# Mixed Convection in a Double Lid-Driven Cavity Filled with Hybrid Nanofluid by Using Finite Volume Method

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## Abstract

**:**

_{2}O

_{3}-Cu-Water) is employed to increase the heat transfer rate in a double lid-driven rectangular cavity. The bottom movable horizontal wall is kept at a high temperature while the top movable horizontal wall is kept at a low temperature. The sidewalls are insulated. The mass, momentum and energy equations are numerically solved using the Finite Volume Method (FVM). The SIMPLE algorithm is used for pressure-velocity coupling. Parameters such as Reynold’s number ($Re$), Richardson number ($Ri$), moving wall direction, solid volume fraction, and cavity length are studied. The results show that the hybrid nanofluid in the rectangular cavity is able to augment the heat transfer significantly. When $Re$ is high, a big size solid body can augment the heat transfer. Heat transfer increases with respect to $Ri$. Meanwhile, the local Nusselt number decreases with respect to the cavity length.

## 1. Introduction

_{2}O

_{3}-Water) nanofluid. The authors have revealed that the increase in both the Rayleigh and Reynolds numbers would increase the energy transport. Mixed convection in a wavy bottom cavity containing a solid inner block has been studied by Azizul et al. [3] and it has been reported that nanofluid would improve the heat transfer in the cavity. Additionally, both Nusselt and Grashof numbers increase with respect to the volume fraction of nanofluids. The heat transfer in a lid-driven rectangular cavity filled with visco-plastic fluid driven by a magnetic field has been investigated by Louaraychi et al. [4]. The authors have found that an increase in Hartmann’s number would decrease the heat transfer regardless of aspect ratio and Reynolds number. Goodarzi et al. [5] have analyzed the mixed convection using (Cu-Water) nanofluid in a shallow rectangular cavity using the two-phase mixture model. They reported that the local Nusselt number, the average Nusselt number and the nanofluid heat transfer coefficient increased with respect to the volume fraction of nanoparticles. Moreover, Karimipour et al. [6] have studied the mixed laminar convection in an inclined shallow lid-driven cavity filled with (Cu-Water) nanofluid by using the Lattice Boltzmann Method (LBM). On the other hand, Mahmoodi [7] studied numerically the mixed convection involving the (Al

_{2}O

_{3}-Water) nanofluid in rectangular lid-driven enclosures. It has been found that the average Nusselt number of the hot wall increased with respect to the volume fraction of nanoparticles. Furthermore, it has been reported that the average Nusselt number at the hot wall of the long enclosure is more than that in the shallow enclosure. In addition, Karimipour et al. [8] investigated numerically the periodic mixed convection flow of (Cu-Water) nanofluid in a 2D rectangular cavity. The authors found that the heat transfer rate could be improved by increasing the volume fraction of nanoparticles. Moreover, Yaseen and Ismael [9] have studied the mixed convection of the (Cu-Water) nanofluid with varying thermo-physical properties in a trapezoidal enclosure saturated with a porous media. They have found that for all Darcy numbers, the average Nusselt number increased as the volume fraction of nanoparticle increased. The movement of nanofluid decreased when the Darcy number decreased, thereby reducing the Nusselt number. Besides that, [9] analyzed the mixed convection of incompressible power-law fluid in an open trapezoidal cavity involving Fluid Structure Interaction (FSI). They have reported that at the lower wall of the channel, the skin friction coefficient decreased with respect to the power-law index. Owing to the superior heat transfer properties of nanofluid, it is widely used in many heat transfer applications such as solar collector, thermal energy storage, material processing, and electronics cooling (see [10,11,12]). Bahiraei [13] has shown that methods such as single-phase model, two-phase model, and LBM [14] can be used to analyze nanofluid. The thermal properties of Al

_{2}O

_{3}nanofluid can be enhanced by combining a small number of metallic nanoparticles with Al

