Introduction
Mechanism | Description | Ref.* | |
---|---|---|---|
1 | deposition of indigenous NMIs | NMI particles, as deoxidation products in the steel melt, are transported toward the SEN wall, and they are deposited on the SEN wall, leading to a buildup of clog layer | |
2 | thermochemical reactions | thermochemical reactions between the SEN refractory material and the steel lead to the (in-situ) formation of clog layer on the SEN wall | |
3 | local oxidation at the SEN wall | negative pressure inside the SEN results in oxygen drawing through the SEN refractory (porous) material into the inner SEN wall; and the oxygen reacts with the steel melt to form oxides on the SEN wall | |
4 | precipitation of NMIs | temperature drop of the melt near the SEN wall leads to lower solubility of oxygen in the steel melt and subsequent precipitation of alumina at SEN-steel interface | |
5 | possible solidification of the steel | if superheat of the melt is low and heat transfer from the SEN is high, the steel may solidify at the SEN wall |
Modeling
Assumptions
Governing Equations for the Flow and Particle Tracking
Conservation Equations | Symbol Definition | ||
---|---|---|---|
Mass |
\( \nabla \cdot (\rho \vec{u}) = 0 \)
| [1] | ρ: density of fluid (kg/m3) μ: viscosity of fluid (kg m−1 s−1) t: time (s) \( \vec{u} \): velocity of fluid (m/s) k: turbulence kinetic energy (m2 s−2) ω: specific dissipation rate of turbulence kinetic energy (s−1) Γk, Γω: diffusivity for k and ω (kg m−1 s−1, kg m−2) \( \tilde{G}_{k} \), Gω: generation of turbulence kinetic energy for k and ω (kg m−1 s−3, kg m−2 s−2) Yk, Yω: dissipation of k and ω (kg m−1 s−3, kg m−2 s−2) Dω: cross-diffusion term of ω (kg m−2 s−2) Sk, Sω, and \( \vec{S}_{u} \): source term due to porous medium of clog (kg m−1 s−3, kg m−2 s−2, and kg m−2 s−2) ϕ: stands for \( \vec{u} \), k, or ω Kper: permeability of the clog Dpore: average diameter of large open pores in the clog \( \bar{f}_{\text{p}} \): average particle fraction fclog: fraction of the clog in the local computational cell n: interpolation correction power |
Momentum |
\( \rho \frac{{\partial \vec{u}}}{\partial t} + \nabla \cdot (\rho \vec{u}\vec{u}) = - \nabla p + \nabla \cdot (\mu \nabla \vec{u}) + \vec{S}_{u} \)
| [2] | |
Turbulence Kinetic Energy |
\( \rho \frac{\partial k}{\partial t} + \nabla \cdot (\rho k\vec{u}) = \nabla \cdot \left( {\varGamma_{k} \nabla k} \right) + \tilde{G}_{k} - Y_{k} + S_{k} \)
| [3] | |
Specific Dissipation Rate |
\( \rho \frac{\partial \omega }{\partial t} + \nabla \cdot (\rho \omega \vec{u}) = \nabla \cdot (\varGamma_{\omega } \nabla k) + G_{\omega } - Y_{\omega } + D_{\omega } + S_{\omega } \)
| [4] | |
\( S_{\phi } = - \frac{\mu }{{K_{\text{per}} }}\phi \)
where \( K_{\text{per}} = \frac{{\left( {1 - \bar{f}_{\text{p}} } \right)\left( {1 - \bar{f}_{\text{p}}^{3} } \right)}}{{108\left( {\bar{f}_{\text{p}}^{1/3} - \bar{f}_{\text{p}} } \right)}}D_{\text{pore}}^{2} \frac{1}{{f_{c\log }^{n} }} \) | [5] |
