# Numerical Modeling of Equal and Differentiated Gas Injection in Ladles: Effect on Mixing Time and Slag Eye

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## Abstract

**:**

## 1. Introduction

## 2. Methodology: Numerical Model Development

^{−3}. Approximately 1200 iterations were required to get the convergence in around 36 h of CPU time in a computer with 8 MB in RAM with an Intel Core

^{®}i7-3770 processor of 3.4 GHZ. Table 2 lists all the numerical simulations performed in this study which is based on a full factorial experimental design at two levels with the three variables, namely, gas flow rate, dual gas injection ratio and (slag) oil thickness, as mentioned in Jardón-Pérez et al. [15].

## 3. Results and Discussion

#### 3.1. Model Validation

#### 3.2. Turbulence Modeling

#### 3.3. Slag Eye Modeling

#### 3.4. Mixing Time Modeling

## 4. Conclusions

- The numerical model using CFD predicts the hydrodynamic behavior of the ladle well, in comparison with the physical model. Turbulent kinetic energy is adequately and qualitatively predicted, although it is somewhat overestimated. It can be said that the model qualitatively predicts the influence of the gas flow, the distribution of the flows and the level of slag on the distribution of velocities and turbulence.
- The predicted slag eye shows a good agreement with the experimental results with slag eye area as a percentage of the total surface. However, due to the interphase interaction, the slag eye from differentiated gas injection is not captured completely by the model.
- The numerical model does not fully predict the effect of differentiated gas injection, since the drag model used does not exactly simulate the interaction between both recirculation zones, hence predicting a smaller area of low-velocity zones.
- There is a deviation in predicted mixing time from experimental mixing time for both equal and differentiated gas injection, which becomes significant at a high gas flow rate and a high slag thickness.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ghosh, A. Secondary Steelmaking: Principles and Applications; CRC Press LLC: Boca Raton, FL, USA, 2001. [Google Scholar]
- Du, S. Improving process design in steelmaking. In Fundamentals of Metallurgy; Elsevier Inc.: Amsterdam, The Netherlands, 2005; pp. 369–398. [Google Scholar]
- Mazumdar, D.; Evans, J.W. Macroscopic Models for Gas Stirred Ladles. ISIJ Int.
**2004**, 44, 447–461. [Google Scholar] [CrossRef] [Green Version] - Amaro-Villeda, A.M.; Ramírez-Argáez, M.A.; Conejo, A.N. Effect of Slag Properties on Mixing Phenomena in Gas-stirred Ladles by Physical Modeling. ISIJ Int.
**2014**, 54, 1–8. [Google Scholar] [CrossRef] [Green Version] - Hoang, Q.N.; Ramírez-Argáez, M.A.; Conejo, A.N.; Blanpain, B.; Dutta, A. Numerical Modeling of Liquid–Liquid Mass Transfer and the Influence of Mixing in Gas-Stirred Ladles. JOM
**2018**, 70, 2109–2118. [Google Scholar] [CrossRef] - Asai, S.; Okamoto, T.; He, J.-C.; Muchi, I. Mixing Time of Refining Vessels Stirred by Gas Injection. Trans. Iron Steel Inst. Jpn.
**1983**, 23, 43–50. [Google Scholar] [CrossRef] [Green Version] - Sano, M.; Mori, K. Fluid flow and mixing characteristics in a gas-stirred molten metal bath. Trans. Iron Steel Inst. Jpn.
**1983**, 23, 169–175. [Google Scholar] [CrossRef] - Joo, S.; Guthrie, R.I.L. Modeling flows and mixing in steelmaking ladles designed for single- and dual-plug bubbling operations. MTB
**1992**, 23, 765–778. [Google Scholar] [CrossRef] - Krishnapisharody, K.; Irons, G.A. An Analysis of Recirculatory Flow in Gas-Stirred Ladles. Steel Res. Int.
**2010**, 81, 880–885. [Google Scholar] [CrossRef] - Khajavi, L.T.; Barati, M. Liquid Mixing in Thick-Slag-Covered Metallurgical Baths—Blending of Bath. Met. Mater. Trans. B
**2010**, 41, 86–93. [Google Scholar] [CrossRef] - Chattopadhyay, K.; Sengupta, A.; Ajmani, S.K.; Lenka, S.N.; Singh, V. Optimisation of dual purging location for better mixing in ladle: A water model study. Ironmak. Steelmak.
**2009**, 36, 537–542. [Google Scholar] [CrossRef] - Liu, H.; Qi, Z.; Xu, M. Numerical Simulation of Fluid Flow and Interfacial Behavior in Three-phase Argon-Stirred Ladles with One Plug and Dual Plugs. Steel Res. Int.
