# Natural Convection and Entropy Generation in Nanofluid Filled Entrapped Trapezoidal Cavities under the Influence of Magnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Physical Model and Mathematical Formulation

Property | Water | CuO | Al${}_{\mathbf{2}}$O${}_{\mathbf{3}}$ |
---|---|---|---|

ρ (kg/m${}^{3}$) | 997.1 | 6500 | 3970 |

$\phantom{\rule{4.pt}{0ex}}{\mathrm{c}}_{p}$ (J/kg·K) | 4179 | 540 | 765 |

$\mathrm{k}$ (W·m${}^{-1}$·K${}^{-1}$) | 0.6 | 18 | 25 |

β (1/K) | $2.1\times {10}^{-4}$ | $0.85\times {10}^{-5}$ | $0.85\times {10}^{-5}$ |

${\sigma}^{-1}$ (Ω·m) | $0.05$ | $2.7\times {10}^{-8}$ | ${10}^{-10}$ |

- For the inclined walls of the trapezoidal enclosures:$U=V=0,\phantom{\rule{4pt}{0ex}}\theta =0$
- For the top and bottom horizontal walls:$U=V=0,\phantom{\rule{4pt}{0ex}}\theta =1$
- Along the interface of the trapezoidal domains (continuity condition):a- ${\left(\frac{\partial \theta}{\partial n}\right)}_{1}={\left(\frac{\partial \theta}{\partial n}\right)}_{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\theta}_{1}={\theta}_{2}$b- ${U}_{1}={U}_{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{V}_{1}={V}_{2}$

## 3. Solution Method and Validation of Code

Grid Name | Grid Size | Nu${}_{\mathbf{m}}$ (L1) | Nu${}_{\mathbf{m}}$ (L2) |
---|---|---|---|

G1 | 938 | 11.92 | 7.67 |

G2 | 2684 | 12.08 | 8.63 |

G3 | 33182 | 14.13 | 11.33 |

G4 | 48772 | 14.56 | 11.77 |

G5 | 64672 | 14.89 | 11.82 |

**Figure 2.**Code validation study, comparison of streamlines and isotherms with Basak et al. [49], $\text{Ra}=5\times {10}^{5},\phantom{\rule{4pt}{0ex}}\text{Da}={10}^{-4},\phantom{\rule{4pt}{0ex}}\text{Pr}=0.015$. (

**a**) streamline, Basak et al. [49]; (

**b**) isotherm, Basak et al. [49]; (

**c**) streamline, present solver; (

**d**) isotherm, present solver.

**Figure 3.**Code validation study, comparison of local Nusselt numbers along the top and bottom wall with Basak et al. [49], $\text{Ra}=5\times {10}^{5},\phantom{\rule{4pt}{0ex}}\text{Da}={10}^{-4},\phantom{\rule{4pt}{0ex}}\text{Pr}=0.015$, (

**a**) local Nusselt numbers, Basak et al. [49]; (

**b**) top wall, present solver; (

**c**) bottom wall, present solver.

## 4. Results and Discussion

**Figure 4.**Effects of varying Rayleigh number on the streamlines for various Hartmann number combinations ($\varphi =0.01$), $(\text{Ra},{\text{Ha}}_{1},{\text{Ha}}_{2})$, (

**a**) $({10}^{3},0,0)$; (

**b**) $({10}^{3},0,20)$ ; (

**c**) $({10}^{3},20,0)$; (

**d**) $({10}^{3},30,30)$; (

**e**) $({10}^{5},0,0)$; (

**f**) $({10}^{5},0,20)$; (

**g**) $({10}^{5},20,0)$; (

**h**) $({10}^{5},30,30)$; (

**i**) $({10}^{6},0,0)$; (

**j**) $({10}^{6},0,20)$; (

**k**) $({10}^{6},20,0)$; (

**l**) $({10}^{6},30,30)$.

**Figure 5.**Isotherms for various Rayleigh numbers and different Hartmann number combinations ($\varphi =0.01$), $(\text{Ra},{\text{Ha}}_{1},{\text{Ha}}_{2})$. (

**a**) $({10}^{3},0,0)$; (

**b**) $({10}^{3},0,20)$ ; (

**c**) $({10}^{3},20,0)$; (

**d**) $({10}^{3},30,30)$; (

**e**) $({10}^{5},0,0)$; (

**f**) $({10}^{5},0,20)$; (

**g**) $({10}^{5},20,0)$; (

**h**) $({10}^{5},30,30)$; (

**i**) $({10}^{6},0,0)$; (

**j**) $({10}^{6},0,20)$; (

**k**) $({10}^{6},20,0)$; (

**l**) $({10}^{6},30,30)$.

**Table 3.**Averaged Nusselt number along the bottom and top wall of the domains for various Rayleigh numbers and different Hartmann number combinations ($\varphi =0.01$).

