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

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 %.

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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

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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.

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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

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Keywords

  • Radiative heat transfer
  • Cellular porous medium
  • Micro-channel
  • Slip regime
  • Temperature jump