# Entropy Generation and Natural Convection of CuO-Water Nanofluid in C-Shaped Cavity under Magnetic Field

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## Abstract

**:**

## 1. Introduction

_{2}O

_{3}nanofluid in the presence of magnetic field and uniform heat generation/absorption. Aghaei et al. [29] evaluated the effects of magnetic field on heat transfer and entropy generation on mixed convection of nanofluids with variable properties in a trapezoidal enclosure. Kefayati [30] analyzed the heat transfer and entropy generation on laminar natural convection of non-Newtonian nanofluids in the presence of an external horizontal magnetic field in a square cavity. Ellahi et al. [31] analyzed a mathematical model in order to study the shape of nanosize particles on entropy generation and natural convection boundary layer flow along an inverted cone. More recently, Ismael et al. [32] studied the entropy generation due to conjugate natural convection conduction heat transfer in a square domain under steady-state condition. They proposed a new criterion for assessment of the thermal performance. As the applications of natural convection in engineering systems have developed, it became necessary to investigate non rectangular/square cavity shape. Cooling of electronic chips/system, micro-electro-mechanical systems (MEMS), solar collectors, and heat exchangers are some of the many important applications that handle different cavity shapes. Biserani et al. [33] used Bejan’s constructal theory to optimize the geometry of H-shaped cavity that intrudes into a solid conducted wall. They optimized other cavities namely, C-shaped and T-shaped cavities and found that H-shaped is superior in thermal performance. Mahmoodi [34] studied free convection in L-shaped cavity filled with Cu-water nanofluid. Mahmood and Hashemi [35] studied the C-shaped cavity filled with a nanofluid. Cho et al. [36] investigated the natural convection enhancement of Al

_{2}O

_{3}-water nanofluid in a U-shaped cavity. Mansour et al. [37] also investigated the natural convection inside U-shaped cavity filled with Cu-water nanofluid but they termed their cavity as C-shaped. Mojumder et al. [38] studied the natural convection in C-shaped cavity filled with Cobalt-kerosene ferrofluid under the effect of externally applied magnetic field. Kasaeipoor et al. [39] studied the convection of Cu-Water nanofluid in a vented T-shaped cavity in the presence of an externally applied magnetic field. Al-Zamily [40] investigated numerically the effect of constant magnetic field on natural convection in a semi-circular enclosure filled with Cu-water nanofluid with the present of heat flux.

## 2. Problem Description and Mathematical Modeling

_{0}), which is shown in Figure 1. The two-dimensional cavity has equal length and height of L. The internal walls of cavity with the length H are maintained at a relatively low temperature T

_{c}. The external right walls are insulated and the other walls are maintained at a relatively high temperature T

_{h}. The flow is assumed to be laminar, steady, and incompressible. The nanofluid is assumed to be Newtonian, incompressible and has a low electrical conductivity. Moreover, the water and CuO nanoparticles are in thermal equilibrium. The Joule heating are assumed to be negligible compared to the applied magnetic field.

Physical properties | Fluid phase | CuO |
---|---|---|

C_{p} (j/kg·K) | 4179 | 540 |

ρ (kg/m3) | 997.1 | 6500 |

Pr | 6.2 | – |

$k\left(\mathrm{w}/\mathrm{m}\xb7\mathrm{K}\right)$ | 0.613 | 18 |

$\beta \left({\mathrm{k}}^{-1}\right)$ | $21\times {10}^{-5}$ | $5.1\times {10}^{-5}$ |

$\sigma $($\mu S/cm$) | 0.05 | $2.7\times {10}^{-8}$ |

_{s}) is 33 nm. The effective viscosity of CuO-water nanofluid, its Brownian-motion velocity and the effect of water temperature on this motion, can be expressed as follows [43]:

_{1}= −1.133 × 10

^{−6}, c

_{2}= −2.771 × 10

^{−6}, c

_{3}= 9 × 10

^{−8}and c

_{4}= −3.93 × 10

^{−7}.

_{2}O

_{3}-water nanofluid with spherical particles:

_{s}is defined as the nanoparticles Reynolds number. In addition, ${l}_{\mathrm{f}}=17\text{}\mathrm{nm}$ is mean free path of water. The accuracy of the above-mentioned equation was also confirmed by Mintsa et al. [45] for CuO and adopted afterwards in many studies, such as by Popa et al. [46].

