166, 2004, pp. 307–319.
Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements
ISBN: 978-960-474-298-1
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