# Second Law Analysis of Laminar Flow In A Channel Filled With Saturated Porous Media

## Abstract

**:**

_{a}) and used to obtain the entropy generation number and the irreversibility ratio. Generally, our result shows that heat transfer irreversibility dominates over fluid friction irreversibility (i.e. 0 ≤ φ < 1), and viscous dissipation has no effect on the entropy generation rate at the centerline of the channel.

## Introduction

## Mathematical Formulation

_{e}is an effective viscosity, µ is the fluid velocity, K is the permeability, and G is the applied pressure gradient.

_{a}the Darcy number, p

_{e}the Peclet number, k the thermal conductivity and ρ the fluid density.

_{a}appear only in the combination of M times D

_{a}, hence, without loss of generality, we take M = 1 in our analysis.

_{a}) yeilds

_{0}is the temperature at the inlet. The dimensionless energy equation is given as

_{0})/qa.

## Entropy Generation Rate

_{r}= G

^{2}a

^{3}/qµ is the Brinkman number, Ω = qa/kT0 the di- mensionless temperature difference. N

_{x}and N

_{y}are the entropy generation by heat transfer due to both axial and transverse heat conduction respectively and N

_{f}is the entropy generation due to fluid friction.

## Results and Discussions

_{r}For a specific case of s = 0.5, and P

_{e}= 20, irreversibility ratio is plotted in Fig7. as a function of transverse distance (y) for different group param-eters (B

_{r}/Ω). The group parameter is an important dimensionless number for irriversibility analysis. It determines the relative importance of viscous effects to temperature gradient entropy generation. Irre- versibility ratio profile is asymmetric about the centerline of the channel due to the asymmetric tem- perature distribution. For all group parameters, each wall acts as a strong concetrator of irriversibility because of the high near-wall gradients of velocity and temperature. Maximum irreversibility ration occurs near the adiabatic wall fo all group parameters. Fluid friction irreversibility is zero at channel centerline(y = 0.5) due to zero velocity gradient (∂u/∂y). Also irreversibility ratio (φ) is independent of the group parameter at y=0.5. Therefore, the magnitude of irreversibility ratio is same at centerline of the channel for all group parameters. Minimum irreversibility ratio occur very near where the temperature gradient is zero. Generally, it is observed that an increase in group parameter strengthens the effect of fluid friction irreversibility, but heat transfer irreversibility dominates over fluid friction irreversibility (i.e. 0 ≤ φ < 1).

**Figure 6.**Entropy generation number for different values of B = B

_{r}Ω

^{−1}( P

_{e}= 20 and s = 0.5)

## Conclusion

_{a}) and group param- eter (B

_{r}Ω

^{−1}). Generally, our result shows that heat transfer irreversibility dominates over fluid friction irreversibility and viscous dissipation has no effect on the entropy generation rate at the centerline of the channel.

