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A Local Thermal Non-Equilibrium Analysis of Fully Developed Forced Convective Flow in a Tube Filled with a Porous Medium

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A local thermal non-equilibrium model has been considered for the case of thermally fully developed flow within a constant heat flux tube filled with a porous medium. Exact temperature profiles for the fluid and solid phases are found after combining the two individual energy equations and then transforming them into a single ordinary differential equation with respect to the temperature difference between the solid phase and the wall subject to constant heat flux. The exact solutions for the case of metal-foam and air combination reveal that the local thermal equilibrium assumption may fail for the case of constant heat flux wall. The Nusselt number is presented as a function of the Peclet number, which shows a significant increase due to both high stagnant thermal conductivity and thermal dispersion resulting from the presence of the metal-foam.

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A :

Surface area (m2)

A int :

Interface between the fluid and solid (m2)

c :

Specific heat (J/kg K)

c p :

Specific heat at constant pressure (J/kg K)

D :

Tube diameter

d m :

Mean pore diameter (m)

h v :

Volumetric heat transfer coefficient (W/m3 K)

k :

Thermal conductivity (W/m K)

K :

Permeability (m2)

n j :

Unit vector pointing outward from the fluid side to solid side (–)

Pr :

Prandtl number (–)

q :

Heat flux (W/m2)

r :

Radial coordinate

T :

Temperature (K)

u D :

Darcian velocity (uniform inlet velocity) (m/s)

u i :

Velocity vector (m/s)

V :

Representative elementary volume (m3)

x i :

Cartesian coordinates (m)

x, y, z :

Cartesian coordinates (m)

ε :

Porosity (–)


Effective porosity (–)

ν :

Kinematic viscosity (m2/s)

ρ :

Density (kg/m2)

\({\tilde {\phi }}\) :

Deviation from intrinsic average

\({\left\langle \phi \right\rangle}\) :

Darician average

\({\left\langle \phi \right\rangle^{\rm f,S}}\) :

Intrinsic average












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Correspondence to A. Nakayama.

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Yang, C., Ando, K. & Nakayama, A. A Local Thermal Non-Equilibrium Analysis of Fully Developed Forced Convective Flow in a Tube Filled with a Porous Medium. Transp Porous Med 89, 237 (2011). https://doi.org/10.1007/s11242-011-9766-1

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  • Porous media
  • Thermal non-equilibrium
  • Aluminum
  • Metal foam
  • Conductivity