# Entropy Generation at Natural Convection in an Inclined Rectangular Cavity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{opt}) increases with (δ) but the optimum width (b

_{opt}) decreases, that means the increase of Minceur ratio:

_{f}) using correlations appropriate to heat transfer and friction in turbulent regime. An analytical study of entropy generation’s problem was established by Sahin [3] who consider a viscous fluid’s turbulent flow in a duct. Results show that entropy generation initially decreases then increases along the duct, it is proportional to the dimensionless temperature difference:

_{w}and T

_{o}represent wall temperature and fluid temperature input, respectively. Baytas [4,5] determined entropy generation in an inclined square cavity with two isotherm walls and two adiabatic walls. He firstly [4] determined optimum angles for which energy losses are reduced. It was shown that entropy generation decreases with the inclination angle of the cavity for low external Rayleigh number (Ra

_{E}), maximum values are obtained for angles between 35° and 55°. Secondly, entropy generation in a porous cavity was studied [5]. It was found that entropy generation is the result of a continuous exchange of energy between fluid and enclosure’s walls. Demirel and Kahraman [6] showed that irreversibility distributions are not continuous through the horizontal walls of a rectangular enclosure that it was differentially heated from its upper side. Magherbi et al. [7] numerically studied entropy generation at the onset of natural convection in a square cavity. They showed that entropy generation depends on thermal Rayleigh number. The effect of irreversibility distribution ratio on entropy generation was also analyzed.

## 2. Governing Equations

## 3. Entropy Generation

## 4. Numerical Procedure

## 5. Results and Discussions

^{3}≤ Gr ≤ 10

^{5}; 10

^{−4}≤ φ

_{D}≤ 10

^{−2}; 1≤ A ≤ 5 and 0° ≤ ϕ ≤ 180°, respectively.

_{D}= 10

^{−2}), transient entropy generation for Gr = 10

^{4}and 10

^{5}at different aspect ratio values is illustrated in Figure 2 and Figure 3. As can be seen, entropy generation increases at the beginning of the transient state where the conduction is the dominant mode of heat transfer, reaches a maximum value which is more important as the aspect ratio of the cavity is more important. As time proceeds, entropy generation decreases and tends towards a constant value at the steady state which depends also on the aspect ratio. For low thermal Grashof number, the decrease of entropy generation is asymptotically showing that the system’s evolution follows the linear branch of thermodynamics for irreversible process according to Prigogine’s theorem. Oscillations of entropy generation are observed for the high values of thermal Grashof number as seen in Figure 4. That is the oscillation behavior obtained before the steady state, corresponds to non linear branch of irreversible processes. In steady state, entropy generation tends towards an asymptotic value which increases with the increase of aspect ratio of the enclosure.

**Figure 2.**Dimensionless total entropy generation versus time for Gr = 10

^{4}; φ

_{D}= 10

^{−2}; ϕ = 90° (a) A = 1; (b) A = 2; (c) A = 3; (d) A = 4; (e) A = 5.

**Figure 3.**Dimensionless total entropy generation versus time for Gr = 10

^{5}; φ

_{D}= 10

^{−2}; ϕ = 90°; (a) A = 1; (b) A = 2; (c) A = 4; (d) A = 5.

**Figure 4.**Dimensionless total entropy generation versus time for Gr = 10

^{5}; φ

_{D}= 10

^{−2}; ϕ = 90°; A = 1 (oscillations).

^{−4}) and thermal Grashof number (10

^{3}, 10

^{4}and 10

^{5}) and for different aspect ratio values of the cavity. As it can be seen, for any fixed aspect ratio and thermal Grashof number values, entropy generation increases with the inclination angle, reaches a maximum value then decreases. Maximum value depends on aspect ratio. This value is obtained at ϕ ≈ 50° for A = 2 and at ϕ ≈ 80° for A = 5. The increase of both aspect ratio and thermal Grashof number induces an increase of entropy generation value. It could be noticed that for the two studied limiting inclination angle values (i.e., ϕ = 0°, 180°), entropy generation value is the same for any fixed aspect ratio.

**Figure 5.**Dimensionless total entropy generation versus inclination angle for Gr = 10

^{3}; φ

_{D}= 10

^{−4}; (a) A = 1; (b) A = 2; (c) A = 3; (d) A = 4; (e) A = 5.

**Figure 6.**Dimensionless total entropy generation versus inclination angle for Gr = 10

^{4}; φ

_{D}= 10

^{−4}; (a) A = 1; (b) A = 2; (c) A = 3; (d) A = 4; (e) A = 5.

**Figure 7.**Dimensionless total entropy generation versus inclination angle for Gr = 10

^{5}; φ

_{D}= 10

^{−4}; (a) A = 1; (b) A = 2; (c) A = 3; (d) A = 5.

