A Model Study of Inclusions Deposition, Macroscopic Transport, and Dynamic Removal at Steel–Slag Interface for Different Tundish Designs

Abstract

This paper presents computational fluid dynamics (CFD) simulation results of inclusions macroscopic transport as well as dynamic removal in tundishes. A novel treatment was implemented using the deposition velocity calculated by a revised unified Eulerian deposition model to replace the widely used Stokes rising velocity in the boundary conditions for inclusions removal at the steel–slag interface in tundishes. In this study, the dynamic removal for different size groups of inclusions at different steel–slag interfaces (smooth or rough) with different absorption conditions at the interface (partially or fully absorbed) in two tundish designs was studied. The results showed that the dynamic removal ratios were higher for larger inclusions than for smaller inclusions. Besides, the dynamic removal ratio was higher for rough interfaces than for smooth interfaces. On the other hand, regarding the cases when inclusions are partially or fully absorbed at a smooth steel–slag interface, the removal ratio values are proportional to the absorption proportion of inclusions at the steel–slag interface. Furthermore, the removal of inclusions in two tundish designs, i.e., with and without a weir and a dam were compared. Specifically, the tundish with a weir and a dam exhibited a better performance with respect to the removal of bigger inclusions (radii of 5, 7, and 9 μm) than that of the case without weir and dam. That was found to be due to the strong paralleling flow near the middle part of the top surface. However, the tundish without weir and dam showed a higher removal ratio of smaller inclusions (radius of 1 μm). The reason could be the presence of a paralleling flow near the inlet zone, where the inclusions deposition velocities were much higher than in other parts.

Appendix: Turbophoretic Model (Unified Eulerian deposiTion Model)

1. Model descriptions

The dimensionless particle deposition velocityV ⁺_dep is defined as follows:

$$ V_{\text{dep}}^{ + } = \frac{{V_{\text{dep}} }}{{u^{*} }} = \frac{{{J \mathord{\left/ {\vphantom {J {C_{\text{P}} }}} \right. \kern-0pt} {C_{\text{P}} }}}}{{u^{*} }}, $$

(A1)

where J is the deposition flux onto the steel–slag interface (hereafter interface) and C _Pis the particle mass concentration (with the unit of mass of particles per unit volume). As a further note, C ⁺_P  = C _P/C ₀ is the dimensionless particle mass concentration and C ₀ is the bulk concentration.

The equation for V ⁺_dep which is derived from particle continuity equation can be expressed as follows by Eq. [A2]:

$$ V_{\text{dep}}^{ + } = - \left( {\frac{{D_{\text{B}} + \varepsilon_{\text{p}} }}{\nu }} \right)\frac{{\partial C_{\text{P}}^{ + } }}{{\partial y^{ + } }} - C_{\text{P}}^{ + } \frac{{D_{\text{T}} }}{\nu }\frac{\partial \ln T}{{\partial y^{ + } }} + C_{\text{P}}^{ + } \overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }}, $$

(A2)

where D _B and ɛ _p are Brownian diffusivity and turbulent diffusivity for particles. The kinematic viscosity for liquid steel ν was 8.5 × 10⁻⁷ m²/s. D _T is the coefficient of diffusion due to temperature gradient. $\overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }} = {{\overline{{V_{{{\text{P}}y}}^{\text{C}} }} } \mathord{\left/ {\vphantom {{\overline{{V_{{{\text{P}}y}}^{\text{C}} }} } {u^{*} }}} \right. \kern-0pt} {u^{*} }}$ is the dimensionless form of the particle convective velocity in the y-direction (perpendicular to the wall).

The Brownian diffusivity for particles D _B can be calculated by the following relationship:

$$ D_{\text{B}} = \frac{kT}{{6\pi \rho_{\text{f}} \nu r}}C_{\text{C}}, $$

(A3)

where k is the Boltzmann constant, T is the absolute temperature: 1803.15 K (1530 °C), ρ _f is the density of fluid (steel): 7020 kg/m³, and r is the radius of the particle. C _c is the Cunningham correction factor[83] which describes the rarefied effect of particles in a gas system[51] and retained the in inclusion-steel system.[52] It can be calculated by the following relationship:

$$ C_{c} = 1 + {\text{Kn}}\left( {2.514 + 0.8 \times \exp \left( { - \frac{0.55}{\text{Kn}}} \right)} \right), $$

