# Numerical Investigation of the Electro-Thermo Convection in an Inclined Cavity Filled with a Dielectric Fluid

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}-water nanofluid. This cavity was subjected to a horizontal, constant, and uniform magnetic field. The vertical walls were differentially heated, and the inclination was varied between 0° and 60°. It was found that the variation of the tilt angle leads to changes in the structure of the flow. Indeed, the best thermal performances were obtained for an angle of inclination of 30°, unlike 60° which has the lowest Nusselt number values.

^{3}, increasing the angle of inclination leads to an improvement in the Nusselt number. For Ra = 10

^{4}and 10

^{5}, the average Nusselt number decreases as the angle of inclination are varied from 30° to 60° and increases again between 60° and 90°.

_{2}O

_{3}/water hybrid nanofluid and partially heated from the top and bottom walls. It has been found that heat transfer and air movement increase with the inclination of the cavity. The direction of the magnetic field and the volume fraction of the added nanoparticles are also parameters that control heat transfer.

^{4}≤ Ra ≤ 10

^{5}) were tested for a total of 20 different conditions. The techniques adopted for the experimental tests are particle image velocimetry (PIV) to determine velocity fields and a set of thermocouples to measure wall temperature. It has been determined that both the average and maximum velocities exhibit an increase as the Rayleigh number and tilting angle increase. The velocity field exhibits two vortices, with a symmetry axis crossing, the center of the enclosure when the tilt angle is 90°; as this angle increases, the symmetry tends to disappear, resulting in the presence of only one vortex for 90°.

## 2. Physical Model and Governing Equations

^{2}(Figure 1). The bottom and top surfaces are thermally insulated, while the side surfaces are differentially heated and held at a constant and homogenous temperature (with θ

_{H}denoting the hot side and θ

_{C}the cold one). To achieve an enhanced heat exchange, a potential difference is created between the horizontal surfaces. The bottom wall serves as the emitting electrode and is maintained at the highest potential (V

_{i}). The top wall is maintained at the lowest potential (V

_{0}) which is considered as the receiving electrode.

_{0}.

## 3. Numerical Method

^{−4}is found adequate to serving this purpose while stills being computationally efficient [32]. The numerical solution can be considered as converged and the SOR algorithm can be stopped when the following convergence criterion is being verified.

## 4. Results and Discussion

^{5}). Moreover, for high Ra values, the optimal heat exchanges are reached at 15–30° cavity inclination, while a reduction of 34% occurs at an angle of 90°.

^{2}= 0.95, a standard error σ = 0.06, and a p-value for the different parameters ranging from 0 to 0.007. All these statistical parameters, such as the p-value, standard deviation, and coefficient of determination, indicating the effectiveness and reliability of the proposed correlation.

## 5. Conclusions

- -
- The flow’s behavior can be significantly impacted by the inclination of the cavity. By increasing this angle, a shift from a unicellular to a bicellular regime was detected.
- -
- At high electric and thermal fields, when the angle α ∈ [0, 40°], increasing the tilt angle results in a 5% improvement in heat transfer. Above this angular range, an oscillation regime could be developed.
- -
- Local Nusselt numbers show that heat exchange occurs mainly in the lower section of the heated wall. A reduction in the convective heat transfer of up to 45% can be recorded when the angle varies from 0 to 90 degrees.
- -
- Increasing the electrical Rayleigh number (up to 800) and the thermal Rayleigh number (to 250,000) results in a respective improvement of 61% and 181% in heat transfer.
- -
- For high electrical Rayleigh values, the dominant electrical forces negate the impact of cavity tilt on heat transfer.
- -
- A multiparametric correlation was suggested to estimate the mean Nusselt number, based on the tilt angle, and both thermal and electrical Rayleigh numbers.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Thermal diffusivity (m^{2}·s^{−1}) |

