# Computational Analysis of Viscoplastic Nanofluid Blending by a Newly Modified Anchorage Impeller within a Stirred Container

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## Abstract

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_{2}O

_{3}) inside a symmetrically stirred tank. A 3D numerical study has been conducted for a stationary laminar flow inside a symmetric cylindrical vessel under influencing parameters, including the inertia parameter ($Re=1,20,100$) and the volume fraction of nanoparticles ($\mathsf{\xd8}=0.02,0.06,0.1$) with different geometric configurations, has been introduced into the stirring system. According to the findings, with high inertia ($Re=100$), the heat transfer inside the stirred tank is enhanced. Furthermore, increasing the nanoparticle fraction volume had a significant impact on the acceleration of heat transfer along the stirred vessel. It has been also found that the geometric configuration of an anchor with added arm blade (Case 2) is more efficient compared with the rest of the anchor agitator.

## 1. Introduction

_{2}O

_{3}). The influence of adding a nanoparticle volume fraction in the range of (0.02, 0.06, 0.1) has been explored as well as the impact of the inertia value on the flow pattern for different values of $Re=1,20,$ and $100$. Three different cases have been used to introduce the effect of the new symmetrical anchor impeller design in this study, which are named Case 1 (classical anchor impeller), Case 2 (anchor impeller with one added blade), and Case 3 (anchor with two added symmetric arm blades).

## 2. Mixing System

- On the impeller and bottom $\frac{\partial T}{\partial n}=0$;
- On the impeller and bottom on the vessel surface $T=1$.

## 3. Mathematical Modeling

_{2}O

_{3}nanoparticles within a Bingham–Papanastasiou base fluid. This study has been performed by using the COMSOL-Multiphysics tools. These software tools solve the system of governing continuity–momentum equation and energy Equations (7)–(11). The finite element method based on the Galerkin’s Discretization was used. The computational domain was defined with an unstructured mesh called tetrahedral mesh, shown in Figure 2, and the residual convergence of the steady state is at least 10

^{−6}for various variables in this simulation.

- Reynolds number:$$Re=\frac{\rho N{d}^{2}}{\mu}$$
- Bingham number:$$Bn=\frac{\tau D}{{\mu}_{p}N}$$
- Nusselt number:$$Nu=\frac{{k}_{nf}}{{k}_{f}}{\displaystyle \int}\frac{\partial T}{\partial X}+\frac{\partial T}{\partial Y}+\frac{\partial T}{\partial Z}$$
- Power consumption:$$Np=\frac{P}{\rho {N}^{3}{d}^{5}}$$$$P={{\displaystyle \int}}_{A}2\pi N\xb7\left(x{F}_{y}-y{F}_{x}\right)\xb7dA$$

## 4. Validation

## 5. Results and Discussions

#### 5.1. Effect of Rheology

#### 5.2. Impeller Design Effect

#### 5.3. Effect of Adding Nanoparticles

## 6. Conclusions

_{2}O

_{3}nanoparticles inside a stirred vessel equipped with different geometric anchor impeller agitator designs. This study aimed to investigate the effect of inertia on the mixing parameters (flow pattern, heat transfer, and power consumption). Various Reynolds numbers ($Re=1,20,100$) and volume fractions of nanoparticles ($\mathsf{\xd8}=0.02,0.06,0.1$) were assessed.

- The flow pattern inside the stirred tank shifted in consequence of the inertia parameters; an increase in the velocity was conducive to an increase in inertia inside the stirred tank.
- The tangential flow was relatively dominant in the stirred tank in the case of inertia absence or low Reynolds value ($Re=1,20$); however, a high Reynolds value ($Re=100$) resulted in shifting the flow pattern from the tangential to the axial direction.
- Heat transfer has been affected by the changing inertia parameter and the volume fraction. Furthermore, with high inertia ($Re=100$), we discovered that the inertia accelerated the heat transfer inside the stirred tank. In addition to the effect of nanoparticle density, however, the high value of the fraction volume led to increasing the heat transfer inside the stirred tank.
- Finally, there was a decrease in energy consumption inside the vessel, which was a consequence of the rise in inertia value.
- Overall, inertia has a high likelihood of improving flow patterns, enhancing heat transfer, and reducing power consumption levels, ultimately being a positive influence on the processes undergone in stirred tanks.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | surface around the impeller (${\mathrm{m}}^{2}$) | $Re$ | Reynolds number (dimensionless) |

