# Experimental and CFD Studies of the Hydrodynamics in Wet Agglomeration Process

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the RVR Agglomeration Reactor

^{®}C-492 and N-300) is provided by the spinning disc powered by a rotating shaft. The rotating disc made from Polyvinyl chloride (PVC) generates both radial swirling flow at the immediate vicinity of the disc and an axial pumping flow at some distance away from the disc towards the stator, which eventually drives the agglomerated flocs towards the shroud of the reactor, as shown schematically in Figure 1b. The axial flow away from the rotating disc towards the shroud facilitates the collection of densified agglomerates on either side of the reactor in batch mode operations.

#### 2.2. Theoretical Analysis of the Fluid Flow

_{ϕ, critical}≈ 2 × 10

^{5}with small patches of turbulence or oscillating vortex disturbance [7,8]. This can be expressed mathematically for the boundary layer near the rotor using the local rotational Reynolds number Re

_{ϕ, local}(Equation (1)). The gap ratio G of the reactor is defined as the ratio of the wheelspace width to the outer disc radius, and is approximately 0.2206 for this reactor system (Equation (2)). The theoretical hydrodynamic characteristic values are given in Table 1.

#### 2.3. Experimental Fluid Flow Measurements

_{ϕ}= 25 as the time at which the system has reached a pseudo-steady state condition [27]. The 2-D PIV measurements, which are described in detail elsewhere [15], consist of an RVR reactor system seeded with tracing particles and distilled water at standard conditions as a working fluid (Figure 2). The geometry of the reactor presented some measurement constraints, which was taken into account in the post-processing and analysis of the experimental data. The laser system for the experimental measurements consist of a LINOS Nano 259-532-100 (Qioptic GmbH & Co. KG, Munich, Germany) providing illumination for a cross section of the reactor under investigation. A high-resolution complementary metal-oxide semiconductor (CMOS) camera was used to capture the steady state fluid flow behaviour for post-processing and cross-correlation in MATLAB R2015a (Mathworks GmbH, Germany). The seeding particles is a mixture (1:1) of silver-coated and hollow glass spheres (Dantec Dynamics, A/S, Denmark) measuring 10 μm in average diameter with a good light scattering efficiency and sufficiently small velocity lag [28,29].

## 3. Numerical Simulation

#### 3.1. Model Description

#### 3.1.1. Governing Equations

#### 3.1.2. Fluid Flow Domain, Mesh and Grid Convergence

^{6}, 0.431 × 10

^{6}, 0.86 × 10

^{6}, and 1.3 × 10

^{6}elements were created for the grid independence study. The grid independence study was conducted by carrying out the numerical simulation on successively finer grids. Thereafter, the results of the circumferential velocities were compared for all the grids to establish that they are grid-independent. Finally, a mesh containing approximately half a million elements was thereafter chosen on the basis of a trade-off between the required computational accuracy and CPU time needed to run the simulation to a converged solution. This optimized mesh was thereafter used for subsequent computations of all parameters of interest with a computation time of approximately 15 h for the highest operating speed. Figure 4 shows the radial and axial circumferential velocity profiles along the Y-Z and X-Z planes at x = 0.003725 and y = 0.045, respectively, for a rotation speed of 70 rpm for all the four grids employed for the convergence study and clearly demonstrates that the results are grid-independent.

#### 3.2. Numerical Methods and Boundary Conditions

## 4. Results and Discussion

#### 4.1. Analysis of the Mean Flow Characteristics

#### 4.1.1. Radial Distribution of the Hydrodynamic Quantities

#### 4.1.2. Axial Distribution of the Hydrodynamic Quantities

_{φ}contours and profiles, and the vorticity amplitude ζ contours and profiles on the X-Z plane are shown in Figure 7a–c and Figure 8a–c respectively. The axial tangential velocity distribution shows a pattern consistent with a Batchelor flow profile, except on plane y = 0 near the shaft where the velocity is zero as expected. Figure 7a–c shows a varying pattern of Batchelor velocity profiles, but with distinct separated boundary layers and a central rotating core. By contrast, the width of the rotor boundary layer is slightly longer and the rotating core slightly shorter when compared to the Batchelor profile (Figure 10b). In terms of the axial vorticity amplitude, all the profiles seem to be identical, except on plane y = 0 at the region close to the shaft where the vorticity amplitude remains constant throughout the plane (Figure 8a). Overall, the velocity and the vorticity amplitude increases with the axial plane height with the plane y = 45 recording the highest magnitude. However, there is a region of high vorticity mostly on the outer edge of the rotating disc close to the reactor wall. This observation is quite interesting as it suggests that the clearance between the disc and the reactor wall might strongly influence the vorticity magnitude. The influence of the gap ratio G on the fluid flow cannot be ruled out, and a detailed geometry parametric investigation prior to the scale-up of the reactor in future can yield some useful data for design optimization.

