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

• Prince K. Singh
• Dipak Mazumdar
Article

## 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 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.

## Nomenclature

BCS

Bloom casting shroud

C

Interface curvature

Cµ

Empirical constant of the turbulence model

C2

Constant

DCN

Collector nozzle diameter

$$D_{{{\text{CN}}_{\text{BCS}} }}$$

Collector nozzle diameter of BCS

$$D_{{{\text{CN}}_{\text{SCS}} }}$$

Collector nozzle diameter of SCS

Dsh

Shroud diameter

$$D_{{{\text{sh}}_{\text{SCS}} }}$$

Diameter of slab casting shroud

$$D_{{{\text{sh}}_{\text{BCS}} }}$$

Diameter of bloom casting shroud

Fσ

Surface tension force per unit volume

Frjet

Jet Froude number

k

Turbulent kinetic energy

Lsh

Length of the shroud

Ljet

Free liquid jet length

Ljet, BCS

Free liquid jet length in BCS

Ljet,SCS

Free liquid jet length in SCS

P

Dynamic pressure referenced to the local hydrostatic pressure

QG

Gas flow rate

QL

Liquid flow rate

r

Radial distance from the centerline of the shroud

Rsh

Radius of the shroud

SCS

Slab casting shroud

vi,m

Time averaged, mixture velocities in the ith direction

vj,m

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

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

Gk

Generation of turbulence kinetic energy

σk

Turbulent Prandtl numbers for k

σk

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).

## References

1. 1.
Prince K. Singh and Dipak Mazumdar, Metallurgical and Materials Trans B, 2018, Vol.48, pp.1945-1962.
2. 2.
Saikat Chatterjee, Donghui Li and Kinnor Chattpopadhyaya: Materials and Metallurgical Transactions B, 2018, 49B, pp 756-766.
3. 3.
D Mazumdar, Prince K Singh, Rohit K Tiwari: ISIJ International, 2018, Vol.58, pp. 1545–1547.
4. 4.
D.Mazumdar and J.W.Evans: Modeling of Steelmaking Processes, CRC Press, Boca Raton, 2009, pp. 217-220.
5. 5.
H.G. Weller: Technical Report TR/HGW/04, Open CFD Ltd., 2008.Google Scholar
6. 6.
Laihua Wang, Hae-Geon Lee, Peter Hayes, Steel Research, 1995, 66, pp.279-286.
7. 7.
L.T. Wang, Q.Y. Zhang, C.H. Deng and Z.B. Li, ISIJ Int., 2005, vol. 45, pp. 1138-44.
8. 8.
R.K. Tiwari: M.Tech Thesis, IIT Kanpur, India, 2018.Google Scholar
9. 9.
K. E. Wardle and H. G. Weller, International Journal of chemical Engineering,2013 Vol.2013, pp.1-13.
10. 10.
ANSYS® FLUENT, 18.0: Theory Guide, Chap. 4, December 2016.Google Scholar
11. 11.
A.K.Bin: Chemical Engineering Science, 1993, Vol.48, pp.3585-3630.
12. 12.
G.M.Evans, G.J.Jameson and C.D.Reilly: Experimental and Thermal Fluid Sciences, 1996, Vol.12, pp.142-149.
13. 13.
P.K. Singh: Unpublished Research, Indian Institute of Technology, Kanpur, 2018.Google Scholar
14. 14.
R.Guthrie: Engineering in Process Metallurgy, Clarendon Press, Oxford, UK,1989, pp.152-157.Google Scholar
15. 15.
MATLAB R2016a: Academic use, February 2016.Google Scholar