# Effect of the Impeller Design on Degasification Kinetics Using the Impeller Injector Technique Assisted by Mathematical Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

#### 2.2. Assumptions

- Physical properties for all the fluids in the system are taken as constants.
- Liquid phase is considered as a continuous phase, while gas is considered an interpenetrated phase in the liquid phase.
- All fluids are considered to be incompressible and Newtonian.
- Gas phase interpenetrated in the liquid is considered to comprise rigid spheres of constant size (1 mm).
- The system is considered to be isothermal without the presence of thermal gradients.
- All walls are considered to be impermeable and the fluid meets the non-slip condition for every mobile or static wall, while the well-known standard wall functions are used to connect the laminar region near the static walls to the turbulent core of the fluid.
- Turbulence in the ladle can be represented by the dispersed RNG k-ε turbulence model and is only present in the liquid phase.
- The volume rate of gas removed from the liquid is negligible in comparison with the input gas flow rate.

#### 2.3. Governing Equations

#### 2.3.1. Mass Conservation for the Liquid and Gas Phase

#### 2.3.2. Momentum Conservation for Liquid and Gas Phases

**P**is the pressure shared by both phases, and $\stackrel{\mathbf{\rightharpoonup}}{\mathit{g}}$, $\stackrel{\mathbf{\rightharpoonup}}{\mathit{F}}$, and ${\mathit{\mu}}_{\mathit{e}\mathit{f}\mathit{f}}$ are the gravitational constant, the momentum exchange between phases, and the effective viscosity, respectively. The effective viscosity is the sum of the liquid molecular viscosity, ${\mathit{\mu}}_{\mathit{l}},$ and the turbulent viscosity, ${\mathit{\mu}}_{\mathit{t}}$, (${\mathit{\mu}}_{\mathit{e}\mathit{f}\mathit{f}}={\mathit{\mu}}_{\mathit{l}}+{\mathit{\mu}}_{\mathit{t}}$). ${\stackrel{\mathbf{\rightharpoonup}}{\mathit{F}}}_{\mathit{l}\mathit{g}}$ and ${\stackrel{\mathbf{\rightharpoonup}}{\mathit{F}}}_{\mathit{g}\mathit{l}}$ have the same value but different sign and can be expressed by the following expression:

#### 2.3.3. RNG k-ε Model

#### 2.4. Boundary and Initial Conditions

#### 2.5. Solution

^{®}version 14.5 for each impeller in transient mode with a time step of 0.001 s until a quasi-steady fluid flow condition was reached at 20 s. A PC with a Core i7, 3.4 GHz processor was used to compute the numerical solutions and every computation took around 5 days, to get the final converged results.

#### 2.6. Experimental Procedure

^{®}(Wayne, NJ, USA), by using the Particle Image Velocimetry equipment (PIV) (Dantec Dynamics© (Skovlunde, Denmark) model LDY 302), which includes a high-speed camera (Vision Research model Dk 2740), and a high-intensity laser using only 45% of the power at 800 Hz in single-frame mode. Image statistics processing was handled by the software Dynamics Studio v4.00© (Dantec Dynamics©, v4.00, c Skovlunde, Denmark). 800 photographs were taken for the purposes of statistical analysis and the fluid was seeded with polyamid particles of 50 micrometers in diameter covered with rodamine B, since the camera was equipped with a 550 nm optical filter in order to discriminate between the tracers light and the light scattered from the bubbles within the flow. The high-speed camera was positioned perpendicularly with respect to the laser sheet. PIV technique details can be seen in [15].

^{®}(Woonsocket, RI, USA) model HI 9146 to get the degassing kinetics for each impeller tested in this work. A complete and detailed description of the technique can be found in [16].

## 3. Results and Discussion

#### 3.1. Validation

#### 3.2. Process Analysis

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

List of symbols | ||

${\mathit{A}}_{\mathit{i}}$ | Interfacial area concentration | $\left[{\mathrm{m}}^{-1}\right]$ |

${\mathit{a}}_{\mathit{k}}$ | inverse Prandtl number for turbulent kinetic energy | |

${\mathit{\alpha}}_{\mathit{s}}$ | Swirl modification constant | |

${\mathit{a}}_{\mathit{\epsilon}}$ | inverse Prandtl number for dissipation rate of turbulent kinetic energy | |

