# MHD Hybrid Nanofluid Mixed Convection Heat Transfer and Entropy Generation in a 3-D Triangular Porous Cavity with Zigzag Wall and Rotating Cylinder

^{1}

^{2}

^{3}

^{4}

^{5}

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## Abstract

**:**

_{3}O

_{4}/MWCNT-water nanofluid are performed using the finite element approach (FEM). The simulation is carried out for a range of parameter values, including the Darcy number (between 10

^{−5}and 10

^{−2}), the Hartmann number (between 0 and 100), the angular speed of the rotation (between −500 and 1000), and the number of zigzags. The stream function, isotherms, and isentropic contours illustrate the impact of many parameters on motion, heat transfer, and entropy formation. The findings indicate that for enhancing the heat transfer rates of hybrid nanofluid in a three-dimensional triangular porous cavity fitted with a rotating cylinder and subjected to a magnetic field, Darcy number > 10

^{−3}, Hartmann number < 0, one zigzag on the hot surface, and rotation speed >500 in flow direction are recommended.

## 1. Introduction

_{2}/water nanoliquids in a photovoltaic/thermal collector and found that it increased electricity production by 7% and heat output by 55%. Singh et al. [10] investigated the machining performance of a graphene nanoplatelet-based water-based nanoliquid while turning an AISI 304. The findings demonstrated that increasing particle concentrations resulted in a considerable decrease in cutting temperature and surface roughness. Zarifi et al. [11] simulated the neutronic behaviour of VVER-1000 reactors using a variety of water-based nanofluids. The effective multiplication factor was decreased in all nanoliquids; however, the effect varied according to the nanoparticle type. Jin et al. [12] enhanced solar energy collection and conveyance by using a water-based Au (gold) and MWCNTs nanoliquid. A significant increase in energy absorption was seen when nanoparticles were added to water; as compared to pure water, MWCNTs nanoliquid harvests about 5.5 times more energy. Additionally, studies have shown a significant increase in the thermal performance of heat pipes using hybrid nanofluids. Martin et al. [13] investigated the thermal performance of a simple heat pipe charged with an aqueous Fe:CuO hybrid nanoliquid at a 50:50 ratio. Triton X-100 was used as a surfactant to increase the stability of the produced sample. The findings indicated a maximum decrease in thermal resistance of 16.91% using the hybrid nanoliquid at a mass fraction of 2% compared to water.

_{3}O

_{4}nanoparticles in the biomedical sector, focusing on their potential for cancer therapy. According to their findings, biosynthesized Fe

_{3}O

_{4}nanoparticles are capable of accumulating and delivering more medicines to the targeted location. More precisely, following injection into the bloodstream, these particles may be manipulated or steered by an external magnetic field [16,17]. Sun et al. [18] studied the cooling performance of a mini-channel heat sink using deionized copper and alumina nanofluids. It was discovered that tested nanofluids performed admirably and significantly improved heat sink cooling performance. When compared to deionized water, a temperature reduction of 4 to 18 °C was seen in the CPU chip. According to the literature about nanofluids, it is well-documented that they are greatly affected by various external factors such the presence of radiation, magnetic fields, porous media, geometrical parameters and operating parameters. Recent research has focused on the effect of spinning cylinders, porous media, and magnetic forces on the mixed convection flow of nanofluids, since these techniques are good for managing the flow and movement of energy within nanofluids.

_{2}O

_{3}-water nanoliquid flow inside a porous media in a side-cooled wavy cavity heated from below using a computer model. Their results indicate that wall waviness, as well as other flow-controlling properties, had a significant effect on thermal convection. Hamzah et al. [42] investigated the MHD thermal transport mechanism of an Al

_{2}O

_{3}-water nanoliquid in a wavy porous multilayer enclosure with varying temperatures. They observed that the waviness and porosity of the porous membrane had a significant effect on heat transfer. Cimpean et al. [43] simulated steady natural convection in an inclined square porous cavity filled with nanofluid and subjected to a sinusoidal temperature distribution using numerical simulations. They focused on the effects of heat and fluid flow on the inclination angle and periodic thermal boundary conditions.

