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Transport in Porous Media

, Volume 4, Issue 3, pp 259–280 | Cite as

On the boundary conditions at the macroscopic level

  • Marc Prat
Article

Abstract

We study the problem of the boundary conditions specified at the boundary of a porous domain in order to solve the macroscopic transfer equations obtained by means of the volume-averaging method. The analysis is limited to the case of conductive transport but the method can be extended to other cases. A numerical study enables us to illustrate the theoretical results in the case of a model porous medium.

Key words

Volume averaging boundary conditions 

Nomenclature

Roman Letters

Åsf

interfacial area of the s-f interface contained within the macroscopic system m2

Asf

interfacial area of the s-f interface contained within the averaging volume m2

Cp

mass fraction weighted heat capacity, kcal/kg/K

ds, df

microscopic characteristic length m

g

vector that maps ▽〈θ〉 to \(\tilde \theta\)s, m

h

vector that maps ▽〈θ〉 to \(\tilde \theta\) f , m

Keff

effective thermal conductivity tensor, kcal/m s K

l

REV characteristic length, m

L

macroscopic characteristic length, m

nfs

outwardly directed unit normal vector for the f-phase at the f-s interface

ne

outwardly directed unit normal vector at the dividing surface

T*

macroscopic temperature field obtained by solving the macroscopic equation (3), K

V

averaging volume, m3

Vs, Vf

volume of the considered phase within the averaging volume, m3

volume of the macroscopic system, m3

s, f

volume of the considered phase within the volume of the macroscopic system, m3

δ∀

dividing surface, m2

Greek Letters

εs, εf

volume fraction

κ

ratio of thermal conductivities

λs, λf

thermal conductivities, kcal/m s K

ϱ

spatial average density, kg/m3

θ

microscopic temperature, K

θ*

microscopic temperature corresponding to T * , K

\(\tilde \theta\)

spatial deviation temperature K

\(\hat \theta\)

error on the temperature due to the macroscopic boundary conditions, K

〈 〉

spatial average

〈 〉s, 〈 〉f

intrinsic phase average

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References

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Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Marc Prat
    • 1
  1. 1.Institut de Mécanique des Fluides de ToulouseToulouseFrance

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