${\mathit{C}}_{\mathit{D}}$ | Drag coefficient | |

${\mathit{C}}_{\mathit{f}}$ | Drag function | |

${\mathit{C}}_{\mathbf{1}\mathit{\epsilon}}$ | RNG k-ε model constant | |

${\mathit{C}}_{\mathbf{2}\mathit{\epsilon}}$ | RNG k-ε model constant | |

${\mathit{C}}_{\mathit{\mu}}$ | RNG k-ε model constant | |

${\mathit{C}}_{\mathit{t}\mathit{d}}$ | Turbulent dispersed model constant | |

${\mathit{C}}_{\mathit{k}\mathit{\epsilon}}$ | Turbulent dispersed model constant | |

$\mathit{d}$ | Impeller diameter | $\left[\mathrm{m}\right]$ |

$\mathit{D}$ | Ladle diameter | $\left[\mathrm{m}\right]$ |

${\mathit{d}}_{\mathit{g}}$ | Bubble diameter | $\left[\mathrm{m}\right]$ |

$\mathit{f}$ | Swirl modification function | |

$\stackrel{\mathbf{\rightharpoonup}}{\mathit{F}}$ | Momentum exchange between phases | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-2}{\text{}\mathrm{s}}^{-2}\right]$ |

$\stackrel{\mathbf{\rightharpoonup}}{\mathit{g}}$ | Gravity acceleration | $\left[{\mathrm{m}\text{}\mathrm{s}}^{-2}\right]$ |

${\mathit{G}}_{\mathit{k}}$ | Generation of turbulent kinetic energy | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-2}{\text{}\mathrm{s}}^{-2}\right]$ |

$\mathit{h}$ | Distance from bottom to impeller line | $\left[\mathrm{m}\right]$ |

$\mathit{H}$ | Height of liquid | $\left[\mathrm{m}\right]$ |

$\mathit{k}$ | Turbulent kinetic energy | $\left[{\mathrm{m}}^{2}{\text{}\mathrm{s}}^{-2}\right]$ |

${\mathit{K}}_{\mathit{i}\mathit{j}}$ | Exchange coefficient between phase $\mathit{i}$ and phase $\mathit{j}$ | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-3}{\text{}\mathrm{s}}^{-1}\right]$ |

$\overrightarrow{\mathit{N}}$ | Angular velocity | $\left[{\mathrm{s}}^{-1}\right]$ |

$\mathit{P}$ | Pressure | $\left[\mathrm{Pa}\right]$ |

$\mathit{Q}$ | Gas flow rate | $\left[{\mathrm{l}\text{}\mathrm{min}}^{-1}\right]$ |

$\overrightarrow{\mathit{r}}$ | Radial position vector | $\left[\mathrm{m}\right]$ |

${\mathit{R}}_{\mathit{l}}$ | Coriolis and centrifugal forces in the rotating frame of reference | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-2}{\text{}\mathrm{s}}^{-2}\right]$ |

${\mathit{R}}_{\mathit{\epsilon}}$ | Term from RNG k-ε model | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-2}{\text{}\mathrm{s}}^{-2}\right]$ |

$\mathit{R}\mathit{e}$ | Reynolds number | |

$\mathit{S}$ | Strain rate magnitude | $\left[{\mathrm{s}}^{-1}\right]$ |

${\mathit{S}}_{\mathit{i}\mathit{j}}$ | Strain rate tensor | $\left[{\mathrm{s}}^{-1}\right]$ |

$\mathit{t}$ | Time | $\left[\mathrm{s}\right]$ |

$\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}$ | Velocity | $\left[{\mathrm{m}\text{}\mathrm{s}}^{-1}\right]$ |

Greek symbols | ||

$\mathit{\alpha}$ | Volume fraction | |

$\mathit{\beta}$ | RNG k-ε model constant | |

$\mathit{\epsilon}$ | Dissipation rate of turbulent kinetic energy | $\left[{\mathrm{m}}^{2}{\text{}\mathrm{s}}^{-3}\right]$ |

$\mathit{\eta}$ | RNG k-ε model relation | |

${\mathit{\eta}}_{\mathbf{0}}$ | RNG k-ε model constant | |

${\mathit{\eta}}_{\mathit{B}}$ | Drag modification of Brucato’s model | |

$\mathit{\mu}$ | Viscosity | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-1}{\text{}\mathrm{s}}^{-1}\right]$ |

${\mathit{\Pi}}_{\mathit{k}}$ | Source term of turbulent kinetic energy | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-2}{\text{}\mathrm{s}}^{-2}\right]$ |

${\mathit{\Pi}}_{\mathit{\epsilon}}$ | Source term of dissipation rate of turbulent kinetic energy | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-2}{\text{}\mathrm{s}}^{-2}\right]$ |