**2011**, 82, 440–458. [Google Scholar] [CrossRef] - Haiyan, T.; Xiaochen, G.; Guanghui, W.; Yong, W. Effect of Gas Blown Modes on Mixing Phenomena in a Bottom Stirring Ladle with Dual Plugs. ISIJ Int.
**2016**, 56, 2161–2170. [Google Scholar] [CrossRef] [Green Version] - Tang, H.; Liu, J.; Zhang, S.; Guo, X.; Zhang, J. A novel dual plugs gas blowing mode for efficient ladle metallurgy. Ironmak. Steelmak.
**2019**, 46, 405–415. [Google Scholar] [CrossRef] - Jardón-Pérez, L.E.; González-Morales, D.R.; Trápaga, G.; González-Rivera, C.; Ramírez-Argáez, M.A. Effect of Differentiated Injection Ratio, Gas Flow Rate, and Slag Thickness on Mixing Time and Open Eye Area in Gas-Stirred Ladle Assisted by Physical Modeling. Metals
**2019**, 9, 555. [Google Scholar] [CrossRef] [Green Version] - Mazumdar, D.; Dhandapani, P.; Sarvanakumar, R. Modeling and Optimisation of Gas Stirred Ladle Systems. ISIJ Int.
**2017**, 57, 286–295. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Ersson, M.; Liu, H.; Jönsson, P.G.; Gan, Y. A Review of Physical and Numerical Approaches for the Study of Gas Stirring in Ladle Metallurgy. Met. Mater. Trans. B
**2018**, 50, 555–577. [Google Scholar] [CrossRef] [Green Version] - Li, B.; Yin, H.; Zhou, C.Q.; Tsukihashi, F. Modeling of Three-phase Flows and Behavior of Slag/Steel Interface in an Argon Gas Stirred Ladle. ISIJ Int.
**2008**, 48, 1704–1711. [Google Scholar] [CrossRef] [Green Version] - Conejo, A.N.; Mishra, R.; Mazumdar, D. Effects of Nozzle Radial Position, Separation Angle, and Gas Flow Partitioning on the Mixing, Eye Area, and Wall Shear Stress in Ladles Fitted with Dual Plugs. Met. Mater. Trans. B
**2019**, 50, 1490–1502. [Google Scholar] [CrossRef] - Villela-Aguilar, J.D.J.; Ramos-Banderas, J.Á.; Hernández-Bocanegra, C.A.; Urióstegui-Hernández, A.; Solorio-Díaz, G. Optimization of the Mixing Time Using Asymmetrical Arrays in Both Gas Flow and Injection Positions in a Dual-plug Ladle. ISIJ Int.
**2020**, 60, 1172–1178. [Google Scholar] [CrossRef] [Green Version] - Shih, T.-H.; Liou, W.W.; Shabbir, A.; Yang, Z.; Zhu, J. A new k-ϵ eddy viscosity model for high reynolds number turbulent flows. Comput. Fluids
**1995**, 24, 227–238. [Google Scholar] [CrossRef] - Troshko, A.A.; Hassan, Y.A. A two-equation turbulence model of turbulent bubbly flows. Int. J. Multiph. Flow
**2001**, 27, 1965–2000. [Google Scholar] [CrossRef] - Krishnakumar, K.; Ballal, N.B.; Sinha, P.K.; Sardar, M.K.; Jha, K.N. Water Model Experiments on Mixing Phenomena in a VOD Ladle. ISIJ Int.
**1999**, 39, 419–425. [Google Scholar] [CrossRef] - González-Bernal, R.; Solorio-Diaz, G.; Ramos-Banderas, A.; Torres-Alonso, E.; Hernández-Bocanegra, C.A.; Zenit, R. Effect of the Fluid-Dynamic Structure on the Mixing Time of a Ladle Furnace. Steel Res. Int.
**2018**, 89, 1700281. [Google Scholar] [CrossRef] - Lou, W.; Zhu, M. Numerical Simulation of Slag-metal Reactions and Desulfurization Efficiency in Gas-stirred Ladles with Different Thermodynamics and Kinetics. ISIJ Int.
**2015**, 55, 961–969. [Google Scholar] [CrossRef] [Green Version] - Zhu, M.-Y.; Inomoto, T.; Sawada, I.; Hsiao, T.-C. Fluid Flow and Mixing Phenomena in the Ladle Stirred by Argon through Multi-Tuyere. ISIJ Int.
**1995**, 35, 472–479. [Google Scholar] [CrossRef] - Ramasetti, E.; Visuri, V.-V.; Sulasalmi, P.; Fabritius, T.; Saatio, T.; Li, M.; Shao, L. Numerical Modeling of Open-Eye Formation and Mixing Time in Argon Stirred Industrial Ladle. Metals
**2019**, 9, 829. [Google Scholar] [CrossRef] [Green Version] - Nunes, R.P.; Pereira, J.A.M.; Vilela, A.C.F.; Laan, F.T.V. Visualisation and analysis of the fluid flow structure inside an elliptical steelmaking ladle through image processing techniques. J. Eng. Sci. Technol.
**2007**, 2, 139–150. [Google Scholar] - Jardón-Pérez, L.E.; Amaro-Villeda, A.; González-Rivera, C.; Trápaga, G.; Conejo, A.N.; Ramírez-Argáez, M.A. Introducing the Planar Laser-Induced Fluorescence Technique (PLIF) to Measure Mixing Time in Gas-Stirred Ladles. Met. Mater. Trans. B
**2019**, 50, 2121–2133. [Google Scholar] [CrossRef] - Ascanio, G. Mixing time in stirred vessels: A review of experimental techniques. Chin. J. Chem. Eng.
**2015**, 23, 1065–1076. [Google Scholar] [CrossRef]

