Analysis of a model for longitudinal electromagnetic stirring in the continuous casting of steel

1.Introduction

Electromagnetic stirring (EMS) is the process by which a high level of stirring efficiency can be achieved through interaction between the magnetic field from a static induction coil and an electrically conducting metal bath. It has long been used in the steel industry as a way to affect the flow of molten metal during the continuous casting process, both in the ladle and during the solidification in the casting machine itself [1–3]; more recently, EMS has also been used in the magnesium and aluminium industries [4,5]. At the same time, mathematical modelling has been used for the purposes of understanding exactly what effect EMS has on the metal flow.

A series of papers by Schwerdtfeger and co-workers [6–11], originally written in the context of the continuous casting of steel, explored both experimentally and theoretically the effect of EMS in the round-billet, rectangular-bloom and slab configurations that are typical for this process; these have formed the basis of modelling in this area and are even cited until the present day [12–20]. The theoretical models developed consist of Maxwell’s equations for the induced magnetic flux density and the Navier Stokes equations for the velocity field of the molten metal; in principle, these equations are two-way coupled, since the alternating magnetic field gives rise to a Lorentz force which drives the velocity field, which can in turn affect the magnetic field. Typically, however, the magnetic Reynolds number is small enough that the velocity field does not affect the spatial and temporal distribution of the magnetic field, implying only one-way coupling. A further often-invoked simplification is that the frequency of the magnetic field is typically large enough to allow the use of the time-averaged value of the Lorentz force as input to the Navier Stokes equations, rather than having to solve for the velocity and magnetic fields simultaneously.

However, on revisiting the problem of rotary EMS in round-billet continuous casting, Vynnycky [21] recently found that the method used originally in [6,11] to determine the components of the Lorentz force did not lead to a unique solution; this was because the normal component of the induced magnetic flux density, rather than the tangential ones, had been prescribed as the boundary condition. Thereafter, Nick and Vynnycky [22] revisited the models for the corresponding problem for longitudinal stirring in rectangular blooms, originally considered in [7–10]; in addition to the issue of the appropriate boundary condition at the interface between the stirrer and the steel were the conditions to be taken at the interfaces between the steel and the surrounding air. Moreover, through nondimensionalization, it was found that the key dimensionless parameter in the model, 𝛥, was the product of the the wave vector and the width of the bloom, and that this parameter was comparatively large (approximately 13) in the models in [7–10]. Interestingly, previous literature had not remarked on the significance of this quantity and its cause is different to the more well-documented skin effect [23], which is related to the frequency of the magnetic field. Furthermore, it was possible to carry out an asymptotic analysis of the problem which identified the structure of the field; this essentially consisted of a boundary layer having a dimensionless width of order of 1∕𝛥 near the surface of the stirrer in which all magnetic flux density vector components decreased rapidly to zero. A follow-on question is then what happens when 𝛥 is not large, as might occur if a smaller bloom and a larger pole pitch is used. Thus, the purpose of this manuscript is to consider, still in the low magnetic Reynolds number limit as in [22], the corresponding analysis when 𝛥 is small; although this can no doubt be done solely numerically using commercially available software, the benefit of the analysis given here is that elucidates more clearly the dependencies of the magnetic flux density, the current density and the Lorentz force components on the applied boundary conditions and key dimensionless parameters. In addition, this gives a way to determine these quantities at a fraction of the computational cost that full 3D simulations require.

The structure of the paper is as follows. In Sections 2 and 3, we recap the mathematical formulation of the problem and the nondimensionalization of the governing equations, respectively. The new analysis, in terms of the asymptotically small parameter 𝛥, is given in Section 4, whilst Section 5 outlines the numerical method used to solve the fully three-dimensional (3D) and reduced two-dimensional (2D) time-dependent problems. The results are presented in Section 6, and conclusions are drawn in Section 7.

2.Model formulation

2.1.Governing equations

We consider, as depicted in Fig. 1, a linear travelling stirrer that is situated at y = 0, −h∕2 ≤ z ≤ h∕2, −l ≤ x ≤ 0, operating on a molten steel region whose cross-section is given by 0 ≤ y ≤ b, −h∕2 ≤ z ≤ h∕2, as shown in Fig. 2; the steel is surrounded by air on the other three sides. As explained in [22], this is a simplification of the actual situation in a continuous casting process, but we have nevertheless retained enough to carry out a meaningful analysis with respect to determining the characteristics of the magnetic flux density.

