Transport in Porous Media

, Volume 126, Issue 1, pp 223–247 | Cite as

Multi-layered Porous Foam Effects on Heat Transfer and Entropy Generation of Nanofluid Mixed Convection Inside a Two-Sided Lid-Driven Enclosure with Internal Heating

  • Sasan Asiaei
  • Ali Zadehkafi
  • Majid SiavashiEmail author


Mixed convection of Cu-water nanofluid inside a two-sided lid-driven enclosure with an internal heater, filled with multi-layered porous foams is studied numerically and its heat transfer and entropy generation number are evaluated. Use of multi-layered porous media instead of homogeneous ones is capable of heat transfer enhancement, by weakening flow where does not impose a pivotal role on heat transfer and amplifying the flow in regions where have more effects on the heat transfer. Eight different arrangements of porous layers are considered and the two-phase mixture model is implemented to simulate nanofluid mixed convection inside the cavity. Results are presented in terms of stream functions, isotherms, Nusselt and entropy generation number for the eight cases considering various Richardson numbers (Ri = 10−4 to 103) and nanofluid concentrations (φ = 0 to 0.04). Results indicate that using the multi-layered porous material can confine flow vortices in the vicinity of the moving walls and could enhance the heat transfer up to 17 percent (with respect to the case using homogeneous porous material with the highest permeability), such that this enhancement is more in lower Ri values (stronger convective effects). Entropy generation number also increases by nanofluid volume fraction increment and Ri decrement. Cases with a higher heat transfer rate also have the higher entropy generation number. In addition, an increase of volume fraction decreases the relative entropy generation number (S*) for low Ri number, while contrary fact observed for high Ri values.


Nanofluid mixed convection porous media multi-layered entropy generation 

List of symbols


Acceleration (m s−2)


Drag coefficient


Specific heat (J kg−1 K−1)


Darcy number


Nanoparticles diameter (m)


Drag function


Acceleration due to gravity (m s−2)


Grashof number


Cavity length


Heat transfer coefficient (W m−2 K−1)


Thermal conductivity (W m−1 K−1)


Boltzmann constant (J K−1)


Nusselt number


Pressure (Pa)


Prandtl number

\( S_{gen,F}^{'''} \)

Friction entropy generation rate (W m−3 K)

\( S_{gen,T}^{'''} \)

Thermal entropy generation rate (W m−3 K)

\( S_{gen,tot}^{'''} \)

Total entropy generation rate (W m−3 K)


Temperature (K)


Ambient temperature


Cold wall temperature


Hot wall temperature

\( \vec{V} \)

Velocity vector (m s−1)

Greek symbols

\( \alpha \)

Thermal diffusivity (m2 s−1)

\( \beta \)

Thermal expansion coefficient (K−1)

\( \varepsilon \)


\( \varphi \)

Volume fraction

\( \kappa \)

Permeability of porous medium (m2)

\( \mu \)

Dynamic viscosity (kg m−1 s−1)

\( \nu \)

Kinematic viscosity (m2 s−1)

\( \rho \)

Density (kg m−3)





Cold wall






Hot wall


Mixture (nanofluid)




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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Sensors and Integrated Bio-Microfluidics/MEMS Laboratory, School of Mechanical EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Applied Multi-Phase Fluid Dynamics Lab, School of Mechanical EngineeringIran University of Science and TechnologyTehranIran
  3. 3.School of Mechanical EngineeringIran University of Science and TechnologyNarmak, TehranIran

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