# Natural Convection in a Circular Enclosure with Four Cylinders under Magnetic Field: Application to Heat Exchanger

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Simplifying Assumptions

- The fluid used is considered incompressible and Newtonian fluid;
- The flow is thought to be stable, laminar, and two-dimensional;
- The effect of radiation considered minimal;
- Neglect is shown also for displacement currents, dissipation, and Joule heating.

#### 2.2. Equations of the Model

#### 2.2.1. Dimensional Form

#### 2.2.2. Nondimensional Form

#### 2.2.3. Boundary Conditions

_{h}temperature ($>{T}_{c}$).

## 3. Results and Discussion

#### 3.1. Validation

#### 3.2. Flow Field

#### 3.3. Temperature Field

#### 3.4. Heat Transfer

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

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

$g$ | Gravitational acceleration, (${\mathrm{m}\mathrm{s}}^{-2}$) |

Ha | Hartmann number |

L | Length of cavity, (m) |

Nu | Nusselt number |

$p$ | Fluid pressure, $Pa={\mathrm{N}\mathrm{m}}^{-2}$ |

${p}^{*}$ | Dimensionless pressure |

Pr | Prandtl number |

Ra | Rayleigh number |

$T$ | Temperature, (K) |

${T}^{*}$ | Dimensionless temperature |

s | Distance between inner cylinders (m) |

S | Dimensionless distance between cylinders |

d | Diameter of inner cylinder |

${T}_{h}$ | Temperature of hot fluid, (K) |

${T}_{c}$ | Temperature of cold fluid, (K) |

MHDNC | magnetohydrodynamic natural convection |

$u,\text{}v,\text{}$ | Velocity components in x, and y directions |

${u}^{*},{\text{}v}^{*},$ | Dimensionless velocity components |

$x,\text{}y,$ | Cartesian coordinates |

${x}^{*},{y}^{*},$ | Dimensionless coordinates |

α | Thermal diffusivity, ${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ |

λ | Thermal conductivity, $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ |

Cp | Specific heat capacity, $\mathrm{J}{\mathrm{K}\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ |

σ | Electrical conductivity, $\mathrm{A}\mathrm{m}{\mathrm{V}}^{-1}$ |

β | Expansion coefficient, ${\mathrm{K}}^{-1}$ |

ρ | Density, ${\mathrm{k}\mathrm{g}\mathrm{m}}^{-3}$ |

υ | Cinematic viscosity, ${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ |

η | Normal direction dimensionless coordinate |

c | Cold |

h | Hot |

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**Table 1.**Validation in numerical code at $\mathrm{R}\mathrm{a}={10}^{5}$ and $\mathrm{R}\mathrm{a}={10}^{6}$.

Ha | Nu Present Work At $\mathbf{R}\mathbf{a}={10}^{5}$ | G.C. Pal et al. [30] At Ra = 10^{5} | Nu Present Work At $\mathbf{R}\mathbf{a}={10}^{6}$ | G.C. Pal et al. [30] At Ra = 10^{6} |
---|---|---|---|---|

20 | $1.3467$ | $1.44$ | $3.0546$ | $3.0$0 |

40 | $1.2980$ | $1.20$ | $2.4429$ | $2.50$ |

60 | $1.1330$ | $1.10$ | $2.1916$ | $2.22$ |

80 | $1.0487$ | $0.98$ | $1.8428$ | $1.77$ |

100 | $1.0039$ | $0.96$ | $1.7782$ | $1.70$ |

120 | $0.9777$ | $0.96$ | $1.5753$ | $1.47$ |

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**MDPI and ACS Style**

Azzouz, R.; Hamida, M.B.B.
Natural Convection in a Circular Enclosure with Four Cylinders under Magnetic Field: Application to Heat Exchanger. *Processes* **2023**, *11*, 2444.
https://doi.org/10.3390/pr11082444

**AMA Style**

Azzouz R, Hamida MBB.
Natural Convection in a Circular Enclosure with Four Cylinders under Magnetic Field: Application to Heat Exchanger. *Processes*. 2023; 11(8):2444.
https://doi.org/10.3390/pr11082444

**Chicago/Turabian Style**

Azzouz, Raouia, and Mohamed Bechir Ben Hamida.
2023. "Natural Convection in a Circular Enclosure with Four Cylinders under Magnetic Field: Application to Heat Exchanger" *Processes* 11, no. 8: 2444.
https://doi.org/10.3390/pr11082444