Fig. 1.

Schematic for the longitudinal stirring of blooms and billets. A travelling wave is passed along the casting (x) direction.

Fig. 2.

Cross-section in the y-z plane of Fig. 1, showing the stirrer, steel and air. O denotes the origin of the coordinates.

We consider the solution of Maxwell’s equations in the magnetohydrodynamic (MHD) approximation, which consist of:

• the magnetic field constraint,

  (1)

  ∇⋅B=0,
  where B is the magnetic flux density vector;

• Ampère’s law,

  (2)

  J+∂D∂t=∇×H,
  where J is the electrical current density vector, H is the magnetic field strength, which is related to B via the magnetic permeability, 𝜂, by B =  𝜂H, and D is the displacement current, although we shall neglect it in what follows;

• Faraday’s law,

  (3)

  ∇×E=−∂B∂t,
  where E is the electric field and t is time;

• Ohm’s law,

  (4)

  J=σ(E+v×B),
  where v is the velocity vector, which we will simply set to zero, and 𝜎 is the electrical conductivity of the medium under consideration, which we take to be constant in both steel and air.

Manipulating (2)–(4), we obtain

(5)

ση∂B∂t=∇2B;

note, however, that this must still be solved together with Eq. (1). Thus, with B_x, B_y and B_z as x-, y- and z-components, respectively, of B, we have

(6)

∂Bx∂x+∂By∂y+∂Bz∂z=0,

(7)

ση∂Bx∂t=∂2Bx∂x2+∂2Bx∂y2+∂2Bx∂z2,

(8)

ση∂By∂t=∂2By∂x2+∂2By∂y2+∂2By∂z2,

(9)

ση∂Bz∂t=∂2Bz∂x2+∂2Bz∂y2+∂2Bz∂z2.

Once B_x, B_y and B_z have been computed, the quantities of principal interest are the components of the Lorentz force, F_x, F_y and F_z, which are given by

(10)

Fx=JyBz−JzBy,Fy=JzBx−JxBz,Fz=JxBy−JyBx,

where (J_x, J_y, J_z) are the components of J and are given by

(11)

(JxJyJz)=1η(∂Bz∂y−∂By∂z∂Bx∂z−∂Bz∂x∂By∂x−∂Bx∂y).

Finally, F_x, F_y and F_z are used to calculate the time-averaged Lorentz force components - F¯x, F¯y and F¯z - via

(12)

F¯k=12π/ω∫02π/ωFkdt′,k=x,y,z,

with the integrals in (12) being taken with respect to time over one period of oscillation, 2𝜋∕𝜔, and with 𝜔 as the angular frequency of the current.

2.2.Boundary and interfacial conditions

At y = 0, as discussed previously [22], we prescribe the tangential components of B; thus,

(13)

Bx={α(z)cos⁡(ωt−λx)for |z|≤h/20for |z|>h/2,

(14)

Bz={β(z)cos⁡(ωt−λx)for |z|≤h/20for |z|>h/2,

where 𝜆 is the wave vector given by 𝜆 = 𝜋∕p, with p as the pole pitch, i.e. the distance from north to south pole in the inductor. For orientation, Fig. 3 shows the common time-dependent part of Eqs (13) and (14), cos(𝜔t −𝜆x), as a function of x for four different values of t during one period of oscillation. For this figure, we have made use of the parameters shown in Table 1 in the following way: 𝜔 is related to the current frequency, f, by 𝜔 =2𝜋f; on the other hand, since 𝜆b turns out to be an important dimensionless parameter in the problem, we have used the value of b in Table 1 and plotted profiles for the cases where 𝜆 = 0.01∕b and 0.1∕b. Note here that the plotted function is sinusoidal in nature, although this may not be apparent from these plots because of the comparatively small values of 𝜆, relative to the stirrer length, l.

Fig. 3.

cos(𝜔t −𝜆x) vs. x for: (a) 𝜆 = 0.01∕b; (b) 𝜆 = 0.1∕b.

