# Entropy Generation Analysis and Natural Convection in a Nanofluid-Filled Square Cavity with a Concentric Solid Insert and Different Temperature Distributions

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Numerical Method and Validation

## 4. Results and Discussion

## 5. Conclusions

- The inner solid insertion tends to influence the flow behavior: temperature distribution and local entropy generation affected by the resistance due to the conductive heat transfer in the solid inner square.
- A very strong enhancement is observed on the heat transfer rate with the increasing of Rayleigh number from ${10}^{4}$ to ${10}^{5}$ as compared to the other values. This is due to a significant increment of the buoyancy forces compared to the viscous forces in this range of Rayleigh number.
- The thermal property and the size of the inner solid square significantly influenced the rate of the heat transfer. A larger solid insert inhibits the convective heat transfer within the square cavity, as well as a low thermal conductivity of the solid insert. Furthermore, the higher thermal conductivity of the solid insert tends to increase the heat transfer rate.
- The global entropy generation increases with the augmentation of the Rayleigh number, while a counteractive behavior was observed on average Bejan number. As Rayleigh number rises, the average Bejan number tends to reduce.
- The global entropy generation is a decreasing function of the size of the inner solid for the case of intensive convection regime ($Ra\ge {10}^{5}$), while average Bejan number is an increasing function of the size of the inner solid of the same regime.
- A larger solid insert with a high thermal resistance inhibits the global entropy generation. However, the global entropy generation is strong at the higher nanoparticles volume fraction. Whereas a relatively bigger solid insert leads to a positive influence on the average Bejan number.
- Some other directions in future work could include non-Newtonian fluids, three dimensional problems and non-uniform heating. Including the non-Darcy effect in the formulation of the porous medium problem is also a considerable future work.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$Be$ | Bejan number |

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

${d}_{bf}$ | equivalent diameter of the a molecule of base fluid |

g | gravitational accleration |

GEG | Dimensionless global entropy generation |

k | thermal conductivity |

${K}_{r}$ | square wall to base fluid thermal conductivity ratio, ${K}_{r}={k}_{w}/{k}_{bf}$ |

L | width and height of enclosure |

d | width and height of inner solid square |

D | ratio of the length of the inner solid to that of the outer hollow, $D=d/L$ |

${N}_{\mu}$ | irreversibility distribution ratio |

$\overline{Nu}$ | average Nusselt number |

Pr | Prandtl number |

$Ra$ | Rayleigh number |

${S}_{gen}$ | Entropy generation rate |

${S}_{GEN}$ | Dimensionless entropy generation rate |

${S}_{\theta}$ | dimensionless entropy generation due to heat transfer irreversibility |

${S}_{\Psi}$ | dimensionless entropy generation nanofluid friction irreversibility |

T | temperature |

${T}_{0}$ | reference temperature (293K) |

u, v | velocity components in the x-direction and y-direction |

U, V | dimensionless velocity components in the X-direction and Y-direction |

x, y and X, Y | space coordinatesand dimensionless space coordinates |

Greek symbols | |

$\alpha $ | thermal diffusivity |

$\beta $ | thermal expansion coefficient |

$\theta $ | dimensionless temperature |

$\mu $ | dynamic viscosity |

$\nu $ | kinematic viscosity |

$\rho $ | density |

$\varphi $ | solid volume fraction |

$\psi \phantom{\rule{0.222222em}{0ex}}\mathrm{and}\phantom{\rule{0.222222em}{0ex}}\Psi $ | stream function and dimensionless stream function |

$\omega \phantom{\rule{0.222222em}{0ex}}\mathrm{and}\phantom{\rule{0.222222em}{0ex}}\Omega $ | vorticity and dimensionless vorticity |

Subscripts | |

c | cold |

$bf$ | base fluid |

h | hot |

$nf$ | nanofluid |

$sp$ | solid nanoparticles |

w | inner solid wall |

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**Figure 2.**streamlines at $Ra={10}^{6}$, isotherms at $Ra=1.836\times {10}^{5}$, isotherms at $Ra={10}^{5}$ and isotherms at $Ra={10}^{6}$ (

