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Transport phenomena in heterogeneous media based on volume averaging theory

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Abstract

Models are developed to describe transport phenomena in a porous medium that take into account the scales and other characteristics of the medium morphology. Equation sets allowing for turbulence and two-temperature or two-concentration diffusion are obtained for non-isotropic porous media with interface exchange. The equations differ from known equations and were developed using an advanced averaging technique, hierarchical modeling methodology, and fully turbulent models with Reynolds stresses and fluxes in the space of every pore. The transport equations are shown to have additional integral and differential terms. The description of the structural morphology determines the importance of these terms and the range of application of the closure schemes. A natural way to transfer from transport equations in a porous media with integral terms to differential equations with coefficients that could be experimentally evaluated and determined is described.Some numerical results that illustrate what occurs when the porosity approaches unity or the porosity approaches zero show that solutions smoothly converge to the transport characteristics of a plane channel. A simple heat sink is modeled as a two-temperature process allowing solution of the conjugate problem to be accomplished and is optimized using design of experiment (DOE) methods. As a final example, acoustic energy absorption in a porous media is addressed.

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Abbreviations

b :

Turbulent fluctuation energy

c d :

Mean drag coefficient in the REV (−)

c dp :

Mean form resistance coefficient in REV (−)

c f :

Mean skin friction coefficient in the REV (−)

c p :

Specific heat (J/kg K)

d pin :

Pin fin diameter (m)

d p :

Porous media hydraulic diameter (m)

dS :

Interphase differential area (m2)

∂S w :

Internal surface in the REV (m2)

E eff :

Heat sink effectiveness

f f :

Fanning friction factor

\( \ifmmode\expandafter\tilde\else\expandafter\~\fi{f}\) :

Value f averaged over Δ Ω f

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

Value f averaged over Δ Ω s

\(\hat f\) :

〈f〉− f, morphologically induced fluctuations of f

〈f 〉f :

Value f, averaged over Δ Ω f in a REV

H :

Height of pin fins (m)

K :

Permeability (m2)

K :

Thermal conductivity (W/m K)

L :

Length of base plate (m)

L :

Turbulence mixing length (m)

m :

Porosity (−)

〈m 〉:

Averaged porosity (−)

P :

Pitch (m), or Pumping power (W)

p :

Pressure (Pa)

q :

Heat flux (W/m2)

Repor :

Pore Reynolds number (−)

S w :

Specific surface ∂ S w/Δ Ω (1/m)

S wp :

\(S_{ \bot } /\Delta \Omega \;(1/{\text{m}})\)

\(S_{ \bot } \) :

Cross flow projected area of obstacles (m2)

T :

Temperature (K)

u, w :

Velocity in x, z-direction (m/s)

V :

Volume (m3)

c:

Characteristic value

f:

fluid phase

i:

Component of turbulent vector

L:

Laminar

m :

scale value

s:

Solid phase

T:

Turbulent

w:

Wall

\(\tilde \alpha_{\text{T}} \) :

Averaged heat transfer coefficient over ∂ S w (W/m2 K)

Δ Ω:

Representative elementary volume (REV) (m3)

Δ Ωf :

Pore volume in a REV (m3)

Δ Ωs :

Solid phase volume in a REV (m3)

ν:

Kinematic viscosity (m2/s)

ρ:

Density (kg/m3)

mu :

Dynamic viscosity (kg/ms)

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Authors and Affiliations

  1. Morin, Martinelli, Gier Memorial Heat Transfer Laboratory, Department of Mechanical and Aerospace Engineering, School of Engineering and Applied Science, University of California, Los Angeles, CA, 90095-1597, USA

    Ivan Catton

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  1. Ivan Catton
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Correspondence to Ivan Catton.

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Catton, I. Transport phenomena in heterogeneous media based on volume averaging theory. Heat Mass Transfer 42, 537–551 (2006). https://doi.org/10.1007/s00231-005-0650-9

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