**Figure 1.**Computational domain of the ladle used for the numerical simulations presented in this study. The two vertical sections indicating the gas injection inlets and the top horizontal section indicating the slag layer are comparatively denser than the remaining mesh domain. The inset shows the mesh density for the slag zone which is different from the melt zone. The top portion of the mesh (half along the symmetry plane) is shown separately.

**Figure 2.**Flow patterns of the eight case studies obtained with the experimental model (particle image velocimetry (PIV) technique) in the longitudinal plane. (

**a**) through (

**h**) are the experiments described in Table 2.

**Figure 3.**Flow patterns of the eight case studies obtained with the numerical model and shown along the same longitudinal plane. (

**a**) through (

**h**) are the experiments described in Table 2. The cases presented in this study are in the same order as in the experimental study of Jardón-Pérez et al. [15].

**Figure 5.**Comparison of experimental (continuous line) and numerical (dotted line) mean velocity radial profiles at h/H = 0.8 for experiments b (

**a**) and g (

**b**); axial profiles at r/R = −0.75 for experiments b (

**c**) and g (

**d**); and axial profiles at r/R = 0.75 for experiments b (

**e**) and g (

**f**).

**Figure 6.**Contours of turbulent kinetic energy (k) of the eight case studies obtained with the numerical model and shown along the same longitudinal plane. (

**a**) through (

**h**) are the experiments described in Table 2. The cases presented in this study are in the same order as in the experimental study of Jardón-Pérez et al. [15].