_{3}O

_{4}and MWCNT/water) inside a 3-D triangular porous chamber under the influence of an internal centrally located spinning circular cylinder and magnetic field. It is worth noting that the examined issue may be seen as a heat transfer performance study for a heat exchanger, an electronic heated element mounted vertically, or a solar collector operating under the effect of a smart working fluid (nanofluid). All of these issues are connected to the enhancement of convective energy transport and the advancement of heat transfer control systems.

## 2. Mathematical Model

_{3}O

_{4}and MWCNT nanoparticles to form the working suspensions. Table 1 summarizes the thermophysical characteristics of both the base fluid and nanoparticles. The one–energy equation model is used, with the assumption that the mixture density is a linear function of the temperature. Taking these assumptions into account, the governing equations for the studied case are as follows:

**Table 1.**Nanoparticles and base fluid thermos-physical properties [45].

Pure Water | Fe_{3}O_{4} | MWCNT | |
---|---|---|---|

$\rho $ (kg/m^{3}) | 997.1 | 5180 | 2100 |

C_{p} (J/kg k) | 4179 | 670 | 710 |

$k$ (W/m k) | 0.613 | 9.7 | 2000 |

## 3. Validation and Mesh Evaluation

_{avg}and the Bejan number are used to establish the independence of heat transfer from the number of grids (see Table 3). Due to the variability of the findings, the fourth grid was chosen as the final grid in all situations, as seen in Figure 1. As a main criterion for reaching findings, Ghasemi et al.’s [46] past investigations were employed to validate our model, as seen in Figure 2.

## 4. Results and Discussion

^{−}

^{5}, 10

^{−}

^{4}, 10

^{−}

^{3}, and 10

^{−}

^{2,}, to demonstrate the ease of flow at higher Da values due to increased permeability, resulting in enhanced heat transfer and minimized entropy generation. It is noted from the figure that flow streamlines pronounce themselves more with higher Da values, which is expected due to low flow resistance (more fluidity) with higher permeability (Da) values. Due to the zigzag feature at the hot surface, the streamlines are less smooth in the left region of the cavity compared to those closer to the colder surface (the right region)—two primary vortices are generated around the cylinder.

^{−}

^{3}. Despite the fact that the cylinder rotation provides a mixed convection heat transfer, the lower value of Da increases the flow resistance. It may restrict the heat transfer to being the natural one. Thus, higher permeability values are recommended to improve flow and heat transfer characteristics.

_{Avg}) and average Bejan number (Be

_{Avg}) values profiles are highlighted.

_{Avg}with the variation in Ha and Ω values. It is clear that the increase in Ha values reduces the heat transfer rates, as discussed in Figure 4. The noticeable trend observed here is that the increase of Ha values up to 22 has a slight effect compared to those > 22, which means the Ha > 22 could show the flow hindering effect of the magnetic field (Lorentz force). However, the decrease in the heat transfer index (Nu

_{Avg}) is not considered high for the mixed heat transfer mechanism due to the cylinder rotation. For example, at Ω = 1000 [rad/s], the Nu

_{Avg}is equal to ~5.78 at Ha = 0, while it is about 5.20 for Ha = 100. The effect of rotation speed and direction on Nu

_{Avg}are similar to those already discussed in Figure 5, in which the higher speed in the same flow direction improves the heat transfer rates.

_{Avg}values slightly increase when Ha < 22 and then gradually rise for Ha > 22, as shown in Figure 8. Better Be

_{Avg}values are obtained with higher cylinder rotation speed, especially when the direction is clockwise, and lower Ha values due to improved mixed-convection heat transfer and decreased irreversible heat losses.

_{Avg}) are increased with increased Da values due to improving the fluidity inside the porous media. Furthermore, because of increased fluidity, the improved heat transfer minimizes the Be values, as shown in Figure 10. Again, better Nu

_{Avg}and lower Be

_{Avg}values are associated with higher Ω values having similar flow direction, as shown in Figure 9 and Figure 10, respectively.

## 5. Conclusions

_{3}O

_{4}/MWCNT-water hybrid nanofluid and equipped with a moving cylinder is investigated for thermal-hydraulic enhancements. The cavity is a trapezoid with a zigzagged hot wall on the left, while the right wall is cold, and the others are well-insulated. Three-dimensional CFD-based simulation was conducted using the Galerkin finite element (GFEM) method with a triangle element shape to simulate the investigated cavity. The developed model was validated against reported experimental data in the literature.

