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Second Law Analysis in Mixed Convection Through an Inclined Porous Channel

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Abstract

The paper provides a numerical investigation of the entropy generation analysis due to mixed convection with viscous dissipation effect of a laminar viscous and incompressible fluid, flowing in an inclined channel filled with a saturated porous medium. The Darcy–Brinkman model is employed. The Navier–Stokes and energy equations are solved by classic Boussinesq incompressible approximation. A special attention is given to the study of the influence of the channel inclination angle on the transient and the steady-state entropy generation. The fluctuations of the transient total entropy generation are investigated when the inclination angle is varied from \(0^{\circ }\) to \(180^{\circ }\). Moreover, the entropy generation and the Bejan number were studied as a function of the inclination angle of the channel, in the steady state of mixed convection. It was found that the total entropy generation is maximum at inclination angle close to \(70^{\circ }\) and minimum at \(0^{\circ }\) and \(180^{\circ }\).

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Abbreviations

c :

Specific heat capacity at constant pressure \((\hbox {m}^{2}{\cdot }\hbox {s}^{-2}{\cdot }\hbox {K}^{-1})\)

Da :

Darcy number \((\mu /h^{2})\)

g :

Gravitational acceleration \((\hbox {m}{\cdot }\hbox {s}^{-2})\)

h :

Channel height (m)

l :

Channel length (m)

Nu :

The local Nusselt number \((\vert \hbox {d}\theta /\hbox {d}Y\vert )\)

\({\overline{Nu}}\) :

The space-averaged Nusselt number

\(\langle {\overline{Nu}}\rangle \) :

Space and time-averaged Nusselt number

p :

Pressure \((\hbox {N}{\cdot }\hbox {m}^{-2})\)

P :

Dimensionless pressure

Pe :

Peclet number \(({\textit{Re}}{\cdot }{\textit{Pr}})\)

Pr :

Prandtl number \((\eta c_{p}/\lambda _\mathrm{m})\)

Ra :

Rayleigh number in porous media \((\beta g\Delta Th^{3}/(\upsilon {\cdot }\alpha _{\mathrm{eff}}))\)

Re :

Reynolds number \(({hu}_{0}/\upsilon )\)

\({\varvec{s}}\) :

Local entropy generation \((\hbox {J}{\cdot }\hbox {m}^{-3}{\cdot }\hbox {s}^{-1}{\cdot }\hbox {K}^{-1})\)

\({\varvec{s}}_\mathrm{t}\) :

Total dimensionless entropy generation \((\hbox {J}{\cdot }\hbox {s}^{-1}{\cdot }\hbox {K}^{-1})\)

\(\langle {\varvec{s}}_\mathrm{t}\rangle \) :

Time-averaged total entropy generation \((\hbox {J}{\cdot }\hbox {s}^{-1}{\cdot }\hbox {K}^{-1})\)

t :

Time (s)

T :

Temperature (K)

\(T_{0}\) :

Mean temperature \([({T}_{\mathrm{h}}+ {T}_{\mathrm{c}})/2]\,(\hbox {K})\)

\(\Delta T\) :

Temperature difference \(({T}_{\mathrm{h}}-{T}_{\mathrm{c}})\)

\(u_{0}\) :

Characteristic velocity \((\hbox {m}{\cdot }\hbox {s}^{-1})\)

u, v :

Velocity components in x and y directions, respectively \((\hbox {m}{\cdot }\hbox {s}^{-1})\)

U, V :

Dimensionless velocity components

x, y :

Cartesian coordinates (m)

X, Y :

Dimensionless Cartesian coordinates

\(\beta _\mathrm{T}\) :

Thermal volumetric expansion coefficient \((\hbox {K}^{-1})\)

\(\varepsilon \) :

Medium porosity

\(\mu \) :

Permeability of the porous media \((\hbox {m}^{2})\)

\(\lambda \) :

Thermal conductivity \((\hbox {kg}{\cdot }\hbox {m}{\cdot }\hbox {s}^{-3}{\cdot }\hbox {K}^{-1})\)

\(\theta \) :

Dimensionless temperature

\(\varTheta \) :

Dimensionless period

\(\uprho \) :

Mass density \((\hbox {kg}{\cdot }\hbox {m}^{-3})\)

\(\uprho _{0}\) :

Reference mass density \((\hbox {kg}{\cdot }\hbox {m}^{-3})\)

\(\upsigma \) :

Specific heat capacities ratio \(((\uprho \hbox {c})_{\mathrm{m}}/(\uprho \hbox {c})_{\mathrm{f}})\)

\(\varLambda \) :

Viscosity ratio \((\eta _{\mathrm{eff}}/\eta )\)

\(\eta \) :

Dynamic viscosity \((\hbox {kg}{\cdot }\hbox {m}^{-1}{\cdot }\hbox {s}^{-1})\)

\(\upalpha \) :

Thermal diffusivity \((\hbox {m}^{2}{\cdot }\hbox {s}^{-1})\)

\(\upsilon \) :

Kinematic viscosity \((\hbox {m}^{2}{\cdot }\hbox {s}^{-1})\)

\(\uptau \) :

Dimensionless time

a:

Dimensionless

c:

Cold wall

eff:

Effective

F:

Fluid friction

f:

Fluid

H:

Heat transfer

h:

Hot wall

l:

Local

m:

Porous media

s:

Solid

References

  1. D.A. Nield, A. Bejan, Convection in Porous Media, 2nd edn. (Springer, New York, 1999)

    Book  MATH  Google Scholar 

  2. A. Bejan, A.D. Kraus, Heat Transfer Handbook (Wiley, New York, 2003)

    Google Scholar 

  3. D.B. Ingham, A. Bejan, E. Mamut, I. Pop, Emerging Technologies and Techniques in Porous Media (Kluwer, Dordrecht, 2004)

