Mathematical Modelling of Gas–Liquid, Two-Phase Flows in a Ladle Shroud

Abstract

Differential and macroscopic models of argon–steel flows in ladle shroud have been developed. In this, argon–steel, two-phase flow phenomena have been formulated via a transient, three dimensional, turbulent flow model, based on the volume of fluid (VOF) calculation procedure. While realizable k–ε turbulence model has been applied to map turbulence, commercial, CFD software ANSYS-Fluent™ (Version 18), has been applied to carry out numerical calculations. Predictions from the model have been directly assessed against experimental measurements across the range of shroud dimensions and volumetric flow rates typically practiced in the industry. It is demonstrated that the two-phase turbulent flow model captures the general features of gas–liquid flows in ladle shroud providing estimates of free jet length and threshold gas flow rates (required to halt air ingression) which are in agreement with corresponding experimental measurements. In the absence of differential solutions, a macroscopic model has been worked out through dimensional analysis embodying multiple non-linear regression. It is shown that dimensionless free jet length in bloom and slab casting shrouds can be estimated reasonably accurately from the following correlation (in SI unit), viz.,

$$ \frac{{L_{{{\text{jet}}}} }}{{D_{{{\text{CN}}}} }} = 2.8 \times 10^{{ - 2}} \left( {\frac{{Q_{{\text{G}}} }}{{Q_{{\text{L}}} }}} \right)^{{1.14}} \left( {\frac{{gD_{{{\text{CN}}}}^{5} }}{{Q_{{\text{L}}}^{2} }}} \right)^{{0.8}} \left( {\frac{{\sigma D_{{{\text{CN}}}}^{3} }}{{\rho _{{\text{L}}} Q_{{\text{L}}}^{2} }}} \right)^{{ - 0.9}} \left( {\frac{{D_{{{\text{sh}}}} }}{{D_{{{\text{CN}}}} }}} \right)^{{2.0}} \left( {\frac{{\rho _{{\text{G}}} }}{{\rho _{{\text{L}}} }}} \right)^{{ - 0.30}} $$

in which, L_jet is the free liquid jet length (m), Q_G is the gas flow rate (m³/s), Q_L is the liquid flow rate (m³/s), D_sh is shroud diameter (m), D_CN is the collector nozzle diameter (m), σ is the interfacial tension (N/m), and ρ_G as well as ρ_L are respectively density of gas and liquid (kg/m³). It is demonstrated that the proposed correlation is consistent with the laws of physical modeling and leads to estimates that are in good agreement with predictions from the differential models, for both air-water as well as argon–steel systems. Numerical simulations as well as macroscopic modeling have indicated that thermo-physical properties of the gas–liquid system are important and exert some influences on the gas–liquid, two-phase, flow in ladle shrouds, albeit not to a large extent. Despite dissimilar thermo-physical properties, full scale water modeling appears to be sufficiently predictive and provides reasonable macroscopic descriptions of the two-phase flow phenomena in industrial ladle shroud systems.