Skip to main content
SpringerLink
Log in

Combined Conduction–Convection–Radiation Heat Transfer of Slip Flow Inside a Micro-Channel Filled with a Porous Material

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

Abstract

Combined conduction–convection–radiation heat transfer is investigated numerically in a micro-channel filled with a saturated cellular porous medium, with the channel walls held at a constant heat flux. Invoking the velocity slip and temperature jump, the thermal behaviour of the porous–fluid system are studied by considering hydrodynamically fully developed flow and applying the Darcy–Brinkman flow model. One energy equation model based on the local thermal equilibrium condition is adopted to evaluate the temperature field within the porous medium. Combined conduction and radiation heat transfer is treated as an effective conduction process with a temperature-dependent effective thermal conductivity. Results are reported in terms of the average Nusselt number and dimensionless temperature distribution, as a function of velocity slip coefficient, temperature jump coefficient, porous medium shape parameter and radiation parameters. Results show that increasing the radiation parameter \((T_{r})\) and the temperature jump coefficient flattens the dimensionless temperature profile. The Nusselt numbers are more sensitive to the variation in the temperature jump coefficient rather than to the velocity slip coefficient. Such that for high porous medium shape parameter, the Nusselt number is found to be independent of velocity slip. Furthermore, it is found that as the temperature jump coefficient increases, the Nusselt number decrease. In addition, for high temperature jump coefficients, the Nusselt number is found to be insensitive to the radiation parameters and porous medium shape parameter. It is also concluded that compared with the conventional macro-channels, wherein using a porous material enhances the rate of heat transfer (up to about 40 % compared to the clear channel), insertion of a porous material inside a micro-channel in slip regime does not effectively enhance the rate of heat transfer that is about 2 %.

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
Fig. 9

Similar content being viewed by others

Abbreviations

\(c_{\hbox {p}}\) :

Specific heat capacity at constant pressure \((\hbox {J Kg}^{-1}\,\hbox {k}^{-1})\)

\(c_{\hbox {v}}\) :

Specific heat capacity at constant volume \((\hbox {J Kg}^{-1}\,\hbox {k}^{-1})\)

\(D_{\hbox {h}}\) :

Hydraulic diameter (m)

Da :

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

ddy:

Grid-size expansion factor

\(F\) :

Momentum accommodation coefficient

\(F_{\hbox {T}}\) :

Thermal accommodation coefficient

\(f_{K}\) :

Friction factor \((=\!\frac{G\sqrt{K}}{\rho u*_m^2 })\)

\(G\) :

Negative of the applied pressure gradient in flow direction \((\hbox {Pa.m}^{-1})\)

H:

Half of the channel height (m)

\(K\) :

Permeability of the medium \((\hbox {m}^{2})\)

\(k\) :

Effective thermal conductivity of the porous medium \((\hbox {W m}^{-1}\ \hbox {K}^{-1})\)

\(k_{\hbox {c}}\) :

Molecular thermal conductivity \((\hbox {W m}^{-1}\ \hbox {K}^{-1})\)

\(k_{\hbox {f}}\) :

Thermal conductivity of the fluid phase \((\hbox {W m}^{-1}\ \hbox {K}^{-1})\)

\(k_{\hbox {r}}\) :

Radiative thermal conductivity \((\hbox {W m}^{-1}\ \hbox {K}^{-1})\)

\(k_{\hbox {s}}\) :

Thermal conductivity of the solid phase \((\hbox {W m}^{-1}\ \hbox {K}^{-1})\)

\(k_{0 }\) :

The effective thermal conductivity at the walls \((\hbox {W m}^{-1}\ \hbox {K}^{-1})\)

Kn :

Knudsen number \((=\!l/D_{h})\)

\(l\) :

Molecular mean-free-path (m)

\(M\) :

Viscosity ratio \((=\!\mu _{\hbox {eff}}/\mu )\)

\(n\) :

Number of iterations

Nu :

Nusselt number

Pe :

Peclet number

\(q_{w}^{\prime \prime }\) :