Ra | Ha${}_{\mathbf{1}}$ | Ha${}_{\mathbf{2}}$ | Nu${}_{\mathbf{m}}$ (L1) | Nu${}_{\mathbf{m}}$ (L2) |
---|---|---|---|---|

10${}^{3}$ | 0 | 0 | 10.638 | 10.602 |

10${}^{3}$ | 0 | 20 | 10.635 | 10.604 |

10${}^{3}$ | 20 | 0 | 10.634 | 10.605 |

10${}^{3}$ | 30 | 30 | 10.632 | 10.607 |

10${}^{4}$ | 0 | 0 | 10.737 | 10.553 |

10${}^{4}$ | 0 | 20 | 10.717 | 10.564 |

10${}^{4}$ | 20 | 0 | 10.659 | 10.599 |

10${}^{4}$ | 30 | 30 | 10.639 | 10.601 |

10${}^{5}$ | 0 | 0 | 13.238 | 9.999 |

10${}^{5}$ | 0 | 20 | 12.942 | 10.143 |

10${}^{5}$ | 20 | 0 | 11.177 | 10.603 |

10${}^{5}$ | 30 | 30 | 10.793 | 10.554 |

10${}^{6}$ | 0 | 0 | 18.061 | 10.172 |

10${}^{6}$ | 0 | 20 | 17.753 | 9.671 |

10${}^{6}$ | 20 | 0 | 16.315 | 10.188 |

10${}^{6}$ | 30 | 30 | 14.995 | 9.521 |

**Figure 6.**Local Nusselt number distributions along the bottom and lower side wall for various Rayleigh numbers ($\varphi =0.01$). (

**a**) L1; (

**b**) L3.

**Figure 7.**Local Nusselt number distributions along the top and upper side wall for various Rayleigh numbers ($\varphi =0.01$). (

**a**) L2; (

**b**) L4.

**Table 4.**Averaged Nusselt number distributions along the bottom and top walls of the trapezoidal domains for various Hartmann numbers of the domains ($\text{Ra}={10}^{5},\varphi =0.015$).

Ha${}_{\mathbf{1}}$ | Ha${}_{\mathbf{2}}$ | Nu${}_{\mathbf{m}}$ (L1) | Nu${}_{\mathbf{m}}$ (L2) |
---|---|---|---|

0 | 0 | 13.245 | 9.996 |

0 | 10 | 13.112 | 9.976 |

0 | 20 | 12.949 | 10.137 |

0 | 30 | 12.877 | 10.242 |

0 | 50 | 12.819 | 10.329 |

10 | 0 | 12.370 | 10.172 |

10 | 10 | 12.300 | 10.124 |

10 | 20 | 12.208 | 10.217 |

10 | 30 | 12.165 | 10.292 |

10 | 50 | 12.130 | 10.363 |

20 | 0 | 11.180 | 10.606 |

20 | 10 | 11.174 | 10.521 |

20 | 20 | 11.163 | 10.464 |

20 | 30 | 11.156 | 10.457 |

20 | 50 | 11.149 | 10.471 |

30 | 0 | 10.816 | 10.748 |

30 | 10 | 10.810 | 10.667 |

30 | 20 | 10.801 | 10.585 |

30 | 30 | 10.794 | 10.555 |

30 | 50 | 10.787 | 10.546 |

50 | 0 | 10.716 | 10.796 |

50 | 10 | 10.708 | 10.720 |

50 | 20 | 10.696 | 10.635 |

50 | 30 | 10.686 | 10.599 |

50 | 50 | 10.675 | 10.583 |

**Figure 8.**Effects of varying Hartmann numbers of the upper and lower domains on the streamline distributions ($\text{Ra}={10}^{5},\varphi =0.015$), (