(A4)

where Kn is the Knudsen number and is defined as follows:

$$ {\text{Kn}} = {\lambda \mathord{\left/ {\vphantom {\lambda {\left( { 2r} \right)}}} \right. \kern-0pt} {\left( { 2r} \right)}}, $$

(A5)

where λ is the mean-free length of the fluid. The value for steel was chosen as 1 × 10⁻¹⁰ m.[52]

The turbulent diffusivity for particles ɛ _p is assumed as equal to the fluid turbulent viscosity ν _turb.[84] The following expression from Johansen[48] was used for the fluid turbulent viscosity:

$$ \frac{{\nu_{\text{turb}} }}{\nu } = \left( {\frac{{y^{ + } }}{11.15}} \right)^{3} \;{\text{when}}\;y^{ + } < 3, $$

$$ \frac{{\nu_{\text{turb}} }}{\nu } = \left( {\frac{{y^{ + } }}{11.4}} \right)^{2} - 0.049774\;\; {\text{when}}\; 3 \le y^{ + } \le 5 2. 10 8, $$

(A6)

$$ \frac{{\nu_{\text{turb}} }}{\nu } = 0.4y^{ + } \;{\text{when}}\;y^{ + } > 5 2. 10 8. $$

The D _T is calculated from the following relationship:[50]

$$ D_{\text{T}} = D_{\text{B}} \left( {1 + {\eta \mathord{\left/ {\vphantom {\eta k}} \right. \kern-0pt} k}T} \right), $$

(A7)

where η is the thermophoretic force coefficient and is expressed by:[50,85]

$$ \eta = \frac{{2.34\left( {6\pi \rho_{\text{f}} \nu^{2} r} \right)\left( {\lambda_{r} + 4.36{\text{Kn}}} \right)}}{{\left( {1 + 6.84{\text{Kn}}} \right)\left( {1 + 8.72{\text{Kn}} + 2\lambda_{r} } \right)}}, $$

(A8)

where λ _r is the ratio of the thermal conductivity of the fluid to that of the particles. This ratio for liquid steel to inclusions was chosen as 37.[52]

The particle momentum equation in the normal direction (y-direction) to the interface is normalized in Eq. [A9] as follows:

$$ \left( {1 + \frac{{\rho_{\text{f}} }}{{2\rho_{\text{P}} }}} \right)\overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }} \frac{{\partial \overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }} }}{{\partial y^{ + } }} + \frac{{\overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }} }}{{\tau_{\text{C}}^{ + } }} = - \frac{{\partial \overline{{V^{\prime}_{{{\text{P}}y}} V^{\prime}_{{{\text{P}}y}} }} }}{{\partial y^{ + } }} - \frac{\nu }{{\left( {u^{*} } \right)^{3} }}\left( {1 - \frac{{\rho_{\text{f}} }}{{\rho_{\text{P}} }}} \right)g, $$

(A9)

where ρ _P is the density of particles (inclusions): 3690 kg/m³, g is the gravity acceleration rate (9.81 m²/s). τ ⁺_C is the normalized form of the Cunningham corrected particle relaxation time, which could be expressed as follows:

$$ \tau_{\text{C}}^{ + } = \frac{{\tau_{\text{C}} }}{{{\nu \mathord{\left/ {\vphantom {\nu {\left( {u^{*} } \right)^{2} }}} \right. \kern-0pt} {\left( {u^{*} } \right)^{2} }}}} = \frac{{\tau_{\text{P}} C_{c} }}{{{\nu \mathord{\left/ {\vphantom {\nu {\left( {u^{*} } \right)^{2} }}} \right. \kern-0pt} {\left( {u^{*} } \right)^{2} }}}} = \frac{{2\rho_{\text{P}} r^{2} \left( {u^{*} } \right)^{2} }}{{9\rho_{\text{f}} \nu^{2} }}C_{c}, $$

(A10)

where τ _P is the particle relaxation time that follows from Stokes drag.

