Transport in Porous Media

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

On the boundary conditions at the macroscopic level

  • Marc Prat


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 


Roman Letters


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


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


mass fraction weighted heat capacity, kcal/kg/K

ds, df

microscopic characteristic length m


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


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


effective thermal conductivity tensor, kcal/m s K


REV characteristic length, m


macroscopic characteristic length, m


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


outwardly directed unit normal vector at the dividing surface


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


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


dividing surface, m2

Greek Letters

εs, εf

volume fraction


ratio of thermal conductivities

λs, λf

thermal conductivities, kcal/m s K


spatial average density, kg/m3


microscopic temperature, K


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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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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