Microsoft Word - 05-p2150-e130140J. Cent. South Univ. (2014) 21:
2150−2159 DOI: 10.1007/s11771-014-2165-3
Numerical simulation of transport phenomena during strip casting
with EMBr in a single belt caster
GONG Hai-jun()1, 2, LI Xin-zhong()1, XU Da-ming()1, GUO
Jing-jie()1
1. School of Materials Science and Engineering, Harbin Institute of
Technology, Harbin 150001, China;
2. College of Mechatronics and Automotive Engineering, Chongqing
Jiaotong University, Chongqing 400074, China
© Central South University Press and Springer-Verlag Berlin
Heidelberg 2014
Abstract: A theoretical investigation of fluid flow, heat transfer
and solidification (solidification transfer phenomena, STP) was
presented which coupled with direct-current (DC) magnetic fields in
a high-speed strip-casting metal delivery system. The bidirectional
interaction between the STP and DC magnetic fields was simplified
as a unilateral one, and the fully coupled solidification transport
equations were numerically solved by the finite volume method
(FVM). While the magnetic field contours for a localized DC
magnetic field were calculated by software ANSYS and then
incorporated into a three-dimensional (3-D) steady model of the
liquid cavity in the mold by means of indirect coupling. A new
FVM-based direct-SIMPLE algorithm was adopted to solve the
iterations of pressure-velocity (P-V). The braking effects of DC
magnetic fields with various configurations were evaluated and
compared with those without static magnetic field (SMF). The
results show that 0.6 T magnetic field with combination
configuration contributes to forming an isokinetic feeding of melt,
the re-circulation zone is shifted towards the back wall of
reservoir, and the velocity difference on the direction of height
decreases from 0.1 m/s to 0. Furthermore, the thickness of
solidified skull increases uniformly from 0.45 mm to 1.36 mm on the
chilled substrate (belt) near the exit. Key words: single belt
casting; electromagnetic brake (EMBr); flow field; direct-SIMPLE
algorithm
1 Introduction
With increasing in demand of high-quality ultrathin slab products
and competition in the global steel market, the strip casting
technology was proposed [1]. Strip casting is a form of
“near-net-shape casting” (NNSC) technique, which potentially offers
an economical, efficient and eco-friendly approach to produce
hot-rolled, coiled steel, and the most typical strip casting
methods are the twin-roll and horizontal single belt casting (HSBC)
processes [2]. Twin-roll process is expected to be competitive
mainly in stainless steel production, however, it has casting speed
limitations caused by the friction force between the stationary
mold and strand, and similarly it has major issues in terms of
productivity and microstructures [3]. The alternative strip casting
process is the HSBC, who is expected to be used in the production
of a large variety of steel grades and only the cooling length of
the machine limits the casting speed in this process [4]. Besides
aforesaid advantages, amorphous, non-crystalline and fine
crystalline structures would be formed in HSBC strips, especially
the
solubility of alloying elements as well as impurities can be
enhanced, and what’s more, macro/micro-segregation would decrease
greatly meanwhile, all of which would lead to desirable
improvements in alloy strip products’ properties. Particularly
worth mentioning is some grades of steel with high strength and
ductility can not be produced by conventional production route in a
steel plant, the HSBC technology of strip is not only necessary but
also the most suitable [5]. So to speak, HSBC is potentially
capable of replacing current direct casting and slab caster
operations in the future because of the advantages like
well-controlled heat transfer rate, flexibility in production rate,
compactness of equipment, and so on [6].
In the HSBC process, the liquid metal from an elevated tundish is
poured under gravity over a back wall and then fed onto a single
horizontal belt, while partially solidified steel is withdrawn
through a narrow gap between the front wall and the moving belt
[4−7]. The liquid steel is solidified in a protective atmosphere,
and then the strip of 8−15 mm in thickness is directly hot rolled
without intermediate reheating [1, 5]. The quality of the alloy
strip produced by HSBC is strongly linked to
Foundation item: Projects(51071062, 51271068, 51274077) supported
by the National Natural Science Foundation of China;
Project(2011CB605504)
supported by the National Basic Research Program (973 Program) of
China Received date: 2013−01−28; Accepted date: 2013−08−25
Corresponding author: GONG Hai-jun, PhD; Tel: +86−13983007545;
E-mail:
[email protected]
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the liquid metal feeding system, which determines how the liquid
metal will be fed onto the chilled belt and is responsible for an
even distribution of the metal across the width of the belt. So, an
optimum metal delivery system should supply melt to the chilling
substrate with a uniform velocity and be beneficial to promote heat
dissipation from the system. In other words, a so-called isokinetic
feeding is necessary due to the high pouring and withdrawing speeds
in this process [3−4, 8−9]. As such, melt flow over the
water-cooled belt needs to be reasonablely controlled.
