# Modeling of Fluid Flow and Residence-Time Distribution in a Five-Strand Tundish

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## Abstract

**:**

## 1. Introduction

## 2. CFD Model Description

- The model is based on a 3D standard set of the Navier–Stokes equations. The continuous phase is treated by a Eulerian framework (using averaged equations);
- The liquid flow was assumed to be isothermal and in steady state;
- Two additional passive scalar-transport equations are solved to separately describe the E-curve and the F-curve. Transient solver is applied to calculate the transportation of the passive scalars;
- The realizable k-ε model was used to describe the turbulence;
- The free surface is flat and kept at a fixed level. The slag layer is not included in the tundish.

#### 2.1. Governing Equation

_{ϕ,eff}is the effective diffusion coefficient, S

_{ϕ}is the source term, x

_{j}are the Cartesian coordinates, u

_{j}are the corresponding average velocity components, t is the time and ρ is the density. The first term expresses the rate of change of ϕ with respect to time, the second term expresses the convection (transport due to fluid-flow), the third term expresses the diffusion (transport due to the variation of ϕ from point-to-point) where Γ

_{ϕ}is the diffusion coefficient of the entity ϕ in the phase and the fourth term expresses the source terms (associated with the creation or destruction of variable ϕ).

#### 2.1.1. Fluid Flow

_{t}is the turbulent viscosity; G

_{k}represents the generation of turbulent kinetic energy due to the mean velocity; Y

_{M}represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate; υ is the kinematic viscosity; σ

_{k}and σ

_{ε}are the turbulent Prandtl numbers for k and ε, respectively.

#### 2.1.2. Tracer Dispersion

_{eff}, is the sum of the molecular and turbulent diffusivity. The velocity field is solved from a steady-state simulation and remained constant during the calculation of the two passive scalars [23].

#### 2.1.3. Characteristic Volumes Calculation from RTD Curves

_{t}is the volume of tundish, and Q is the volumetric flow rate.

_{i}) and time (θ) are given by

_{0}is the concentration that corresponds to the condition where the added tracer is uniformly distributed in the tundish.

_{p}/V

_{t}), the mixed-flow volume fraction (V

_{m}/V

_{t}) and the dead-volume fraction (V

_{d}/V

_{t}) have been calculated through Equations (11)–(13) [25].

_{min}is the minimal dimensionless time at the tundish outlet and θ

_{peak}is the peak dimensionless time at the tundish outlet.

#### 2.2. Geometry and Mesh

#### 2.2.1. Tundish Geometry

- Case 1—bare tundish;
- Case 2—tundish with turbulence inhibitor (TI);
- Case 3—tundish with U-baffle with deflector holes(UB);
- Case 4—tundish with U-baffle with deflector holes and turbulence inhibitor (UB + TI).

#### 2.2.2. Computational Domain and Mesh

#### 2.3. Initial and Boundary Conditions

#### 2.3.1. Liquid Phase

^{−3}and 8.9 × 10

^{−4}Pa∙s. No-slip conditions were applied at all solid surfaces for the liquid phase. Free-slip condition was applied at the free surface. A constant inlet velocity was used. At the tundish outlet, the outflow boundary condition was applied. A wall function was applied to bridge the viscous sublayer and provide the near-wall boundary conditions for the average flow and the turbulence transport equations. The wall conditions were connected by means of empirical formulae to the first grid node close to the solid surfaces. Table 2 presents the parameters and boundary conditions used in the CFD simulations.

#### 2.3.2. Tracer

#### 2.4. Solution Procedure

^{−4}, together with the stability of the velocity and the turbulence at the key monitor points. The under-relaxation parameter of flow calculations for the pressure, the velocity and the turbulence were 0.2, 0.8 and 0.8, respectively.

## 3. Water Model

_{m}/l

_{p}= λ

^{2}/gl

_{m}/u

_{p}= λ

^{1/2}

_{m}/Q

_{p}= λ

^{5/2}

^{−1}); g is the acceleration of gravity (m∙s

^{−2}); l is the length (m); Q is the volumetric flow rate (m

^{3}∙s

^{−1}) and λ is the length scale (1/3 in this study). The subscript m and p represent model and prototype, respectively. The volume of water model (V

_{m}) was 0.2 m

^{3}and the volumetric flow rate of water model (Q

_{m}) was 0.00028 m

^{3}∙s

^{−1}.

## 4. Results and Discussion

#### 4.1. Validation of CFD Model

#### 4.1.1. Independent of Computational Mesh

#### 4.1.2. Numeric vs. Physical Modeling

#### 4.2. Liquid Flow in Tundish with Different FCD

- View A: Longitudinal plane of inlet;
- View B: Horizontal plane (close to bottom);
- View C: Longitudinal plane of all the outlets.