**Figure 8.**Comparison of the time-averaged photograph of the slag eye obtained with the (

**a**) experimental model (operating condition of differentiated dual gas injection with 5% (slag) oil thickness and 2.22 L/min gas flow rate) and the corresponding time-averaged (

**b**) numerical model prediction. Dimensions of measured and predicted open slag eyes are also indicated.

**Figure 9.**Comparison between the model prediction (dotted line) and experimental mixing time (black dots) obtained by Jardón-Pérez et al. [15].

**Figure 10.**Effect of the gas flow rate on the mixing time predicted by the model and experimental measurements for (

**a**) low gas flow and (

**b**) differentiated gas flow.

Boundary | Mass Transport Condition | Momentum Transport Condition |
---|---|---|

Inlets | velocity inlet of air with turbulent intensity | velocity inlet of air with turbulent intensity |

Outlet | pressure outlet with air backflow | pressure outlet with air backflow |

bottom wall | impermeable boundary | no slip with standard wall functions |

lateral wall | impermeable boundary | no slip with standard wall functions |

**Table 2.**Design of experiment with high and low values of the three variables, namely, gas flow rate, dual gas injection ratio and (slag) oil thickness for the eight case studies presented in this study.

Cases | Experiment Number | (Slag) Oil Thickness (hs) (%) | Gas Flow Rate (Q) (L/min) | Dual Gas Injection Ratio (P) (%/%) |
---|---|---|---|---|

1 | a | 3 | 1.54 | 50/50 |

2 | b | 3 | 2.22 | 50/50 |

3 | c | 3 | 1.54 | 25/75 |

4 | d | 3 | 2.22 | 25/75 |

5 | e | 5 | 1.54 | 50/50 |

6 | f | 5 | 2.22 | 50/50 |

7 | g | 5 | 1.54 | 25/75 |

8 | h | 5 | 2.22 | 25/75 |

**Table 3.**Mean values of velocity v (× 10

^{−2}m/s) for the experimental and numerical model along the longitudinal plane (symmetry plane) for the eight cases presented in this study.

Low 1.54 L/min Gas Flow Rate (Q) | High 2.22 L/min Gas Flow Rate (Q) | |||
---|---|---|---|---|

(Slag) Oil Thickness (hs) | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio |

3% oil thickness | ||||

experimental | 4.36 ± 2.61 | 5.33 ± 3.17 | 4.18 ± 2.72 | 4.53 ± 3.16 |

numerical | 4.91 ± 4.94 | 5.98 ± 6.01 | 5.03 ± 5.24 | 5.91 ± 6.14 |

difference (%) | 12.81 | 12.13 | 20.46 | 30.37 |

5% oil thickness | ||||

experimental | 4.62 ± 2.86 | 4.74 ± 3.09 | 3.60 ± 2.27 | 4.53 ± 2.77 |

numerical | 5.07 ± 5.15 | 5.30 ± 5.38 | 4.68 ± 5.00 | 5.52 ± 5.87 |

difference (%) | 9.85 | 11.87 | 30.07 | 22.01 |

**Table 4.**Mean values of turbulent kinetic energy k (× 10

^{−3}m

^{2}/s

^{2}) for the experimental and numerical model along the longitudinal plane (symmetry plane) for the eight cases presented in this study.

