# Convective Heat Transfer and Entropy Generation for Nano-Jet Impingement Cooling of a Moving Hot Surface under the Effects of Multiple Rotating Cylinders and Magnetic Field

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## Abstract

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## 1. Introduction

## 2. Numerical Model

- Jet inlet, $u=0,v=-{u}_{c},T={T}_{c}$
- Outlet: $\frac{\partial u}{\partial x}=\frac{\partial T}{\partial x}=0,\phantom{\rule{4pt}{0ex}}v=0$
- At the target plate: $u={u}_{0},v=0,\phantom{\rule{4pt}{0ex}}T={T}_{h}$
- Upper plate walls: $u=v=0,\phantom{\rule{4pt}{0ex}}\frac{\partial T}{\partial x}=0$
- At the rotating cylinder walls: $u=-\omega (y-yci),\phantom{\rule{4pt}{0ex}}v=\omega (x-xci),\phantom{\rule{4pt}{0ex}}\frac{\partial T}{\partial n}=0$

## 3. Results and Discussion

## 4. Conclusions

- Rotations of the CCs near the jet inlet have negative impacts of the HT enhancement for both stationary and moving hot wall cases. Reductions in the average Nu of up to 20% and 12.5% are obtained by using rotations at Rew = −500 for stationary (V = 0) and moving wall (VR = 0.25) cases.
- The wall movement contributes positively to the cooling performance while HT enhancements up to 14% are achieved by wall velocity at the highest speed (VR = 0.25).
- Depending upon the activation of cylinder rotations, the impacts of MGF strength on the HT characteristics are different. For non-rotating CCs, cooling performance is reduced by about 28% until Ha = 10, while by using rotations at Rew = −200, it is increased by about 18.6%.
- For the rotating CC case, average Nu variations up to 12% can be achieved by varying the horizontal location of the CCs while away from the inlet, higher cooling performances are obtained.
- When the hot wall starts to move at VR = 0.25, the EG increments up to 6.25% and $5.25\%$ are obtained for non-rotating and rotating CC cases compared with the stationary wall (VR = 0) configuration.
- The MGF acts to reduce the EG by about 23% for non-rotating cylinders while increment of EG by about 66% is obtained for rotating cylinders at Rew = 200. Away from the jet inlet, the EG rises.
- The best configuration for the case of non-rotating CCs is achieved at (Ha, VR, Mx) = (0, 0.25, −4), and HT increment becomes 8.7% compared with the reference case of (Rew = 0, VR = 0, Ha = 0, Mx = 0). For the non-rotating CC case, the optimum set of parameters is achieved at (Ha, VR, Mx) = (20, 0.25, −8) with HT enhancement of 34.2% compared with the reference.
- The maximum EG is obtained for configuration with (Ha, VR, Mx) = (20, 0.25, −8) with non-rotating CCs, while the value is 15% higher than the reference case. When rotations are active at Rew = −500, the case (Ha, VR, Mx) = (0, 0.25, −4) has only 1% higher EG when compared to reference.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

B${}_{0}$ | Magnetic field strength |

Ha | Hartmann number |

H | separating distance |

h | heat transfer coefficient |

k | thermal conductivity |

L | plate length |

n | unit normal vector |

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

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

p | pressure |

Pr | Prandtl number |

r | cylinder radius |

Re | Reynolds number |

Rew | rotational Reynolds number |

T | temperature |

u, v | x-y velocity components |

uc | jet velocity |

uw | wall velocity |

VR | velocity ratio |

wj | slot width |

x, y | Cartesian coordinates |

Greek Characters | |

$\alpha $ | thermal diffusivity |

$\gamma $ | magnetic field inclination |

$\theta $ | non-dimensional temperature |

$\nu $ | kinematic viscosity |

$\rho $ | density of the fluid |

$\sigma $ | electrical conductivity |

$\varphi $ | solid volume fraction |

$\omega $ | rotational speed |

Subscripts | |

c | cold |

h | hot |

m | average |

nf | nanofluid |

w | wall |

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**Figure 1.**Different available methods that can be used for flow control and thermal performance improvement of jet impingement cooling.