Transport Equation | Symbol Definition | ||
---|---|---|---|
\( m_{p} \frac{{d\vec{u}_{p} }}{dt} = \vec{F}_{B} + \vec{F}_{D} + \vec{F}_{L} + \vec{F}_{\text{VM}} + \vec{F}_{\text{press}} \)
| [6] | \( \vec{g} \): gravity (m s−2) ρp: density of particle (kg m−3) dp: diameter of particle (m) CD: drag coefficient (—) Rep: particle Reynolds number (—) Us: difference between instantaneous streamwise velocities for fluid and particle (m s−1) G: local velocity gradients (s−1) J: correction factor of the lift force (—) | |
Buoyancy Force |
\( \vec{F}_{\text{B}} = \frac{{(\rho_{\text{p}} - \rho )\pi d_{\text{p}}^{3} }}{6}\vec{g} \)
| [7] | |
Drag Force |
\( \vec{F}_{\text{D}} = \frac{1}{8}\pi d_{\text{p}}^{2} \rho C_{\text{D}} \left| {\overrightarrow {{u^{'} }} - \vec{u}_{\text{p}} } \right|\left( {\overrightarrow {{u^{'} }} - \vec{u}_{\text{p}} } \right) \)
\( C_{\text{D}} = \left\{ {\begin{array}{*{20}c} {\frac{24}{{{\text{Re}}_{\text{p}} }}, } & {{\text{Re}}_{\text{p}} < 0.1} \\ {\frac{24}{{{\text{Re}}_{\text{p}} }}\left( {1 + 0.15{\text{Re}}_{\text{p}}^{0.687} } \right),} & { 0. 1\le {\text{Re}}_{p} \le 10^{3} } \\ {0.44,} & {{\text{Re}}_{p} > 10^{3} } \\ \end{array} } \right. \)
\( \text{Re} _{{\text{p}}} = \frac{{|\vec{u} - \vec{u}_{{\text{p}}} |d_{{\text{p}}} \rho }}{\mu } \)
| [8] | |
Lift Force |
\( \vec{F}_{\text{L}} = - \frac{9}{4\pi }\mu d_{\text{p}}^{2} U_{\text{s}} {\text{sgn}}(G)\left( {\frac{\rho }{\mu }|G|} \right)^{1/2} J \)
| [9] | |
Virtual Mass Force |
\( \vec{F}_{\text{VM}} = \frac{{\rho \pi d_{\text{p}}^{3} }}{12}\frac{\text{d}}{{{\text{d}}t}}\left( {\vec{u} - \vec{u}_{\text{p}} } \right) \)
| [10] | |
Pressure Gradient Force |
\( \vec{F}_{\text{press}} = \frac{{\rho \pi d_{\text{p}}^{3} }}{6}\frac{{D\vec{u}}}{Dt} \)
| [11] |
Clog Growth Algorithm
Simulation Settings
Density of Steel Melt (kg m−3) | 7200 |
Density of Particle (kg m−3) | 3700 |
Viscosity of Steel Melt (kg m−1 s−1) | 0.006 |
Diameter of Particles (µm) | 2–10 |
Time-Step for Flow Calculation (s) | 0.01 |
Average Volume Fraction of Solid Particles (\( \bar{f}_{\text{p}} \)) | 0.55 |
Average Pore Diameter in the c log (µm) | 20 |
Number of Representative Particles (N-Factor) | 1–1257 |
Parameter Studies
Mesh Type
Test cases
Results
Transient clog growth
Clog growth without flow (case 1)
Clog growth with flow (case 2)
N-Factor
Test cases
dp (µm) | N-Factor | Particle Number Injection Rate (Million Particles Per Second) |
---|---|---|
2 | 1, 10, 25, 78, 157, 314, 628, 1257 | 860, 85.5, 34.2, 10.944, 5.472, 2.736, 1.368, 0.684 |
4 | 1, 6, 10, 31, 78, 157 | 102.6, 17.1, 10.26, 3.42, 1.368, 0.684 |
6 | 1, 3, 6, 9, 23, 47 | 30.78, 10.26, 5.472, 3.42, 1.368, 0.684 |
8 | 1, 3, 5, 10, 20 | 12.996, 4.788, 2.736, 1.368, 0.684 |
10 | 1, 3, 5, 10 | 6.84, 2.052, 1.368, 0.684 |