Abstract
The study of boundary effects initiated in a previous paper is continued. New assumptions regarding the geometrical structure of the boundary surface are introduced. Under these assumptions, it is shown that macroscopic Neumann conditions do not generally affect the determination of the macroscopic field in the case of the transport process considered — heat conduction. For this type of boundary condition, the boundary effect is generally confined within a thin layer near the boundary. When heat sources are taken into account within the porous domain, the result is different. In this case, making use of a Neumann boundary condition, expressed in terms of macroscopic variables, amounts to introducing an extra flux. Under normal circumstances, however, this additional flux is negligible.
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Abbreviations
- A :
-
cross-sectional area of a unit cell
- A e :
-
cross-sectional area of a unit cell at the boundary surface
- A sf :
-
interfacial area of the s-f interface contained within the averaging volume
- \(\mathbb{A}_\upsilon\) :
-
surface area per unit volume (A sf/\(\mathbb{V}\))
- A sf :
-
interfacial area of the s-f interface contained within the macroscopic system
- g:
-
closure vector
- h:
-
closure vector
- k:
-
heat transfer coefficient at the s-f interface
- Keff :
-
effective thermal conductivity tensor
- ℓ x :
-
unit cell length
- n:
-
unit vector
- ne :
-
outwardly directed unit normal vector at the boundary
- nsf :
-
outwardly directed unit normal vector for thes-phase at f-s interface
- q:
-
heat flux density
- T * :
-
macroscopic temperature defined by the macroscopic problem
- s :
-
closure variable
- V :
-
volume of the macroscopic system
- ∂V :
-
boundary surface of the macroscopic domain
- ∂V 1 :
-
macroscopic sub-surface of the boundary surface
- x :
-
local coordinate
- εs,f :
-
volume fraction
- λs, glf :
-
microscopic thermal conductivities
- θ:
-
‘true’ microscopic temperature
- θ* :
-
microscopic temperature corresponding toT *
- \(\hat \theta\) :
-
microscopic error temperature
- ζ:
-
vector defined by Equation (34)
- < >:
-
spatial average
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Prat, M. Some refinements concerning the boundary conditions at the macroscopic level. Transp Porous Med 7, 147–161 (1992). https://doi.org/10.1007/BF00647394
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DOI: https://doi.org/10.1007/BF00647394