# Computational Fluid Dynamics Numerical Simulation on Flow Behavior of Molten Slag–Metal Mixture over a Spinning Cup

^{*}

## Abstract

**:**

## 1. Introduction

## 2. CFD Model Development

#### 2.1. Definition of Computational Domain

#### 2.2. Model Assumptions

- The mass flow rate of the molten slag–metal mixture at the inlet remains constant;
- Molten slag and metallic iron are well mixed at the inlet;
- The flow is isothermal, incompressible, and in a steady state;
- In the vicinity of the spinning cup, the airflow is influenced only by the spinning cup and the liquid motion;
- In order to limit the computation amount without losing accuracy, the computation domain is confined to covering appropriate portions of the pouring liquid stream and the air above the spinning cup;
- The center axis of the spinning cup precisely coincides with that of the cylindrical pouring stream of the molten slag–metal mixture.

#### 2.3. Computation Grids, Boundary Conditions, and Material Properties

^{−1}, 1.7 N·m

^{−1}, and 1.3 N·m

^{−1}, respectively [31].

#### 2.4. Governing Equations

- Continuity Equation:

- 2.
- Momentum Equation:

- 3.
- SST turbulence equations:

- 4.
- Volume fraction equation:

^{−3}); u is the velocity vector (m·s

^{−1}); p is the pressure (Pa); μ is the dynamic viscosity (Pa·s); μ

_{t}is the turbulent viscosity (Pa·s); F

_{g}is the source term due to gravity (N·m

^{−3}); F

_{s}is the source term due to surface tension force (N·m

^{−3}), which is obtained by using the continuous surface force (CSF) method reported by Brackbill et al. [35]; k is the turbulence kinetic energy (m

^{2}·s

^{−2}); σ

_{k}

_{3}is the Prandtl number for turbulence kinetic energy; ω is the turbulence eddy frequency (s

^{−1}); σ

_{ω}

_{2}is the Prandtl number for turbulence eddy frequency in the transformed k–ω turbulence model; σ

_{ω3}is the Prandtl number for turbulence eddy frequency; P

_{k}is the turbulent kinetic energy generation rate (W·m

^{−3}); N

_{p}is the total number of fluid phases; F

_{1}is the mixing function; α

_{3,}β′ and β

_{3}are the turbulence model constants; r is the volume fraction; and, subscripts α and β are the fluid phase identification indices.

#### 2.5. Solution Method and Computation Scheme

## 3. Results and Discussion

#### 3.1. High-Temperature Experiments on Centrifugal Granulation of Molten Slag–Metal Mixtures

#### 3.1.1. Experimental Condition

_{2}, 13.02 wt.%Al

_{2}O

_{3}, 5.83 wt.%MgO, and 0.15 wt.%MnO.

#### 3.1.2. Experimental Results

^{−1,}and the cup spinning speed is 2500 RPM, the measured diameter of the slag particles mainly falls in the range between 0.27 mm and 2.8 mm, and nearly 98% of slag particles have a diameter smaller than 0.88 mm. These slag particles are fine enough for efficient heat exchange with air and also possess enough high glass content (>90%) for use in making cement. Also, from the relationship between the Sauter mean diameter (SMD) of the experimentally granulated slag particles and the metallic iron content in the slag, as can be seen from the figure, the SMD decreases gradually with the increase in the metallic iron content. For the liquid feeding rate of 0.5 kg·min

^{−1}and the cup spinning speed of 2500 RPM, when the metal iron content increases from 5% to 15%, the SMD of the slag particles decreases by 13.77%, indicating that the participation of the metal in the slag promoted the breakup of the slag to a certain extent.

#### 3.2. Spreading Flow Behavior of Molten Slag–Metal Mixture Inside a Spinning Cup

#### 3.2.1. Behavior of Molten Slag Flow on Spinning Cup Inner Face

#### 3.2.2. Behavior of Liquid Metal Flow on Spinning Cup Inner Face

#### 3.3. Effect of Design and Operating Parameters on Liquid Film Thickness at the Edge of Spinning Cup

#### 3.3.1. Effect of Spinning Cup Size on Liquid Film Thickness

#### 3.3.2. Effect of Cup Spinning Speed on Liquid Film Thickness

#### 3.3.3. Effect of Liquid Feeding Rate on Liquid Film Thickness

^{−1}increase in the slag feeding rate, the liquid film thickness increases by about 28.15%. As the liquid feeding rate increases, more volume of slag passes over the surface of the spinning cup per unit of time. Since the spinning cup speed and diameter of the cup remain constant, the effect of centrifugal force is relatively constant, so more slag will form a thicker liquid film on the inner surface of the spinning cup.