**a**) $({\text{Ha}}_{1}=0,{\text{Ha}}_{2}=0)$ ; (

**b**) $({\text{Ha}}_{1}=0,{\text{Ha}}_{2}=20)$; (

**c**) $({\text{Ha}}_{1}=0,{\text{Ha}}_{2}=50)$; (

**d**) $({\text{Ha}}_{1}=20,{\text{Ha}}_{2}=0)$; (

**e**) $({\text{Ha}}_{1}=20,{\text{Ha}}_{2}=20)$; (

**f**) $({\text{Ha}}_{1}=20,{\text{Ha}}_{2}=50)$; (

**g**) $({\text{Ha}}_{1}=50,{\text{Ha}}_{2}=0)$; (

**h**) $({\text{Ha}}_{1}=50,{\text{Ha}}_{2}=20)$; (

**i**) $({\text{Ha}}_{1}=50,{\text{Ha}}_{2}=50)$.

**Figure 9.**Effects of varying Hartmann numbers of the upper and lower domains on the isotherm distributions ($\text{Ra}={10}^{5},\varphi =0.015$), (

**a**) $({\text{Ha}}_{1}=0,{\text{Ha}}_{2}=0)$ ; (

**b**) $({\text{Ha}}_{1}=0,{\text{Ha}}_{2}=20)$; (

**c**) $({\text{Ha}}_{1}=0,{\text{Ha}}_{2}=50)$; (

**d**) $({\text{Ha}}_{1}=20,{\text{Ha}}_{2}=0)$; (

**e**) $({\text{Ha}}_{1}=20,{\text{Ha}}_{2}=20)$; (

**f**) $({\text{Ha}}_{1}=20,{\text{Ha}}_{2}=50)$; (

**g**) $({\text{Ha}}_{1}=50,{\text{Ha}}_{2}=0)$; (

**h**) $({\text{Ha}}_{1}=50,{\text{Ha}}_{2}=20)$; (

**i**) $({\text{Ha}}_{1}=50,{\text{Ha}}_{2}=50)$.

**Figure 10.**Local Nusselt number distributions along the bottom and lower side wall for various Hartmann numbers of the upper and lower domains ($\text{Ra}={10}^{5},\varphi =0.015$), (

**a**) L1; (

**b**) L3.

**Figure 11.**Local Nusselt number distributions along the top and upper side wall for various Hartmann numbers of the upper and lower domains ($\text{Ra}={10}^{5},\varphi =0.015$), (

**a**) L2; (

**b**) L4.

**Table 5.**Effects of nanoparticle volume fraction on the average Nusselt number for various Hartmann numbers of the upper and lower trapezoidal domains ($\text{Ra}={10}^{4}$).

φ | Ha${}_{\mathbf{1}}$ | Ha${}_{\mathbf{2}}$ | Nu${}_{\mathbf{m}}$ (L1) | Nu${}_{\mathbf{m}}$ (L2) |
---|---|---|---|---|

0 | 0 | 0 | 10.993 | 9.954 |

0.01 | 0 | 0 | 11.316 | 10.213 |

0.02 | 0 | 0 | 11.644 | 10.478 |

0.03 | 0 | 0 | 11.977 | 10.749 |

0.04 | 0 | 0 | 12.315 | 11.025 |

0 | 0 | 20 | 10.921 | 10.00 |

0.01 | 0 | 20 | 11.241 | 10.263 |

0.02 | 0 | 20 | 11.565 | 10.531 |

0.03 | 0 | 20 | 11.893 | 10.804 |

0.04 | 0 | 20 | 12.227 | 11.082 |

0 | 20 | 0 | 10.478 | 10.163 |

0.01 | 20 | 0 | 10.776 | 10.438 |

0.02 | 20 | 0 | 11.079 | 10.717 |

0.03 | 20 | 0 | 11.388 | 11.001 |

0.04 | 20 | 0 | 11.702 | 11.292 |

0 | 30 | 30 | 10.438 | 10.139 |

0.01 | 30 | 30 | 10.734 | 10.409 |

0.02 | 30 | 30 | 11.035 | 10.685 |

0.03 | 30 | 30 | 11.342 | 10.965 |

0.04 | 30 | 30 | 11.655 | 11.251 |

**Table 6.**Effects of nanoparticle volume fraction on the average Nusselt number for various Hartmann numbers of the upper and lower trapezoidal domains ($\text{Ra}=5\times {10}^{5}$).