#### 4.3. E-Curve

#### 4.4. F-Curve

_{tracer}= 1 at the inlet) flows along a short path to the Outlet 1 as soon as it enters the tundish. The deviation for the F-curve among three outlets in Case 1 and Case 2 are bigger than that in Case 3 and Case 4. Figure 12 suggests that the predicted intermixing time related to the four studied cases. In Case 4, the tundish equipped with U-baffle and turbulence inhibitor generates the shortest intermixing time and the lowest deviation among the three outlets. This means that an optimum flow control in the tundish can shorten the intermixing zones, thereby increasing the steel yields during the mixed grade continuous casting process.

## 5. Conclusions

- A combination of the U-baffle with deflector holes and turbulence inhibitor was proposed for a five-strand tundish. The existence of turbulence inhibitor impaired the turbulence zone in the outlet chamber due to the redirection of the incoming flow. Additionally, the U-type baffle with deflector holes could reorient the flow and extend the flow path, which was predicted by the numeric flow simulation and visualized through tracer dispersion in the water modeling;
- A sharp increase in the tracer concentration suggests the short-circuiting phenomena in the bare tundish, resulting in a relatively high dead volume fraction, up to 27%. High dead volume fraction was an undesirable feature in the tundish design;
- The tundish equipped with the U-baffle with deflector holes could improve the flow characteristics in the E-curve analysis. The dead volume fractions were less than 10% and the plug volume fractions were around 20% for all the outlets. The deviation around E-curves indicated a lowered difference of the flow characteristics among the outlets. The comparison of two U-baffle cases showed that the existence of turbulence inhibitor delays the breakthrough time, but shortened the mean residence time;
- Intermixing time of the mixed grade casting were numerically investigated for the ladle changeover operation by the analysis of the F-curve. A slope change of F-curve was observed when there was a short-circuiting phenomenon. The tundish equipped with U-baffle and turbulence inhibitor generated the shortest intermixing time and the lowest deviation at the outlets.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Dimensions of in-plant 5-strand tundish including turbulence inhibitor and U-baffle with deflector holes (unit: mm); (

**b**) turbulence Inhibitor (unit: mm); (

**c**) U-baffle with deflector holes (unit: mm); (

**d**) Schematics of the four tundish configurations (Case 1–4).

**Figure 3.**(

**a**) Computational domain of numeric model (one-half of the 5-strand tundish); (

**b**) computational fluid dynamic (CFD) mesh on the symmetrical plane and zoom-in view.

**Figure 10.**E-curves of the numeric and experimental model. (

**a**) Case 1—bare; (

**b**) Case 2—TI; (

**c**) Case 3—UB; (

**d**) Case 4—UB + TI.

**Figure 11.**F-curve of the numeric model. (

**a**) Case 1—bare; (

**b**) Case 2—TI; (

**c**) Case 3—UB; (

**d**) Case 4—UB + TI.

**Table 1.**Summary of mathematical modeling investigations on residence-time distribution (RTD) in the tundish.

Reference | Model ^{1} | Code | Design | Numeric Model | Parameter Study ^{2} | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Strand | Fluid ^{3} | FCD ^{4} | Gas | Fluid ^{5} | Turb. ^{6} | Inclu. ^{7} | RTD ^{8} | ||||

S. López-Ramirez (1998) [5] | N | - | 2 | S | B, TI | - | - | k-ε | - | E | SFR, FCD, TC |

Vargas-Zamora (2004) [6] | N, P | - | 1 | W | TI, D | - | - | - | - | F | GFR |

Zhong (2008) [7] | P | - | 2 | W | TI, D, W | N_{2} | - | - | - | E | TC, FCD, GFR |

Bensouici (2009) [8] | N, P | Fluent | 1 | W | W, D | - | - | k-ε | - | E | MS, FCD |

Zheng (2011) [9] | N, P | CFX | 2 | S | TI, B | Ar | Eu | k-ε | La | E | TC, GFR, IS |

Chen (2013) [10] | N, P | Fluent | 1 | S, W | W | Ar | Eu | k-ε | La | E | TC, FCD, IS |

Chen (2015) [11] | N, P | Phoenics | 1 | W | SR, D, W, TI | - | - | k-ε | - | E | MS, TS, TP |

Chang (2015) [12] | N, P | Fluent | 7 | S, W | TI, B | Ar | Eu | k-ε | La | E | GFR, FCD |

Devi (2015) [13] | N, P | Fluent | 2 | S, W | D | Ar | Eu | k-ε | - | E | FCD, GFR |

He (2016) [14] | N, P | Fluent | 5 | S, W | TI, B | - | - | E | TC, SFR | ||

Neves (2017) [15] | N, P | CFX | 2 | W | SR, D, W | Air | Eu | k-ε | - | E | GFR, FCD |