#### 3.3.4. Effect of Metal Content in Slag on Liquid Film Thickness and Slag Particle Size

#### 3.4. Model Validation

^{−1}); μ is the liquid dynamic viscosity (Pa·s); R is the radius of the disc or inner radius of the spinning cup (mm); Ω is the spinning speed (RPM); and ρ is the liquid density (kg·m

^{−3}).

^{−1}) is likely caused by the shape difference between the cup and the disc. For the same diameter and under the same operating conditions, a spinning cup has a larger wetting area of the liquid film and thus produces a thinner film than a disc of the same diameter does, and this tendency becomes more obvious for low liquid feeding rates.

## 4. Conclusions

- The molten slag–metal mixture exhibits an immiscible state between the slag and the liquid metal on the inner face of the spinning cup, forming layering or wrapping patterns. Due to the action of interfacial tension, the metallic iron is unevenly dispersed inside the liquid slag film layer.
- The results of the high-temperature centrifugal granulation experiments indicate that an increase in cup spinning speed increases the proportion of smaller-sized slag particles. For a constant liquid feeding rate and cup spinning speed, an increase in metallic iron content in slag increases the Sauter mean diameter of the slag particles. For the cup spinning speed of 2500 RPM, when the metallic iron content in slag increases from 5% to 15%, the Sauter mean diameter of the slag particles decreases by 13.77%.
- The liquid film thickness at the spinning cup edge increases with the increase in liquid feeding rate but decreases with the increase in cup spinning speed. For the spinning cup with a diameter of 30 mm, the liquid film thickness decreases by 10.88% when the cup spinning speed is increased from 2000 RPM to 3000 RPM. For every 1 kg·min
^{−1}increase in the liquid feeding rate, the liquid film thickness increases by about 28.15%. When the spinning cup diameter is increased from 30 mm to 50 mm, the liquid film thickness can be reduced by 19.26%. - The diameter of slag particles correlates positively with the increase in the liquid film thickness at the spinning cup edge. The ratio between the arithmetic mean diameter of slag particles and the liquid film thickness decreases nearly linearly with the increase in the metallic iron content in slag, and, on average, the mean diameter of the slag particles is approximately 4.25 times the liquid film thickness at the spinning cup edge. Therefore, this ratio can be utilized for estimating the slag particle size based on the liquid film thickness at the spinning cup edge predicted using the two-dimensional CFD model developed in this work.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Process flow diagram of a two-step process of steel slag treatment through a combination of top-submerged lance bath smelting reduction with dry and centrifugal granulation of molten slag–metal mixture with waste heat recovery. (Red and blue colours stand for molten slag–metal mixture and gaseous phase, respectively.)

**Figure 2.**Schematic illustration of the cross-section shows the flow process of liquid in a spinning cup.

**Figure 3.**Schematic illustration of the computational domain defined in the two-dimensional CFD model.

**Figure 4.**Schematic diagram of computation grid division of the computational domain (

**right-hand side**) and relevant boundary condition types

**(left-hand side**) for the two-dimensional CFD model (the letters A to G represent the junction points of various boundaries of the two-dimensional computation domain, c.f., Table 1).

**Figure 5.**Schematic diagram of high-temperature experimental set-up for centrifugal granulation of molten slag–metal mixture using a spinning cup.

**Figure 6.**Slag particle size distribution in terms of mass fraction and accumulative mass fraction measured from high-temperature experiments on centrifugal granulation of molten slag–metal mixtures using a spinning cup.

**Figure 7.**Behavior of molten slag flow on the inner face of the spinning cup simulated by the two-dimensional CFD model: (

**a**) volume fraction contour plot, (

**b**) velocity vector field (liquid feeding rate = 0.5 kg·min

^{−1}, cup spinning speed = 2500 RPM, cup sidewall height = 5 mm, and metallic iron content in slag = 10%).

**Figure 8.**Plot of slag and metal volume fraction contours showing flow behavior of metal on the inner surface of spinning cup simulated by CFD model: (

**a**) volume fraction contour plot, (

**b**) velocity vector field (liquid feeding rate = 0.5 kg·min

^{−1}, cup spinning speed = 3000 RPM, cup sidewall height = 5 mm, and metallic iron content in slag = 10%).

**Figure 9.**CFD model predicted velocity vector fields (

**left**) and phase volume fraction contours (

**right**) of molten slag–metal mixture flow inside the spinning cup with different radius (R): (

**a**) R = 15 mm, (

**b**) R = 17.5 mm, (

**c**) R = 20 mm, (

**d**) R = 22.5 mm, (

**e**) R = 25 mm (liquid feeding rate = 0.5 kg·min

^{−1}, cup spinning speed = 2500 RPM, cup sidewall height = 5 mm, and metallic iron content in slag = 10%).