φ | Ha${}_{\mathbf{1}}$ | Ha${}_{\mathbf{2}}$ | Nu${}_{\mathbf{m}}$ (L1) | Nu${}_{\mathbf{m}}$ (L2) |
---|---|---|---|---|

0 | 0 | 0 | 16.283 | 10.043 |

0.01 | 0 | 16.758 | 16.299 | 10.318 |

0.02 | 0 | 0 | 17.240 | 10.599 |

0.03 | 0 | 0 | 17.728 | 10.885 |

0.04 | 0 | 0 | 18.222 | 11.176 |

0 | 0 | 20 | 15.918 | 9.739 |

0.01 | 0 | 20 | 16.394 | 9.990 |

0.02 | 0 | 20 | 16.876 | 10.247 |

0.03 | 0 | 20 | 17.364 | 10.510 |

0.04 | 0 | 20 | 17.857 | 10.779 |

0 | 20 | 0 | 14.438 | 10.180 |

0.01 | 20 | 0 | 14.863 | 10.461 |

0.02 | 20 | 0 | 15.293 | 10.747 |

0.03 | 20 | 0 | 15.727 | 11.038 |

0.04 | 20 | 0 | 16.166 | 11.336 |

0 | 30 | 30 | 13.022 | 9.992 |

0.01 | 30 | 30 | 13.411 | 10.248 |

0.02 | 30 | 30 | 13.803 | 10.509 |

0.03 | 30 | 30 | 14.199 | 10.777 |

0.04 | 30 | 30 | 14.599 | 11.052 |

**Figure 12.**Normalized entropy generation for the upper and lower trapezoidal domains, (

**a**) for $\text{Ra}={10}^{5},\varphi =0.015$ and various Hartmann numbers; (

**b**) for $\text{Ra}={10}^{5}$, Ha${}_{1}$ = Ha${}_{2}$ = 10 and various nanoparticle volume fractions.

**Table 7.**Normalized entropy generation for lower and upper trapezoidal domains for various Hartmann numbers of the domains ($\text{Ra}={10}^{5},\varphi =0.015$).

Ha${}_{\mathbf{1}}$ | Ha${}_{\mathbf{2}}$ | S*${}_{\mathbf{g}}$ (D1) | S*${}_{\mathbf{g}}$ (D2) |
---|---|---|---|

0 | 0 | 1.0000 | 1.0000 |

0 | 10 | 1.0029 | 0.9811 |

0 | 20 | 1.0087 | 0.9716 |

0 | 30 | 1.0145 | 0.9703 |

0 | 50 | 1.0174 | 0.9685 |

10 | 0 | 0.9449 | 0.9905 |

10 | 10 | 0.9478 | 0.9716 |

10 | 20 | 0.9536 | 0.9653 |

10 | 30 | 0.9594 | 0.9621 |

10 | 50 | 0.9652 | 0.9590 |

20 | 0 | 0.8922 | 0.9685 |

20 | 10 | 0.8928 | 0.9621 |

20 | 20 | 0.8942 | 0.9527 |

20 | 30 | 0.8957 | 0.9495 |

20 | 50 | 0.8971 | 0.9464 |

30 | 0 | 0.8725 | 0.9685 |

30 | 10 | 0.8725 | 0.9590 |

30 | 20 | 0.8725 | 0.9502 |

30 | 30 | 0.8739 | 0.9464 |

30 | 50 | 0.8754 | 0.9432 |

50 | 0 | 0.8693 | 0.9685 |

50 | 10 | 0.8690 | 0.9590 |

50 | 20 | 0.8684 | 0.9495 |

50 | 30 | 0.8681 | 0.9461 |

50 | 50 | 0.8681 | 0.9435 |

**Table 8.**Effects of nanoparticle volume fraction on the entropy generation ratio for various Hartmann numbers of the upper and lower trapezoidal domains ($\text{Ra}=5\times {10}^{5}$).

φ | Ha${}_{\mathbf{1}}$ | Ha${}_{\mathbf{2}}$ | S*${}_{\mathbf{g}}$ (D1) | S*${}_{\mathbf{g}}$ (D2) |
---|---|---|---|---|