Heat flux at the channel walls \((\hbox {W m}^{-2})\)

\(Re_{K}\) :

Modified Reynolds number \((=\!\rho u^{*}\sqrt{K}/\mu )\)

\(s\) :

Porous media shape parameter \((=\!\frac{1}{\sqrt{DaM}})\)

\(T\) :

Temperature (K)

\(T_{m }\) :

Bulk mean temperature (K)

\(T_{r}\) :

Temperature variation parameter (Eq. 22)

\(T_{w}\) :

Channel wall temperature (K)

\(u\) :

Dimensionless velocity \((=\!\frac{\mu u^{*}}{GH^{2}})\)

\(u^{*}\) :

Velocity \((\hbox {m s}^{-1})\)

:

Normalized velocity \((=\!u/u_m =u^{*}/u_m^{*} )\)

\(u_{m}^{*}\) :

Mean velocity \((\hbox {m s}^{-1})\)

\(x^{*},y^{*}\) :

Dimensional coordinates (m)

\(y\) :

Dimensionless \(y^{*}\) coordinate

\(\alpha ^{*}\) :

Slip coefficient (m)

\(\alpha \) :

Dimensionless slip coefficient \((=\!\frac{\alpha ^{*}}{H})\)

\(\beta ^{*}\) :

Jump coefficient (m)

\(\beta \) :

Dimensionless jump coefficient \((=\!\frac{\beta ^{*}}{H})\)

\(\beta _{R}\) :

Rosseland mean extinction coefficient \((\hbox {m}^{-1})\)

\(\gamma \) :

The specific-heat ratio \((=\!c_p /c_v )\)

\(\theta \) :

Dimensionless temperature \((=\!\frac{T-T_w }{T_m -T_w })\)

\(\lambda \) :

Radiation parameter (Eq. 18)

\(\mu \) :

Fluid viscosity \((\hbox {Kgm}^{-1}\hbox {s}^{-1})\)

\(\mu _{\hbox {eff}}\) :

Effective viscosity in the Brinkman term \((=\!\mu /\phi , \hbox { Kgm}^{-1}\hbox {s}^{-1})\)

\(\nu \) :

An arbitrary dependent parameter used in Eq. (25)

\(\sigma \) :

Stefan-Boltzmann coefficient \((\hbox {Wm}^{-2}\hbox {K}^{-4})\)

\(\rho \) :

Fluid density \((\hbox {kgm}^{-3})\)

\(\phi \) :

Porosity of the porous medium

eff :

Effective

\(f\) :

Fluid phase

\(i\) :

Index

\(m\) :

Mean

\(s\) :

Solid phase

\(w\) :

Wall

References

  • Alazmi, B., Vafai, K.: Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions. Int. J. Heat Mass Transfer 45, 3071–3087 (2002)

    Article  Google Scholar 

  • Al-Nimr, M.A., Haddad, O.M.: Comments on ‘forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas’. Transp. Porous Media 64, 161–170 (2006)

    Article  Google Scholar 

  • Al-Nimr, M.A., Haddad, O.M.: Comments on ‘forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas’. Transp. Porous Media 67, 165–167 (2007)

    Article  Google Scholar 

  • Andreozzi, A., Bianco, N., Manca, O., Naso, V.: Numerical analysis of radiation effects in a metallic foam by means of the radiative conductivity model. Appl. Therm. Eng. 49, 14–21 (2012)

    Article  Google Scholar 

  • Badran, B., Albayyari, J.M., Gerner, F.M., Ramadas, P., Henderson, H.T., Baker, K.W.: Liquid-metal micro heat pipes. ASME HTD 236, 71–85 (1993)

    Google Scholar 

  • Bejan, A.: Convection Heat Transfer. Wiley, NY (2004)

    Google Scholar 

  • Bovand, M., Rashidi, S., Dehghan, M., Esfahani, J.A., Valipour, M.S.: Control of wake and vortex shedding behind a porous bluff-body by exerting an external magnetic field. J. Magn. Magn. Mater. (2015). doi:10.1016/j.jmmm.2015.03.012