## 3. Numerical Solution and Grid Dependency Test

^{5}for different Hartman number. The comparison results are tabulated in Table 3, and they further guarantee the validity of the present numerical code.

Grid points | $\mathbf{40}\mathbf{\times}\mathbf{40}$ | $\mathbf{60}\mathbf{\times}\mathbf{60}$ | $\mathbf{80}\mathbf{\times}\mathbf{80}$ | $\mathbf{100}\mathbf{\times}\mathbf{100}$ | $\mathbf{120}\mathbf{\times}\mathbf{120}$ | |
---|---|---|---|---|---|---|

AR = 0.1 | Ra = 1000 | 0.6884 | 0.6771 | 0.6333 | 0.6328 | 0.6328 |

Ra = 15,000 | 0.6851 | 0.6674 | 0.6617 | 0.6605 | 0.6604 | |

AR = 0.7 | Ra = 1000 | 5.2215 | 5.2069 | 5.2018 | 5.2008 | 5.2007 |

Ra = 15,000 | 5.2433 | 5.2163 | 5.2052 | 5.2026 | 5.2024 |

**Figure 3.**Validation of the present code against Mahmoodi and Hashemi [35] for a Cu-water nanofluid natural convection in cavity.

## 4. Results and Discussion

#### 4.1. Effect of Raylirgh Number

**Figure 4.**The effects of Rayleigh on the Streamlines, Isotherms and Isentropic lines for nanofluid $\varphi =0.04$) at Ha = 30 and AR = 0.3.

_{m}with the nanoparticles volume fraction φ is recorded with all studied Ra values. This can be surely attributed to the enhanced thermal energy transport due to the enhanced nanofluid thermal conductivity.

**Figure 5.**Variation of local Nusselt number for various Rayleigh numbers at Ha = 30, AR = 0.3 and $\varphi =0.04$.

**Figure 6.**Variation of average Nusselt number for various Rayleigh numbers with $\varphi $ for AR = 0.3 and Ha = 30.

**Figure 7.**Variation of global entropy generation ratio (${S}_{m,\text{*}}={S}_{m}/{S}_{m,\phi =0}$) for various Rayleigh numbers with $\varphi $ for AR = 0.3 and Ha = 30.

#### 4.2. Effect of Hartman Number

_{m},

_{φ = 0}) is presented in Figure 10. The noticed increase of the normalized Nusselt number with Ha reflects that the drag action of the magnetic field on the nanofluid is less than that on pure fluid.

**Figure 8.**The effects of Hartmann numbers on the Streamlines, Isotherm and Entropy generation for pure-water (—) and nanofluid $\varphi =0.04$ (- - -) at Ra = 10,000 and AR = 0.3.

**Figure 9.**Variation of local Nusselt number for various Hartman numbers at Ra = 10,000, AR = 0.3 and $\varphi =\mathrm{0.04.}$

**Figure 10.**Variation of average Nusselt number ratio (${\text{Nu}}_{\mathrm{m},*}={\text{Nu}}_{\mathrm{m}}/{\text{Nu}}_{\mathrm{m},\mathsf{\phi}=0}$) for various Hartman number with $\varphi $ at Ra = 10,000 and AR = 0.3.

**Figure 11.**Variation of global entropy generation for various Hartman numbers with Ra for AR = 0.3, and $\varphi =0.04$.

**Figure 12.**Variation of $\epsilon $ ($\mathsf{\epsilon}={\mathrm{S}}_{\mathrm{m}}/{\text{Nu}}_{\mathrm{m}}$) for various Rayleigh numbers and Hartman numbers for AR = 0.3 and $\varphi =0.04$.

#### 4.3. Effect of the Aspect Ratio

**Figure 13.**The effects of AR on the Streamlines, Isotherms and Isentropic lines for nanofluid at φ = 0, 0.04, Ra = 10,000 and Ha = 30.

**Figure 14.**Variation of local Nusselt number for various AR at $\varphi =0.04$, Ra = 10,000 and Ha = 30.

**Figure 16.**Variation of average Entropy generation for various AR with Hartman numbers for Ra = 10,000 and $\varphi =0.04$.