0 | 0 | 0 | 1.0000 | 1.0000 |

0.01 | 0 | 0 | 1.0292 | 1.0277 |

0.02 | 0 | 0 | 1.0584 | 1.0563 |

0.03 | 0 | 0 | 1.0882 | 1.0856 |

0.04 | 0 | 0 | 1.1185 | 1.1154 |

0 | 0 | 20 | 1.0022 | 0.9961 |

0.01 | 0 | 20 | 1.0308 | 1.0236 |

0.02 | 0 | 20 | 1.0639 | 1.0514 |

0.03 | 0 | 20 | 1.0904 | 1.0802 |

0.04 | 0 | 20 | 1.1185 | 1.1088 |

0 | 20 | 0 | 0.9713 | 0.9956 |

0.01 | 20 | 0 | 0.9989 | 1.0235 |

0.02 | 20 | 0 | 1.0271 | 1.0518 |

0.03 | 20 | 0 | 1.0552 | 1.0806 |

0.04 | 20 | 0 | 1.0844 | 1.1099 |

0 | 30 | 30 | 0.9708 | 0.9901 |

0.01 | 30 | 30 | 0.9978 | 1.0172 |

0.02 | 30 | 30 | 1.0260 | 1.0448 |

0.03 | 30 | 30 | 1.0541 | 1.0729 |

0.04 | 30 | 30 | 1.0833 | 1.1016 |

## 5. Conclusions

- At the highest value of the Rayleigh number, the magnetic field is more effective on the reduction of the natural convection of the upper and lower cavities.
- When the Hartmann number of the lower cavity incraeses, the local heat transfer along the bottom wall is deteriorated and it is enhanced partly for locations at $0\le X\le 0.3$ and $0.7\le X\le 1$ along the top wall of the upper cavity.
- In the absence of magnetic field in the lower cavity, the averaged heat transfer increases with increasing values of the Hartmann number of the upper cavity. When the Hatmann number of the lower cavity is greater than 10, convection is reduced for the upper domain.
- The averaged Nusselt number increments are in the range of 10% and 12% for the highest solid volume fraction of the nanoparticle compared to base fluid. The heat transfer enhancement rates with nanofluids are not influenced by the presence of the magnetic field.
- The normalized entropy generation increases with increasing values of solid volume fraction of the nanoparticles and decreasing values of magnetic field strength for both domains. The entropy generation ratio decreases for the lower trapezoidal domain is more pronounced compared to upper one with increasing Hartmann number.

## Author Contributions

## Conflicts of Interest

## Abbreviations

B${}_{0}$ magnetic field strength |

Gr Grashof number, $\frac{g{\beta}_{f}({T}_{h}-{T}_{c}){H}^{3}}{{\nu}_{f}^{2}}$ |

h local heat transfer coefficient, (W/m${}^{2}\xb7$K) |

Ha Hartmann number, ${B}_{0}H\sqrt{\frac{{\sigma}_{nf}}{{\rho}_{nf}{\nu}_{f}}}$ |

k thermal conductivity, (W/m·K) |

H length of the enclosure, (m) |

n unit normal vector |

Nu local Nusselt number |

p pressure, (Pa) |

P non-dimensional pressure |

Pr Prandtl number, $\frac{{\nu}_{f}}{{\alpha}_{f}}$ |

T temperature, (K) |

u, v x-y velocity components, (m/s) |

x, y Cartesian coordinates, (m) |

X, Y dimensionless coordinates |

Greek Characters |

α thermal diffusivity, (m${}^{2}$/s) |

β expansion coefficient, (1/K) |

ϕ nanoparticle volume fraction |

θ non-dimensional temperature, $\frac{T-{T}_{c}}{{T}_{h}-{T}_{c}}$ |

ν kinematic viscosity, (m${}^{2}$/s) |

ρ density of the fluid, (kg/m${}^{3}$) |

γ strength of the dipole |

σ electrical conductivity, (S/m) |

Subscripts |

c cold wall |

m average |

h hot wall |

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**MDPI and ACS Style**

Selimefendigil, F.; Öztop, H.F.; Abu-Hamdeh, N. Natural Convection and Entropy Generation in Nanofluid Filled Entrapped Trapezoidal Cavities under the Influence of Magnetic Field. *Entropy* **2016**, *18*, 43.
https://doi.org/10.3390/e18020043

**AMA Style**

Selimefendigil F, Öztop HF, Abu-Hamdeh N. Natural Convection and Entropy Generation in Nanofluid Filled Entrapped Trapezoidal Cavities under the Influence of Magnetic Field. *Entropy*. 2016; 18(2):43.
https://doi.org/10.3390/e18020043

**Chicago/Turabian Style**

Selimefendigil, Fatih, Hakan F. Öztop, and Nidal Abu-Hamdeh. 2016. "Natural Convection and Entropy Generation in Nanofluid Filled Entrapped Trapezoidal Cavities under the Influence of Magnetic Field" *Entropy* 18, no. 2: 43.
https://doi.org/10.3390/e18020043