  • Buonomo, B., Manca, O., Lauriat, G.: Forced convection in micro-channels filled with porous media in local thermal non-equilibrium conditions. Int. J. Therm. Sci. 77, 206–222 (2014)

    Article  Google Scholar 

  • Dehghan, M., Basirat Tabrizi, H.: On near-wall behavior of particles in a dilute turbulent gas-solid flow using kinetic theory of granular flows. Powder Technol. 224, 273–280 (2012)

    Article  Google Scholar 

  • Dehghan, M., Basirat Tabrizi, H.: Turbulence effects on the granular model of particle motion in a boundary layer flow. Can. J. Chem. Eng. 92, 189–195 (2014)

    Article  Google Scholar 

  • Dehghan, M., Mirzaei, M., Mohammadzadeh, A.: Numerical formulation and simulation of a non-Newtonian magnetic fluid flow in the boundary layer of a stretching sheet. J. Model. Eng. 11(34), 73–82 (2013)

    Google Scholar 

  • Dehghan, M., Jamal-Abad, M.T., Rashidi, S.: Analytical interpretation of the local thermal non-equilibrium condition of porous media imbedded in tube heat exchangers. Energy Convers. Manag. 85, 264–271 (2014a)

    Article  Google Scholar 

  • 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 Media 102(2), 139–152 (2014b)

    Article  Google Scholar 

  • Dehghan, M., Daneshipour, M., Valipour, M.S., Rafee, R., Saedodin, S.: Enhancing heat transfer in microchannel heat sinks using converging flow passages. Energy Convers. Manag. 92, 244–250 (2015a)

    Article  Google Scholar 

  • Dehghan, M., Rahmani, Y., Ganji, D.D., Saedodin, S., Valipour, M.S., Rashidi, S.: Convection-radiation heat transfer in solar heat exchangers filled with a porous medium: homotopy perturbation method versus numerical analysis. Renew. Energy 74, 448–455 (2015b)

    Article  Google Scholar 

  • Dehghan, M., Valipour, M.S., Saedodin, S.: Temperature-dependent conductivity in forced convection of heat exchangers filled with porous media: A perturbation solution. Energy Convers. Manag. 91, 259–266 (2015c)

    Article  Google Scholar 

  • Deng, B., Qiu, Y., Kim, C.N.: An improved porous medium model for microchannel heat sinks. Appl. Therm. Eng. 30, 2512–2517 (2010)

    Article  Google Scholar 

  • Duncan, A.B., Peterson, G.P.: Review of microscale heat transfer. ASME Appl. Mech. Rev. 47, 397–428 (1994)

    Article  Google Scholar 

  • Haddad, O.M., Abu-Zaid, M., Al-Nimr, M.A.: Developing free convection gas flow in a vertical open-ended micro-channel filled with porous media. Numer. Heat Transfer A 48, 693–710 (2005)

    Article  Google Scholar 

  • Haddad, O.M., Al-Nimr, M.A., Al-Omary, JSh: Forced convection of gaseous slip flow in porous microchannels under local thermal non-equilibrium conditions. Transp. Porous Media 67, 453–471 (2007)

    Article  Google Scholar 

  • Haddad, O.M., Al-Nimr, M.A., Taamneh, Y.: Hydrodynamic and thermal behavior of gas flow in microchannels filled with porous media. J. Porous Media 9, 403–414 (2006)

    Article  Google Scholar 

  • Harley, J.C., Huang, H., Bau, H.H., Zemel, J.N.: Gas flow in microchannels. J. Fluid Mech. 284, 257–274 (1995)

    Article  Google Scholar 

  • Hashemi, S.M.H., Fazeli, S.A., Shokouhmand, H.: Fully developed non-Darcian forced convection slip-flow in a micro-annulus filled with a porous medium: analytical solution. Energy Convers. Manag. 52, 1054–1060 (2011)

    Article  Google Scholar 

  • Hooman, K.: Entropy generation for microscale forced convection: effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry. Int. J. Heat Mass Transf. 34, 945–957 (2007)