The variable $\overline{{V^{\prime}_{{{\text{P}}y}} V^{\prime}_{{{\text{P}}y}} }}$ is the ensemble-averaged squared particle fluctuating velocity (V ^′_Py ) normal to the interface.[48] The variable $\overline{{V^{\prime}_{{{\text{P}}y}} V^{\prime}_{{{\text{P}}y}} }}$ was assumed to be equal to the Reynolds normal stress of the fluid $\overline{{V^{\prime}_{{{\text{f}}y}} V^{\prime}_{{{\text{f}}y}} }}$.[52] The latter could be obtained by the relationship $\overline{{V^{\prime}_{{{\text{f}}y}} V^{\prime}_{{{\text{f}}y}} }} = \left( {\sqrt {\overline{{V^{\prime 2}_{{{\text{f}}y}} }} } } \right)^{2}$, where the RMS velocity of fluid was modified by Guha[50,51] from the result of Kallio and Reeks[86] and are expressed as follows:

$$ \sqrt {\overline{{V^{\prime 2}_{{{\text{f}}y}} }} } = \frac{{0.005\left( {y^{ + } } \right)^{2} }}{{1 + 0.002923\left( {y^{ + } } \right)^{2.128} }} $$

(A11)

2. Boundary conditions, numerical solution, and validation

At the upper^‡ boundary y ⁺ = 60, the dimensionless particle convective velocity $\overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }} = 0$ and the dimensionless particle mass concentration C ⁺_P  = 1, i.e., the same with the bulk concentration. The location of the lower⁴ boundary is affected by the interface roughness.[50] The expression of Wood[87] was used at the lower boundary as follows:

$$ y^{ + } = 0.45k_{S}^{ + } + r^{ + } = \left[ {0.45k_{S} + r} \right]\left( {{{u^{*} } \mathord{\left/ {\vphantom {{u^{*} } \nu }} \right. \kern-0pt} \nu }} \right), $$

(A12)

where k _S and k ⁺ _S are the roughness height and its normalized form, respectively. At the lower boundary, the dimensionless particle mass concentration C ⁺_P  = 0, which represents that no inclusions exist at the interface.

The treatment of the thermophoretic force by Guha[50] and the temperature file at the turbulent BL from Kay and Nedderman[88] are retained in this study. Specifically, the temperature can be obtained as follows:

$$ \frac{{T - T_{I} }}{\Delta T} = \frac{{{ \Pr } \cdot y^{ + } }}{{\Delta T_{60}^{ + } }}\;{\text{when}}\;y^{ + } < 5, $$

$$ \frac{{T - T_{I} }}{\Delta T} = \frac{{ 5 {\text{Pr}} + 5 {\text{ln}}\left( { 0. 2 {\text{Pr}} \cdot y^{ + } + 1 - { \Pr }} \right)}}{{\Delta T_{60}^{ + } }}\;{\text{when}}\; 5 < y^{ + } < 30, $$

(A13)

$$ \frac{{T - T_{I} }}{\Delta T} = \frac{{ 5 {\text{Pr}} + 5 {\text{ln}}\left( {1 + 5{ \Pr }} \right) + 2.5\ln ({{y^{ + } } \mathord{\left/ {\vphantom {{y^{ + } } {30}}} \right. \kern-0pt} {30}})}}{{\Delta T_{60}^{ + } }}\;{\text{when}}\; 30 < y^{ + } < 60, $$

where ΔT ⁺₆₀  = 5Pr + 5ln(1 + 5Pr) + 2.5ln(60/30). T _I is the temperature at the interface: 1818 K (1545 °C), ΔT is the temperature difference between the upper boundary and the interface: 5 K (5 °C), and Pr is Prandtl number: 0.08.

The numerical solution procedure is provided in Reference 52. Essentially, the one-dimensional turbulent BL was discretized into 10,000 grids. First, $\frac{\partial }{{\partial y^{ + } }}\left\{ {{\text{Eq}} .\left[ {\text{A2}} \right]} \right\} = 0$ with the BCs were solved and the inclusion mass concentration profile were obtained. Second, the $\overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }}$ of all the grids were obtained by solving the discretized Eq. [A9] with the BCs. Finally, the deposition velocity was calculated by using Eq. [A2] with the inputs of inclusion mass concentration and $\overline{{V_{{{\text{P}}y}}^{{{\text{C}} + }} }}$.

The experimental data of the gravity-driven aerosol particle deposition onto horizontal walls at floor were chosen for validation. The deposition process is similar to the buoyancy-driven inclusion particle deposition onto top slags. Since in these cases, the buoyancy and the gravity play a similar role for inclusions and aerosol particles deposition. The input parameters of the aerosol particle and air system are the same as those used in a previous study.[52] The validation with the experimental data of Sehmel[89] is shown in Figure A1. The agreement is good for the extension of the turbophoretic model to study the inclusion-liquid steel system.

Fig. A1

Comparison of the calculated deposition velocities with the experimental data (Sehmel[89]) for aerosol particle deposition on floors