Skip to main content
SpringerLink
Log in

Free Convection Heat Transfer From A Sphere In A Porous Medium Using A Thermal Non-equilibrium Model

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

This work studies the free convection heat transfer from a sphere with constant wall temperature embedded in a fluid-saturated porous medium using a thermal non-equilibrium model. The governing equations are transformed into boundary-layer partial differential equations by the coordinate transform, and the obtained governing equations are then solved by the cubic spline collocation method. The temperature distributions for fluid and solid phases are shown for different values of the porosity scaled thermal conductivity ratio, the interphase heat transfer parameter, and the streamwise coordinate. The effects of the porosity scaled thermal conductivity ratio and the interphase heat transfer parameter between solid and fluid phases on the local Nusselt numbers for fluid and solid phases are examined. Results show the local Nusset number for the porous medium can be increased by increasing the porosity scaled thermal conductivity ratio. Moreover, the thermal non-equilibrium effect is more significant for low values of the porosity scaled thermal conductivity ratio or the interphase heat transfer parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Abbreviations

\(a\) :

Radius of the sphere

\(A\) :

Angle made by the outward normal from the sphere with the downward vertical

\(C_p \) :

Constant-pressure specific heat

\(f\) :

Dimensionless stream function

\(g\) :

Acceleration due to gravity

\(h\) :

Interphase heat transfer coefficient between solid and fluid phases

\(H\) :

Interphase heat transfer parameter between solid and fluid phases

\(K\) :

Permeability of the porous medium

\(Nu\) :

Local Nusselt number

\(Ra\) :

Darcy–Rayleigh number

\(T\) :

Temperature

\(u,v\) :

Dimensional velocity components along \(x\) and \(y\) axes

\(x,y\) :

Dimensional Cartesian coordinates along and normal to the sphere

\(\alpha _\mathrm{f}\) :

Thermal diffusivity of the fluid

\(\beta \) :

Coefficient of thermal expansion

\(\gamma \) :

Porosity scaled thermal conductivity ratio

\(\varepsilon \) :

Porosity

\(\theta \) :

Dimensionless temperature

\(\nu _\mathrm{f} \) :

Fluid kinematic viscosity

\(\xi ,\eta \) :

Dimensionless coordinates

\(\rho \) :

Density

\(\bar{\psi }\) :

Stream function

f:

Fluid phase

s:

Solid phase

w:

Condition at wall

\(\infty \) :

Condition at infinity

References

  • Barletta, A., Celli, M.: Local thermal non-equilibrium flow with viscous dissipation in plane horizontal porous layer. Int. J. Therm. Sci. 50, 53–60 (2011)

    Article  Google Scholar 

  • Baytaş, A.C., Pop, I.: Free convection in a square porous cavity using a thermal non-equilibrium model. Int. J. Therm. Sci. 41, 861–870 (2002)

    Article  Google Scholar 

  • Baytaş, A.C.: Thermal non-equilibrium natural convection in a square enclosure with a heat-generating solid phase, non-Darcy porous medium. Int. J. Energy Res. 27, 975–988 (2003)

    Article  Google Scholar 

  • Bhadauria, B.S., Agarwal, S.: Convective transport in a nanofluid saturated porous layer with thermal non-equilibrium model. Transp. Porous Med. 88, 107–131 (2011)

    Article  Google Scholar 

  • Chamkha, A., Gorla, R.S.R., Ghodeswar, K.: Non-similar solution for convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid. Transp. Porous Med. 86, 13–22 (2011)

    Google Scholar 

  • Cheng, C.Y.: A boundary layer analysis of heat transfer by free convection from permeable horizontal cylinders of elliptic cross-section in porous media using a thermal non-equilibrium model. Int. Commun. Heat Mass Transf. 34, 612–613 (2007)

    Google Scholar 

  • Ganapathy, R.: Time dependent free convection motion and heat transfer in an infinite porous medium induced by a heated sphere. Int. J. Heat Mass Transf. 40, 1551–1557 (1997)

    Article  Google Scholar 

  • Kimura, S., Pop, I.: Conjugate free convection from a sphere in a porous medium. Int. J. Heat Mass Transf. 37, 2187–2192 (1994)

    Article  Google Scholar 

  • Merkin, J.H.: Free convection boundary layers on axi-symmetric and two-dimensional bodies of arbitrary shape in a saturated porous medium. Int. J. Heat and Mass Transf. 22, 1461–1462 (1979)