    Article  Google Scholar 

  • Hooman, K.: Heat transfer and entropy generation for forced convection through a microduct of rectangular cross-section: effects of velocity slip, temperature jump, and duct geometry. Int. Commun. Heat Mass Transf. 35, 1065–1068 (2008a)

    Article  Google Scholar 

  • Hooman, K.: Heat and fluid flow in a rectangular microchannel filled with a porous medium. Int. J. Heat Mass Transfer 51, 5804–5810 (2008b)

    Article  Google Scholar 

  • Hooman, K.: A perturbation solution for forced convection in a porous-saturated duct. J. Comput. Appl. Math. 211, 57–66 (2008c)

    Article  Google Scholar 

  • Hooman, K.: Slip flow forced convection in a microporous duct of rectangular cross-section. Appl. Therm. Eng. 29, 1012–1019 (2009)

    Article  Google Scholar 

  • Hooman, K., Ejlali, A.: Effects of viscous heating, fluid property variation, velocity slip, and temperature jump on convection through parallel plate and circular microchannels. Int. Commun. Heat Mass Transf. 37, 34–38 (2010)

    Article  Google Scholar 

  • Imani, G.R., Maerefat, M., Hooman, K.: Pore-scale numerical experiment on the effect of the pertinent parameters on heat flux splitting at the boundary of a porous medium. Transp. Porous Media 98, 631–649 (2013)

    Article  Google Scholar 

  • Jennings, S.G.: The mean free path in air. J. Aerosol Sci. 19, 159–166 (1988)

    Article  Google Scholar 

  • Jiang, P.X., Ren, Z.P., Wang, B.X.: Numerical simulation of forced convection heat transfer in porous plate channels using thermal equilibrium and nonthermal equilibrium models. Numer. Heat Transf. Part A Appl. 35, 99–113 (1999)

    Article  Google Scholar 

  • Jiang, P.X., Xu, R.N., Gong, W.: Particle-to-fluid heat transfer coefficients in miniporous media. Chem. Eng. Sci. 61, 7213–7222 (2006)

    Article  Google Scholar 

  • Karimi, N., Mahmoudi, Y., Mazaheri, K.: Temperature fields in a channel partially-filled with a porous material under local thermal non-equilibrium condition-an exact solution. J. Mech. Eng. Sci. Part C. (2014). doi:10.1177/0954406214521800

  • Khaled, A.R.A., Vafai, K.: Cooling augmentation using microchannels with rotatable separating plates. Int. J. Heat Mass Transf. 54, 3732–3739 (2011)

    Article  Google Scholar 

  • Kim, S.J.: Methods for thermal optimization of microchannel heat sinks. Heat Transf. Eng. 25, 37–49 (2004)

    Article  Google Scholar 

  • Kim, S.J., Kim, D.: Forced convection in microstructures for electronic equipment cooling. J. Heat Transf. 121, 639–645 (1999)

    Article  Google Scholar 

  • Kuznetsov, A.V., Nield, D.A.: Thermally developing forced convection in a porous medium occupied by a rarefied gas: parallel plate channel or circular tube with walls at constant heat flux. Transp. Porous Media 76, 345–362 (2009)

    Article  Google Scholar 

  • Koh, J.C.Y., Colony, R.: Heat transfer of microstructure for integrated circuits. Int. Commun. Heat Mass Transf. 13, 89–98 (1986)

    Article  Google Scholar 

  • Kockmann, N.: Transport Phenomena in Micro Process Engineering. Springer, Berlin (2008)

    Google Scholar 

  • Lee, D.Y., Vafai, K.: Analytical characterization and conceptual assessment of solid and fluid temperature differentials in porous media. Int. J. Heat Mass Transfer 31, 423–435 (1999a)

    Article  Google Scholar 

  • Lee, D.Y., Vafai, K.: Comparative analysis of jet impingement and microchannel cooling for high heat flux applications. Int. J. Heat Mass Transf. 42, 1555–1568 (1999b)