Low 1.54 L/min Gas Flow Rate (Q) | High 2.22 L/min Gas Flow Rate (Q) | |||
---|---|---|---|---|

(Slag) Oil Thickness (hs) | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio |

3% oil thickness | ||||

experimental | 0.74 ± 0.51 | 1.18 ± 0.75 | 0.83 ± 0.55 | 1.05 ± 0.75 |

numerical | 0.78 ± 1.02 | 1.15 ± 1.44 | 1.00 ± 1.37 | 1.24 ± 1.66 |

difference (%) | 5.86 | 2.56 | 19.88 | 18.73 |

5% oil thickness | ||||

experimental | 0.97 ± 0.61 | 1.11 ± 0.72 | 0.60 ± 0.39 | 0.78 ± 0.57 |

numerical | 0.91 ± 1.17 | 1.14 ± 1.43 | 0.89 ± 1.23 | 1.11 ± 1.49 |

difference (%) | 6.64 | 1.98 | 46.87 | 41.88 |

**Table 5.**Slag eye area as a percentage of the total surface area for different operating conditions experimentally obtained through image analysis from both experimental and numerical models.

Low 1.54 L/min Gas Flow Rate (Q) | High 2.22 L/min Gas Flow Rate (Q) | |||
---|---|---|---|---|

(Slag) Oil Thickness (hs) | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio |

3% oil thickness | ||||

experimental | 39.40 ± 2.27 | 45.40 ± 3.53 | 51.47 ± 1.49 | 58.35 ± 1.97 |

numerical | 45.35 | 49.75 | 49.84 | 55.21 |

difference (%) | 15.10 | 9.59 | 3.17 | 5.38 |

5% oil thickness | ||||

experimental | 34.13 ± 1.79 | 34.21 ± 2.96 | 43.99 ± 2.06 | 49.88 ± 3.03 |

numerical | 30.07 | 38.31 | 33.89 | 38.93 |

difference (%) | 11.90 | 11.99 | 22.95 | 21.94 |

**Table 6.**Mixing time in seconds for different operating conditions experimentally (obtained through the planar laser-induced fluorescence (PLIF) method) and compared with the numerical model.

Low 1.54 L/min Gas Flow Rate (Q) | High 2.22 L/min Gas Flow Rate (Q) | |||
---|---|---|---|---|

(Slag) Oil Thickness (hs) | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio | 50%:50% Dual Gas Injection Ratio | 25%:75% Dual Gas Injection Ratio |

3% oil thickness | ||||

experimental | 8.04 ± 0.57 | 7.24 ± 0.80 | 6.84 ± 0.26 | 6.57 ± 0.40 |

numerical | 9.67 | 8.71 | 8.18 | 7.07 |

difference (%) | 20.28 | 20.31 | 19.56 | 7.67 |

5% oil thickness | ||||

experimental | 10.09 ± 1.04 | 9.35 ± 1.13 | 7.18 ± 0.62 | 5.92 ± 0.45 |

numerical | 12.53 | 9.49 | 8.82 | 8.16 |

difference (%) | 24.15 | 1.47 | 22.79 | 37.85 |

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**MDPI and ACS Style**

Jardón-Pérez, L.E.; González-Rivera, C.; Ramirez-Argaez, M.A.; Dutta, A.
Numerical Modeling of Equal and Differentiated Gas Injection in Ladles: Effect on Mixing Time and Slag Eye. *Processes* **2020**, *8*, 917.
https://doi.org/10.3390/pr8080917

**AMA Style**

Jardón-Pérez LE, González-Rivera C, Ramirez-Argaez MA, Dutta A.
Numerical Modeling of Equal and Differentiated Gas Injection in Ladles: Effect on Mixing Time and Slag Eye. *Processes*. 2020; 8(8):917.
https://doi.org/10.3390/pr8080917

**Chicago/Turabian Style**

Jardón-Pérez, Luis E., Carlos González-Rivera, Marco A. Ramirez-Argaez, and Abhishek Dutta.
2020. "Numerical Modeling of Equal and Differentiated Gas Injection in Ladles: Effect on Mixing Time and Slag Eye" *Processes* 8, no. 8: 917.
https://doi.org/10.3390/pr8080917