# The Effect of Nanoparticle Shape and Microchannel Geometry on Fluid Flow and Heat Transfer in a Porous Microchannel

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## Abstract

**:**

_{2}O

_{3}nanofluids (NF) were numerically investigated considering the nanoparticle shape and different cross-sections of a porous microchannel. Spherical, cubic, and cylindrical shapes of the nanoparticle as well as circular, square, and triangular cross-sections of the microchannel were considered in the simulation. The finite volume method and the SIMPLE algorithm have been employed to solve the conservation equations numerically, and the k-ε turbulence model has been used to simulate the turbulence fluid flow. The models were simulated at Reynolds number ranging from 3000 to 9000, the nanoparticle volume fraction ranging from 1 to 3, and a porosity coefficient of 0.7. The results indicate that the average Nusselt number (Nu

_{ave}) increases and the friction coefficient decreases with an increment in the Re for all cases. In addition, the rate of heat transfer in microchannels with triangular and circular cross-sections is reduced with growing Re values and concentration. The spherical nanoparticle leads to maximum heat transfer in the circular and triangular cross-sections. The heat transfer growth for these two cases are about 102.5% and 162.7%, respectively, which were obtained at a Reynolds number and concentration of 9000 and 3%, respectively. However, in the square cross-section, the maximum heat transfer increment was obtained using cylindrical nanoparticles, and it is equal to 80.2%.

## 1. Introduction

_{ave}can be enhanced by a porosity coefficient increment. Goodarzi et al. [19] numerically studied the effect of slip velocity and temperature jump on the NF flow in the microchannel filled with porous materials. Their results show that in higher Reynolds numbers, the local Nusselt number (Nu) enhanced significantly. Moreover, a lower permeability leads to a higher local Nu.

_{ave}increases by a higher nanoparticle volume fraction and Re enhancement.

_{ave}.

_{ave}. Gao and Jian [28] proposed an analytical solution of magnetohydrodynamic fluid flow in a microchannel with a circular cross-section. According to their results, the volumetric flow rate grew and then decreased with the Hartmann number. El Mghari [29] conducted an experimental and numerical investigation on the effect of condensation on the thermal performance of a microchannel with a square cross-section. Their results indicate that the average and local Nu are strongly dependent on the heat flux through the wall.

_{ave}rises happened by Re and nanoparticle volumetric percentage growth. Ferrari et al. [31] conducted a numerical study on the effect of boiling inside a two-dimensional microchannel with a square cross-section. They found that for all Res, the bubble velocity in the channel with the square cross-section is higher than the other ones. Bahmanpour et al. [32] numerically studied the influence of various teeth on the heat transfer and NF flow in a microchannel. Their results show that an increment in the velocity of the NF leads to a significant improvement of the Nu

_{ave}and the thermal hydraulic performance. Weng et al. [33] carried out an experimental and numerical study on the effect of ribs on heat transfer in the microchannel. According to their findings, the presence of vortex generators will improve the thermal performance of the microchannel.

## 2. Problem Statement and Numerical Simulation

#### 2.1. Problem Definition

#### 2.2. Formulation

## 3. Meshing

#### 3.1. Boundary Conditions

#### 3.2. Grid Independence

#### 3.3. Validation

_{ave}have been compared to the results of their work, as shown in Figure 4. It can be seen that there is merely an inconsiderable difference between the Nu

_{ave}obtained in this paper and that of Arjun and Rakesh [55]. The error is just about 4.832%, which confirms the results’ validity.

## 4. Results and Discussion

#### 4.1. Nusselt Number in the Microchannel with the Circular Cross-Section

_{ave}for the cylindrical nanoparticle is smaller than that for the spherical and cubic nanoparticles at a concentration of 0.03. It is also can be concluded that the Nu

_{ave}for the spherical nanoparticle is larger than the cubic nanoparticles, and this difference increases for larger Re values. This observation was the same for a concentration of 0.02.

#### 4.2. Friction Coefficient for the Microchannel with the Circular Cross-Section

#### 4.3. Average Nusselt Number in the Microchannel with the Square Cross-Section

#### 4.4. Friction Coefficient for the Microchannel with the Square Cross-Section

#### 4.5. Average Nusselt Number in the Microchannel with the Triangular Cross-Section

#### 4.6. Friction Coefficient for the Microchannel with the Triangular Cross-Section

## 5. Conclusions

- The Nusselt number increases with growing Re for a porosity coefficient of 0.7, all three microchannel cross-sections, and all three nanoparticle shapes. Moreover, the Nusselt number decreases with growing NF concentration for the mentioned study cases.
- The Nusselt number for the cylindrical nanoparticle is smaller than those for the spherical and cubic nanoparticles for all volume fractions studied.
- For the cylindrical nanoparticle, the friction coefficient is reduced by Re enhancement. Furthermore, the friction coefficient is observed to be a growing function of nanoparticle volume fraction in the NF. This is exhibited more prominently at smaller Re values.
- In summary, this study shows that for a microchannel filled with porous media, the use of spherical nanoparticles results in a higher Nusselt number and a lower friction factor compared to other shapes of nanoparticles.
- In terms of the geometrical effect of the microchannel cross-section, the best heat transfer rate in the microchannel with triangular, rectangular, and circular cross-sections is recommended, respectively. Circular and triangular microchannels also have the lowest and highest friction coefficients, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Nomenclature | |

