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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
Article

Abstract

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.

Keywords

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

List of Symbols

Variables

\(a_\mathrm{sf}\)

Specific surface area

\(c_\mathrm{p}\)

Specific heat at constant pressure

\(C_\mathrm{F}\)

Inertial constant (Eq. 12)

\(d_\mathrm{p}\)

Particle diameter

Da

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

\(F\)

Forchheimer number

\(G\)

Negative of the applied pressure gradient in flow direction

\(H\)

Half of the channel gap

\(h_\mathrm{sf}\)

Fluid-solid heat transfer coefficient

\(K\)

Permeability of the medium

\(k\)

Conductivity ratio

\(k_\mathrm{f}\)

Conductivity of fluid phase

\(k_\mathrm{f,eff}\)

Effective conductivity of fluid phase

\(k_\mathrm{m}\)

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

\(k_\mathrm{s}\)

Conductivity of solid phase

\(k_\mathrm{s,eff}\)

Effective conductivity of solid phase

\(M\)

Viscosity ratio

Nu

Nusselt number

\(O\)

Order of magnitude

Pr

Prandtl number

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

Heat flux at the wall

\(s\)

Porous media shape parameter

\(T\)

Temperature

\(T_\mathrm{m}\)

Bulk mean temperature

\(T_\mathrm{w}\)

Wall temperature

\(u\)

Dimensionless velocity

\(u^*\)

Velocity

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

Normalized velocity

\(u^*_\mathrm{m}\)

Mean velocity

\(x^*,y^*\)

Dimensional coordinates

\(y\)

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

Subscripts

0,1,2

Coordinate identifier

f

Fluid phase

s

Solid phase

Notes

Acknowledgments

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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