_{D}≥ 10

^{−3}. This is due to the predominance of convective irreversibility inside the enclosure, which results from the increase of both thermal and velocity gradients of the fluid. The aspect ratio of the cavity has a considerable effect on entropy generation as illustrated in Figure 9. Obtained results show that entropy generation increases with aspect ratio and Grashof number. As it can be seen, a linear behavior of entropy generation versus aspect ratio is obtained for A ≥ 3.

**Figure 8.**Dimensionless total entropy generation versus inclination angle for Gr = 10

^{3}; ϕ = 90°; (a) A = 1; (b) A = 2; (c) A = 3; (d) A = 4; (e) A = 5.

**Figure 9.**Dimensionless total entropy generation versus aspect ratio for ϕ = 90°; (a) Gr = 10

^{3}and φ

_{D}= 10

^{−2}; (b) Gr = 10

^{4}and φ

_{D}= 10

^{−3}; (c) Gr = 10

^{5}and φ

_{D}= 10

^{−4}.

**Figure 10.**Variation of Bejan number versus inclination angle for Gr = 10

^{4}; φ

_{D}=10

^{−4}; (a) A = 1; (b) A = 2; (c) A = 3; (d) A = 4; (e) A = 5.

^{3 }≤ Gr ≤ 10

^{4}. This is due to thermal and velocity gradients in the above mentioned regions as indicated by isothermal lines and stream lines. For relatively higher Grashof number value (i.e., Gr ≥ 10

^{5}), entropy generation chart shows that lines of irreversibility are practically located through the active sides (heated and cooled walls). This is due to considerable thermal and velocity gradients as described by isothermal and stream lines for Gr = 10

^{5}.

**Figure 11.**Isotherm, stream and isentropic lines for A = 2; φ

_{D}=10

^{−2}; ϕ = 90° (a) Gr = 10

^{3}; (b) Gr = 10

^{4}; (c) Gr = 10

^{5}.

## 6. Conclusions

## Nomenclature

A | aspect ratio of the cavity |

A_{x}; A_{y} | aspect ratios along x and y, respectively |

g | gravitational acceleration (m·s ^{−2}) |

Gr | thermal Grashof number |

H’ (L’) | length (width) of the cavity (m) |

J_{z} | dimensionless flux vector (z = u, v, T) |

L^{*} | characteristic length (m) |

P | dimensionless pressure |

P’ | pressure (kg·m ^{−1}·s^{−2}) |

Pr | Prandtl number |

Ra | Rayleigh number |

${\dot{S}}_{L}$ | dimensionless local entropy generation |

S | dimensionless total entropy generation |

T | dimensionless temperature |

T’ | temperature (K) |

T’_{c} | cold temperature (K) |

T_{h}’ | hot temperature (K) |

T’_{o} | reference temperature (K) |

t | dimensionless time |

t’ | time (s) |

U^{*} | characteristic velocity (m·s ^{−1}) |

V | dimensionless velocity vector |

V’ | velocity vector (m·s ^{−1}) |

u, v | dimensionless velocity components |

u’, v’ | velocity components (m·s ^{−1}) |

x, y | dimensionless Cartesian coordinate system |

x’, y’ | cartesian coordinates (m) |

## Greek Letters

α | thermal diffusivity (m ^{2}·s^{−1}) |

β_{T} | thermal bulk expansion coefficient (K ^{−1}) |

ΔT^{’} | temperature difference (K) |

υ | kinematic viscosity (m ^{2}·s^{−1}) |

μ | dynamic viscosity (kg·m ^{−1}·s^{−1}) |

ϕ | inclination angle of the cavity (°) |

φ_{D} | irreversibility distribution ratio |

ρ | fluid density (kg·m ^{−3}) |

λ | thermal conductivity (J·m ^{−1}·s^{−1}·K^{−1}) |

${\dot{\sigma}}_{L}$ | volumetric local entropy generation (J·m ^{−3}·s^{−1}·K^{−1}) |

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**MDPI and ACS Style**

Bouabid, M.; Magherbi, M.; Hidouri, N.; Brahim, A.B.
Entropy Generation at Natural Convection in an Inclined Rectangular Cavity. *Entropy* **2011**, *13*, 1020-1033.
https://doi.org/10.3390/e13051020

**AMA Style**

Bouabid M, Magherbi M, Hidouri N, Brahim AB.
Entropy Generation at Natural Convection in an Inclined Rectangular Cavity. *Entropy*. 2011; 13(5):1020-1033.
https://doi.org/10.3390/e13051020

**Chicago/Turabian Style**

Bouabid, Mounir, Mourad Magherbi, Nejib Hidouri, and Ammar Ben Brahim.
2011. "Entropy Generation at Natural Convection in an Inclined Rectangular Cavity" *Entropy* 13, no. 5: 1020-1033.
https://doi.org/10.3390/e13051020