**a**) numerical and experimental results of Calcagni et al. [32] and (

**b**) numerical results of the present study for $\varphi =0$ and $D=0$.

**Figure 3.**Streamlines, isotherms, local entropy generation due to heat transfer and due to fluid friction (

**a**) Kaluri and Basak [16] and (

**b**) present study for $Ra={10}^{6}$, $Da={10}^{-3}$, $\varphi =0$ and $D=0$.

**Figure 4.**Variations of the streamlines (

**left**); isotherms (

**middle**); and isentropic (

**right**) evolution by the solid volume fraction ($\varphi $) for $Ra={10}^{5}$, ${K}_{r}=1$ and $D=0.25$. (

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

**b**) $\varphi =0.03$; (

**c**) $\varphi =0.06$; (

**d**) $\varphi =0.09$.

**Figure 5.**Variations of the streamlines (

**left**); isotherms (

**middle**); and isentropic (

**right**) evolution by Rayleigh number ($Ra$) for $\varphi =0.04$, ${K}_{r}=1$ and $D=0.25$. (

**a**) $Ra={10}^{3}$; (

**b**) $Ra={10}^{4}$; (

**c**) $Ra={10}^{5}$; (

**d**) $Ra={10}^{6}$.

**Figure 6.**Variations of the streamlines (

**left**); isotherms (

**middle**); and isentropic (

**right**) evolution by thermal conductivity ratio (${K}_{r}$) for $Ra={10}^{5}$, $\varphi =0.04$ and $D=0.25$. (

**a**) ${K}_{r}=0.44$; (

**b**) ${K}_{r}=2.40$; (

**c**) ${K}_{r}=9.90$; (

**d**) ${K}_{r}=23.8$.

**Figure 7.**Variations of the streamlines (

**left**); isotherms (

**middle**); and isentropic (

**right**) evolution by the length of the inner solid (D) for $Ra={10}^{5}$, $\varphi =0.04$ and ${K}_{r}=1$. (

**a**) $D=0$; (

**b**) $D=0.2$; (

**c**) $D=0.4$; (

**d**) $D=0.6$.

**Figure 8.**Variations of local Nusselt number interfaces with X for different (

**a**) $Ra$ and (

**b**) $\varphi $ at ${K}_{r}=1$ and $D=0.25$.

**Figure 9.**Variations of local Nusselt number interfaces with X for different (

**a**) ${K}_{r}$ and (

**b**) D at $Ra={10}^{5}$ and $\varphi =0.04$.

**Figure 10.**Variations of average Nusselt number with $Ra$ for different (

**a**) $\varphi $; (

**b**) ${K}_{r}$ and (

**c**) D.

**Figure 11.**Variations of average Nusselt number with $\varphi $ for different (

**a**) $Ra$; (

**b**) ${K}_{r}$ and (

**c**) D.

**Figure 12.**Variations of average Nusselt number with D for different (

**a**) $Ra$; (

**b**) $\varphi $ and (

**c**) ${K}_{r}$.

**Figure 13.**Variations of (

**a**) the global entropy generation (GEG) and (

**b**) Bejan number ($Be$) with $Ra$ for different values of $\varphi $ at ${K}_{r}=1$ and $D=0.25$.

**Figure 14.**Variations of (

**a**) the global entropy generation (GEG) and (

**b**) Bejan number ($Be$) with $Ra$ for different values of D at $\varphi =0.04$ and ${K}_{r}=1$.

**Figure 15.**Variations of (

**a**) the global entropy generation (GEG) and (

**b**) Bejan number ($Be$) with $\varphi $ for different values of ${K}_{r}$ at $Ra={10}^{5}$ and $D=0.25$.

**Figure 16.