Abstract
We study the modelling of purely conductive heat transfer between a porous medium and an external fluid within the framework of the volume averaging method. When the temperature field for such a system is classically determined by coupling the macroscopic heat conduction equation in the porous medium domain to the heat conduction equation in the external fluid domain, it is shown that the phase average temperature cannot be predicted without a generally negligible error due to the fact that the boundary conditions at the interface between the two media are specified at the macroscopic level.
Afterwards, it is presented an alternative modelling by means of a single equation involving an effective thermal conductivity which is a function of point inside the interfacial region.
The theoretical results are illustrated by means of some numerical simulations for a model porous medium. In particular, temperature fields at the microscopic level are presented.
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Abbreviations
- Å sf :
-
interfacial area of thes-f interface contained within the macroscopic system m2
- A sf :
-
interfacial area of thes-f interface contained within the averaging volume m2
- C p :
-
mass fraction weighted heat capacity, kcal/kg/K
- 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 s,l f :
-
microscopic characteristic length m
- L :
-
macroscopic characteristic length, m
- n fs :
-
outwardly directed unit normal vector for thef-phase at thef-s interface
- n :
-
outwardly directed unit normal vector at the dividing surface.
- R 0 :
-
REV characteristic length, m
- T i :
-
macroscopic temperature at the interface, K
- \(\hat T_f \) :
-
error on the external fluid temperature due to the macroscopic boundary condition, K
- 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.
- ∀mp :
-
volume of the porous medium domain, m3
- ∀ex :
-
volume of the external fluid domain, m3
- ∀ s , ∀ f :
-
volume of the considered phase within the volume of the macroscopic system, m3
- ∂∀:
-
dividing surface, m2
- x, z :
-
spatial coordinates
- ε s,ε f :
-
volume fraction
- κ :
-
ratio of the effective thermal conductivity to the external fluid thermal conductivity
- λ * :
-
macroscopic thermal conductivity (single equation model) kcal/m s K
- λ s,λ f :
-
microscopic thermal conductivities, kcal/m s K
- 〈ϱ〉:
-
spatial average density, kg/m3
- θ :
-
microscopic temperature, K
- θ * :
-
microscopic temperature corresponding toT *, K
- \(\tilde \theta \) :
-
spatial deviation temperature K
- \(\hat \theta \) :
-
error in the temperature due to the macroscopic boundary conditions, K
- 〈θ *〉 i :
-
macroscopic temperature at the interface given by the single equation model, K
- 〈 〉:
-
spatial average
- 〈 〉s, 〈 〉f :
-
intrinsic phase average.
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Prat, M. Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid. Transp Porous Med 5, 71–95 (1990). https://doi.org/10.1007/BF00714629
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DOI: https://doi.org/10.1007/BF00714629