^{−5}, 10

^{−4}, 10

^{−3}, and 10

^{−2}), Hartmann number (0, 20, 50 and 100), rotating cylinder speed (−500, 0, 500, 1000), and the number of zigzags of the hot wall (1, 2, and 4). Based on the investigated parameters, the following conclusions are drawn:

- -
- Higher Darcy number values enhance the fluid flow, temperature distribution, and heat transfer and minimize heat transfer’s contribution to the total entropy generation.
- -
- The effect of the Hartmann number on the flow characteristic is not high for mixed heat transfer flow in a porous cavity having a zigzagged wall; however, values of >22 (Hartmann number) could be avoided.
- -
- Clockwise rotation of the embedded cylinder enhances the heat transfer and fluidity better than a counter-clockwise one. However, both directions lead to a better heat transfer mechanism than the stationary cylinder (natural convection).
- -
- For the investigated geometries, one zigzag on the hot wall is sufficient to enhance the heat transfer rates and reduce the entropy generation contribution from the heat losses.

^{−3}, Hartmann number <0, one zigzag on the hot surface, and cylinder rotation of >500 in flow direction are recommended.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

List of Symbols | |

${C}_{p}$ | Specific heat capacity (Jkg^{−1} K^{−1}) |

$H,L$ | Dimensionless of triangular cavity (m) |

$p$ | Pressure (Pa) |

$T$ | Temperature (K) |

$X,Y$ | Dimensionless Cartesian coordinates |

$Ha$ | Hartmann number |

$Pr$ | Prandtl number |

$Pr$ | Permeability |

${B}_{0}$ | Intensity of magnetic field |

$g$ | Gravitational acceleration (ms^{−2}) |

$g$ | Thermal conductivity (Wm^{−1} K^{−1}) |

$P$ | Dimensionless pressure |

$u,v$ | Velocity components (ms^{−1}) |

$x,y$ | Dimensional Cartesian coordinates (m) |

$Ra$ | Rayleigh number |

Nu | Nusselt number |

$r$ | Radius of cylinder |

${F}_{c}$ | Forchheimer coefficient |

Greeks symbols | |

$\epsilon $ | Porosity |

Subscripts | |

${f}_{np}$ | Fluid |

np | Solid particles |

loc | Local |

c | Cold |

hnf | Hybrid nanofluid |

avg | Average |

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**Figure 1.**The computational domain: (

**a**) 3D view of the enclosure; (

**b**) 2D view of the geometry with illustrations of the boundary conditions. (

**c**) Grid mesh.

**Figure 2.**Comparisons of present model with previous works [46].

**Figure 3.**Three-dimensional surface plots to show the effect of Darcy number (permeability) on the streamlines, temperature in terms of dimensionless isotherms, and Bejan number.

**Figure 4.**Three-dimensional surface plots to show the effect of the Hartmann number on the streamlines, temperature in terms of dimensionless isotherms, and Bejan number.

**Figure 5.**Three-dimensional surface plots to show the effect of the cylinder rotation direction on the streamlines, temperature in terms of dimensionless isotherms, and Bejan number.

**Figure 6.**Three-dimensional surface plots to show the effect of the zigzag number on the streamlines, temperature in terms of dimensionless isotherms, and Bejan number.

**Figure 7.**Effect of the Hartmann number on the average Nusselt number at different values and directions of the cylinder rotation.

**Figure 8.**Effect of the Hartmann number on the average Bejan number at different values and directions of the cylinder rotation.

**Figure 9.**Effect of the Darcy number on the average Nusselt number at different values and directions of the cylinder rotation.

**Figure 10.**Effect of the Darcy number on the average Bejan number at different values and directions of the cylinder rotation.

**Figure 11.**Effect of the values and direction of the cylinder rotation on the average Nusselt number for three zigzags number of the hot wall.

**Figure 12.**Effect of the values and direction of the cylinder rotation on the average Bejan number for three zigzags number of the hot wall.