$\mathit{\rho}$ | Density | $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-3}\right]$ |

${\mathit{\tau}}_{\mathit{g}}$ | Particle relaxation time | $\left[\mathrm{s}\right]$ |

$\mathit{\Omega}$ | Characteristic swirl number | |

Subscripts | ||

$\mathit{l}$ | Liquid phase | |

$\mathit{g}$ | Gas phase | |

$\mathit{t}$ | Turbulent | |

$\mathit{e}\mathit{f}\mathit{f}$ | Effective |

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**Figure 1.**(

**a**) Schematic representation of the batch degassing ladle, indicating the rotating speed, $\overrightarrow{\mathit{N}}$, and the gas flow rate,

**Q**and the geometric features of the ladle; (

**b**) Interphase between the rotating frame of reference and the fixed frame of reference showing the two regions of the ladle (the mobile frame of reference and the fixed frame of reference).

**Figure 2.**Three impeller designs tested in this research. (

**a**) Commercial impeller design A; (

**b**) Commercial impeller design B; (

**c**) New design proposed in previous work [16] design C.

**Figure 4.**Measured (

**left**) and computed (

**right**) velocity fields at 400 rpm and 10 L/min for impeller A (

**a**,

**d**), for impeller B (

**b**,

**e**) and for impeller C (

**c**,

**f**).

**Figure 5.**Comparison between measured and computed gas holdup fields. Gas holdup in the r-z plane of the ladle for the three impellers tested at 400 rpm and 10 L/min. Experimental: (

**a**) Impeller A; (

**b**) impeller B; (

**c**) impeller C. Computed: (

**d**) Impeller A; (

**e**) impeller B; (

**f**) impeller C.

**Figure 7.**Predicted angular velocity component fields in a θ-r plane at the height of the impeller for the three impellers tested at 400 rpm and 10 L/min. (

**a**) impeller A; (

**b**) impeller B; (

**c**) impeller C.

**Figure 8.**Pressure fields in the r-z plane of the ladle for the three impellers tested at 400 rpm and 10 L/min. (

**a**) Impeller A; (

**b**) impeller B; (

**c**) impeller C.

**Figure 9.**Water eddy viscosity fields in the r-z plane of the ladle for the three impellers tested at 400 rpm and 10 L/min. (

**a**) Impeller A; (

**b**) impeller B; (

**c**) impeller C.

Characteristic | Description |
---|---|

Fluids | Incompressible and Newtonian, water and air |

Flow regime | Turbulent (Re = 1,160,000) |

Rotating speed ($\overrightarrow{\mathit{N}}$) | 400 rpm |

Gas flow rate ($\mathit{Q}$) | 10 L/min |

Ladle diameter ($\mathit{D}$) | 0.5 m |

Impeller diameter ($\mathit{d}$) | 0.166 m |

Height of liquid ($\mathit{H}$) | 0.5 m |

Distance from bottom to impeller line ($\mathit{h}$) | 0.166 m |

Impeller design | Commercial designs A and B and a new design C (Figure 2) |

Geometry | Elements | Total | |
---|---|---|---|

Hexahedric | Tetrahedric | ||

Impeller design A | 29,270 | 34,030 | 63,300 |

Impeller design B | 29,270 | 31,610 | 60,880 |

Impeller design C | 31,520 | 21,000 | 52,520 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abreu-López, D.; Amaro-Villeda, A.; Acosta-González, F.A.; González-Rivera, C.; Ramírez-Argáez, M.A.
Effect of the Impeller Design on Degasification Kinetics Using the Impeller Injector Technique Assisted by Mathematical Modeling. *Metals* **2017**, *7*, 132.
https://doi.org/10.3390/met7040132

**AMA Style**

Abreu-López D, Amaro-Villeda A, Acosta-González FA, González-Rivera C, Ramírez-Argáez MA.
Effect of the Impeller Design on Degasification Kinetics Using the Impeller Injector Technique Assisted by Mathematical Modeling. *Metals*. 2017; 7(4):132.
https://doi.org/10.3390/met7040132

**Chicago/Turabian Style**

Abreu-López, Diego, Adrián Amaro-Villeda, Francisco A. Acosta-González, Carlos González-Rivera, and Marco A. Ramírez-Argáez.
2017. "Effect of the Impeller Design on Degasification Kinetics Using the Impeller Injector Technique Assisted by Mathematical Modeling" *Metals* 7, no. 4: 132.
https://doi.org/10.3390/met7040132