#### 4.1.3. Spatial Distribution of Mixing and Turbulence Intensities

#### 4.1.4. Theoretical Validation of the CFD Model

## 5. Conclusions

## 6. Patents

## Author Contributions

## Funding

## Conflicts of Interest

## Notations

Re_{φ} | Rotational Reynolds number (-) |

ζ | Vorticity magnitude (s^{−1}) |

G | Gap ratio (-) |

β | Swirl ratio (-) |

ρ | Density (kg·m^{−3}) |

δ | Boundary layer thickness (m) |

n | Disc rotational speed (s^{−1}) |

x | Radial coordinate (m) |

z | Axial coordinate (m) |

r_{d} | Rotating disc outer radius (m) |

s | Wheelspace or cavity width (m) |

µ | Dynamic viscosity (kg·m^{−1}·s^{−1}) |

ω | Disc or plate angular velocity (rad·s^{−1}) |

ω′ | Angular velocity of the rotating core (m·s^{−1}) |

Ω | Disc tangential or tip velocity (m·s^{−1}) |

U_{φ} | Tangential velocity component (m·s^{−1}) |

U_{r} | Radial velocity component (m·s^{−1}) |

U_{z} | Axial velocity component (m·s^{−1}) |

r | Distance along the r-axis (m) |

r_{s} | Shaft radius (m) |

r_{r} | Reactor shroud radius (m) |

_{ϕ} | Initial particle volume fractions (-) |

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**Figure 1.**Schematic illustration of the reactor configuration and flow profiles: (

**a**) RVR reactor, and (

**b**) rotor-stator flow boundary layers showing the tangential velocity and vorticity amplitude profiles on the rotating disc (Reproduced from [8] with permissions © 2011 Elsevier).

**Figure 3.**Graphical illustration of the simplified reactor solid model and the generated computational grid for the numerical simulation. (

**a**) CAD model; (

**b**) Hex mesh.

**Figure 4.**Influence of the computational grid size on the computed tangential velocity profiles on radial Y-Z and axial X-Z planes for the RNG k-ε model at 70 rpm for (

**a**) x = 0.003725; (

**b**) y = 0.045.

**Figure 5.**Reactor cross sections showing the velocity vector fields and contours for the RNG-k-ε model and the radial tangential velocity profiles along the Y-Z plane at N = 70 rpm: (

**a**) rotating disc boundary layer; (

**b**) x = 0.0015; (

**c**) x = 0.003725; (

**d**) x = 0.006.

**Figure 6.**Reactor cross sections showing the vorticity vector maps and contours RNG-k-ε model and radial vorticity magnitude profiles along Y-Z plane at N = 70 rpm (

**a**) rotating disc boundary layer; (

**b**) x = 0.0015; (

**c**) x = 0.003725; (

**d**) x = 0.006.

**Figure 7.**Reactor cross-sections showing the velocity contours and axial tangential velocity profiles along three horizontal lines in the X-Z plane at N = 70 rpm with the RNG model: (

**a**) y = 0; (

**b**) y = 0.025; (

**c**) y = 0.045.

**Figure 8.**Reactor cross-sections showing the vorticity contours and axial vorticity magnitude profiles along three horizontal lines on X-Z plane at N = 70 rpm with the RNG model: (

**a**) y = 0; (

**b**) y = 0.025; (

**c**) y = 0.045.

**Figure 9.**Spatial distribution of the turbulence and mixing intensities (

**a**) Reynolds stress distributions on the Y-Z plane at x = 0.0015, 0.003725, and 0.006, and X-Z plane parallel to the shaft at y = 0, 0.025, 0.045, and 0.068. (

**b**) Turbulent kinetic energy distribution on the Y-Z plane at x = 0.003725 and X-Z plane parallel to the shaft at y = 0.025 for N = 70–130 rpm.

**Figure 10.**Flow streamlines for a cross sections of the reactor wheelspace and normalized axial tangential velocity profiles on X-Z plane at N = 70 rpm: (

**a**) cross sections of the CFD and Batchelor models; (

**b**) tangential velocity profile of the CFD and Batchelor models at y = 0.068 [8].

**Table 1.**Theoretical values of the flow quantities at different operating speeds near the rotating disc boundary layer.

Operating Condition | Hydrodynamic Parameters | ||
---|---|---|---|

Operating Speeds (rpm) | Disc Rotational Reynolds Number (-) | Disc Tip Velocity (m·s^{−1}) | Disc Vorticity (s^{−1}) |

70 | 2.59 × 10^{4} | 0.4985 | 14.662 |

90 | 3.33 × 10^{4} | 0.6410 | 18.852 |

110 | 4.07 × 10^{4} | 0.7834 | 23.041 |

130 | 4.81 × 10^{4} | 0.9258 | 27.231 |

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**MDPI and ACS Style**

Oyegbile, B.; Akdogan, G.; Karimi, M. Experimental and CFD Studies of the Hydrodynamics in Wet Agglomeration Process. *ChemEngineering* **2018**, *2*, 32.
https://doi.org/10.3390/chemengineering2030032

**AMA Style**

Oyegbile B, Akdogan G, Karimi M. Experimental and CFD Studies of the Hydrodynamics in Wet Agglomeration Process. *ChemEngineering*. 2018; 2(3):32.
https://doi.org/10.3390/chemengineering2030032

**Chicago/Turabian Style**

Oyegbile, Benjamin, Guven Akdogan, and Mohsen Karimi. 2018. "Experimental and CFD Studies of the Hydrodynamics in Wet Agglomeration Process" *ChemEngineering* 2, no. 3: 32.
https://doi.org/10.3390/chemengineering2030032