Transport in Porous Media

, Volume 102, Issue 2, pp 139–152 | Cite as

Perturbation Analysis of the Local Thermal Non-equilibrium Condition in a Fluid-Saturated Porous Medium Bounded by an Iso-thermal Channel

  • Maziar Dehghan
  • Mohammad Sadegh ValipourEmail author
  • Seyfollah Saedodin


This study focuses analytically on the local thermal non-equilibrium (LTNE) effects in the developed region of forced convection in a saturated porous medium bounded by isothermal parallel-plates. The flow in the channel is described by the Brinkman–Forchheimer-extended Darcy equation and the LTNE effects are accounted by utilizing the two-equation model. Profiles describing the velocity field obtained by perturbation techniques are used to find the temperature distributions by the successive approximation method. A fundamental relation for the temperature difference between the fluid and solid phases (the LTNE intensity) is established based on a perturbation analysis. It is found that the LTNE intensity (\(\Delta \textit{NE}\)) is proportional to the product of the normalized velocity and the dimensionless temperature at LTE condition. Also, it depends on the conductivity ratio, Da number, and the porosity of the medium. The intensity of LTNE condition (\(\Delta \textit{NE}\)) is maximum at the middle of the channel and decreases smoothly to zero by moving to the wall. Finally, the established relation for the intensity of LTNE condition is simple and fundamental for estimating the importance of LTNE condition and validation of numerical simulation results.


Local thermal non-equilibrium Perturbation Saturated porous medium Channel flow Brinkman–Forchheimer-extended Darcy model Conductivity ratio 

List of Symbols



Specific surface area


Specific heat at constant pressure


Inertial constant (Eq. 12)


Particle diameter


Darcy number (K/H\(^{2}\))


Forchheimer number


Negative of the applied pressure gradient in flow direction


Half of the channel gap


Fluid-solid heat transfer coefficient


Permeability of the medium


Conductivity ratio


Conductivity of fluid phase


Effective conductivity of fluid phase


Effective conductivity of the medium (\(k_\mathrm{f,eff}+ k_\mathrm{s,eff})\)


Conductivity of solid phase


Effective conductivity of solid phase


Viscosity ratio


Nusselt number


Order of magnitude


Prandtl number

\(q^{\prime \prime }_\mathrm{w}\)

Heat flux at the wall


Porous media shape parameter




Bulk mean temperature


Wall temperature


Dimensionless velocity



\(\mathop {u}\limits ^{{\frown }}\)

Normalized velocity


Mean velocity


Dimensional coordinates


Dimensionless coordinate

Greek Letters

\(\Delta \textit{NE}\)

Complex representing the intensity of LTNE condition

\(\varepsilon \)

Small parameter (\(1/ h_\mathrm{sf} \, a_\mathrm{sf})\)

\(\theta \)

Dimensionless temperature

\(\mu \)

Fluid viscosity

\(\mu _\mathrm{eff}\)

Effective viscosity in the Brinkman term

\(\rho \)

Fluid density

\(\upphi \)

Porosity of the medium



Coordinate identifier


Fluid phase


Solid phase



The authors are grateful to Prof. D.A. Nield for his valuable help in completing the article.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Maziar Dehghan
    • 1
  • Mohammad Sadegh Valipour
    • 1
    Email author
  • Seyfollah Saedodin
    • 1
  1. 1.Faculty of Mechanical EngineeringSemnan UniversitySemnanIran

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