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Transport in Porous Media

, Volume 67, Issue 3, pp 453–471 | Cite as

Forced convection of gaseous slip-flow in porous micro-channels under Local Thermal Non-Equilibrium conditions

  • O. M. HaddadEmail author
  • M. A. Al-Nimr
  • J. Sh. Al-Omary
Original Paper

Abstract

Steady laminar forced convection gaseous slip-flow through parallel-plates micro-channel filled with porous medium under Local Thermal Non-Equilibrium (LTNE) condition is studied numerically. We consider incompressible Newtonian gas flow, which is hydrodynamically fully developed while thermally is developing. The Darcy–Brinkman–Forchheimer model embedded in the Navier–Stokes equations is used to model the flow within the porous domain. The present study reports the effect of several operating parameters on velocity slip and temperature jump at the wall. Mainly, the current study demonstrates the effects of: Knudsen number (Kn), Darcy number (Da), Forchheimer number (Γ), Peclet number (Pe), Biot number (Bi), and effective thermal conductivity ratio (K R) on velocity slip and temperature jump at the wall. Results are given in terms of skin friction (C f Re *) and Nusselt number (Nu). It is found that the skin friction: (1) increases as Darcy number increases; (2) decreases as Forchheimer number or Knudsen number increases. Heat transfer is found to (1) decreases as the Knudsen number, Forchheimer number, or K R increases; (2) increases as the Peclet number, Darcy number, or Biot number increases.

Keywords

Forced convection Slip-flow Porous media Micro-channel Graetz problem Local thermal non-equilibrium 

Nomenclature

Bi

Biot number \(({h_{\rm sf} L^2}\mathord{\left/ {\vphantom {{h_{\rm sf} L^2} {\varepsilon k_{\rm f}}}} \right. \kern-\nulldelimiterspace} {\varepsilon k_{\rm f}})\)

cf

Coefficient in the Forchheimer term

Cf

Skin friction coefficient

Cp

Constant pressure specific heat

Cv

Constant volume specific heat

D

Pore diameter

Da

Darcy number \((K/\varepsilon L^2)\)

h

Local heat transfer coefficient

hsf

Interstitial heat transfer coefficient

k

Thermal conductivity

K

Intrinsic permeability of the porous medium

Kn

Modified Knudsen number \(\left(\frac{\lambda}{D}\,\frac{D}{L}\right)\)

KR

Effective thermal conductivity ratio \((\varepsilon {k_{\rm f}} \mathord{\left/ {\vphantom {{k_{\rm f}}{(1-\varepsilon)k_{\rm s}}}}\right. \kern-\nulldelimiterspace} {(1-\varepsilon)k_{\rm s}})\)

L

Half channel width

Nu

Nusselt number \((hL/\varepsilon k_{\rm f})\)

p

Pressure

Pe

Peclet number \(({u_{\rm o} L} \mathord{\left/ {\vphantom {{u_o L} {\varepsilon \alpha}}} \right. \kern-\nulldelimiterspace} {\varepsilon \alpha)}\)

Pr

Prandtl number (μ /α ρf)

qw

Heat transfer rate from the plate wall

Re*

Modified Reynolds number in porous media \((\rho_{\rm f} u_o L/\mu \varepsilon)\)

t

Time

t0

Reference time(ρ L 2/μ)

T

Temperature

u

Axial velocity

u0

Reference axial velocity \((\varepsilon L^2/\mu (-{\rm d}p/{\rm d}x))\)

U

Non-dimensional axial velocity (u/u o)

x

Axial coordinate

X

Dimensionless axial coordinate (x/L)

y

Transverse coordinate

Y

Dimensionless transverse coordinate (y/L)

Greek symbols

 

α

Thermal diffusivity

γ

Specific heat ratio (C p /C ν)

Γ

Dimensionless coefficient of Forchheimer \((\rho_{\rm f} c_{\rm f} \varepsilon^2(-{\rm d}p/{\rm d}x)L^4/\mu^2\sqrt k)\)

λ

Mean free path of the gas molecules

\(\varepsilon \)

Porosity of the porous medium

μ

Dynamic viscosity

ρf

Fluid density

σT

Thermal accommodation coefficient

σv

Tangential momentum accommodation coefficient

θ

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

τ

Non-dimensional time (t/t o)

τw

Shear stress at the wall \((-\mu (\partial u/\partial y)\left.)\right|_{\rm w} \)

Subscripts

 

f

Fluid

mf

Mean value for the fluid

s

Solid

w

Wall

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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • O. M. Haddad
    • 1
    Email author
  • M. A. Al-Nimr
    • 1
  • J. Sh. Al-Omary
    • 1
  1. 1.Department of Mechanical EngineeringJordan University of Science and TechnologyIrbidJordan

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