# Forced Convection of Pulsating Nanofluid Flow over a Backward Facing Step with Various Particle Shapes

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

**Boundary Conditions and Nusselt Number Calculation**

- At the channel inlet:$U=1+sin\left(2\pi \mathrm{St}\tau \right),\phantom{\rule{4pt}{0ex}}V=0,\phantom{\rule{4pt}{0ex}}\theta =1$
- On the bottom wall downstream of the step:$U=V=0,\phantom{\rule{4pt}{0ex}}\theta =1$
- For the remaining walls:$U=V=0,\phantom{\rule{4pt}{0ex}}\frac{\partial \theta}{\partial n}=0$
- At the channel exit, outflow conditions were imposed:$\frac{\partial U}{\partial X}=0,\phantom{\rule{4pt}{0ex}}V=0,\phantom{\rule{4pt}{0ex}}\frac{\partial \theta}{\partial X}=0$

**Nanofluid Effective Thermophysical Properties**

## 3. Solution Methodology

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

A | amplitude of pulsation |

f | frequency of pulsation |

h | local heat transfer coefficient |

k | thermal conductivity |

H | step size |

n | unit normal vector |

Nu${}_{x}$ | local Nusselt number |

Nu${}_{m}$ | averaged Nusselt number |

p | pressure |

Pr | Prandtl number |

Re | Reynolds number |

St | Strouhal number |

T | temperature |

t | time |

u, v | x-y velocity components |

x, y | Cartesian coordinates |

Greek Characters | |

$\alpha $ | thermal diffusivity |

$\beta $ | expansion coefficient |

$\varphi $ | solid volume fraction |

$\nu $ | kinematic viscosity |

$\theta $ | non-dimensional temperature |

$\rho $ | density of the fluid |

Subscripts | |

c | cold |

h | hot |

m | average |

nf | nanofluid |

p | solid particle |

st | static |

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**Figure 2.**Particle size effect on the reattachment length for various nanoparticle volume fractions.

**Figure 3.**Variation of the x component velocity along the y direction for various particle shapes with $\varphi =0.02$.

**Figure 4.**Variation of the x component velocity along the y direction for various nanoparticle volume fractions with cylindrically-shape particles.

**Figure 5.**Effects of the particle shape and volume fraction on the local Nusselt number distribution along the hot bottom wall downstream of the step.

**Figure 6.**Variation of streamlines for several time instances within a period for the base fluid and for the nanofluid with cylindrical nanoparticles.

**Figure 7.**Isotherm distributions for several time instances within a period for the base fluid and for the nanofluid with cylindrical nanoparticles.

**Figure 8.**Effects of particle shape on the spatially-average Nusselt number along the hot wall for various pulsating frequencies.

**Figure 9.**Average Nusselt numbers versus solid particle volume fractions for various pulsating frequencies and nanoparticle shapes.

**Figure 10.**The effects of the Strouhal number on the spatially-averaged Nusselt number along the hot wall for water and the spherical particle shape with $\varphi =0.02$.

**Figure 11.**The effects of the Strouhal number on the spatially-averaged Nusselt number along the hot wall for the blade and cylindrical particle shapes with $\varphi =0.02$.

**Figure 12.**The average Nusselt number versus the Strouhal number for different nanoparticle volume fractions with spherically-shaped particles.

Property | Water | SiO${}_{2}$ |
---|---|---|

$\rho \phantom{\rule{4pt}{0ex}}(\mathrm{kg}/{\mathrm{m}}^{3})$ | 998.2 | 2200 |

$\phantom{\rule{4.pt}{0ex}}{\mathrm{c}}_{p}\phantom{\rule{4pt}{0ex}}(\mathrm{J}/\mathrm{kg}\phantom{\rule{4.pt}{0ex}}\mathrm{K})$ | 4812 | 703 |

$\mathrm{k}\phantom{\rule{4pt}{0ex}}(\mathrm{W}/\mathrm{mK})$ | 0.61 | 1.2 |

$\mu \phantom{\rule{4pt}{0ex}}(\mathrm{N}\phantom{\rule{4.pt}{0ex}}\mathrm{s}/{\mathrm{m}}^{2})$ | 0.001003 | - |

Nanoparticle Shape | C${}_{\mathit{k}}$ | A${}_{\mathbf{1}}$ | A${}_{\mathbf{2}}$ |
---|---|---|---|

cylindrical | 3.95 | 13.5 | 904.4 |

bricks | 3.37 | 1.9 | 471.4 |

blades | 2.74 | 14.6 | 123.3 |

**Table 3.**Grid independence test with spherical particles at two different solid particle volume fractions.

Grid Name | Number of Elements | Nu${}_{\mathit{m}}$ ($\mathit{\varphi}\mathbf{=}\mathbf{0}\mathbf{\%}$) | Nu${}_{\mathit{m}}$ ($\mathit{\varphi}\mathbf{=}\mathbf{4}\mathbf{\%}$) |
---|---|---|---|

G1 | 632 | 0.965 | 1.345 |

G2 | 1430 | 1.243 | 1.594 |

G3 | 6240 | 1.436 | 1.795 |

G4 | 9848 | 1.455 | 1.836 |

G5 | 67,256 | 1.467 | 1.865 |

**Table 4.**Comparison of the recirculation lengths with the results of Khandelwal et al. [28].

Re | L${}_{\mathit{R}}$/D (Khandelwal et al. [28]) | L${}_{\mathit{R}}$/D (Present Solver) | Difference (%) |
---|---|---|---|

100 | 2.73 | 2.64 | −3.29 |

150 | 3.39 | 3.36 | −0.88 |

200 | 3.99 | 3.96 | −0.75 |

**Table 5.**Spatial-temporal average Nusselt number for various particle shapes at a Strouhal number of 0.1 and $\varphi =0.04$. HTE, Heat Transfer Enhancement.

Particle Type | Nu${}_{\mathit{s}}$ (Steady) | Nu${}_{\mathit{m}}$ (Pulsating) | HTE in Steady (%) | HTE in Pulsating (%) |
---|---|---|---|---|

Spherical | 1.635 | 2.092 | 12.37 | 27.95 |

Blade | 1.836 | 1.950 | 26.185 | 19.26 |

Cylindrical | 1.895 | 1.953 | 30.24 | 19.44 |

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**MDPI and ACS Style**

Chamkha, A.J.; Selimefendigil, F.
Forced Convection of Pulsating Nanofluid Flow over a Backward Facing Step with Various Particle Shapes. *Energies* **2018**, *11*, 3068.
https://doi.org/10.3390/en11113068

**AMA Style**

Chamkha AJ, Selimefendigil F.
Forced Convection of Pulsating Nanofluid Flow over a Backward Facing Step with Various Particle Shapes. *Energies*. 2018; 11(11):3068.
https://doi.org/10.3390/en11113068

**Chicago/Turabian Style**

Chamkha, Ali J., and Fatih Selimefendigil.
2018. "Forced Convection of Pulsating Nanofluid Flow over a Backward Facing Step with Various Particle Shapes" *Energies* 11, no. 11: 3068.
https://doi.org/10.3390/en11113068