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

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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.

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Abbreviations

Bi:

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

c f :

Coefficient in the Forchheimer term

Cf :

Skin friction coefficient

C p :

Constant pressure specific heat

C v :

Constant volume specific heat

D :

Pore diameter

Da :

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

h :

Local heat transfer coefficient

h sf :

Interstitial heat transfer coefficient

k :

Thermal conductivity

K :

Intrinsic permeability of the porous medium

Kn :

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

K R :

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)

q w :

Heat transfer rate from the plate wall

Re * :

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

t :

Time

t 0 :

Reference time(ρ L 2/μ)

T :

Temperature

u :

Axial velocity

u 0 :

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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Authors and Affiliations

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

    O. M. Haddad, M. A. Al-Nimr & J. Sh. Al-Omary

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  1. O. M. Haddad
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  2. M. A. Al-Nimr
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  3. J. Sh. Al-Omary
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Haddad, O.M., Al-Nimr, M.A. & Al-Omary, J.S. Forced convection of gaseous slip-flow in porous micro-channels under Local Thermal Non-Equilibrium conditions. Transp Porous Med 67, 453–471 (2007). https://doi.org/10.1007/s11242-006-9036-9

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