**Figure 10.**Effect of spinning cup diameter on liquid film thickness (slag feeding rate = 0.5 kg·min

^{−1}, cup spinning speed = 2500 RPM, cup sidewall height = 5 mm, metallic iron content in slag = 10%).

**Figure 11.**Effect of cup spinning speed on liquid film thickness (liquid feeding rate = 0.5 kg·min

^{−1}, cup sidewall height = 5 mm, metallic iron content in slag = 10%).

**Figure 12.**Effect of liquid feeding rate on liquid film thickness (cup spinning speed = 2500 RPM, cup sidewall height = 5 mm, metallic iron content in slag = 10%).

**Figure 14.**Comparison of the liquid film thickness predicted in this work with that calculated using Equation (9) (spinning cup diameter = 30 mm, cup sidewall height = 5 mm, metallic iron content in slag = 10%).

Boundary | Boundary Name | Boundary Type | Condition |
---|---|---|---|

ABC | Spinning cup bottom wall inner face | Wall | Non-slip spinning wall |

CD | Spinning cup sidewall inner face | Wall | Non-slip spinning wall |

DE | Side boundary | Opening | Fixed pressure (=0 Pa) |

EF | Top boundary | Opening | Fixed pressure (=0 Pa) |

FG | Liquid inlet | Inlet | Fixed mass flowrate |

AG | Center axis | Rotational symmetrical axis | Zero flux |

Material | Density (kg·m^{−3}) | Viscosity (Pa·s) |
---|---|---|

Liquid blast furnace slag | 2590 | 0.5 |

Air | 1.185 | 1.831 × 10^{−5} |

Steel | 7200 | 0.0065 |

Simulation Case Number | Liquid Feeding Rate (kg·min ^{−1}) | Cup Spinning Speed (RPM) | Spinning Cup Diameter (mm) | Metallic Iron Content in Slag (%) |
---|---|---|---|---|

1 | 0.5 | 2000 | 30 | 10 |

2 | 0.5 | 2250 | 30 | 10 |

3 | 0.5 | 2500 | 30 | 10 |

4 | 0.5 | 2750 | 30 | 10 |

5 | 0.5 | 3000 | 30 | 10 |

6 | 1 | 2500 | 30 | 10 |

7 | 1 | 2750 | 30 | 10 |

8 | 1 | 3000 | 30 | 10 |

9 | 1.5 | 2500 | 30 | 10 |

10 | 1.5 | 2750 | 30 | 10 |

11 | 1.5 | 3000 | 30 | 10 |

12 | 2 | 2500 | 30 | 10 |

13 | 2 | 2750 | 30 | 10 |

14 | 2 | 3000 | 30 | 10 |

15 | 0.5 | 2500 | 35 | 10 |

16 | 0.5 | 2500 | 40 | 10 |

17 | 0.5 | 2500 | 45 | 10 |

18 | 0.5 | 2500 | 50 | 10 |

19 | 0.5 | 2500 | 30 | 5 |

20 | 1 | 2500 | 30 | 5 |

21 | 1.5 | 2500 | 30 | 5 |

22 | 0.5 | 2500 | 30 | 15 |

23 | 1 | 2500 | 30 | 15 |

24 | 1.5 | 2500 | 30 | 15 |

**Table 4.**Ratios between experimental slag particle mean diameter and CFD model predicted liquid film thickness for different metallic iron content in slag (liquid feeding rate = 0.5 kg·min

^{−1}, cup spinning speed = 2500 RPM, spinning cup diameter = 30 mm).

Metallic Iron Content in Slag (%) | Slag Particle Diameter to Liquid Film Thickness Ratio |
---|---|

5 | 4.57 |

10 | 4.24 |

15 | 3.94 |

Average | 4.25 |

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**MDPI and ACS Style**

Wang, J.; Pan, Y.; Zhao, M.; Ma, P.; Lv, S.; Huang, Y.
Computational Fluid Dynamics Numerical Simulation on Flow Behavior of Molten Slag–Metal Mixture over a Spinning Cup. *Processes* **2024**, *12*, 372.
https://doi.org/10.3390/pr12020372

**AMA Style**

Wang J, Pan Y, Zhao M, Ma P, Lv S, Huang Y.
Computational Fluid Dynamics Numerical Simulation on Flow Behavior of Molten Slag–Metal Mixture over a Spinning Cup. *Processes*. 2024; 12(2):372.
https://doi.org/10.3390/pr12020372

**Chicago/Turabian Style**

Wang, Jun, Yuhua Pan, Ming Zhao, Ping Ma, Shali Lv, and Yawei Huang.
2024. "Computational Fluid Dynamics Numerical Simulation on Flow Behavior of Molten Slag–Metal Mixture over a Spinning Cup" *Processes* 12, no. 2: 372.
https://doi.org/10.3390/pr12020372