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: 331ghj@163.com
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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.
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