**Figure 17.**Variation of $\epsilon $ ($\mathsf{\epsilon}={\mathrm{S}}_{\mathrm{m}}/{\text{Nu}}_{\mathrm{m}}$) for various AR with Hartman numbers for Ra = 10,000 and $\varphi =0.04$.

## 5. Conclusions

- (1)
- The addition of nanoparticles enhances the convective heat transfer inside the C-shaped cavity at all Rayleigh numbers, whereas the entropy generation increases with increasing the volume fraction of the nanoparticles. This increase becomes fast at higher Rayleigh number.
- (2)
- The average Nusselt number increases considerably when the hot and cold walls become narrower, i.e., at higher aspect ratio.
- (3)
- The nanofluid utilization becomes more pronounced at lower aspect ratio.
- (4)
- The applied magnetic field is an inactive process at lower Rayleigh number.
- (5)
- The entropy generation rate decreases rapidly with the applied magnetic field.
- (6)
- A threshold value of Hartman number equal to 30 can give the best thermal performance in the C-shaped cavity.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

B_{0} | Magnetic field strength, T | Greek symbols | |

${C}_{p}$ | Specific heat, J·kg^{−1}·K^{−1} | $\alpha $ | Thermal diffusivity, m^{2}·s^{−1} |

$g$ | Gravitational acceleration, m·s^{2} | $\beta $ | Thermal expansion coefficient, K^{−1} |

$H$ | Length of heat source, m | ε | performance criterion (${S}_{m}/N{u}_{m}$) |

$Ha$ | Hartmann number, ${B}_{0}L\sqrt{{\sigma}_{f}/{\rho}_{f}{\nu}_{f}}$ | $\varphi $ | Solid volume fraction |

k | Thermal conductivity, W·m^{−1}·K^{−1} | $\sigma $ | Effective electrical conductivity, μ·S/cm |

$L$ | Length of cavity, m | ${\kappa}_{b}$ | Boltzmann constant, J·K^{−1} |

$Nu$ | Local Nusselt number | $\theta $ | Dimensionless temperature, $(T-{T}_{c})/({T}_{h}-{T}_{c})$ |

Nu_{m} | Average Nusselt number of heat source | $\mu $ | Dynamic viscosity, N·S·m^{−2} |

$p$ | Fluid pressure, Pa | $\nu $ | Kinematic viscosity, m^{2}·s^{−1} |

$P$ | Dimensionless pressure, $pL/{\rho}_{nf}{\alpha}_{f}^{2}$ | $\rho $ | Density, kg·m^{3} |

$\mathrm{Pr}$ | Prandtl number, ν_{f}/α_{f} | Subscripts | |

$T$ | Temperature, K | $c$ | Cold |

T_{c} | Cold wall temperature, K | $f$ | Pure fluid |

${T}_{h}$ | Hot wall temperature, K | $h$ | hot wall |

$u,v$ | Velocity components in x, y directions, m·s^{−1} | $m$ | Average |

$U,V$ | Dimensionless Velocity components, $\left(u,v\right)L/{\alpha}_{f}$ | $nf$ | Nanofluid |

$x,y$ | Cartesian coordinates, m | $s$ | Nanoparticle |

X,Y | Dimensionless coordinates, (x,y)/L |

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**MDPI and ACS Style**

Chamkha, A.; Ismael, M.; Kasaeipoor, A.; Armaghani, T.
Entropy Generation and Natural Convection of CuO-Water Nanofluid in C-Shaped Cavity under Magnetic Field. *Entropy* **2016**, *18*, 50.
https://doi.org/10.3390/e18020050

**AMA Style**

Chamkha A, Ismael M, Kasaeipoor A, Armaghani T.
Entropy Generation and Natural Convection of CuO-Water Nanofluid in C-Shaped Cavity under Magnetic Field. *Entropy*. 2016; 18(2):50.
https://doi.org/10.3390/e18020050

**Chicago/Turabian Style**

Chamkha, Ali, Muneer Ismael, Abbas Kasaeipoor, and Taher Armaghani.
2016. "Entropy Generation and Natural Convection of CuO-Water Nanofluid in C-Shaped Cavity under Magnetic Field" *Entropy* 18, no. 2: 50.
https://doi.org/10.3390/e18020050