For realizing the purpose of an even and stable melt flow, a porous
flow modifier in the reservoir was proposed by scholars for
ensuring uniform flow of liquid steel onto the substrate, which was
proved to be effective [10−11]. However, it is likely to result in
practical problems meanwhile, such as clog for aluminum killed
steel grades [9]. Fortunately, appropriate in-mold electromagnetic
(EM) fields have a stabilizing and optimizing effect on liquid
metal flows [8−9, 12], and their remarkable feature has no contact
with melt by which EM-force improves flow and no pollution or clog
occurs at the same time. In view of the advantages of EM-fields,
scholars proposed various level local static magnetic fields in the
width direction of a mold with fictitious configurations and
values, which were used to investigate the effects of EM-brake
(EMBr) on melt flow, and their superiority of the significant
effects on fluid flow field were confirmed [13]. However, these
hypothetical static magnetic fields are neither precise nor
realistic for forecasting an actual process. Up to now, though
various modeling studies have been performed for EMBr of melt flow
within the strip/slab caster’s mould, most of these have been
limited to melt flow without considering solidification [9, 14],
and few reports on the modeling of flow coupled with solidification
in HSBC under EMBr. The aim of this work is a theoretical study on
the potential effects of EMBr on metal flow patterns and
solidification in the single belt casting process with authors’
uniform numerical model and direct-SIMPLER algorithm [15], in which
three configurations of authentic magnetic field loads were
calculated by finite element method (FEM) software ANSYS.
2 Mathematical modeling 2.1 Problem statement
The physical model of a metal melt delivery system for a single
belt caster is depicted schematically in Fig. 1 [9]. Liquid steel
from an elevated tundish is delivered under gravity over the back
wall, and then the molten steel is fed into a reservoir with a
continuous moving water-chilled belt acting as bottom, while the
partially
solidified steel adhering to the belt is withdrawn at uniform speed
(V0) through a narrow gap (thickness of slab) between the front
wall and the moving substrate. The corresponding geometrical
parameters of the melt
Fig. 1 Schematic of single belt caster and cross section of
metal
delivery system [9]: (a) Schematic of single belt strip caster
and
extended nozzle metal delivery system; (b) Schematic of
longitudinal cross section of metal delivery system
delivery system are given in Table 1.
The quality of the produced strip is mainly governed by two key
factors, liquid flow to the substrate and solidification on the
substrate [1, 4]. The derivation of a thin-strip caster requires
relatively high casting speeds, and accordingly the flow adjacent
to the moving belt and exit as well as the whole extended nozzle
(i.e., reservoir) is high. In order to control the flow for meeting
the iso-kinetical feeding in the reservoir, the DC magnetic field
is used as shown in Fig. 2 [9, 13]. It should be noted that the
applied magnetic fields are Table 1 Geometrical parameters of metal
delivery system
Parameter Value
Strip thickness, d/m 0.01
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model (Unit: mm): (a) Vertical and horizontal magnets;
(b) Combination magnet with vertical and horizontal [9, 13]
different from other researches [9, 13], which is assumed to be
constant across the width of the reservoir in the electrode region.
In this work, the applied magnetic fields are derived by software-
ANSYS.
Some empirical interfacial heat transfer coefficients are taken to
represent the complex set of factors when the melt contacts a
cooling substrate, and an adiabatic condition is assumed for the
walls. The physical properties of the low carbon steel used in
present model and simulation conditions are given in Table 2. 2.2
Governing equations
In order to develop the governing equations for the simulation of
flow and solidification with EMBR in the metal delivery system
considered for a single belt caster, the following assumptions are
made:
1) EM characteristics of steel melt are uniform and
isotropic;
2) The induced magnetic field is negligible compared to the imposed
magnetic field;
3) Surface tension effects are negligible; 4) A uniform velocity
profile for inlet flow, and a
fully developed flow was imposed at the exit; 5) Newtonian and
laminar liquid flow presents; 6) The model alloy is a binary
system, or can be
simplified to a pseudo-binary system.
