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Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

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Abstract

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.

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Abbreviations

A fs :

solid /fluid interfacial area

\(\vec{b}\) :

function vector

c p :

specific heat capacity at constant pressure

d :

solid particle diameter

D :

total effective thermal diffusivity tensor

Da :

Darcy number, K/l 2

k :

thermal conductivity tensor

K :

permeability

l :

length of unit cell

\(\bar{\bar{I}}\) :

identity matrix

\(\vec{n}_{\rm fs}\) :

outward unit normal vector from the fluid phase

Pe l :

Peclet number, 〈u〉l/αf

P :

local pressure

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P}\) :

periodic pressure

ΔP :

macroscopic pressure drop across a unit cell

Pr :

Prandtl number, ν/α

Re l :

Reynolds number, 〈u〉l/ν

T :

microscopic temperature

\(\vec{u}\) :

microscopic velocity vector

u :

x-velocity component

v :

y-velocity component

V :

volume of the unit cell

V f :

fluid phase volume in the unit cell

V s :

solid phase volume in the unit cell

α:

thermal diffusivity

ɛ:

porosity

θ:

non-dimensional temperature

ν:

kinematic viscosity

ψ:

stream function

ρ:

density

f:

fluid phase

s:

solid phase

eff:

effective

fs:

solid/fluid interface

//:

longitudinal direction

⊥:

transverse direction

〈 〉:

volume average value

d:

thermal dispersion

f:

fluid phase

s:

solid phase

*:

non-dimensional variable

′:

fluctuation

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Authors and Affiliations

  1. Département de Génie Mécanique, USTHB, B. P. 32, El Alia, Bab Ezzouar, 16111, Algeria

    Mebrouk Ait Saada & Salah Chikh

  2. Department of Mechanical Engineering, The University of Vermont, Burlington, VT, 05405, USA

    Antonio Campo

Authors
  1. Mebrouk Ait Saada
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  2. Salah Chikh
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  3. Antonio Campo
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Correspondence to Antonio Campo.

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Saada, M.A., Chikh, S. & Campo, A. Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach. Heat Mass Transfer 42, 995–1006 (2006). https://doi.org/10.1007/s00231-005-0061-y

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  • DOI: https://doi.org/10.1007/s00231-005-0061-y

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