**Figure 2.**Schematic description of the SJ-I cooling system for a moving hot wall under combined effects of MGF and multiple rotating CCs.

**Figure 3.**Test results for grid independence: Average Nu for different grid sizes considering rotating and non-rotating CCs (

**a**) (Ha = 5, Mx = −1) and grid distribution near the CCs (

**b**).

**Figure 4.**Code validation 1: Average Nu comparisons at three different Ha considering the convection in a differentially heated enclosure. Results in [42] are used, and Grashof number is fixed to Gr = $2\times {10}^{5}$ (

**a**). Code validation 2: Stagnation point Nu comparison for SJ-I cooling of an isothermal surface by using the reference results in [43,44] at Reynolds number 100 and 300 (

**b**).

**Figure 5.**Effects of Rew on streamline variations considering two different velocities of hot wall (Ha = 5, Mx = −1).

**Figure 6.**Effects of hot wall velocity on streamline variations for rotating and non-rotating CC cases (Ha = 5, Mx = −1).

**Figure 8.**Impacts of MGF strength on the streamline distributions considering rotating (

**a**–

**d**), non-rotating CC cases (

**e**–

**h**) and on the average Nu variations (

**i**) (VR = 0.25, Mx = −1).

**Figure 9.**Variations of streamlines with changes in the horizontal location of the multiple CCs considering rotational (

**a**–

**c**) and stationary (

**d**–

**f**) cases of CCs and impacts of horizontal location of CCs on the variation of average Nu considering rotating and non-rotating CC cases (

**g**) (Ha = 5, Mx = −1).

**Figure 10.**Impacts of rotational Re on the variation of EG for stationary (VR = 0) and moving wall (VR = 0.25) cases (

**a**) and impacts of VR on EG for rotating (Rew = −500) and non-rotating CC cases (Rew = 0) CC cases (

**b**) (Ha = 5, Mx = −1).

**Figure 11.**Effects of MGF strength (

**a**) horizontal location of the CCs (

**b**) on the variation of EG considering rotating and non-rotating CC cases (VR = 0.25, Mx = −1).

**Figure 12.**Comparison of best cases in terms of thermal performance (

**a**) and EG (

**b**) considering both rotating and non-rotating CC case.

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## Share and Cite

**MDPI and ACS Style**

Kolsi, L.; Selimefendigil, F.; Larguech, S.; Ghachem, K.; Albalawi, H.; Alshammari, B.M.; Labidi, T.
Convective Heat Transfer and Entropy Generation for Nano-Jet Impingement Cooling of a Moving Hot Surface under the Effects of Multiple Rotating Cylinders and Magnetic Field. *Mathematics* **2023**, *11*, 1891.
https://doi.org/10.3390/math11081891

**AMA Style**

Kolsi L, Selimefendigil F, Larguech S, Ghachem K, Albalawi H, Alshammari BM, Labidi T.
Convective Heat Transfer and Entropy Generation for Nano-Jet Impingement Cooling of a Moving Hot Surface under the Effects of Multiple Rotating Cylinders and Magnetic Field. *Mathematics*. 2023; 11(8):1891.
https://doi.org/10.3390/math11081891

**Chicago/Turabian Style**

Kolsi, Lioua, Fatih Selimefendigil, Samia Larguech, Kaouther Ghachem, Hind Albalawi, Badr M. Alshammari, and Taher Labidi.
2023. "Convective Heat Transfer and Entropy Generation for Nano-Jet Impingement Cooling of a Moving Hot Surface under the Effects of Multiple Rotating Cylinders and Magnetic Field" *Mathematics* 11, no. 8: 1891.
https://doi.org/10.3390/math11081891