Properties | Classical Nanofluid | Hybrid Nanofluid |
---|---|---|

Density | ${\rho}_{hnf}=\left(1-\phi \right){\rho}_{fluid}+\phi {\rho}_{np}$ | ${\rho}_{np}=\frac{{\phi}_{{}_{F{e}_{3}{O}_{4}}}{\rho}_{{}_{F{e}_{3}{O}_{4}}}+{\phi}_{MWCNT}{\rho}_{{}_{MWCNT}}}{\phi}$ |

Heat capacity | ${\left(\rho {c}_{p}\right)}_{hnf}=\left(1-\phi \right){\left(\rho {c}_{p}\right)}_{fluid}+\phi {\left(\rho {c}_{p}\right)}_{np}$ | ${\left({c}_{p}\right)}_{np}=\frac{{\phi}_{{}_{F{e}_{3}{O}_{4}}}{\left({c}_{p}\right)}_{{}_{F{e}_{3}{O}_{4}}}+{\phi}_{MWCNT}{\left({c}_{p}\right)}_{{}_{MWCNT}}}{\phi}$ |

Coefficient of thermal expansion | ${\left(\rho \beta \right)}_{hnf}=\left(1-\phi \right){\left(\rho \beta \right)}_{fluid}+\phi {\left(\rho \beta \right)}_{np}$ | ${\beta}_{np}=\frac{{\phi}_{{}_{F{e}_{3}{O}_{4}}}{\beta}_{{}_{F{e}_{3}{O}_{4}}}+{\phi}_{MWCNT}{\beta}_{{}_{MWCNT}}}{\phi}$ |

Electrical conductivity | ${\sigma}_{hnf}=\left(1-\phi \right){\sigma}_{fluid}+\phi {\sigma}_{np}$ | ${\sigma}_{np}=\frac{{\phi}_{{}_{F{e}_{3}{O}_{4}}}{\sigma}_{{}_{F{e}_{3}{O}_{4}}}+{\phi}_{MWCNT}{\sigma}_{{}_{MWCNT}}}{\phi}$ |

Thermal conductivity | ${k}_{hnf}=\frac{{k}_{np}+(n-1){k}_{f}-(n-1)({k}_{f}-{k}_{np})\phi}{{k}_{np}+(n-1){k}_{f}+({k}_{f}-{k}_{np})\phi}{k}_{f}$ | ${k}_{np}=\frac{{\phi}_{{}_{F{e}_{3}{O}_{4}}}{k}_{{}_{F{e}_{3}{O}_{4}}}+{\phi}_{MWCNT}{k}_{{}_{MWCNT}}}{\phi}$ |

Viscosity | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-\phi \right)}^{2.5}}$ |

No. of Grid Elements | 8207 | 14,676 | 31,957 | 124,306 | 72,629 |

Nu_{avg} | 3.2612 | 3.2598 | 3.3491 | 3.3493 | 3.3492 |

Be_{avg} | 0.4456 | 0.42162 | 0.43049 | 0.43025 | 0.43025 |

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**MDPI and ACS Style**

Abderrahmane, A.; Qasem, N.A.A.; Younis, O.; Marzouki, R.; Mourad, A.; Shah, N.A.; Chung, J.D. MHD Hybrid Nanofluid Mixed Convection Heat Transfer and Entropy Generation in a 3-D Triangular Porous Cavity with Zigzag Wall and Rotating Cylinder. *Mathematics* **2022**, *10*, 769.
https://doi.org/10.3390/math10050769

**AMA Style**

Abderrahmane A, Qasem NAA, Younis O, Marzouki R, Mourad A, Shah NA, Chung JD. MHD Hybrid Nanofluid Mixed Convection Heat Transfer and Entropy Generation in a 3-D Triangular Porous Cavity with Zigzag Wall and Rotating Cylinder. *Mathematics*. 2022; 10(5):769.
https://doi.org/10.3390/math10050769

**Chicago/Turabian Style**

Abderrahmane, Aissa, Naef A. A. Qasem, Obai Younis, Riadh Marzouki, Abed Mourad, Nehad Ali Shah, and Jae Dong Chung. 2022. "MHD Hybrid Nanofluid Mixed Convection Heat Transfer and Entropy Generation in a 3-D Triangular Porous Cavity with Zigzag Wall and Rotating Cylinder" *Mathematics* 10, no. 5: 769.
https://doi.org/10.3390/math10050769