# Unified Analysis of Multi-Chamber Contact Tanks and Mixing Efficiency Based on Vorticity Field. Part I: Hydrodynamic Analysis

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Governing Equations of Fluid Flow

_{i}and x

_{j}represent the Cartesian coordinates. The overbar indicates the time-averaged component for RANS and the resolved turbulence scales for LES. The term $\overline{{\mathsf{\tau}}_{ij}}$ represents viscous stress, which describes the momentum transport due to molecular motion, given as

#### 2.2. Simulation Setup and Boundary Conditions

^{−5}. The overall convergence of the steady state solver is controlled by the maximum allowable continuity equation residual, which is selected as 1 × 10

^{−19}throughout this study. Note that the convergence criteria for the steady state solution are selected to be the same as in [6]. In the LES, unsteady simulation of the problem was carried out using the solver pimpleFoam available in OpenFoam, which is a time-implicit solver for large-time step calculation of transient flows. A second-order accurate implicit scheme (backward) is used for temporal discretization of the governing equations. The maximum allowable time step size for the stability was adjusted according to the Courant–Friedrich–Levy (CFL) number, which was set to 0.5 in this study. The time step size was adjusted internally so that the maximum CFL = 0.5 was maintained during the simulation. This resulted in a time step in the order of about 0.001 s, even though the preferred implicit solver could perform a stable transient solution for the CFL = 5.

## 3. Model Validation

#### Hydraulic Simulations

_{2}and 2

_{3}) and small (4

_{3}) recirculating regions are evolving in the chambers, separated from each other by water jets (3

_{2}and 3

_{3}). Here the capital number shows the zone number and the subscript shows the chamber number. The separation layer between the jet region and recirculation region develops at the inlet of the chamber and impinges on the right baffle at approximately the same elevation as the inlet, which can also be seen in Figure 7b. Similar behavior has been observed in the adjacent flow pattern between the jet region and recirculation region in Chamber 2, in which the flow tends to separate from the mean flow when it approaches the outlet opening. This observation shows that the baffle opening has a positive effect on the efficiency of the chamber at reducing the areas of the recirculation regions. The flow field in the downward and upward directions in Chambers 2 and 3 can theoretically be separated by a layer that splits the region into recirculation and jet flow regions. The separation layer could be identified based on the flow characteristics. Different definitions are available in the literature to distinguish rotational (recirculation zones) from the mean flow (jet region), as mentioned earlier.

_{2}and 1

_{3}) correspond to the energetically dynamic eddies in the vicinity of the sharp corner of the baffles, which could be captured by LES only. In this study, gradient of vorticity and flexion product concepts will be employed to identify the separation layer.

## 4. Vorticity Analysis for RANS and LES

#### 4.1. Separation of Flow Zones in RANS

**G**is a locally non-dimensionalized vorticity gradient with respect to the magnitude of the vorticity gradient in order to capture the change of sign of

**G**since the gradient is identically zero on the separation layer for a three-dimensional flow field. Each component of the tensor

**G**is computed at the center of the computational grid during the simulation. In order to identify the separation layers on the x–y plane (Figure 9a), the gradients of ${\mathsf{\omega}}_{z}$ will be evaluated since this vorticity component is perpendicular to the x–y plane.

#### 4.2. Separation of Flow Zones in LES

_{10}/τ. Here t

_{10}is the time required for 10% of the injected concentration to exit the contactor and τ is the theoretical residence time, which is computed as the ratio of the total volume of the contactor to the flow rate. Baffling factor of the present contactor was computed as 0.334 using RANS in [4] and as 0.3 using LES in [10], which can be classified as a poor baffling condition for the present reactor. Overall volumetric efficiencies are computed as 0.370 for RANS and 0.234 for LES in this study, which are close to the values obtained from the tracer analysis in [4,10]. This comparison shows that the volumetric efficiency coefficients proposed in this study can also be used to assess the baffling performance of the contactor without need of performing a tracer transport analysis, which is a time-consuming process, especially for LES.

#### 4.3. Evaluation of the Lamb Vector for the Identification of Vortex Core Lines and Separation Layers

_{3}–2

_{3}, 1

_{3}–3

_{3}, and 3

_{3}–4

_{3}) are captured by the use of the Lamb vector. This analysis should also include flow zone separation, beyond what is discussed earlier.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Distribution of vertical velocity component along x direction at different elevations: (

**a**) y/H = 0.27; (

**b**) y/H = 0.5; (

**c**) y/H = 0.72.

**Figure 4.**Variation of Euclidean norm with time on each sampling line: (

**a**) mean velocity magnitude; (

**b**) pressure.

**Figure 5.**Point vertical velocities: (

**a**) variation of vertical velocity at three points as function of time; (

**b**) comparisons of different averaging strategies for the vertical velocity at Point 4.

**Figure 7.**Dimensionless velocity vectors and recirculation regions in Chambers 2 and 3 on the x–y plane at z = L/2: (

**a**) steady state flow field in RANS; (

**b**) time-averaged flow field in LES.

**Figure 9.**Visualization of streamline patterns and vorticity field on the x–y plane at mid span (z = L/2) for RANS: (

**a**) dimensionless velocity vectors and vorticity contours. The dashed line corresponds to the separation layers; (

**b**) Separated boundaries of the jet region using vorticity gradient for a vertical cross section.

**Figure 10.**Three-dimensional configuration of jet regions of each chamber in the mixing tank for RANS. Jet flow region was identified using vorticity gradient.

**Figure 11.**Visualization of the jet region using the flexion product for RANS: (

**a**) on the x–y plane at mid span (z = L/2); (

**b**) a three-dimensional view of the jet regions in each chamber.

**Figure 12.**Visualization of jet region using a combination of vorticity gradient and flexion product for LES: (

**a**) on the x–y plane at mid span (z = L/2); (

**b**) a three-dimensional view of the jet regions of each chamber in the mixing tank.

**Figure 13.**Contour plot of the magnitude of the dimensionless Lamb vector on the x–y plane at mid span (z = L/2) for RANS.

Mesh | nx × ny × nz | Max (y^{+}) |
---|---|---|

Present Study | 210 × 90 × 90 | 2.21 |

Kim et al. [10] | 195 × 88 × 82 | 3.00 |

Zhang et al. [20] | 208 × 101 × 83 | 2.00 |

Chamber (m) | ${\left({\mathsf{\eta}}_{\mathit{e}\mathit{f}\mathit{f}}\right)}_{\mathit{m}}$ | |
---|---|---|

RANS | LES | |

1 | 0.348 | 0.242 |

2 | 0.396 | 0.230 |

3 | 0.350 | 0.252 |

4 | 0.396 | 0.211 |

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**MDPI and ACS Style**

Demirel, E.; Aral, M.M.
Unified Analysis of Multi-Chamber Contact Tanks and Mixing Efficiency Based on Vorticity Field. Part I: Hydrodynamic Analysis. *Water* **2016**, *8*, 495.
https://doi.org/10.3390/w8110495

**AMA Style**

Demirel E, Aral MM.
Unified Analysis of Multi-Chamber Contact Tanks and Mixing Efficiency Based on Vorticity Field. Part I: Hydrodynamic Analysis. *Water*. 2016; 8(11):495.
https://doi.org/10.3390/w8110495

**Chicago/Turabian Style**

Demirel, Ender, and Mustafa M. Aral.
2016. "Unified Analysis of Multi-Chamber Contact Tanks and Mixing Efficiency Based on Vorticity Field. Part I: Hydrodynamic Analysis" *Water* 8, no. 11: 495.
https://doi.org/10.3390/w8110495