$Bn$ | Bingham number (dimensionless) | $S$ | rate-of-deformation tensor |

$D$ | vessel diameter ($\mathrm{m}$) | $V{t}^{*}$ | dimensionless velocity (${\mathit{Vt}}^{*}=\mathit{Vt}/\pi \mathit{ND}$) |

$Da$ | anchor impeller diameter ($\mathrm{m}$) | $w$ | blade width ($\mathrm{m}$) |

$Ds$ | shaft diameter ($\mathrm{m}$) | Greek Symbols | |

${F}_{x}$ | $x$—Force direction ($\mathrm{kg}/{\mathrm{ms}}^{2}$) | $\dot{\gamma}$ | shear rate ($1/\mathrm{s}$) |

${F}_{y}$ | $y$—Force direction ($\mathrm{kg}\xb7\mathrm{m}/\mathrm{s}2$) | $\mu $ | dynamic viscosity ($\mathrm{kg}/\mathrm{ms}$) |

$H$ | tank height ($\mathrm{m}$) | ${\mu}_{0}$ | plastic viscosity ($\mathrm{kg}/\mathrm{ms}$) |

$m$ | stress growth parameter ($\mathrm{s}$) | $\Gamma $ | torque ($\mathrm{kg}\xb7{\mathrm{m}}^{2}/{\mathrm{s}}^{2}$) |

$N$ | rotational speed impeller ($\mathrm{m}/\mathrm{s}$) | $\rho $ | fluid density, ($\mathrm{kg}/{\mathrm{m}}^{3}$) |

$Np$ | power number (dimensionless) | $\tau $ | shear stress ($\mathrm{kg}/{\mathrm{ms}}^{2}$) |

$Nu$ | Nusselt number average (dimensionless) | $\Omega $ | impeller rotational speed ($1/\mathrm{s}$) |

$R$ | vessel radius ($\mathrm{m}$) |

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**Figure 3.**Comparison of tangential velocity distribution outcomes along the impeller plan for $Re=13.8$ with the findings of Ref. [27].

**Figure 4.**Streamline distribution inside (

**a**) the vessel volume and (

**b**) along the horizontal section.

**Figure 5.**Velocity distribution in (

**a**) the impeller plane, (

**b**) the median plane, and (

**c**) along the horizontal section.

**Figure 9.**Velocity contour for different positions of (

**a**) vertical section and (

**b**) median plane impeller plane.

$D$ | $H$ | $Da$ | $w$ | $Ds$ | $c$ |

0.3 | 0.3 | 0.15 | 0.03 | 0.02 | 0.06 |

Properties | Density ($\mathrm{kg}/{\mathrm{m}}^{3}$) | Thermal Capacity ($\mathrm{J}/\mathrm{kg}\mathrm{K}$) | Thermal Conductivity ($\mathrm{W}/\mathrm{m}\mathrm{K}$) | Dynamic Viscosity ($\mathrm{Pa}\xb7\mathrm{s}$) | Thermal Expansion ($1/\mathrm{K}$) |

Al_{2}O_{3} | 3970 | 765 | 40 | - | 0.85 × 10^{−5} |

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## Share and Cite

**MDPI and ACS Style**

Brahim, M.; Benhanifia, K.; Jamshed, W.; Al-Farhany, K.; Redouane, F.; Eid, M.R.; Hussain, S.M.; Akram, M.; Kamel, A.
Computational Analysis of Viscoplastic Nanofluid Blending by a Newly Modified Anchorage Impeller within a Stirred Container. *Symmetry* **2022**, *14*, 2279.
https://doi.org/10.3390/sym14112279

**AMA Style**

Brahim M, Benhanifia K, Jamshed W, Al-Farhany K, Redouane F, Eid MR, Hussain SM, Akram M, Kamel A.
Computational Analysis of Viscoplastic Nanofluid Blending by a Newly Modified Anchorage Impeller within a Stirred Container. *Symmetry*. 2022; 14(11):2279.
https://doi.org/10.3390/sym14112279

**Chicago/Turabian Style**

Brahim, Mebarki, Kada Benhanifia, Wasim Jamshed, Khaled Al-Farhany, Fares Redouane, Mohamed R. Eid, Syed Modssir Hussain, Mohammad Akram, and Alwaleed Kamel.
2022. "Computational Analysis of Viscoplastic Nanofluid Blending by a Newly Modified Anchorage Impeller within a Stirred Container" *Symmetry* 14, no. 11: 2279.
https://doi.org/10.3390/sym14112279