_{2}O

_{3}, or better known as the hybrid nanofluid. The use of hybrid nanofluid enhances the thermal conductivity and the stability of nanofluid. Suresh et al. [15] have studied experimentally the fully developed laminar flow through a straight and heated circular tube using hybrid nanofluid (Al

_{2}O

_{3}-Cu-Water). Lately, Sarkar et al. [16] and Babu et al. [17] have reviewed different hybrid nanofluids to identify various heat transfer properties, thermo-physical characteristics and synthesis techniques for different heat transfer applications. These review studies have revealed that hybrid nanofluid is better than conventional nanofluid in terms of heat transfer. From the literature review, the most widely used model for modelling nanofluids is the homogeneous single-phase model, whereby the accuracy is dependent on the employed thermo-physical models. Takabi and Salehi [18] have studied numerically the convective heat transfer in a sinusoidal cavity filled with (Al

_{2}O

_{3}-Cu-Water) hybrid nanofluid and (Al

_{2}O

_{3}-Water) single nanofluid. It has been reported that the use of hybrid nanofluid would lead to higher heat transfer rate as compared to the single nanofluid. Chamkha et al. [19] have studied the unsteady conjugate natural convection inside a semi-circular enclosure filled with (Al

_{2}O

_{3}-Cu-Water) hybrid nanofluid. As reported, the use of hybrid nanofluid led to higher values of thermal conductivity and Rayleigh number. Some experimental studies have been conducted by using hybrid nanofluids [20,21,22,23,24]. From the literature review, the study of mixed convection with hybrid nanofluid within a rectangular cavity has not been studied yet. Hence, the present study examines the fluid flow and heat transfer rate in the cavity. The goal of this research is to assess the impacts of Richardson number, moving wall direction, Reynolds number, solid volume fraction, and cavity length on the mixed convection performance in a rectangular cavity filled with (Al

_{2}O

_{3}-Cu-Water) hybrid nanofluid. We have chosen the aluminum oxide and copper nanoparticles in this work because copper and aluminum oxide are high in thermal conductivity and cheaper as compared to other types.

## 2. Mathematical Formulation

_{2}O

_{3}and Cu nanoparticles. The mathematical models governing the mixed convection can be written as:

_{2}O

_{3}-Cu-Water) hybrid nanofluid for particle sizes 33 nm and 29 nm in the ambient condition as:

## 3. Numerical Method

## 4. Grid-Independence Test

## 5. Results and Discussion

_{2}O

_{3}nanoparticles are shown in Table 2.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${C}_{p}$ = specific heat capacity; |

$df$ = diameter of the base fluid molecule; |

$dp$ = diameter of the nanoparticle; |

$g$ = gravitational acceleration; |

$k$ = thermal conductivity; |

${k}_{b}$ = Boltzmann’s constant $\left(1.380648\times {10}^{-23}\right)$; |

$\mathrm{L}$ = side length of enclosure; |

$Gr$ = Grashof number; |

$p\&P$ = pressure and dimensionless pressure, |

$\mathrm{Pr}$ = Prandtl number; |

$Re$ = Reynolds number; |

$R{e}_{B}$ = Brownian motion Reynolds number; |

$Ri$ = Richardson number, $Ri=Gr/R{e}^{2}$; |

$T$ = temperature; |

${T}_{0}$ = reference temperature ($310\mathrm{K}$); |

${T}_{fr}$ = freezing point of the base fluid ($273.15\mathrm{K}$); |

$v\&V$ = velocity anddimensionless velocity, |

${u}_{B}$ = Brownian velocity of the nanoparticle; and |

$x,y\&X,Y$ = space coordinates anddimensionless space coordinates. |

Greek symbol |

$\theta $ = dimensionless temperature; |

$\beta $ = thermal expansion coefficient; |

$\mu $ = dynamic viscosity; |

$\nu $ = kinematic viscosity; |

$\rho $ = density; |

$\phi $ = solid volume fraction; |

$\alpha $ = thermal diffusivity; |

$\beta $ = Thermal expansion coefficient |

Subscript |

$c$ = cold; |

$f$ = base fluid; |

$h$ = hot; |

$hnf$ = hybrid nanofluid; |

$p$ = solid nanoparticles; |

$\lambda $ = lid-driven direction |

b = bottom wall |

t = top wall |

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**Figure 3.**Variation of streamlines (left) and isotherms (right) with respect to solid volume fraction (

