Transport in Porous Media

, Volume 98, Issue 1, pp 35–57 | Cite as

Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

  • Eren Ucar
  • Moghtada Mobedi
  • Ioan PopEmail author


A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.


Porous medium Partially filled porous media Channel flow Fully developed. 

List of Symbols

\(\text{ Da}\)

Darcy number


Friction coefficient


Pressure gradient in \(x\)- direction


Convective heat transfer coefficient, \(\text{ W/m}^{2}\) K


Half height of channel

\(k_m \)

The overall thermal conductivity, W/m K


Permeability, \(\text{ m}^{2}\)


The ratio of effective conductivity over fluid conductivity


Length of the channel, m


Dimensionless viscosity ratio parameter

\(Nu_m \)

Overall Nusselt number


Reynolds number

\(q^{{\prime }{\prime }}\)

heat flux, \(\text{ W/m}^{2}\)


Porous media shape parameter


Temperature, \(\text{ K}\)

\(T_i \)

Initial temperature, \(\text{ K}\)


Velocity component along \(x\)- direction, m/s


Dimensionless velocity component along dimensionless \(X\)- direction, m/s


Dimensionless interface velocity component along dimensionless \(X\)- direction


Dimensionless normalized velocity


Coordinate along the axis of the channel, m


Coordinate normal to the surfaces of the channel, m


Dimensionless coordinates

Greek letters

\(\beta \)

Coefficient for the walls

\(\varepsilon \)


\(\mu \)

Dynamic viscosity, kg/m s


Kinematic viscosity of fluid, \(\text{ m}^{2}\)/s

\(\theta \)

Dimensionless temperature

\(\theta _i \)

The dimensionless interface temperature

\(\rho \)

Density, \(\text{ kg/m}^{3}\)

\(\xi \)

Porous thickness



Clear fluid region


















Upper wall


Lower wall

\(w\,\mu \)

Average wall


Belongs to the upper wall


Belongs to the lower wall


  1. Al-Nimr, M.A., Alkam, M.K.: A modified tubeless solar collector partially filled with porous substrate. Renew. Energy 13, 165–173 (1997)CrossRefGoogle Scholar
  2. Forooghi, P., Abkar, M., Saffar-Avval, M.: Steady and unsteady heat transfer in a channel partially filled with porous media under thermal non-equilibrium condition. Transp. Porous Media 86, 177–198 (2010)CrossRefGoogle Scholar
  3. Guo, Z., Kim, S.Y., Sung, H.J.: Pulsating flow and heat transfer in a pipe partially filled with a porous medium. Int. J. Heat Mass Transf. 40, 4209–4218 (1997)CrossRefGoogle Scholar
  4. Jen, T.-C., Yan, T.Z.: Developing fluid flow and heat transfer in a channel partially filled with porous medium. Int. J. Heat Mass Transf. 48, 3995–4009 (2005)CrossRefGoogle Scholar
  5. Kuznetsov, A.V.: Analytical investigation of couette flow in a composite channel partially filled with a porous medium and partially with a clear fluid. Int. J. Heat Mass Transf. 41, 2556–2560 (1998)CrossRefGoogle Scholar
  6. Kuznetsov, A.V.: Fluid flow and heat transfer analysis of Couette flow in a composite duct. Acta Mechanica 140, 163–170 (1999)CrossRefGoogle Scholar
  7. Kuznetsov, A.V., Nield, D.A.: Forced convection in a channel partly occupied by a bidisperse porous medium: asymmetric case. Int. J. Heat Mass Transf. 53, 5167–5175 (2010)CrossRefGoogle Scholar
  8. Mohamad, A.A.: Heat transfer enhancements in heat exchangers fitted with porous media part I: constant wall temperature. Int. J. Thermal Sci. 42, 385–395 (2003)CrossRefGoogle Scholar
  9. Morosuk, T.V.: Entropy generation in conduits filled with porous medium totally and partially. Int. J. Heat Mass Transf. 48, 2548–2560 (2005)CrossRefGoogle Scholar
  10. Nield, D.A., Kuznetsov, A.V.: Forced convection in a channel partly occupied by a bidisperse porous medium: symmetric case. ASME J. Heat Transf. 133, 072601 (2011)CrossRefGoogle Scholar
  11. Poulikakos, D., Kazmierczak, M.: Forced convection in a duct partially filled with a porous material. ASME J. Heat Transf. 109, 653–662 (1987)CrossRefGoogle Scholar
  12. Satyamurty, V.V., Bhargavi, D.: Forced convection in thermally developing region of a channel partially filled with a porous material and optimal porous fraction. Int. J. Thermal Sci. 49, 319–332 (2010)CrossRefGoogle Scholar
  13. Sayehvand, H., Shokouhmand, H.: Study of forced convection in a pipe partially filled with a porous medium. The 4th Wseas International Conference on Heat Transfer, Thermal Engineering and Environment, pp. 292–300. Elounda (2006)Google Scholar
  14. Shokouhmand, H., Jam, F., Salimpour, M.R.: The effect of porous insert position on the enhanced heat transfer in partially filled channels. Int. Commun. Heat Mass Transf. 38, 1162–1167 (2011)CrossRefGoogle Scholar
  15. Teamah, M.A., El-Maghlany, W.M., Khairat Dawood, M.M.: Numerical simulation of laminar forced convection in horizontal pipe partially or completely filled with porous material. Int. J. Thermal Sci. 50, 1512–1522 (2011)CrossRefGoogle Scholar
  16. Yang, C., Liu, W., Nakayama, A.: Forced convective heat transfer enhancement in a tube with its core partially filled with a porous medium. Open Transport Phenom. J. 1, 1–6 (2009)CrossRefGoogle Scholar
  17. Yang, C., Nakayama, A., Liu, W.: Heat transfer performance assessment for forced convection in a tube partially filled with a porous medium. Int. J. Thermal Sci. 54, 98–108 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentIzmir Institute of TechnologyUrlaTurkey
  2. 2.Department of MathematicsBabeş -Bolyai UniversityCluj-NapocaRomania

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