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

## Abstract

*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.*,

*L*

_{jet}is the free liquid jet length (m),

*Q*

_{G}is the gas flow rate (m

^{3}/s),

*Q*

_{L}is the liquid flow rate (m

^{3}/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/m

^{3}). 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.

## Nomenclature

- BCS
Bloom casting shroud

*C*Interface curvature

**C**_{µ}Empirical constant of the turbulence model

**C**_{2}Constant

*D*_{CN}Collector nozzle diameter

- \( D_{{{\text{CN}}_{\text{BCS}} }} \)
Collector nozzle diameter of BCS

- \( D_{{{\text{CN}}_{\text{SCS}} }} \)
Collector nozzle diameter of SCS

*D*_{sh}Shroud diameter

- \( D_{{{\text{sh}}_{\text{SCS}} }} \)
Diameter of slab casting shroud

- \( D_{{{\text{sh}}_{\text{BCS}} }} \)
Diameter of bloom casting shroud

*F*_{σ}Surface tension force per unit volume

*Fr*_{jet}Jet Froude number

**k**Turbulent kinetic energy

*L*_{sh}Length of the shroud

*L*_{jet}Free liquid jet length

*L*_{jet, BCS}Free liquid jet length in BCS

*L*_{jet,SCS}Free liquid jet length in SCS

*P*Dynamic pressure referenced to the local hydrostatic pressure

*Q*_{G}Gas flow rate

*Q*_{L}Liquid flow rate

*r*Radial distance from the centerline of the shroud

*R*_{sh}Radius of the shroud

- SCS
Slab casting shroud

*v*_{i,m}Time averaged, mixture velocities in the

*i*th direction*v*_{j,m}Time averaged, mixture velocities in

*j*th direction- \( \rho_{\text{L}} \)
Density of liquid

- \( \rho_{\text{G}} \)
Density of gas

- \( \sigma_{{}} \)
Surface tension

- \( \alpha_{\text{G}} \)
Critical gas flow rate

*λ*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

*G*_{k}Generation of turbulence kinetic energy

*σ*_{k}Turbulent Prandtl numbers for

*k**σ*_{k}Turbulent Prandtl numbers for

*ɛ*- \( \mu_{\text{t}} \)
Turbulent viscosity

- \( \varepsilon \)
Dissipation rate of turbulent kinetic energy

## Notes

### Acknowledgment

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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