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Forced convection gaseous slip flow in circular porous micro-channels

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Abstract

Laminar forced convection of gaseous slip flow in a circular micro-channel filled with porous media under local thermal equilibrium condition is studied numerically using the finite difference technique. Hydrodynamically fully developed flow is considered and the Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous domain. The present study reports the effect of several operating parameters (Knudsen number (Kn), Darcy number (Da), Forchhiemer number (Γ), and modified Reynolds number \((Re_D^\ast)\)) on the velocity slip and temperature jump at the wall. Results are given in terms of the velocity distribution, temperature distribution, skin friction \((C_{\rm f} Re_D^\ast)\), and the Nusselt number (Nu). It is found that the skin friction is increased by (1) decreasing Knudsen number, (2) increasing Darcy number, and (3) decreasing Forchheimer number. Heat transfer is found to (1) decrease as the Knudsen number, or Forchheimer number increase, (2) increase as the Peclet number or Darcy number increase.

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Abbreviations

C:

Coefficient in the Forchheimer term

C f :

Skin friction coefficient

C p :

Constant pressure specific heat

C v :

Constant volume specific heat

D :

Diameter of the circular channel

Da :

Darcy number (\(K/\varepsilon\,r_0 ^2)\)

h :

Local heat transfer coefficient

K :

Intrinsic permeability of the porous medium

Kn :

Knudsen number (λ/r 0)

k :

Thermal conductivity

k m :

Overall thermal conductivity ((1−ɛ)k s + ɛ k f)

p :

Pressure

Pe :

Peclet number (Re * D Pr)

Pr :

Prandtl number (μ C p /k m)

\(Re_D^{\ast}\) :

Modified Reynolds number in porous media (ρf u 0 D/μ ɛ)

R :

Non-dimensional transverse coordinate (r/r 0)

r :

Transverse (radial) coordinate

r 0 :

Channel radius

T :

Temperature of the fluid

T m :

Mixing cup temperature

T w :

Temperature of the wall

T i :

Temperature at the inlet of the channel

t :

Time

t 0 :

Reference time (\(\rho\,r_0 ^2/\mu )\)

U :

Non-dimensional axial velocity (u/u 0)

u :

Axial velocity

u 0 :

Reference axial velocity \((\varepsilon\,r_0 ^2/\mu (-\hbox{d}p/\hbox{d}z))\)

z :

Axial coordinate

Z :

Non-dimensional axial coordinate (z/r 0)

λ:

Mean free path of the gas molecules

σ v :

Tangential momentum accommodation coefficient

σ T :

Thermal accommodation coefficient

γ:

Specific heat ratio (C p /C v )

Γ:

Non-dimensional Forchheimer coefficient (\(\rho_{\rm f}\,C\,\varepsilon ^2\,\left({-\hbox{d}p/\hbox{d}z}\right)\,r_0 ^4/\mu ^2\,\sqrt K)\)

ɛ:

Porosity of the porous medium

μ:

Dynamic viscosity

ρ:

Density

θ:

Non-dimensional temperature (T−T i/T w−T i)

θm :

Non-dimensional mixing cup temperature (T m−T i/T w−T i)

τ:

Non-dimensional time (t/t 0)

τw :

Shear stress at the wall (−μ (∂u/∂r)w)

f:

Fluid

m:

Mean value for the fluid

s:

Solid

w:

Wall condition

References

  • Araki T., Kim M., Iwai H. and Suzuki K. (2002). An experimental investigation of gaseous flow characteristics in microchannels. Microscale Thermophys. Eng. 6: 117–130

    Article  Google Scholar 

  • Arkilic E.B., Breuer K.S. and Schmidt M.A. (1994). Gaseous flow in microchannels, application of microfabrication to fluid mechanics. ASME FED 197: 57–66

    Google Scholar 

  • Burmeister L.C. (1993). Convective Heat Transfer. John Wiley & Sons, New York

    Google Scholar 

  • Duncan G.P. and Peterson G.P. (1994). Review of microscale heat transfer. Appl. Mech. Rev. 47(9): 397–428

    Article  Google Scholar 

  • Eckert E.G.R., Drake R.M. (1972) Analysis of Heat and Mass Transfer. McGraw-Hill, New York, pp. 467–486

    Google Scholar 

  • Gad-El-Hak M. (1999). The fluid mechanics of microdevices—the freeman scholar lecture. J. Fluids Eng. 121: 5–33

    Google Scholar 

  • Haddad O.M., Al-Nimr M.A. and Abu-Zaid M. (2004). Entropy generation due to laminar forced convection fluid flow through microchannel. Entropy J 6(5): 413–426

    Google Scholar 

  • Haddad O.M., Al-Nimr M.A. and AbuZaid M.M. (2005a). The effect of frequency of fluctuating driving force on basic gaseous micro-flows. Acta Mech 179: 249–259

    Article  Google Scholar 

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

    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(5), 403–414 (2006a)

  • Haddad, O.M., Al-Nimr, M.A., AbuZaid, M.M.: Effect of periodically oscillating driving force on basic microflows in porous media. J. Porous Media 9(7), 695–707 (2006b)

  • Haddad, O.M., Al-Nimr, M.A., Al-Omary, J.Sh.: Forced convection of gaseous slip flow in porous micro-channels under local thermal non-equilibrium conditions. Transp. Porous Media, published online doi:10.1007/s11242-006-9036-9 (2006c)

  • Hoffman J.D. (1992). Numerical Methods for Engineers and Scientists. McGraw Hill, New York

    Google Scholar 

  • Karniadakis G. and Beskok A. (2002). Micro Flows Fundamentals and Simulation. Springer-Verlag, New York

    Google Scholar 

  • Li W., Lin J., Lee S. and Chen M. (2002). Effect of roughness on rarefied gas flow in long microtubes. J. Micromech. Microeng. 12: 149–156

    Article  Google Scholar 

  • Liu, J., Tai, Y., Ho, C.: MEMS for pressure distribution studies of gaseous flows in micro-channels. Proc. IEEE Micro-Electro-Mechanical Syst. 209–215 (1995)

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

    Google Scholar 

  • Nield D. and Kuznetsov A. (2006). 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

    Article  Google Scholar 

  • Obot N.T. (2002). Toward a better understanding of friction and heat transfer in microchannels – Literature review. Microscale Thermophys. Eng. 6: 155–173

    Article  Google Scholar 

  • Qu W., Mala Gh. and Li D. (2000). Heat transfer for water flow in trapezoidal silicon microchannels. Int. J. Heat Mass Transfer 43: 3925–3936

    Article  Google Scholar 

  • Tang, G.H., Tao, W.Q., He, Y.L.: Gas slippage effect on microscale porous flow using the lattice Boltzmann method. Phys. Rev. E 72(5), No. 56301 (2005)

    Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mechanical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbed, 22110, Jordan

    O. M. Haddad, M. A. Al-Nimr & M. S. Sari

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  1. O. M. Haddad
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  2. M. A. Al-Nimr
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  3. M. S. Sari
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Correspondence to O. M. Haddad.

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Haddad, O.M., Al-Nimr, M.A. & Sari, M.S. Forced convection gaseous slip flow in circular porous micro-channels. Transp Porous Med 70, 167–179 (2007). https://doi.org/10.1007/s11242-006-9093-0

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  • DOI: https://doi.org/10.1007/s11242-006-9093-0

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