    Article  Google Scholar 

  • Leroy, V., Goyeau, B., Taine, J.: Coupled upscaling approaches for conduction, convection, and radiation in porous media: theoretical developments. Transp. Porous Media 98, 323–347 (2013)

    Article  Google Scholar 

  • Lin, L., Pisano, A.P.: Bubble formation on a micro line heater. ASME DSC 32, 147–163 (1991)

    Google Scholar 

  • Mahmoudi, Y.: Effect of thermal radiation on temperature differential in a porous medium under local thermal non-equilibrium condition. Int. J. Heat Mass Transfer 76, 105–121 (2014)

    Article  Google Scholar 

  • Mahmoudi, Y.: Constant wall heat flux boundary condition in micro-channels filled with a porous medium with internal heat generation under local thermal non-equilibrium condition. Int. J. Heat Mass Transf. 85, 524–542 (2015)

    Article  Google Scholar 

  • Mahmoudi, Y., Karimi, N.: Numerical investigation of heat transfer enhancement in a pipe partially filled with a porous material under local thermal non-equilibrium condition. Int. J. Heat Mass Transfer 68, 161–173 (2014)

    Article  Google Scholar 

  • Mahmoudi, Y., Karimi, N., Mazaheri, K.: Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition: effects of different thermal boundary conditions at the porous-fluid interface. Int. J. Heat Mass Transfer 70, 875–891 (2014)

    Article  Google 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. Therm. Sci. 50(12), 2386–2401 (2011)

    Article  Google 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)

    Article  Google Scholar 

  • Mohammad, A.A.: Heat transfer enhancement in heat exchangers fitted with porous media. Part I: constant wall temperature. Int. J. Therm. Sci. 42, 385–395 (2003)

    Article  Google Scholar 

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

    Google Scholar 

  • Nield, D.A., Kuznetsov, A.V.: Forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas. Transp. Porous Media 64, 161–170 (2006)

    Article  Google Scholar 

  • Nield, D.A., Kuznetsov, A.V.: Reply to comments on ‘Forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas’. Transp. Porous Media 67, 169–170 (2007)

    Article  Google Scholar 

  • Nield, D.A., Kuznetsov, A.V.: Forced convection in cellular porous materials: effect of temperature-dependent conductivity arising from radiative transfer. Int. J. Heat Mass Transf. 53, 2680–2684 (2010)

    Article  Google Scholar 

  • Ouyang, X., Vafai, K., Jiang, P.: Analysis of thermally developing flow in porous media under local thermal non-equilibrium conditions. Int. J. Heat Mass Transf. 67, 768–775 (2013)

    Article  Google Scholar 

  • Rad, P.M., Aghanajafi, C.: The effect of thermal radiation on nanofluid cooled microchannels. J. Fusion Energy 28, 91–100 (2009)

    Article  Google Scholar 

  • Rashidi, S., Bovand, M., Pop, I., Valipour, M.S.: Numerical simulation of forced convective heat transfer past a square diamond-shaped porous cylinder. Transp. Porous Media 102(2), 207–225 (2014)

    Article  Google Scholar 

  • Rashidi, S., Dehghan, M., Ellahi, R., Riaz, M., Jamal-Abad, M.T.: Study of streamwise and transverse magnetic fields on fluid flow and heat transfer around an obstacle embedded in a porous medium. J. Magn. Magn. Mater. 378, 128–137 (2015)

    Article  Google Scholar 

  • Rosseland, S.: Theoretical Astrophysics: Atomic Theory and the Analysis of Stellar Atmosphere and Envelopes. Clarendon press, Oxford (1936)

    Google Scholar 

  • Siegel, R., Howell, J.R.: Thermal Radiation Heat Transfer. Taylor and Francis, London (1993)

    Google Scholar 

  • Shokouhmand, H., MeghdadiIsfahani, A.H., Shirani, E.: Friction and heat transfer coefficient in micro and nano channels filled with porous media for wide range of Knudsen number. Int. Commun. Heat Mass Transf. 37, 890–894 (2010)