    Book  MATH  Google Scholar 

  4. D.B. Ingham, I. Pop, P. Cheng, Transp. Porous Media 5, 381–398 (1990)

    Article  Google Scholar 

  5. A.K. Al-Hadhrami, L. Elliott, D.B. Ingham, Transp. Porous Media 49, 265–289 (2002)

    Article  MathSciNet  Google Scholar 

  6. D.A. Nield, A.V. Kuznetsov, M. Xiong, Int. J. Heat Mass Transf. 46, 643–651 (2003)

    Article  MATH  Google Scholar 

  7. O.T. Gideon, E.A. Owoloko, O.E. Osafile, Int. J. Adv. Sci. Technol. 3, 65–85 (2011)

    Google Scholar 

  8. E.M.A. Elbashbeshy, Appl. Math. Comput. 136, 139–149 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. O.D. Makinde, E. Osalusi, Mech. Res. Commun. 33, 692–698 (2006)

    Article  MATH  Google Scholar 

  10. M. Nourollahi, M. Farhadi, K. Sedighi, Therm. Sci. 14, 329–340 (2010)

    Article  Google Scholar 

  11. A. Mchirgui, N. Hidouri, M. Magherbi, A. Ben Brahim, Comput. Fluids 96, 105–115 (2014)

    Article  MathSciNet  Google Scholar 

  12. M.S. Malashetty, J.C. Umavathi, J.P. Kumar, Heat Mass Transf. 40, 871–876 (2004)

    Article  ADS  Google Scholar 

  13. K. Hooman, H. Gurgency, Transp. Porous Media 68, 301–319 (2007)

    Article  MathSciNet  Google Scholar 

  14. A.K. Al-Hadhrami, L. Elliott, D.B. Ingham, Transp. Porous Media 53, 117–122 (2003)

    Article  MathSciNet  Google Scholar 

  15. K. Hooman, F. Hooman, S.R. Mohebpour, Int. J. Exergy 5, 78–96 (2008)

    Article  Google Scholar 

  16. T. Chinyoka, O.D. Makinde, A.S. Eegunjobi, J. Porous Media 16, 823–836 (2013)

    Article  Google Scholar 

  17. O.D. Makinde, T. Chinyoka, A.S. Eegunjobi, Int. J. Exergy 12, 279–297 (2013)

    Article  Google Scholar 

  18. A. Bejan, Convection Heat Transfer (Wiley, New York, 1984)

    MATH  Google Scholar 

  19. D.A. Nield, Transp. Porous Media 41, 349–357 (2000)

    Article  MathSciNet  Google Scholar 

  20. D.A. Nield, Modeling fluid flow in saturated porous media and at interfaces, in Transport Phenomena in Porous Media II, ed. by D.B. Ingham, I. Pop (Elsevier, Oxford, 2002)

    Google Scholar 

  21. D.A. Nield, A. Bejan, Convection in Porous Media, 3rd edn. (Springer, New York, 2006)

    MATH  Google Scholar 

  22. D.A. Nield, A.V. Kuznetsov, M. Xiong, Transp. Porous Media 56, 351–367 (2004)

    Article  Google Scholar 

  23. E. Magyari, D.A.S. Rees, B. Keller, Effect of viscous dissipation on the flow in fluid saturated porous media, in Handbook of Porous Media, 2nd edn., ed. by K. Vafai (Taylor and Francis, New York, 2005), pp. 373–407

    Google Scholar 

  24. A. Bejan, Entropy Generation Through Heat and Fluid Flow (Wiley, New York, 1982)

    Google Scholar 

  25. S.H. Tasnim, S. Mahmud, M.A.H. Mamun, Int. J. Exergy 4, 300–308 (2002)

    Article  Google Scholar 

  26. S. Mahmud, R.A. Fraser, Int. J. Therm. Sci. 44, 21–32 (2005)

    Article  Google Scholar 

  27. H.J. Saabas, B.R. Baliga, Numer. Heat Transf. B 26, 381–407 (1994)

    Article  ADS  Google Scholar 

  28. A.A. Karamallah, W.S. Mohammad, W.H. Khalil, Eng. Technol. J. 29, 1721–1736 (2011)

    Google Scholar 

  29. H. Abassi, S. Turki, S. Ben Nasrallah, Numer. Heat Transf. A 39, 307–320 (2001a)

    Article  ADS  Google Scholar 

  30. H. Abassi, S. Turki, S. Ben Nasrallah, Int. J. Therm. Sci. 40, 649–658 (2001b)

    Article  Google Scholar 

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Authors and Affiliations

  1. Applied Thermodynamics Unit, Chemical and Process Engineering Department, Engineers National School of Gabes, Gabes University, Omar Ibn El Khattab Street, 6029, Gabès, Tunisia

    Amel Tayari & Ammar Ben Brahim

  2. Civil Engineering Department, High Institute of Applied Sciences and Technology, Gabes University, Omar Ibn El Khattab Street, 6029, Gabès, Tunisia

    Mourad Magherbi

Authors
  1. Amel Tayari
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  2. Ammar Ben Brahim
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  3. Mourad Magherbi
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Correspondence to Amel Tayari.

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Tayari, A., Brahim, A.B. & Magherbi, M. Second Law Analysis in Mixed Convection Through an Inclined Porous Channel. Int J Thermophys 36, 2881–2896 (2015). https://doi.org/10.1007/s10765-015-1925-0

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