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

  • Prince K. SinghEmail author
  • Dipak Mazumdar


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 kε 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, Ljet is the free liquid jet length (m), QG is the gas flow rate (m3/s), QL is the liquid flow rate (m3/s), Dsh is shroud diameter (m), DCN is the collector nozzle diameter (m), σ is the interfacial tension (N/m), and ρG as well as ρL 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.



Bloom casting shroud


Interface curvature


Empirical constant of the turbulence model




Collector nozzle diameter

\( D_{{{\text{CN}}_{\text{BCS}} }} \)

Collector nozzle diameter of BCS

\( D_{{{\text{CN}}_{\text{SCS}} }} \)

Collector nozzle diameter of SCS


Shroud diameter

\( D_{{{\text{sh}}_{\text{SCS}} }} \)

Diameter of slab casting shroud

\( D_{{{\text{sh}}_{\text{BCS}} }} \)

Diameter of bloom casting shroud


Surface tension force per unit volume


Jet Froude number


Turbulent kinetic energy


Length of the shroud


Free liquid jet length

Ljet, BCS

Free liquid jet length in BCS


Free liquid jet length in SCS


Dynamic pressure referenced to the local hydrostatic pressure


Gas flow rate


Liquid flow rate


Radial distance from the centerline of the shroud


Radius of the shroud


Slab casting shroud


Time averaged, mixture velocities in the ith direction


Time averaged, mixture velocities in jth direction

\( \rho_{\text{L}} \)

Density of liquid

\( \rho_{\text{G}} \)

Density of gas

\( \sigma_{{}} \)

Surface tension

\( \alpha_{\text{G}} \)

Critical gas flow rate


Scaling factor

\( \rho_{m} \)

Mixture density

\( \alpha_{1} \)

Volume fraction of phase 1

\( \alpha_{2} \)

Volume fraction of phase 2

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} \)

Normal vector to the interface

\( \delta_{\text{s}} \)

Dirac delta function

\( \overrightarrow {{u_{\text{c}} }} \)

Velocity is applied normal to the interface


Generation of turbulence kinetic energy


Turbulent Prandtl numbers for k


Turbulent Prandtl numbers for ɛ

\( \mu_{\text{t}} \)

Turbulent viscosity

\( \varepsilon \)

Dissipation rate of turbulent kinetic energy



The authors gratefully acknowledge Mr. Rohit K. Tiwari, a former graduate student in the Process and Steel research Laboratory, IIT Kanpur for full scale computational results presented in Figure 11(a).


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Copyright information

© The Minerals, Metals & Materials Society and ASM International 2019

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringIndian Institute of TechnologyKanpurIndia

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