**a**) $\varphi =0$ and (

**b**) $\varphi =0.04$.

**Figure 4.**Variation of streamlines (left) and isotherms (right) with respect to $Ri$ (

**a**) $Ri=0.01$, (

**b**) $Ri=0.1,$ (

**c**) $Ri=1,$ and (

**d**) $Ri=10$.

**Figure 5.**Variation of streamlines (left) and isotherms (right) with respect to $Re$ (

**a**) $Re=10$, (

**b**) $Re=50,$ (

**c**) $Re=250,$ and (

**d**) $Re=500$.

**Figure 6.**Variation of streamlines (left) and isotherms (right) with respect to the direction of moving walls (

**a**) ${\lambda}_{t}=1,{\lambda}_{b}=-1$, (

**b**) ${\lambda}_{t}=-1,{\lambda}_{b}=1,$ (

**c**) ${\lambda}_{t}=1,{\lambda}_{b}=1,$ and (

**d**) ${\lambda}_{t}=-1,{\lambda}_{b}=-1$.

**Figure 7.**Variation of streamlines (left) and isotherms (right) with respect to cavity length L (

**a**) $L=0.5$, (

**b**) $L=1,$ (

**c**) $L=1.5,$ and (

**d**) $L=2.5$.

Size | Average Nusselt Number $\overline{\mathit{N}\mathit{u}}$ |
---|---|

60 × 30 | 12.237437 |

80 × 40 | 13.041916 |

100 × 50 | 13.457630 |

120 × 60 | 13.714399 |

140 × 70 | 13.892601 |

**Table 2.**Thermo-physical properties of water, Cu nanoparticle and Al

_{2}O

_{3}nanoparticle at $T=310\mathrm{K}$ [29]

Physical Properties | Fluid (Water) | Copper | Al_{2}O_{3} |
---|---|---|---|

$k\left({\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}\right)$ | 0.628 | 400 | 40 |

$\mu \times {10}^{6}\left(\mathrm{kg}/\mathrm{ms}\right)$ | 695 | - | - |

$\rho \left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | 993 | 8933 | 3970 |

${C}_{p}\left(\mathrm{J}/\mathrm{kgK}\right)$ | 4178 | 385 | 765 |

$\beta \times {10}^{-5}\left(1/\mathrm{K}\right)$ | 36.2 | 1.67 | 0.85 |

${d}_{p}\left(\mathrm{nm}\right)$ | 0.385 | 29 | 33 |

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**MDPI and ACS Style**

Ali, I.R.; Alsabery, A.I.; Bakar, N.A.; Roslan, R.
Mixed Convection in a Double Lid-Driven Cavity Filled with Hybrid Nanofluid by Using Finite Volume Method. *Symmetry* **2020**, *12*, 1977.
https://doi.org/10.3390/sym12121977

**AMA Style**

Ali IR, Alsabery AI, Bakar NA, Roslan R.
Mixed Convection in a Double Lid-Driven Cavity Filled with Hybrid Nanofluid by Using Finite Volume Method. *Symmetry*. 2020; 12(12):1977.
https://doi.org/10.3390/sym12121977

**Chicago/Turabian Style**

Ali, I.R., Ammar I. Alsabery, N.A. Bakar, and Rozaini Roslan.
2020. "Mixed Convection in a Double Lid-Driven Cavity Filled with Hybrid Nanofluid by Using Finite Volume Method" *Symmetry* 12, no. 12: 1977.
https://doi.org/10.3390/sym12121977