Table 1

Model parameters

Parameter	Symbol	Value
Steel width	b	0.25 m
Characteristic magnetic flux density	B0	0.02 T
Current frequency	f	50 Hz
Steel breadth	h	0.35 m
Stirrer length	l	1.2 m
Air magnetic permeability	𝜂a	1.2566 ×10−6  Vs A−1 m−1
Steel magnetic permeability	𝜂s	1.2566 ×10−6  Vs A−1 m−1
Air electrical conductivity	𝜎a	3 ×10−15 −8 ×10−15  A V−1 m−1
Steel electrical conductivity	𝜎s	7.14 ×105  A V−1 m−1

For the steel-air interfaces at y = b, |z| ≤ h∕2 and |z| = h∕2,  0 ≤ y ≤ b, we apply the conventional electromagnetic interface conditions between two media. These are the continuity of the tangential components of the magnetic field strength, the continuity of the normal component of the magnetic flux density and the continuity of the tangential components of the electric field; these are written, respectively, as

(15)

[B⋅tη]−+=0,

(16)

[B⋅n]−+=0,

(17)

[E⋅t]−+=0,

where t and n are the unit tangential and normal vectors, respectively, at these interfaces. In (15)–(17), [ ]−+ is taken to mean the difference in the value of a function in the air (+) and in the steel (−).

In the air far from the steel, we should also expect the magnetic flux density to vanish, so we take

(18)

B→0as r→∞,

where r=x2+y2+z2.

3.Nondimensionalization

We nondimensionalize the equations by setting

The boundary conditions (13)–(18) are then: at Y = 0,

4.Analysis

As in [22], equation (20) and the boundary conditions in (27) imply that the solutions for B_X, B_Y and B_Z will have the form

Our earlier work [22] considered the case 𝛥 ≫ 1, and we identified a boundary layer of thickness 1∕𝛥 near the surface of the stirrer. This time, we consider the other extreme when 𝛥 ≪ 1, whilst treating the parameters 𝛺_a and 𝛺_s as being of O (1). Note also that if we had had 𝛺_s ≫ 1, this would have led to the skin-effect case; however, 𝛺_s ≫ 1 still does not lead to the 𝛥 ≫ 1 case considered in [22], since there is no term in 𝛺_s in (33). This further emphasizes the fact that the problem treated in [22] was not that of the classical skin-effect type.

We first determine the solutions for B_X, B_Y and B_Z; thereafter, these are used to constitute expressions for J_X, J_Y, J_Z, F_X, F_Y and F_Z.

5.Numerical solution

5.1.3D computations

In order to verify the correctness of the asymptotic analysis, equations (6)–(9), subject to (13)–(18), were also solved numerically. For this, the Magnetic Fields module of the finite-element software Comsol Multiphysics was used, as in [22]. Second-order Lagrangian quadrilateral elements were employed on a mesh having approximately 55000 elements, corresponding to around 1.3 million degrees of freedom. As regards the size of the computational domain, its outer edge was taken to be a half-cylinder with a radius of 0.4 m and a height of 1.2 m. The geometry and the mesh are shown in Figs 4–6. Note that the mesh is somewhat different to that used in [22], since there is no need to refine the mesh near y = 0, as there is no boundary layer for the values of 𝛥 used; on the other hand, the magnetic flux density in the steel does not decay rapidly to zero away from the stirrer, meaning that a more or less uniform mesh is necessary throughout the steel region.

Fig. 4.

Geometry and mesh used for three-dimensional computations.

Fig. 5.

Mesh used for three-dimensional computations, adjacent to the stirrer face at y = 0.

Fig. 6.

A cross-section in the y-z plane of the mesh used for three-dimensional computations.

Adaptive time-stepping was used for the numerical integration, and the convergence criterion at each time step was taken as

(100)

(1Ndof∑i=1Ndof⁡(|Ei|Ai+R|Ui|)2)12<1,

where (U_i) is the solution vector corresponding to the solution at each time step, E_i is the estimated error in the latest approximation to the ith component of the true solution vector, A_i is the absolute tolerance for the ith degree of freedom, and R is the relative tolerance; for the computations, R = 10⁻², A_i = 10⁻³ for i = 1, …, N_dof were used, where N_dof is the number of degrees of freedom. Note that all computations were carried out on an architecture with an Intel^® Core^TM i7 CPU 970 @ 3. 20 GHz × 6 processor with a 11.6 GB RAM. In order to collect data that is appropriate for computing the time-averaged Lorentz force, it is necessary to run the simulation long enough to overcome the initial transient. For this purpose, the model was run for three periods of oscillation, corresponding to 0.06 s; as consequence, typical computation times for one simulation were found to be of the order of 9–11 hours.