Table 2 Physical properties of used low carbon steel
Parameter Value
Thermal conductivity (S), KS/(W·m−1·K−1) 40
Thermal conductivity (L), KL/(W·m−1·K−1) 33
Specific heat (S), cps/(kJ·kg−1·K−1) 672
Specific heat (L), cpL/kJ·(kg·K)−1 781
Viscosity, μ/kg·(m·s)−1 0.0068
Latent heat, H/ (kJ·kg−1) 280
Pouring temperature, Tin/ºC 1555
Liquidus temperature, TL/ºC 1535
Solidus temperature, TS/ºC 1492
(1492−T)/157
Based on the above assumptions, the time-averaged
transport equations governing the system can be represented by the
following partial differential equations:
1) Heat transfer equation
t t
V = + T
(1)
In this work, a simple relation for liquid fraction function of
enthalpy is used. It is convenient to introduce the total system
enthalpy defined as
0
L S
T T H c T f H f T T
T T
T T
( ) [( ) ] [ ( )]
t
V F (4) where FB denotes body force term.
B L L L= +f ρF g F (5)
And the Lorentz force FL acting on the flowing melt can be further
expressed as
L L L L= { [( )] ( ) }f J F B + V B B B B V (6)
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Herein, the Darcy’s law is utilized in the mushy zone. Assuming
that Darcy’s law can be taken as a momentum equation for
interdendritic fluid flow in the mushy zone, the set of equations
governing the flow in the mushy zone is put in the following
form
L L L
The mushy zone is modeled as a porous medium
with either an isotropic or anisotropic permeability that is a
function of the liquid volume fraction (fL). An isotropic
permeability is modeled using the Kozeny-Karman relationship given
by
3
2.3 Initial conditions and boundary
The mold is assumed to be filled by melt with uniform temperature
initially. A uniform velocity and temperature profile at the inlet
are prescribed. At the free surface and outlet,
zero-normal-gradient conditions are imposed for the velocity and
enthalpy. The no-strip condition is imposed for the velocity. The
boundary conditions of single-belt casting with the configuration
in Fig. 1 can be written as follows.
1) Inlet. The uniform profiles for all variables were used at the
inlet.
inflow , 0v v u w (9)
2) Outlet. Fully developed conditions are adopted at the
outlet.
0 u v w
3) Free surface. The normal gradients of all
variables are set to be zero. Surface tension effects are assumed
to be negligible.
0, =0 u v
4) Substrate. Assuming no slip condition at the
moving belt for solid-phase, the velocity in the direction of
movement will be equal to the belt velocity and the other two are
set to zero.
00, u w v V (12)
5) Walls of reservoir. With the assumption of no slip on the walls,
all the velocities of the grids adhering to walls are set to
zero.
2.4 Solution method
The fully coupled transport equations associated with the boundary
conditions are solved by the finite volume method on a staggered
grid system. In order to
couple the velocity field and pressure in the momentum equations,
the Direct-SIMPLER algorithm [15] is adopted. Based on the momentum
transport and mass conservation equation, a corresponding discrete
equation for solving pressure can be given as Eq. (13) in a time
step and, mass residuum RES is limited to less than 2×10−5 as the
convergence criterion in the present computation. The computer code
developed is based on our previous works which has been
successfully used to simulate 2D EM-brake in a slab continuous
casters [15−16]. The computations are carried out over 100×65×40
grids for x, y, and z-directions, respectively.
1 1 1 1 1
- - +⋅ - ⋅ - ⋅
1 1 1 1 L 1, , , 1/ 2, L , 1, , 1/ 2,( ) ( )n n n n
i j k i j k i j k i j kf P a f P a+ + + + + - - +- ⋅ - ⋅
1 1 1 1 L , 1, , , 1/ 2 L , , 1 , , 1/ 2( ) ( )n n n n
i j k i j k i j k i j kf P a f P a+ + + + + - - +- ⋅ - ⋅ 1 1
L , , 1 , ,( ) ( , , 1, 2,3, )n n i j k i j kf P b i j k + +
+ = = (14)
where the coefficients and their significations are similar to
those of SIMPLE algorithm [17], the only difference as well as the
important characteristic of this coefficient matrix is substituting
Δ(fLPL) for ΔP. 3 Computational results and discussion 3.1
Preparation of SMF-load files for indirect coupled
EM-STP For obtaining transverse SMF in the x-direction, the
magnetic field generator devices are located at the sidewalls of
the tundish pool, as shown in Fig. 2 and Fig. 3(a).