${c}_{p}$ | Specific heat capacity, J/K |

Nu | Nusselt number |

Re | Reynolds number |

k | Thermal conductivity coefficient, W/m·K |

Greek Symbols | |

$\rho $ | Density, kg/m^{3} |

$\varphi $ | Nanoparticles volume fraction, Pa·s |

$\mu $ | Dynamic viscosity |

ε | Porosity coefficient |

Subscripts | |

$f$ | Base fluid |

$avg$ | Average |

$np$ | Nanoparticle |

$nf$ | Nanofluid |

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**Figure 4.**Validation of the present study with Arjun and Rakesh [55] results.

**Figure 5.**Nu

_{ave}vs. Re for a porosity coefficient of 0.7 and for different volume fractions in a microchannel with a circular cross-section.

**Figure 6.**Friction coefficient vs. Re for a porosity coefficient of 0.7 and for different volume fractions in a microchannel with a circular cross-section.

**Figure 7.**Nu

_{ave}vs. Re for a porosity coefficient of 0.7 and for different volume fractions in a microchannel with a square cross-section.

**Figure 8.**Friction coefficient vs. Re for a porosity coefficient of 0.7 and for different volume fractions in a microchannel with a square cross-section.

**Figure 9.**Nu

_{ave}vs. Re for a porosity coefficient of 0.7 and for different volume fractions in a microchannel with a triangular cross-section.

**Figure 10.**Friction coefficient vs. Re for a porosity coefficient of 0.7 and for different volume fractions in a microchannel with a triangular cross-section.

Inlet Velocity | Reynolds |
---|---|

6.03 | 3000 |

10.05 | 5000 |

14.07 | 7000 |

18.09 | 9000 |

Two-Dimensional (Flow Modeling with Circular Cross-Section) Three-Dimensional (Flow Modeling with Square and Triangular Cross-Sections) | Computational Domain |
---|---|

Steady-state | Time-dependence |

Nonlinear $k-\epsilon $ | Turbulence model |

Use of standard wall function | Behavior near the wall |

Active | Equation of conservation of energy |

Quantity | Equations |
---|---|

1 | Continuity |

2 (for flow in the circular cross-section) | Navier–Stokes |

3 (for flow in the triangular and square cross-sections) | |

1 | Energy |

2 | Turbulence |

Value | Equations |
---|---|

0.3 | Pressure |

0.7 | Momentum |

0.8 | Turbulence energy |

0.8 | Turbulence energy dissipation rate |

1 | Energy |

Method | Term in Equations |
---|---|

Standard | Pressure |

Upwind | Momentum |

Upwind | Turbulence energy |

Upwind | Turbulence energy dissipation rate |

Upwind | Energy |

**Table 6.**Physical properties of water and nanoparticles [54].

Water | Aluminum Oxide | Physical Properties |
---|---|---|

997.1 | 3970 | $\rho (kg/{m}^{3})$ |

0.00089 | - | $\mu (Pa.s)$ |

4179 | 765 | ${c}_{P}(J/kg.K)$ |

0.6 | 40 | $k(W/m.K)$ |

Number of Cells | Nusselt Number |
---|---|

35 × 250 | 0.85515696 |

40 × 250 | 0.8424636 |

45 × 250 | 55.56332 |

50 × 250 | 52.42397 |

55 × 250 | 51.91109 |

Number of Cells | Nusselt Number |
---|---|

35 × 35 × 250 | 1.567 |

40 × 40 × 250 | 15.67 |

45 × 45 × 250 | 32.11 |

50 × 50 × 250 | 34.987 |

55 × 55 × 250 | 35.678 |

Number of Cells | Nusselt Number |
---|---|

35 × 35 × 250 | 8.674 |

40 × 40 × 250 | 23.12 |

45 × 45 × 250 | 65.14 |

50 × 50 × 250 | 73.98 |

55 × 55 × 250 | 74.134 |

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**MDPI and ACS Style**

Abdelmalek, Z.; D’Orazio, A.; Karimipour, A.
The Effect of Nanoparticle Shape and Microchannel Geometry on Fluid Flow and Heat Transfer in a Porous Microchannel. *Symmetry* **2020**, *12*, 591.
https://doi.org/10.3390/sym12040591

**AMA Style**

Abdelmalek Z, D’Orazio A, Karimipour A.
The Effect of Nanoparticle Shape and Microchannel Geometry on Fluid Flow and Heat Transfer in a Porous Microchannel. *Symmetry*. 2020; 12(4):591.
https://doi.org/10.3390/sym12040591

**Chicago/Turabian Style**

Abdelmalek, Zahra, Annunziata D’Orazio, and Arash Karimipour.
2020. "The Effect of Nanoparticle Shape and Microchannel Geometry on Fluid Flow and Heat Transfer in a Porous Microchannel" *Symmetry* 12, no. 4: 591.
https://doi.org/10.3390/sym12040591