Numerical Simulation on Flow Characteristic of Molten Steel in the Mold with Freestanding Adjustable Combination Electromagnetic Brake

Abstract

The general flow characteristics of the molten steel and the behavior of molten steel and liquid slag interface (level fluctuation) in the mold with a new type of electromagnetic brake, freestanding adjustable combination electromagnetic brake (FAC-EMBr), was investigated by numerical simulation. The influences of the magnetic induction intensity, the submerged entry nozzle (SEN) immersion depth, the SEN port angle, and casting speed on the effect of FAC-EMBr are investigated. The results show that electromagnetic field formed by FAC-EMBr can cover the jet flow impact region, the upward backflow region, and the meniscus region simultaneously. With the increase of magnetic flux density, the jet flow impact intensity and the molten steel velocity in the upward and downward backflow regions are effectively suppressed, the meniscus wave height also decreases gradually, and the level fluctuation tends to be stable. For the influence of SEN port angle and SEN depth on the effect of FAC-EMBr mold, the results show that the molten steel velocity in the three main influence regions are all effectively depressed with increase of the value of SEN port angle and SEN depth. Even though under the condition of high casting speed (V_C = 2.2 m/min), the application of FAC-EMBr can achieve effective braking effect.

Appendix A

In the mold, the molten metal is governed by the following equations:

$$ \nabla \cdot u{ = }0 $$

(A1)

$$ \partial_{t} u + (u \cdot \nabla )u + \frac{\nabla p}{\rho } = \nu \nabla^{2} u + \frac{1}{\rho }(F_{g} + F_{m} ), $$

(A2)

where u, ∇p, $\nu$and F_g and F_m are velocity, pressure gradient, kinematic viscosity, gravity and the electromagnetic (Lorentz) forces, respectively. As the fluid flow in the mold is fully turbulent, the standard k−ε model (two equations) is adopted in the simulation, where k and ε denote the turbulence kinetic energy and its dissipation rate, respectively. Therefore, the Reynolds-Average–Navier stokes (RANS) equations can be written as follows:

$$ \partial_{{x_{i} }} U_{i} = 0 $$

(A3)

$$ \partial_{t} U_{i} + \rho \partial_{{x_{j} }} (U_{i} U_{j} ) = - \partial_{{x_{i} }} P + \partial_{{x_{j} }} (2\mu S_{ij} - \rho \overline{{u_{i}^{'} u_{j}^{'} }} ) + F_{m} + \rho g_{i}, $$

(A4)

where S_ij is defined as that mean strain-rate tensor:

$$ S_{ij} = \frac{1}{2}(\partial_{{x_{j} }} U_{i} + \partial_{{x_{i} }} U_{j} ) $$

(A5)

$- \overline{{u_{i}^{'} u_{j}^{'} }}$ is the Reynolds stress tensor. U and P are the mean parts of u and p, respectively. u′ is the fluctuation part of u. The k−ε transport equations can be expressed as

$$ \partial_{t} (\rho k) + \partial_{{x_{j} }} (\rho kU_{j} ) = \tau_{ij} \partial_{xj} U_{i} - \varepsilon + \partial_{xj} [(\nu + \frac{{\nu_{t} }}{{\sigma_{k} }})\partial_{xj} k] $$

(A6)

$$ \partial_{t} (\rho \varepsilon ) + \partial_{{x_{j} }} (\rho \varepsilon U_{j} ) = C_{1\varepsilon } \frac{\varepsilon }{k}\tau_{ij} \partial_{{x_{j} }} U_{i} - C_{2\varepsilon } \frac{{\varepsilon^{2} }}{k} + \partial_{{x_{j} }} [(\nu + \frac{{\nu_{t} }}{{\sigma_{\varepsilon } }})\partial_{{x_{j} }} \varepsilon ], $$

(A7)

where σ_k and σ_ε denote the turbulent Prandtl numbers for k and ε, respectively. C_1ε and C_2ε are constant values: 1.44 and 1.92 in the present simulation. $\nu_{t}$is the kinetic eddy viscosity:

$$ \nu_{t} = C_{\mu } \frac{{k^{2} }}{\varepsilon }, $$

(A8)

where C_µ is a constant value, 0.09.

In Eq. [A4], F_m is determined by

$$ F_{m} = j \times B, $$

(A9)

where j is the induced current, which is determined by Ohm’s law:

$$ j = \sigma ( - \nabla \phi + u \times B), $$

(A10)

where B is the magnetic flux density.

Volume of Fluid (VOF) model is selected as the multiphase model to capture the steel/slag interface behavior. The steel/slag interface is captured by solving Eq. [A3]. For steel phase,

$$ \partial_{t} (\alpha_{st} \rho ) + \nabla \cdot (\alpha_{st} \rho U_{i} ) = S_{st} + \sum\limits_{n = 1}^{2} {(m_{st \to sl} - m_{sl \to st} )}, $$

(A11)

where m_st→sl denotes the mass transfer from the steel phase to the slag face, and vice versa. α_st denotes the volume of fraction of phase steel in the cell. α_st = 0, α_st = 1, and 0 < α_st < 1 denote the cell is empty of phase steel, full of phase steel, and the cell contains the interface between the phase steel and other phase, respectively. S_st is the source term of steel phase. In the present research, the continuity equation is shared by steel and slag phase. The mass transfer between different phases is neglected. Equation A11 can be simplified as

$$ \partial_{t} (\alpha_{\text{st}} \rho ) + \nabla \cdot (\alpha_{\text{st}} \rho U_{i} ) = 0. $$

(A12)

For the slag phase, the volume of fraction is obtained by the constrain condition:

$$ \alpha_{\text{st}} + \alpha_{\text{sl}} = 1. $$

(A13)