    Article  Google Scholar 

  • Tseng, C.C., Sikorski, R.L., Viskanta, R., Chen, M.Y.: Effect of radiation on heat transfer in open-cell foams at high temperature. Proceedings of the ASME 2011 International Mechanical Engineering Congress & Exposition (IMECE2011), November 11–17, Denver, Colorado, USA (2011)

  • Tso, C.P., Mahulikar, S.P.: Combined evaporating meniscus-driven convection and radiation in annular microchannels for electronics cooling application. Int. J. Heat Mass Transf. 43, 1007–1023 (2000)

    Article  Google Scholar 

  • Tunc, G., Bayazitoglu, Y.: Heat transfer in rectangular microchannels. Int. J. Heat Mass Transf. 45, 765–773 (2002)

    Article  Google Scholar 

  • Vafai, K., Khaled, A.R.A.: Analysis of flexible microchannel heat sink systems. Int. J. Heat Mass Transf. 48, 1739–1746 (2005)

    Article  Google Scholar 

  • Vafai, K., Zhu, L.: Analysis of two-layered micro-channel heat sink concept in electronic cooling. Int. J. Heat Mass Transf. 42, 2287–2297 (1999)

    Article  Google Scholar 

  • Valipour, M.S., Masoodi, R., Rashidi, S., Bovand, M., Mirhosseini, M.: A numerical study of convection around a square porous cylinder using Al2O3-H2O nanofluid. Therm. Sci. 18(4), 1305–1314 (2014)

    Article  Google Scholar 

  • Viskanta, R.: Overview of radiative transfer in cellular porous materials. Proceedings of ASME 2009 heat transfer summer conference, San Francisco, CA, July 19–23 (2009)

  • Wang, F., Tan, J., Yong, S., Tan, H., Chu, S.: Thermal performance analyses of porous media solar receiver with different irradiative transfer models. Int. J. Heat Mass Transf. 78, 7–16 (2014)

    Article  Google Scholar 

  • White, F.M.: Viscous Fluid Flow., 3rd edn. McGraw-Hill, NY (2006)

    Google Scholar 

  • Yang, K., Vafai, K.: Analysis of temperature gradient bifurcation in porous media: an exact solution. Int. J. Heat Mass Transf. 53, 4316–4325 (2010)

    Article  Google Scholar 

  • Yang, K., Vafai, K.: Restrictions on the validity of the thermal conditions at the porous-fluid interface: an exact solution. J. Heat Transf. 133, 112601-1–112601-12 (2011a)

    Google Scholar 

  • Yang, K., Vafai, K.: Analysis of heat flux bifurcation inside porous media incorporating inertial and dispersion effects - an exact solution. Int. J. Heat Mass Transf. 54, 5286–5297 (2011b)

    Article  Google Scholar 

  • Zhao, C.Y., Lu, T.J., Hodson, H.P.: Thermal radiation in metal foams with open cells. Int. J. Heat Mass Transf. 47, 2927–2939 (2004)

    Article  Google Scholar 

  • Zhao, C.Y., Tassou, S.A., Lu, T.J.: Analytical considerations of thermal radiation in cellular metal foams with open cells. Int. J. Heat Mass Transf. 51, 929–940 (2008)

    Article  Google Scholar 

Download references

Acknowledgments

The first author is grateful to Prof. Hassan Basirat Tabrizi from Amirkabir University of Technology for his constructive helps during the preparation of this work.

Author information

Authors and Affiliations

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

    Maziar Dehghan, Mohammad Sadegh Valipour & Seyfolah Saedodin

  2. Department of Engineering, University of Cambridge, Cambridge, UK

    Yasser Mahmoudi

Authors
  1. Maziar Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yasser Mahmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammad Sadegh Valipour
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Seyfolah Saedodin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maziar Dehghan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dehghan, M., Mahmoudi, Y., Valipour, M.S. et al. Combined Conduction–Convection–Radiation Heat Transfer of Slip Flow Inside a Micro-Channel Filled with a Porous Material. Transp Porous Med 108, 413–436 (2015). https://doi.org/10.1007/s11242-015-0483-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-015-0483-z

Keywords

Search

Navigation