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.
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Abbreviations
- Å sf :
-
interfacial area of the s-f interface contained within the macroscopic system m2
- A sf :
-
interfacial area of the s-f interface contained within the averaging volume m2
- C p :
-
mass fraction weighted heat capacity, kcal/kg/K
- d s , d f :
-
microscopic characteristic length m
- g :
-
vector that maps ▽〈θ〉 to \(\tilde \theta\) s, m
- h :
-
vector that maps ▽〈θ〉 to \(\tilde \theta\) f , m
- K eff :
-
effective thermal conductivity tensor, kcal/m s K
- l :
-
REV characteristic length, m
- L :
-
macroscopic characteristic length, m
- n fs :
-
outwardly directed unit normal vector for the f-phase at the f-s interface
- n e :
-
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
- V s , V f :
-
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
- ε s , ε f :
-
volume fraction
- κ :
-
ratio of thermal conductivities
- λ s , λ f :
-
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
References
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Prat, M. On the boundary conditions at the macroscopic level. Transp Porous Med 4, 259–280 (1989). https://doi.org/10.1007/BF00138039
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DOI: https://doi.org/10.1007/BF00138039