EM-characteristics of the melt are assumed to be uniform and
isotropic in the present model. For small magnetic Reynolds
numbers, the induced magnetic field can be neglected and hence, the
magnetic field is uncoupled with the velocity field [16]. Under the
assumption of constant EM-property of the solidifying steel, the
STP will not exert influences on the EM-fields. Therefore, the
coupling between the EM-fields and the STP-related fields can be
simplified as one-way influence, and the EM-fields of the
solidification system can be prior to that of the STP
calculations.
In the present numerical simulation, the computations of the
EM-fields are performed using finite element method (FEM)-based
commercial software ANSYS. Nevertheless, the simulations of EM-STP
are implemented with a FVM-based method. To joint the two different
numerical simulation schemes, a data-conversion algorithm proposed
early [18] is used to convert the EM-files output from ANSYS, and
these EM-data files are converted into those available data for
FVM-based STP computer simulation in EMBR acting as SMF-loads.
Figure 3 presents the contour of horizontal-type magnetic field
with 3×103 at DC load
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pictures output by ANSYS and those data displayed by authors’
post-treatment software after data format conversion:
(a) FEM-format; (b) FVM-format
output by ANSYS and the data displayed by the authors’
post-treatment software after data format conversion. The
configurations and modules of B are all consistent before and after
data format conversion from FEM to FVM correspondingly. 3.2 Effect
of magnetic field configurations on flow
pattern During the single-belt casting process, the melt
close to the bottom of reservoir is cooled and solidified primarily
and then dragged along the direction of the moving belt by viscous
forces. The impinging flow on the substrate and backward velocity
are so high under the circumstance that can result in remelting the
solidified shell. So, the delivery of liquid steel onto the cooling
substrate should be controlled in such a manner that the normal
velocity to the belt at outlet to be so small that prevents from
remelting the solid layer. Figure 4(a) shows the flow pattern
within the delivery system in the absence of DC magnetic brake at
x=99 mm in the reservoir, which was deemed to be symmetry plane
approximately in x-coordinate orientation. The inflow is separated
into two parts while moving downwards, part
Fig. 4 Flow pattern in symmetry plane of reservoir in absence
of magnetic brake: (a) Result given by present model;
(b) Ref. [10]; (c) Ref. [11]
of the melt strikes the substrate strongly is close to the outlet
and then dragged towards the exit, and the rest flow towards the
back wall and forms a large re-circulation zone, which is well in
accordance with results of Refs. [9−11].
In the present computational study, three different DC magnetic
configurations are used to control the fluid flow in the reservoir
of the single belt caster. The various magnetic fields in this work
are induced by exerting the same DC-load on coils with 3×104 A·T.
The effects of various configurations of DC magnetic fields on the
velocity fields in the symmetry plane of the reservoir are shown in
Figs. 5(a)−(c). It is shown that the horizontal- type as well as
the combination configuration magnetic field make the flow more
uniform on the substrate and near the exit than get the
vertical-type magnetic field be studied. In fact, the vertical-type
magnetic field is counterproductive here for braking flow at exit
due to its strong motion-impeding action near the side-wall of the
reservoir, see Fig. 5(b).
A more clear explanation for this phenomenon can be derived from
Fig. 6. As shown in Fig. 6(a), in the case
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Fig. 5 Effect of magnetic field configurations on flow
pattern:
(a) Horizontal-type magnetic field; (b) Vertical-type
magnetic
field; (c) Combination-type magnetic field
Fig. 6 Flow pattern near side-wall (x=3 mm): (a) Without
magnetic field; (b) Vertical-type magnetic field with 3×104
A·T
of without DC-magnetic field, the fluid flow near the side-wall of
the reservoir is unblocked, which is similar to that at symmetry
plane, while the flow was almost braked once a vertical-type
magnetic field is exerted (Fig. 6(b)). In other words, the melt
adjacent to the two side-walls is extruded toward the center of the
reservoir, and then the flow near the exit of the symmetry plane
has a hydraulic jump.
