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

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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.

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Abbreviations

\(a_\mathrm{sf}\) :

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

\(\Delta \textit{NE}\) :

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

0,1,2:

Coordinate identifier

f:

Fluid phase

s:

Solid phase

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Acknowledgments

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

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Semnan University, P.O. Box 35131-19111, Semnan, Iran

    Maziar Dehghan, Mohammad Sadegh Valipour & Seyfollah Saedodin

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  1. Maziar Dehghan
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  2. Mohammad Sadegh Valipour
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  3. Seyfollah Saedodin
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Correspondence to Mohammad Sadegh Valipour.

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Dehghan, M., Valipour, M.S. & Saedodin, S. Perturbation Analysis of the Local Thermal Non-equilibrium Condition in a Fluid-Saturated Porous Medium Bounded by an Iso-thermal Channel. Transp Porous Med 102, 139–152 (2014). https://doi.org/10.1007/s11242-013-0267-2

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