C | Dimensionless number of the injection strength |

$\overrightarrow{\mathrm{E}}$ | Electric field (V·cm^{−1}) |

$\overrightarrow{\mathrm{g}}$ | Acceleration of gravity (m·s^{−2}) |

K | Ionic mobility (m^{2}·V^{−1}·s^{−1}) |

L | Enclosure width (m) |

M | Dimensionless number which characterizes EHD properties of the liquid |

Pr | Prandlt number |

q | Electric charge density (C·m^{−3}) |

Ra | Thermal Rayleigh number |

T | Dimensionless electric Rayleigh number |

t | Time (s) |

$\overrightarrow{\mathrm{U}}$ | Velocity (m·s^{−1}) |

V | Electric potential (V) |

x,y | Cartesian coordinate (m) |

Greek symbols | |

α | cavity inclination (°) |

β | Coefficient of thermal expansion of fluid (K^{−1}) |

ε | Permittivity of the fluid (F·m^{−1}) |

θ | Dimensionless temperature (K) |

μ | Dynamic viscosity (Pa·s) |

ν | Kinematic viscosity (m^{2}·s^{−1}) |

ρ | Density (kg.m^{−3}) |

ψ | Stream function (m^{2}·s^{−1}) |

ω | Vorticity (s^{−1}) |

Subscript | |

H | Hot |

C | Cold |

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**Figure 2.**Charge density distributions (

**left**), streamlines (

**middle**), and isothermal lines (

**right**) for Ra = 50,000, T = 600, Pr = 10, C = 10, for different tilt angles.

**Figure 3.**(

**a**) Time profile of the average Nusselt number; (

**b**) local Nusselt number on the hot plate for Ra = 50,000 and T = 600 and for various inclinations.

**Figure 4.**Distributions of charge density for different tilt angles and for different values of the electric Rayleigh number, Ra = 10,000, Pr = 10, C = 10, and M = 10.

**Figure 5.**Local Nusselt number on the hot wall for Ra = 50,000, Pr = 10, C = 10, M = 10, and various cavity inclination angles.

**Figure 6.**Average Nusselt number according to tilt cavity angle for Ra = 10,000, Pr = 10, C = 10, and M = 10 and various T values (

**a**) 0 ≤ T ≤ 200 (

**b**) 400 ≤ T ≤ 800.

**Figure 7.**Average Nusselt number according to tilt cavity angle for T = 200, Pr = 10, C = 10, M = 10, and various Ra values.

**Figure 8.**Average Nusselt number according to tilt cavity angle for various Ra and T numbers lines: Correlation values; Symbols: Numerical results.

Study Setup | Ra | T | α |
---|---|---|---|

Effect of Tilt angle | 50,000 | 600 | 0–90 |

Effect of Tilt angle for various electric field | 10,000 | 0–800 | |

Effect of Tilt angle for various thermal gradient | 5000–250,000 | 200 |

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**MDPI and ACS Style**

Akrour, D.; Elkhazen, M.I.; Hassen, W.; Kriaa, K.; Maatki, C.; Hadrich, B.; Kolsi, L.
Numerical Investigation of the Electro-Thermo Convection in an Inclined Cavity Filled with a Dielectric Fluid. *Processes* **2023**, *11*, 2506.
https://doi.org/10.3390/pr11082506

**AMA Style**

Akrour D, Elkhazen MI, Hassen W, Kriaa K, Maatki C, Hadrich B, Kolsi L.
Numerical Investigation of the Electro-Thermo Convection in an Inclined Cavity Filled with a Dielectric Fluid. *Processes*. 2023; 11(8):2506.
https://doi.org/10.3390/pr11082506

**Chicago/Turabian Style**

Akrour, Dalila, Mohamed Issam Elkhazen, Walid Hassen, Karim Kriaa, Chemseddine Maatki, Bilel Hadrich, and Lioua Kolsi.
2023. "Numerical Investigation of the Electro-Thermo Convection in an Inclined Cavity Filled with a Dielectric Fluid" *Processes* 11, no. 8: 2506.
https://doi.org/10.3390/pr11082506