# Investigation of Mixed Convection in a Cylindrical Lid Driven Cavity Filled with Water-Cu Nanofluid

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{4}, 10

^{5}, 10

^{6}) numbers are studied. In addition, the effect of concentration of nano materials ($\varphi $ = 0%, 1%, 5%), the Height Ratio (HR = 1, 0.5, 2) on Nusselt number, isotherm lines and streamlines are studied. The results show that Reynolds number also can change the effect of nano particles on the heat transfer rate. It is observed that the height ratio increase can improve the Nusselt number since the number and the size of vortices inside the cavity changes. In addition, increase of Ra number can change the flow structure inside the cavity which can help in increasing of Nusselt number.

## 1. Introduction

_{2}O

_{3}, SiO

_{2}or Fe

_{2}O

_{3}, which can enhance the thermal conductivity and heat capacity [12,13,14,15,16,17,18,19,20,21]. Choi et al. introduced nano-particle usage in utilizing heat transfer for the first time and showed that the usage of nanofluid could enhance thermal conductivity by 7 percent [22].

_{2}, Al

_{2}O

_{3}, TiO

_{2,}and CuO water mixture. They also found that the Nusselt number increases as the volume fraction of nano-particles increases but it’s value decreases when the diameter of nano-particle increases.

_{2}O

_{3}and the test case was examined for several Reynolds, Darcy, Hartman numbers and nano-particles concentrations. They found that the increase in Reynolds, Darcy number and the nanoparticle volume fraction could increase the heat transfer while the increase in the Hartman number can decrease it. Alrashed et. al studied the entropy generation and mixed convection in an open cavity that an isothermal block was placed at the center of the cavity to heat up the nanofluid [30]. The flow could enter from the top of the cavity, pass through the domain, and leave the cavity from the bottom. They checked various Richardson numbers and block sizes and found that the nano-particles can be effective just for small block size. Their study also revealed that the entropy generation is highly dependent on the Ri number.

_{2}O

_{3}was studied for different inclination by Hussain et al. [31]. The most important objective of their research was to find the influence of nanoparticles volume fraction and Reynolds number on the patterns of temperature field. They realized that increasing of volume fraction to 8% could increase the local Nusselt number up to 50%. Khanafer and Aithal used a spectral element method to study the influence of a rotating cylinder on the mixed convection in a lid-driven cavity [32]. They simulated both clockwise and counterclockwise rotation and they found that there is critical velocity at which the increase of Nusselt number stops; then the Nusselt number starts decreasing, thereafter. Mixed convection in a 3D cavity with two rotational adiabatic cylinders was analyzed to find the highest heat transfer rate for different nanoparticles [11]. The cavity was filled with three different nanoparticles Cu, Al

_{2}O

_{3}and TiO

_{2}. It was shown that the Cu and water have the highest heat transfer rate and can enhance the heat transfer rate up to 38.1%. All these studies were conducted to analysis the heat transfer rate in different conditions that can help us find optimum design for industrial systems by optimization algorithm like genetic algorithm or artificial algorithm which is explained in detail in [33]. Recently, researchers study the new type of nanofluid like carreau or Wilson and the application of nanofluids as phase change material [34,35,36].

## 2. Problem Configuration

## 3. Materials and Methods

## 4. The Numerical Approach

^{−6}. The $\varphi $ in the Equation (17) is the generic variable and it represents $U,\text{}V,W\text{}\mathrm{or}\text{}\theta $ in each equation. The $\left(i,j,k\right)$ represents the coordinates of each node in the xyz coordinate system (x, y and z direction respectively) and the $n$ shows the iteration numbers.

## 5. Verification and Validation

## 6. Result and Discussion

^{4}to 10

^{6}. The nano-particles concentration also varied from 0% to 5% in this research. All material properties were assumed to be constant. In addition, the aspect ratios were selected between 0.5, 1 and 2 to cover all range of geometry from shallow cavity to deep cavity. In this study, the ratio of radius difference to the inner radius was kept unity and constant for all cases.