**Variations of (

**a**) the global entropy generation (GEG) and (

**b**) Bejan number ($Be$) with $\varphi $ for different values of D at $Ra={10}^{5}$ and ${K}_{r}=1$.

**Figure 17.**Variations of (

**a**) the global entropy generation (GEG) and (

**b**) Bejan number ($Be$) with D for different values of $Ra$ at $\varphi =0.04$ and ${K}_{r}=1$.

**Figure 18.**Variations of (

**a**) the global entropy generation (GEG) and (

**b**) Bejan number ($Be$) with D for different values of $\varphi $ at $Ra={10}^{5}$ and ${K}_{r}=1$.

**Figure 19.**Variation of (

**a**) the global entropy generation (GEG) and (

**b**) Bejan number ($Be$) with D for different values of ${K}_{r}$ at $Ra={10}^{5}$ and $\varphi =0.04$.

**Table 1.**Grid testing for ${\Psi}_{\mathrm{min}}$ and ${\overline{Nu}}_{nf}$ at different grid size for $Ra={10}^{5}$, $\varphi =0.04$, ${K}_{r}=1$ and $D=0.25$.

Grid Size | ${\mathsf{\Psi}}_{\mathbf{min}}$ | ${\overline{\mathit{Nu}}}_{\mathit{nf}}$ |
---|---|---|

$10\times 10$ | −0.76597 | 3.7233 |

$20\times 20$ | −0.76613 | 3.7569 |

$40\times 40$ | −0.76653 | 3.7802 |

$60\times 60$ | −0.76658 | 3.7979 |

$80\times 80$ | −0.76761 | 3.8014 |

$100\times 100$ | −0.76878 | 3.8016 |

$120\times 120$ | −0.76897 | 3.8018 |

$140\times 140$ | −0.76956 | 3.8019 |

$160\times 160$ | −0.76968 | 3.8019 |

**Table 2.**Thermo-physical properties of the base fluid (water) with Al${}_{2}$O${}_{3}$ nanoparticles at $T=295$ K [27].

Physical Properties | Fluid Phase (Water) | Al${}_{2}$O${}_{3}$ |
---|---|---|

$Cp\phantom{\rule{0.166667em}{0ex}}(\mathrm{J}/\mathrm{kgK})$ | 4179 | 765 |

$\rho \phantom{\rule{0.166667em}{0ex}}(\mathrm{kg}/{\mathrm{m}}^{3})$ | 997.1 | 3970 |

$k\phantom{\rule{0.166667em}{0ex}}({\mathrm{Wm}}^{-1}\xb7{\mathrm{K}}^{-1})$ | 0.613 | 25 |

$\beta \times {10}^{5}\phantom{\rule{0.166667em}{0ex}}(1/\mathrm{K})$ | 21 | 0.85 |

${d}_{p}\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{nm}\right)$ | 3.85 | 36 |

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**MDPI and ACS Style**

Alsabery, A.I.; Ishak, M.S.; Chamkha, A.J.; Hashim, I. Entropy Generation Analysis and Natural Convection in a Nanofluid-Filled Square Cavity with a Concentric Solid Insert and Different Temperature Distributions. *Entropy* **2018**, *20*, 336.
https://doi.org/10.3390/e20050336

**AMA Style**

Alsabery AI, Ishak MS, Chamkha AJ, Hashim I. Entropy Generation Analysis and Natural Convection in a Nanofluid-Filled Square Cavity with a Concentric Solid Insert and Different Temperature Distributions. *Entropy*. 2018; 20(5):336.
https://doi.org/10.3390/e20050336

**Chicago/Turabian Style**

Alsabery, Ammar I., Muhamad Safwan Ishak, Ali J. Chamkha, and Ishak Hashim. 2018. "Entropy Generation Analysis and Natural Convection in a Nanofluid-Filled Square Cavity with a Concentric Solid Insert and Different Temperature Distributions" *Entropy* 20, no. 5: 336.
https://doi.org/10.3390/e20050336