    Article  Google Scholar 

  • Minkowycz, W.J., Cheng, P., Chang, C.H.: Mixed convection about a nonisothermal cylinder and sphere in a porous medium. Numer. Heat Transf. 8, 349–359 (1985)

    Google Scholar 

  • Mohamad, A.A.: Non-equilibrium, natural convection in a differentially heated cavity filled with a saturated porous matrix. ASME J. Heat Transf. 122, 380–384 (2000)

    Article  Google Scholar 

  • Mohamad, A.A.: Natural convection from a vertical plate in a saturated porous medium: non-equilibrium theory. J. Porous Media 4, 181–186 (2001)

    Google Scholar 

  • Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)

    Google Scholar 

  • Nilson, R.H.: Natural convective boundary layer on two-dimensional and axisymmetric surfaces in high-Pr fluids or in fluid saturated porous media. ASME J. Heat Transf. 103, 803–807 (1981)

    Article  Google Scholar 

  • Pop, I., Ingham, D.B.: Convective natural convection about a heated sphere in a porous medium. In: Proceedings of the 9th International Heat Transfer Conference, vol. 5-NC-15, pp. 567–572, Jerusalem, 19–24 Aug 1990

  • Rees, D.A.S., Pop, I.: Free convective stagnation-point flow in a porous medium using a thermal nonequilibrium model. Int. Commun. Heat Mass Transf. 26, 945–954 (1999)

    Article  Google Scholar 

  • Rees, D.A.S., Pop, I.: Vertical free convective boundary-layer flow in a porous medium using a thermal non-equilibrium model. J. Porous Media 3, 31–44 (2000)

    Google Scholar 

  • Rees, D.A.S., Bassom, A.P., Pop, I.: Forced convection past a heated cylinder in a porous medium using a thermal nonequilibrium model: boundary layer analysis. Eur. J. Mech. B/Fluids 22, 473–486 (2003)

    Article  Google Scholar 

  • Sano, T.: Unsteady forced and natural convection around a sphere immersed in a porous medium. J. Eng. Math. 30, 515–525 (1996)

    Article  Google Scholar 

  • Saeid, N.H.: Analysis of mixed convection in a vertical porous layer using non-equilibrium model. Int. J. Heat Mass Transf. 47, 5619–5627 (2004)

    Article  Google Scholar 

  • Saeid, N.H., Mohamad, A.A.: Periodic free convection from a vertical plate in a saturated porous medium, non-equilibrium model. Int. J. Heat Mass Transf. 48, 3855–3863 (2005)

    Article  Google Scholar 

  • Saeid, N.H.: Analysis of free convection about a horizontal cylinder in a porous media using a thermal non-equilibrium model. Int. Commun. Heat Mass Transf. 33, 158–165 (2006)

    Article  Google Scholar 

  • Salman Ahmed, N.J., Badruddin, I.A., Kanesan, J., Zainal, Z.A., Nazim Ahamed K.S.: Study of mixed convection in annular vertical cylinder filled with saturated porous medium, using thermal non-equilibrium model. Int. J. Heat Mass Transf. 54, 3822–3825 (2011)

    Google Scholar 

  • Shakeri, E., Nazari, M., Kayhani, M.H.: Free convection heat transfer over a vertical cylinder in a saturated porous medium using a local thermal non-equilibrium model. Transp. Porous Med. 93, 453–460 (2012)

    Google Scholar 

  • Saravanan, S.: Thermal non-equilibrium porous convection with heat generation and density maximum. Transp. Porous Med. 7(6), 35–43 (2009)

    Article  Google Scholar 

  • Wang, P., Kahawita, R.: Numerical integration of partial differential equations using cubic spline. Int. J. Comput. Math. 13, 271–286 (1983)

    Article  Google Scholar 

  • Yan, B., Pop, I., Ingham, D.B.: A numerical study of unsteady free convection from a sphere in a porous medium. Int. J. Heat Mass Transf. 40, 893–903 (1997)

    Article  Google Scholar 

  • Yih, K.A.: Viscous and Joule heating effects on non-Darcy MHD natural convection flow over a permeable sphere in porous media with internal heat generation. Int. Commun. Heat Mass Transf. 27, 591–600 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Southern Taiwan University of Science and Technology, Yungkang, 71005, Taiwan

    Ching-Yang Cheng

Authors
  1. Ching-Yang Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ching-Yang Cheng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, CY. Free Convection Heat Transfer From A Sphere In A Porous Medium Using A Thermal Non-equilibrium Model. Transp Porous Med 98, 209–221 (2013). https://doi.org/10.1007/s11242-013-0141-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-013-0141-2

Keywords

Search

Navigation