Skip to main content
SpringerLink
Log in

Analytical Study of Heat Flux Splitting in Micro-channels Filled with Porous Media

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

Abstract

The conditions of occurring heat flux splitting (bifurcation) phenomenon in a micro-channel filled with a porous medium including internal heat generations within both the solid and fluid phases under the local thermal non-equilibrium (LTNE) condition is analytically studied in the slip regime. The channel walls are subjected to a constant heat flux. Exact solutions for both the dimensionless temperatures of the two phases and the Nusselt number are obtained. Effects of the pertinent parameters such as heat generation parameter (\(\omega \)), the interphase heat transfer parameter (Vadasz in J Porous Media 15:249–258, 2012) or Biot number (\(\textit{Bi}\)), the fluid-to-solid effective conductivity ratio (k), and the temperature jump coefficient (\(\beta \)) on the dimensionless temperature profile (\(\theta \)) of the two phases as well as the Nusselt number are investigated. Moreover, the validity of one-equation model (the local thermal equilibrium assumption) is analyzed by comparing the Nusselt number obtained by one-equation model (LTE) with that obtained by the two-equation model (LTNE). Results reveal that the conditions at which the heat flux bifurcation (splitting) occurs in the slip regime is the same as those of the no-slip regime. In addition, a kind of heat flux bifurcation in which the solid and fluid phases have the same dimensionless temperature sign is observed in the slip regime, while it was not previously observe in the no-slip regime. It is discussed that the Nusselt number can increase or decrease with respect to \(\omega \) and may have either positive or negative values in both the no-slip and slip regimes. The presence of internal heat generation intensifies the role of \(\beta \) in the Nusselt number reduction. In addition, the accuracy of LTE model increases with increased \(\textit{Bi}\) and with decreased \(\beta \), while it is not a monotonic function of k in the presence of internal heat generation.

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

Similar content being viewed by others

Abbreviations

\(a_{\mathrm{sf}}\) :

Specific surface area \((\hbox {m}^{-1})\)

Bi :

The interphase heat transfer parameter or the Biot number (defined by Eq. 14)

\(c_\mathrm{p}\) :

Specific heat at constant pressure \((\hbox {J}\,\hbox {kg}^{-1}\,\hbox {K}^{-1})\)

\(D_\mathrm{h}\) :

Hydraulic diameter (m)

\(d_\mathrm{p}\) :

Particle diameter (m)

\(\hbox {Err}_{Nu}\) :

Error of Nusselt number obtained from LTE model (defined by Eq. 37)

H :

Half of the channel height (m)

\(h_\mathrm{sf}\) :

Fluid–solid heat transfer coefficient \((\hbox {W}\,\hbox {m}^{-2}\,\hbox {K}^{-1})\)

k :

Effective conductivity ratio (\(k_\mathrm{f,eff}/ k_\mathrm{s,eff})\)

\(k_\mathrm{f}\) :

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

\(k_\mathrm{f,eff}\) :

Effective thermal conductivity of fluid phase

\(k_\mathrm{m}\) :

Effective thermal conductivity of the medium (\(k_\mathrm{f,eff}+ k_\mathrm{s,eff})\)

\(k_\mathrm{s}\) :

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

\(k_\mathrm{s,eff }\) :

Effective thermal conductivity of solid phase

Kn :

Knudsen number, \(l/D_\mathrm{h}\) (used in Eq. 14)

l :

Molecular mean-free-path of the fluid phase (used in Eq. 9)

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

Nusselt number based on \(k_\mathrm{f,eff}\) (defined by Eq. 25)

Nu \(_\mathrm{m}\) :

Nusselt number based on \(k_\mathrm{m}\) (defined by Eq. 24)

Pr :

Prandtl number

q :

Internal heat transfer between the fluid and solid phases \((\hbox {W}\,\hbox {m}^{-2}\), defined by Eq. 35)

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

Heat flux at the wall (\(\hbox {W}\,\hbox {m}^{-2}\))

\(S_\mathrm{f}\) :

Internal heat generation within the fluid phase \((\hbox {W}\,\hbox {m}^{-3})\)

\(S_\mathrm{s}\) :

Internal heat generation within the solid phase \((\hbox {W}\,\hbox {m}^{-3})\)

T :

Temperature (K)

\(u^{*}\) :

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

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

Dimensional coordinates (m)

x, y :

Dimensionless coordinates

\(\beta \) :