From the comparison of velocity fields under different magnetic
fields, it can be concluded that the horizontal-type and
combination-type magnetic field can result in a better flow control
at the substrate close to the outlet, which is different from the
conclusions of Refs. [9, 13]. In those references, the applied
magnetic field is assumed to be constant in the region of magnet
located while decays exponentially on either sides of the magnet,
as shown in Fig. 7(a). Actually, the penetration depth of magnetic
field is restricted by the strength of DC-load and magnetic
permittivity of the material. The exponential function and the
penetration depth of magnetic field assumed in Ref. [9] are far
from the real values calculated by ANSYS in this work, (Figs. 7(b)
and (c)).
According to the theory of magnetohydrodynamics, the Lorentz force
is induced when an electrically conducting melt flows across the
DC-magnetic field. This external force acts in the opposite
direction of the melt stream and then results in suppression of the
melt flow. Ignoring displacement current, the braking force in
every orientation per unit volume is calculated by Eq. (14).
2 2
L L L L L{ ( ) ( ) }y y z z x y z xf B + B B B B F = V V V i
2 2 L L L L{ [( ) ( ) ]}x x z z y x z yf B + B B B B V V V j
2 2 L L L L{ [( ) ( ) ]}x x y y z x y zf B + B B B B V V V k
(15)
where FL represents the induced braking force, VL represents the
liquid velocity, σ is the electrical conductivity and B is the
magnetic flux density.
The induced FL vectors in the symmetry plane for the
combination-type magnetic field braking are shown in Fig. 8. As can
be seen from this figure, the distribution of FL in the upper part
of the reservoir clearly shows the braking effect on the flow
towards the front wall, and the circulation zone is shifted
drastically towards the back wall and compressed, while the
intensity of the impinging flow onto the substrate has been
suppressed at the exit and enhanced at the meniscus.
As mentioned above, the liquid metal delivery system is one of the
key aspects concerning single-belt casting, consequently a
so-called iso-kinetical feeding is necessary and important for this
process. Figure 9 shows the effect of a combination-type magnetic
field on the velocity field of the symmetry plane. One can find
that
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(a) Fictitious magnetic field in Ref. [9]; (b) Characteristic
curve
vertical-type magnetic field in x-section; (c) Characteristic
curve vertical-type magnetic field in y-section
Fig. 8 Lorentz force vectors at symmetry plane of reservoir
for
combination-type magnetic configurations with 3×104 A·T-load
Fig. 9 Flow velocity at symmetry plane of reservoir:
(a) Without magnetic field; (b) Combination-type magnetic
field 3×104 A·T-load
the velocity of outflow at the exit has a saltation without the
action of magnetic field, which shows a good agreement with Ref.
[9] and the measured results in water model by JEFFERIES [11]. The
saltation immediately become flat and smooth once the
combination-type magnetic field applied, and the velocity
difference on the direction of height decreased from 0.1 m/s to 0,
as the highlight areas with ellipse dotted line shown in Figs. 9
(a) and (b). 3.3 Effect of magnetic flux density on velocity
field
As shown in Fig. 5, by using a horizontal-type DC magnetic field or
combination-type magnetic field, the recirculation zone is
contracted greatly and shifted towards the back wall, and the
impinging flow on the substrate is suppressed meanwhile. The effect
of the two DC magnetic field configurations in damping convection
is confirmed. In this section, the effect of magnetic flux density
B on the velocity field is discussed. Figure 10 presents the effect
of magnetic flux density on the velocity field of symmetry plane
with a combination- type magnetic field. It shows that by
increasing B, the circulation zone shrinks more and the flow
becomes
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Fig. 10 Effect of magnetic flux density on velocity field at
different symmetry planes: (a) B=0.2 T; (b) B=0.4 T; (c) B=0.6 T;
(d) B=
0.8 T; (e) B=1.0 T; (f) Influence of magnetic flux density on
velocity at exit by using a combination-type magnetic field
more uniform over the substrate. Moreover, the velocity of impinge
flow normal to the substrate at the exit of reservoir decreases
significantly up to |B|=0.6 T, and with increment of magnetic flux
density, the impinge flow remains constant for the magnetic flux
density more than 0.6 T, (Figs. 10(d) and (f)). Since the velocity
changes after the magnetic intensity of 0.6 T is very slow, one can
conclude that an intensity of 0.6 T could be the optimum value for
designing the magnetic system. 3.4 Heat transfer in single belt
casting
In a strip casting process, any surface defect cannot be allowed
because product has almost final contour to be used. Therefore, an
exact knowledge of the heat flux from a solidifying metal to a mold
is of interest to control the initial solidification [19]. In
single-belt casting, when melt is fed onto the cooling belt, heat
is lost from the top free surface and the substrate by radiation
and conduction to air and the single-belt simultaneously. Ignoring
all other factors that influence heat transfer, such as shrinkage,
air gap, roughness and etc, the
interfaces on the top and the bottom are simplified and counted
with two equivalent coefficients of heat transfer, Ksurface (0.045
W·m−1·K−1) and Kbottom (10 kW·m−1·K−1) in this work.