^{4}(See Figure 4a), there is just a large vortex in the cavity and a small vortex at the left corner. As the Rayleigh number increases to 10

^{5}, the buoyant force can push up the large vortex and make more space for another vortex which direction of rotation is against the first vortex. By reaching to Ra = 10

^{6}, the buoyancy force becomes more dominant and it can make a flow similar to Taylor-Couette flow. In this case, we have four vortices rotating against each other pair wisely. Increasing the Rayleigh number to more than 10

^{6}makes the flow turbulent and it can make many vortices in the flow.

^{4}and low Reynolds numbers, the Nusselt number is almost a straight line but for the high Reynolds (10

^{3}), the Nusselt number increases as the concentration of nano-particles increases in the fluid and still, the growth rate is linear. At Ra = 10

^{5}, the Nusselt number does not increase by concentration change for Re = 100 but for Re = 400 and 1000, the increase of concentration helps to increase the Nusselt number. At Ra = 10

^{6}, the concentration of nano-particles affects Nusselt numbers for all Reynolds numbers. For the case with HR = 0.5, as all curves show, the Nusselt number is affected by the change of nano particle concentrations. The change in Nusselt number for the Reynolds number is about 6% when the concentration changes from 0% to 5%. This value for the high Reynolds number (Re = 1000) reaches to 25%. Figure 9 presents the values of Nusselt number for the cavity with the aspect ratio of 2. Compare to the other aspect ratios, the Nusselt values decrease due to the bigger space between the bottom and the top wall. The change of Nusselt values for the Re = 100 is about 4% although this change for higher Reynolds reaches to 46%. By comparing Nusselt values for different aspect ratios, it can be concluded that when the Rayleigh or aspect ratio increases, the nano-particles becomes more effective. The reason is that the nano-particles enhance the heat capacity of the nanofluid mixture more than other parameters like density, viscosity or conductivity and for the high Re flow, the effect of advection terms are more than diffusion terms which helps to increase heat transfer in the flow.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | Area (${m}^{2}$) |

$U,V,W$ | Dimensionless velocities |

$u,v,w$ | Velocity components ($m/s$) |

$X,Y,Z$ | Dimensionless Cartesian coordinates |

$x,y,z$ | Cartesian coordinates ($m$) |

$T$ | Temperature ($K$) |

$\theta $ | Dimensionless Temperature |

$Re$ | Reynolds number |

$Ri$ | Richardson number |

$Pr$ | Prandtl number |

$Nu$ | Nusselt number |

$p$ | Pressure ($Pa$) |

$P$ | Dimensionless pressure |

$k$ | Thermal conductivity ($W/m\xb7K$) |

h | Heat transfer coefficient ($W/{m}^{2}\xb7K$) |

n | Normal direction |

L | Reference length |

C_{p} | Heat capacity ($\raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$K$}\right.$) |

g | Gravity acceleration ($\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{${s}^{2}$}\right.$) |

Subscripts | |

$f$ | fluid |

$nf$ | Nano fluid |

s | Solid nano particle |

Greek letters | |

$\varphi $ | Volume fraction of nano particles |

$\beta $ | Thermal expansion coefficient ($1/K$) |

$\mu $ | dynamic viscosity ($Pa\xb7s$) |

$\rho $ | Density ($kg/{m}^{3}$) |

$\psi $ | dimensionless stream function |

$\mathsf{\Phi}$ | Primitive variables |

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**Figure 3.**Streamlines of middle plane for different aspect ratios of cavity at Re = 100, Ra = 10

^{4}, φ = 0%; (

**a**) HR = 0.5, (

**b**) HR = 1, (

**c**) HR = 2.

**Figure 4.**Streamlines of middle plane for different Rayleigh numbers at the Re = 100, φ = 0% and HR=2. (

**a**) Ra = 10

^{4}, (

**b**) Ra = 10

^{5}, (

**c**) Ra = 10

^{6}.