Dimensionless temperature jump coefficient (defined in Eq. 17)

\(\beta ^{*}\) :

Temperature jump coefficient (m, defined in Eq. 9)

\(\theta \) :

Dimensionless temperature (defined in Eq. 11)

\(\lambda \) :

A parameter (used in Eqs. 22, 23)

\(\mu \) :

Fluid viscosity \((\hbox {kg}\,\hbox {m}^{-1}\,\hbox {s}^{-1}\), used in Eq. 6)

\(\rho \) :

Fluid density

\(\phi \) :

Porosity of the medium

\(\omega \) :

Heat generation parameter (defined by Eq. 14)

1–3:

Identifiers

f:

Fluid phase

LTE:

Local thermal equilibrium

m:

Mean value

s:

Solid phase

w:

Wall

References

  • 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 

  • Chen, X., Xia, X.L., Meng, X.L., Dong, X.H.: Thermal performance analysis on a volumetric solar receiver with double-layer ceramic foam. Energy Convers. Manag. 97, 282–289 (2015)

    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)

  • Dehghan, M., Mahmoudi, Y., Valipour, M.S., Saedodin, S.: Combined conduction–convection–radiation heat transfer of slip flow inside a micro-channel filled with a porous material. Transp. Porous Media 108, 413–436 (2015b)

  • 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., Saedodin, S., Valipour, M.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., 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 

  • 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 

  • 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 Transf. A 48, 693–710 (2005)

    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. Commun. 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 (2008)

    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.: Estimation of heat flux bifurcation at the heated boundary of a porous medium using a pore-scale numerical simulation. Int. J. Therm. Sci. 54, 109–118 (2012)

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

    Article  Google Scholar 

  • Jiang, P.X., Lu, X.C.: Numerical simulation and theoretical analysis of thermal boundary characteristics of convection heat transfer in porous media. Int. J. Heat Fluid Flow 28, 1144–1156 (2007)

    Article  Google Scholar 

  • Kaviany, M.: Principles of heat transfer in porous media. Springer, New York (1995)

    Book  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.: Thermal interaction at the interface between a porous medium and an impermeable wall. J. Heat Transf. 123, 527–533 (2001)

    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 

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

    Article  Google Scholar 

  • Lindner, F., Mundt, C., Pfitzner, M.: Fluid flow and heat transfer with phase change and local thermal non-equilibrium in vertical porous channels. Transp. Porous Media 106, 201–220 (2015)

    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 Transf. 70, 875–891 (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., 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 

  • Nield, D.A.: A note on local thermal non-equilibrium in porous media near boundaries and interfaces. Transp. Porous Media 95, 581–584 (2012)

    Article  Google Scholar 

  • Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, New York (2013)

    Book  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 

  • 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 

  • Torabi, M., Zhang, K., Yang, G., Wang, J., Wu, P.: Heat transfer and entropy generation analyses in a channel partially filled with porous media using local thermal non-equilibrium model. Energy 82, 922–938 (2015)

    Article  Google Scholar 

  • Vadasz, P.: Small Nield number convection in a porous layer heated from below via a constant heat flux and subject to lack of local equilibrium. J. Porous Media 15, 249–258 (2012)

    Article  Google Scholar 

  • Vafai, K., Yang, K.: A note on local thermal non-equilibrium in porous media and heat flux bifurcation phenomenon in porous media. Transp. Porous Media 96, 169–172 (2013)

    Article  Google Scholar 

  • White, F.M.: Viscous Fluid Flow, 3rd edn. McGraw-Hill, New York (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.: Analysis of heat flux bifurcation inside porous media incorporating inertial and dispersion effects—an exact solution. Int. J. Heat Mass Transf. 54, 5286–5297 (2011a)

    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 (2011b)

    Article  Google Scholar 

  • Yang, K., Vafai, K.: Transient aspects of heat flux bifurcation in porous media: an exact solution. J. Heat Transf. 133, 052602 (2011c)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Maziar Dehghan, Mohammad Sadegh Valipour & Seyfolah Saedodin

Authors
  1. Maziar Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Sadegh Valipour
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. 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., Valipour, M.S. & Saedodin, S. Analytical Study of Heat Flux Splitting in Micro-channels Filled with Porous Media. Transp Porous Med 109, 571–587 (2015). https://doi.org/10.1007/s11242-015-0536-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-015-0536-3

Keywords

Search

Navigation