The influences of various magnetic configurations on the isothermal
curve of melt on symmetry plane are shown in Fig. 11. It means that
the processes of heat transfer to the moving belt after exerting
various magnetic fields are uniformed than that without DC
magnetic-braking. Only judging from the uniformity degree of melt
solidification, all the magnetic flow modifiers are effective in
this work, and a ridge temperature contour appears in these cases
with magnetic-braking adjacent to the back wall, which indicates
that fluid convection exists here (Fig. 5), thus ensures that no
solidification occurs here and belt conveyer can run smoothly
consequently. What’s more should noted is, the temperature gradient
in the direction of z-coordinate near free surface decreased after
using DC magnetic-braking, especially in the case of exerting
horizontal-type magnetic field, this would be beneficial
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Fig. 11 Isothermal curve of melt on symmetry plane before and after
exerting 0.6 T DC-magnetic fields with various
configurations:
(a) No magnetic field; (b) Horizontal-type magnetic field; (c)
Vertical-type magnetic field; (d) Combination-type magnetic
field
to avoid forming skull at free surface.
Broadly speaking, DC magnetic-braking is favorable in solving the
skull formation problem at the meniscus and free surface as well as
the re-melting problem at the substrate and the exit, in view of
the homogeneity of the solidification skull on the substrate near
the exit, the effect of horizontal and combination magnetic
configurations are of the most optimal among these magnetic
contours.
Using the combination-type magnetic flow modifier proposed results
in promoting a more even skull along the chilled substrate as shown
in Fig. 12. As can be seen from Fig. 12, the development of the
mushy region (fraction of solid phase, Fs) along the belt is in a
more regular and reasonable manner than that observes in the
absence of a magnetic flow modifier, and the thickness of
solidified skull increased from 0.45 mm to 1.36 mm near the exit.
NETTO and GUTHRIE [10] gave an empirical solution for the thickness
of the solidified shell, expressed as
0( )z y K y / V (16) where z(y) denotes thickness of the solidified
shell, K is the solidification constant varies between 10 and 20
mm·min−1/2, and y denotes longitudinal position (direction of V0).
In fact, these factors influence the thickness of the solidified
shell should include thermal conductivity of the solid shell,
overall heat transfer
Fig. 12 Contour line of mushy region in substrate: (a)
Absence
DC magnetic flow modifier; (b) With combination-type
magnetic flow modifier
coefficient, initial melt temperature and ambient temperature,
latent heat of fusion, and so on. A specific relation among these
values still remains subsequent researches.
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4 Conclusions
1) The result show that three magnetic fields with different
configurations can shift the re-circulation zone towards the back
wall of the reservoir and lower the intensity flow impingement over
the chilled substrate at the exit and enhancing it at the
meniscus.
2) The horizontal and the combination magnetic devices bring better
flow modifications in the metal delivery system among the three
magnetic fields in this work, and 0.6 T magnetic field is the
optimum value for getting a uniform flow, the velocity difference
on the direction of height decreases from 0.1 to 0 m/s.
3) The DC magnetic flow modifier with horizontal or the combination
magnetic field has the potential of controlling the flow and
avoiding the problem of skull formation at the meniscus and free
surface as well as the re-melting problem on the substrate, the
thickness of solidified skull can increase uniformly from 0.45 mm
to 1.36 mm near the exit.
4) The present numerical model and method can be used to optimize
the design of a magnetic flow control device for the proposed
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