**Figure 5.**Isotherm lines of the middle plane for different aspect ratios of the cavity at Re = 100, Ra = 10

^{4}, φ = 0%. (

**a**) HR = 0.5, (

**b**) HR = 1, (

**c**) HR = 2.

**Figure 6.**Isotherm lines of the middle plane for different Rayleigh numbers at the Re = 100, Ra = 10

^{4}and HR = 2. (

**a**) φ = 0%, (

**b**) φ = 1%, (

**c**) φ = 5%.

**Figure 7.**Variation of averaged Nusselt number, at different nanoparticles volume concentration at HR = 1. (

**a**) Ra = 10

^{4}, (

**b**) Ra = 10

^{5}, (

**c**) Ra = 10

^{6}.

**Figure 8.**Variation of averaged Nusselt number, at different nanoparticles volume concentration at HR = 0.5. (

**a**) Ra = 10

^{4}, (

**b**) Ra = 10

^{5}, (

**c**) Ra = 10

^{6}.

**Figure 9.**Variation of averaged Nusselt number, at different nanoparticles volume concentration at HR = 2. (

**a**) Ra = 10

^{4}, (

**b**) Ra = 10

^{5}, (

**c**) Ra = 10

^{6}.

$\mathit{\rho}\left(\mathit{k}\mathit{g}{\mathit{m}}^{-3}\right)$ | ${\mathit{C}}_{\mathit{P}}\left(\mathit{J}\mathit{k}{\mathit{g}}^{-1}{\mathit{K}}^{-1}\right)$ | $\mathit{k}\left(\mathit{W}{\mathit{m}}^{-1}{\mathit{K}}^{-1}\right)$ | $\mathit{\beta}\left({\mathit{K}}^{-1}\right)$ | $\mathit{\mu}$ | |
---|---|---|---|---|---|

${\mathrm{H}}_{2}\mathrm{O}$ | 997.1 | 4179 | 0.613 | 21 × 10^{−5} | 9.09 × 10^{−5} |

$\mathrm{Cu}$ | 8954 | 383 | 400 | 1.67 × 10^{−5} | - |

Grid | ${\mathit{G}}^{1}$ | ${\mathit{G}}^{2}$ | ${\mathit{G}}^{3}$ | ${\mathit{G}}^{4}$ |
---|---|---|---|---|

nodes | $31\times 31\times 51$ | $51\times 51\times 71$ | $71\times 71\times 91$ | $101\times 101\times 121$ |

Nu | 15.56 | 15.98 | 16.041 | 16.049 |

% of deviation | 3 | 0.5 | ≈0 | 0 |

Computational time(s) | 8190 | 11756 | 14435 | 16280 |

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**MDPI and ACS Style**

Niazmand, A.; Fathi Sola, J.; Alinejad, F.; Rahimi Dehgolan, F. Investigation of Mixed Convection in a Cylindrical Lid Driven Cavity Filled with Water-Cu Nanofluid. *Inventions* **2019**, *4*, 60.
https://doi.org/10.3390/inventions4040060

**AMA Style**

Niazmand A, Fathi Sola J, Alinejad F, Rahimi Dehgolan F. Investigation of Mixed Convection in a Cylindrical Lid Driven Cavity Filled with Water-Cu Nanofluid. *Inventions*. 2019; 4(4):60.
https://doi.org/10.3390/inventions4040060

**Chicago/Turabian Style**

Niazmand, Amirreza, Jalal Fathi Sola, Farhad Alinejad, and Foad Rahimi Dehgolan. 2019. "Investigation of Mixed Convection in a Cylindrical Lid Driven Cavity Filled with Water-Cu Nanofluid" *Inventions* 4, no. 4: 60.
https://doi.org/10.3390/inventions4040060