# 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.

## 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.

## References

- Alazmi, B., Vafai, K.: Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions. Int. J. Heat Mass Transf.
**45**, 3071–3087 (2002)CrossRefGoogle Scholar - Hooman, K., Merrikh, A.A.: Analytical solution of forced convection in a duct of rectangular cross section saturated by a porous medium. ASME J. Heat Transf.
**128**, 596–600 (2006)CrossRefGoogle Scholar - Hooman, K., Gurgenci, H.: Effects of viscous dissipation and boundary conditions on forced convection in a channel occupied by a saturated porous medium. Transp. Porous Med.
**68**, 301–319 (2007)CrossRefGoogle Scholar - Hooman, K.: A perturbation solution for forced convection in a porous-saturated duct. J. Comput. App. Math.
**211**, 57–66 (2008)CrossRefGoogle Scholar - Hung, Y., Tso, C.P.: Effects of viscous dissipation on fully developed forced convection in porous media. Int. Comm. Heat Mass Transf.
**36**, 597–603 (2009)CrossRefGoogle Scholar - Kaviany, M.: Laminar flow through a porous channel bounded by isothermal parallel plates. Int. J. Heat Mass Transf.
**28**, 851–858 (1985)CrossRefGoogle Scholar - Khandelwal, M.K., Bera, P.: A thermal non-equilibrium perspective on mixed convection in a vertical channel. Int. J. Thermal Sci.
**56**, 23–34 (2012)CrossRefGoogle Scholar - Kuznetsov, A.V.: A perturbation solution for a nonthermal equilibrium fluid flow through a three-dimensional sensible heat storage packed bed. ASME J. Heat Transf.
**118**, 508–510 (1996)CrossRefGoogle Scholar - Kuznetsov, A.V.: A perturbation solution for heating a rectangular sensible heat storage packed bed with a constant temperature at the walls. Int. J. Heat Mass Transf.
**40**, 1001–1006 (1997a)CrossRefGoogle Scholar - Kuznetsov, A.V.: Thermal nonequilibrium, non-Darcian forced convection in a channel filled with a fluid saturated porous medium: a perturbation solution. Appl. Sci. Res.
**57**, 119–131 (1997b)CrossRefGoogle Scholar - Mahmoudi, Y., Maerefat, M.: Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition. Int. J. Thermal Sci.
**50**, 2386–2401 (2011)CrossRefGoogle Scholar - Mahmud, S., Fraser, R.A.: Conjugate heat transfer inside a porous channel. Heat Mass Transf.
**41**, 568–575 (2005a)CrossRefGoogle Scholar - Mahmud, S., Fraser, R.A.: Flow, thermal, and entropy generation characteristics inside a porous channel with viscous dissipation. Int. J. Thermal Sci.
**44**, 21–32 (2005b)CrossRefGoogle Scholar - Mirzaei, M., Dehghan, M.: Investigation of flow and heat transfer of nanofluid in microchannel with variable property approach. Heat Mass Transf.
**49**, 1803–1811 (2013)CrossRefGoogle Scholar - Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)Google Scholar
- Nield, D.A., Junqueira, S.L.M., Lage, J.L.: Forced convection in a fluid saturated porous medium channel with isothermal or isoflux boundaries. J. Fluid Mech.
**322**, 201–214 (1996)CrossRefGoogle Scholar - Nield, D.A.: Effects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel. J. Porous Media
**1**(2), 181–186 (1998)Google Scholar - Nield, D.A., Kuznetsov, A.V.: Local thermal nonequilibrium effects in forced convection in a porous medium channel: a conjugate problem. Int. J. Heat Mass Transf.
**42**, 3245–3252 (1999)CrossRefGoogle Scholar - Nield, D.A., Kuznetsov, A.V.: The interaction of thermal nonequilibrium and heterogeneous conductivity effects in forced convection in layered porous channels. Int. J. Heat Mass Transf.
**44**, 4369–4373 (2001)CrossRefGoogle Scholar - Nield, D.A., Kuznetsov, A.V., Xiong, M.: Effect of local thermal non-equilibrium on thermally developing forced convection in a porous medium. Int. J. Heat Mass Transf.
**45**, 4949–4955 (2002)CrossRefGoogle Scholar - Nield, D.A., Kuznetsov, A.V.: Forced convection in a helical pipe filled with a saturated porous medium. Int. J. Heat Mass Transf.
**47**, 5175–5180 (2004)CrossRefGoogle Scholar - Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, New York (2006)Google Scholar
- Qu, Z.G., Xu, H.J., Tao, W.Q.: Fully developed forced convective heat transfer in an annulus partially filled with metallic foams: an analytical solution. Int. J. Heat Mass Transf.
**55**, 7508–7519 (2012)CrossRefGoogle Scholar - Schumann, T.E.W.: Heat transfer: liquid flowing through a porous prism. J. Franklin Inst.
**208**, 405–416 (1929)CrossRefGoogle Scholar - Spiga, G., Spiga, M.: A rigorous solution to a heat transfer two-phase model in porous media and packed beds. Int. J. Heat Mass Transf.
**24**, 355–364 (1981)CrossRefGoogle Scholar - Vafai, K., Kim, S.J.: Forced convection in a channel filled with a porous medium: an exact solution. ASME J. Heat Transf.
**111**, 1103–1106 (1989)CrossRefGoogle Scholar - Vafai, K., Kim, S.J.: Discussion of the paper by A. Hadim “Forced convection in a porous channel with localized heat sources”. ASME J. Heat Transf.
**117**, 1097–1098 (1995)CrossRefGoogle Scholar - White, H.C., Korpela, S.A.: On the calculation of the temperature distribution in a packed bed for solar energy applications. Solar Energy
**23**, 141–144 (1979)CrossRefGoogle Scholar