Wang (2017) [16] | N, P | Fluent | 8 | S | TI, F | - | - | k-ε | La | E | TC, FCD, IS |

Aguilar–Rodriguez (2018) [17] | N | Fluent | 1 | S | - | Ar | VOF | k-ε | La | E | GFR, TC, FCD |

Yang (2019) [18] | N | CFX | 2 | S | D, TI | - | - | k-ε | La | E | FCD, TC |

Wang (2020) [19] | N | Fluent | 2 | S | W, TI, F | - | Eu | k-ε | La | E | IS, FCD, TC |

^{1}N—numeric model; P—physical model;

^{2}SFR—steel flow rate; GFR—gas flow rate; TC—tundish configuration; MS—mesh size; TS—time step; ID—inlet depth; IS—inclusion size; FCD—flow control devices;

^{3}W—water; S—steel;

^{4}FCD—flow control devices; SR—stop rod; D—dam; W; weir; TI—Turbulence inhibitor; F—filter; B—baffles;

^{5}Eu—Eulerian; VOF—volume of fluid;

^{6}Turb—turbulence;

^{7}Inclu—inclusion; La—Lagrangian;

^{8}E—E-curve; F—F-curve.

Water density | 998 kg/m^{3} |

Water viscosity | 0.00089 Pa·s |

Reference Pressure | 101,325 Pa |

Inlet flow rate | 0.00028 m^{3}/s |

Outlet (outflow ratio) | 0.2:0.4:0.4 (Outlet 1/2/3) |

Wall | No slip |

Free surface | Free slip |

Tracer inlet (E-curve) | 1 (t <= 0–2 s), 0 (t > 2 s) |

Tracer inlet (F-curve) | 1 |

Mesh | Mesh Number | Mesh Size (m) | t_{theo} (s) ^{1} | t_{min} (s) | t_{max} (s) | t_{mean} (s) | V_{p}/V (%) | V_{m}/V (%) | V_{d}/V (%) |
---|---|---|---|---|---|---|---|---|---|

1 | 4 Million | 0.002 | 749 | 31 | 222 | 685 | 17 | 75 | 9 |

2 | 2 Million | 0.003 | 749 | 26 | 229 | 669 | 17 | 72 | 11 |

3 | 1 Million | 0.004 | 749 | 30 | 208 | 664 | 16 | 73 | 11 |

^{1}t

_{theo}: theoretical residence time; t

_{min}: minimum breakthrough time; t

_{max}: time corresponding to peak concentration; t

_{mean}: mean residence time. V

_{p}/V: plug flow volume fraction. V

_{d}/V: dead volume fraction; V

_{m}/V: mixed flow volume fraction.

Case | t_{theo} (s) | t_{min} (s) | t_{max} (s) | t_{mean} (s) | V_{p}/V (%) | V_{m}/V (%) | V_{d}/V (%) |
---|---|---|---|---|---|---|---|

1—Outlet 1 | 749 | 4 | 36 | 544 | 3 | 70 | 27 |

1—Outlet 2 | 749 | 13 | 239 | 733 | 17 | 81 | 2 |

1—Outlet 3 | 749 | 78 | 155 | 711 | 16 | 79 | 5 |

2—Outlet 1 | 749 | 22 | 46 | 482 | 5 | 60 | 36 |

2—Outlet 2 | 749 | 28 | 65 | 673 | 6 | 84 | 10 |

2—Outlet 3 | 749 | 69 | 123 | 748 | 13 | 86 | 1 |

3—Outlet 1 | 749 | 27 | 252 | 696 | 19 | 74 | 7 |

3—Outlet 2 | 749 | 32 | 304 | 716 | 22 | 73 | 4 |

3—Outlet 3 | 749 | 15 | 274 | 690 | 19 | 73 | 8 |

4—Outlet 1 | 749 | 44 | 250 | 692 | 20 | 73 | 8 |

4—Outlet 2 | 749 | 44 | 291 | 707 | 22 | 72 | 6 |

4—Outlet 3 | 749 | 27 | 261 | 682 | 19 | 72 | 9 |

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**MDPI and ACS Style**

Sheng, D.-Y.; Yue, Q.
Modeling of Fluid Flow and Residence-Time Distribution in a Five-Strand Tundish. *Metals* **2020**, *10*, 1084.
https://doi.org/10.3390/met10081084

**AMA Style**

Sheng D-Y, Yue Q.
Modeling of Fluid Flow and Residence-Time Distribution in a Five-Strand Tundish. *Metals*. 2020; 10(8):1084.
https://doi.org/10.3390/met10081084

**Chicago/Turabian Style**

Sheng, Dong-Yuan, and Qiang Yue.
2020. "Modeling of Fluid Flow and Residence-Time Distribution in a Five-Strand Tundish" *Metals* 10, no. 8: 1084.
https://doi.org/10.3390/met10081084