## Acknowledgement

## Nomenclature

a | channel width |

B_{r} | Brinkman number |

c_{p} | specific heat at constant pressure |

D_{a} | Darcy number |

G | applied pressure gradient |

k | fluid thermal conductivity |

K | Permeability |

M | µ _{e}/µ |

P_{e} | Peclet number |

q | fluid flux rate |

s | (M D _{a})^{−1/2}) |

T_{0} | wall temperature |

T | absolute temperature |

U | dimensionless fluid velocity |

u | dimensionless fluid velocity as s → 0 |

$\overline{u}$ | fluid velocity |

x | dimensionless axial coordinate |

y | dimensionless transverse coordinate |

$\overline{x}$ | axial coordinate |

$\overline{y}$ | transverse coordinate |

Greek symbols | |

µ | fluid viscosity |

µ_{e} | effective viscosity in the Brinkman term |

θ | dimensionless temperature |

Ω | dimensionless temperature difference qa/kT0 |

ρ | fluid density |

## References

- Arpaci, V. S. Radiative entropy production - lost heat into entropy. Int. J. Heat Mass Transfer
**1993**, 36, 4193–4197. [Google Scholar] - Bejan, A. Second law analysis in heat transfer. Energy - The Int. J.
**1980**, 5, 721–732. [Google Scholar] [CrossRef] - Bejan, A. Entropy Generation Through Heat and Fluid Flow. John Wiley & Sons. Inc.: Canada, 1994; Chapter 5; p. 98. [Google Scholar]
- Bejan, A. Entropy Generation Minimization, CRC Press Flow. In CRC Press Flow; John Wiley & Sons. Inc.: Canada, 1996; Chapter 5; p. 98. [Google Scholar]
- Bejan, A. Entropy Generation Minimization. CRC Press: USA, 1996. [Google Scholar]
- Erbay, L.B.; Altaç, Z.; Sülü, B. Entropy Generation in a Square Enclosure Heated From a Vertical Lateral Wall. In Proceedings of the 15th International Symposium on Efficiency, Costs, Optimization, Simulation and Environmental Aspects of Energy Systems; 2002; 3, pp. 1609–1616. [Google Scholar]
- Krane, R. J. A. Second law analysis of the optimum design and operation of thermal energy storage systems. Int. J. Heat Mass Transfer
**1987**, 30, 43–57. [Google Scholar] [CrossRef] - Latife, B. E.; Mehmet, S. E.; Birsen, S.; Yalcum, M. M. Entropy generation during fl flow between two parallel plates with moving bottom plate. Entropy
**2003**, 5, 506–518. [Google Scholar] - Mahmud, S.; Fraser, R. A. Thermodynamic analysis of flow and heat transfer inside channel with two parallel plates. Exergy, an International Journal
**2002**, 2, 140–146. [Google Scholar] [CrossRef] - Mahmud, S.; Fraser, R. A. The second law analysis in fundamental convective heat transfer problems. Int. J. of Thermal Sciences
**2002**, 42(2), 177–186. [Google Scholar] [CrossRef] - Nag, P.K.; Kumar, N. Second law optimization of convective heat transfer through a duct with constant heat flux. Int. J. Energy Research
**1989**, 13(5), 537–543. [Google Scholar] - Makinde, O. D. Exothermic explosions in a slab: A case study of series summation technique. Int. Comm. Heat and Mass Transfer
**2004**, 31(8), 1227–1231. [Google Scholar] [CrossRef] - Narusawa, U. The second law analysis of mixed convection in rectangular ducts. Heat and Mass Transfer
**2001**, 37, 197–203. [Google Scholar] [CrossRef] - Nield, D. A.; Kuznetsov, A. V.; Ming, X. Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects. Int. J. Heat and Mass Transfer
**2003**, 46, 643–651. [Google Scholar] [CrossRef] - Rott, N. Themoacoustics. Adv. Appl. Mech.
**1980**, 20, 135–175. [Google Scholar] - Sahin, A.Z. Irreversibilities in various duct geometries with constant wall heat flux and laminar flow. Energy, The International J.
**1998**, 23(6), 465–473. [Google Scholar] [CrossRef] - Sahin, A.Z. Entropy generation in turbulent liquid flow through a smooth duct subjected to constant wall temperature. Int. J. Heat and Mass Transfer
**2000**, 43, 1469–1478. [Google Scholar] [CrossRef] - Sahin, A.Z. Entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to constant heat flux. Exergy, an International Journal
**2002**, 2, 314–321. [Google Scholar] [CrossRef] - Salah, S.; Soraya, A. Second law analysis of laminar falling liquid fi along an inclined heated plate. Int. Comm. Heat Mass Transfer
**2004**, 31(No. 6), 879–886. [Google Scholar] - Swift, G. W. Themoacoustics: A unifying perspective for some engines and refrigerators. ASA Publication: New York, 2002. [Google Scholar]
- Syeda, H. T.; Shohel, M. Entropy generation in a verical concentric channel with temperature dependent viscosity. Int. Comm. Heat Mass Transfer
**2002**, 29(No. 7), 907–918. [Google Scholar]

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**MDPI and ACS Style**

Makinde, O.D.; Osalusi, E.
Second Law Analysis of Laminar Flow In A Channel Filled With Saturated Porous Media. *Entropy* **2005**, *7*, 148-160.
https://doi.org/10.3390/e7020148

**AMA Style**

Makinde OD, Osalusi E.
Second Law Analysis of Laminar Flow In A Channel Filled With Saturated Porous Media. *Entropy*. 2005; 7(2):148-160.
https://doi.org/10.3390/e7020148

**Chicago/Turabian Style**

Makinde, O. D., and E. Osalusi.
2005. "Second Law Analysis of Laminar Flow In A Channel Filled With Saturated Porous Media" *Entropy* 7, no. 2: 148-160.
https://doi.org/10.3390/e7020148