Abstract
A new analytical derivation for momentum transport during laminar flow through granular porous media is discussed and some of its implied results described. In the very low Reynolds number regime fully developed laminar flow is assumed and in the higher laminar Reynolds number regime the Forchheimer (non-Darcy) effect is modelled through reference to form drag induced by the solid constituents of the porous medium. The results are compared to the Ergun equation, which is empirically based on experimental measurements, and the correspondence is shown to be remarkably close.
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Abbreviations
- A :
-
Ergun equation linear term coefficient
- A p :
-
cross-sectional pore area
- B :
-
Ergun equation non-linear term coefficient
- d :
-
microscopic characteristic length
- d x :
-
total tortuous flow length within RUC,V f /A p
- d s :
-
solid cube side width
- e q :
-
streamwise unit vector,q/q
- F :
-
microscopic shear factor
- f :
-
parallel plate friction factor
- g :
-
gravitational body force per unit mass
- I :
-
shear force in RUC
- K :
-
Darcy hydrodynamic permeability
- p :
-
pressure
- p f :
-
〈p〉 f
- q :
-
magnitude ofq
- q :
-
specific discharge, 〈v〉
- Re‖ :
-
parallel plate pore Reynolds number, 2ρv p (d−d s )/Μ
- Re qs :
-
cube Reynolds number,ρqd s /Μ
- S :
-
surface area
- S fs :
-
fluid solid interface
- t :
-
time
- V f :
-
fluid filled ‘void’ volume with RUC
- V s :
-
solid volume within RUC
- V 0 :
-
total volume of RUC
- v :
-
fluid velocity field withinV f
- v i :
-
fluid velocity at streamline inflection point
- v f :
-
intrinsic average velocity withinV f , 〈v〉 f
- v p :
-
cross-sectional mean fluid speed in streamwise pore section
- v ⊥ :
-
cross-sectional mean fluid speed in transverse pore section
- ε :
-
porosity (void fraction),V f /V 0
- Μ :
-
fluid dynamic viscosity
- Ν :
-
normal vector on solid surface and pointing into fluid
- ρ :
-
fluid mass density
- Φ :
-
generic variable
- χ :
-
pore structure tortuosity
- ·:
-
tensor inner product
- 〈Φ〉:
-
volumetric phase average ofΦ,
$$\frac{1}{{V_0 }}\iiint_{V_f } {\phi {\text{d}}V}$$ - 〈Φ〉 f :
-
volumetric intrinsic phase average ofΦ,
$$\frac{1}{{V_f }}\iiint_{V_f } {\phi {\text{d}}V}$$ - \(\mathop \phi \limits^ \circ\) :
-
deviation,Φ−〈Φ〉 f
- ⊥:
-
as subscript, transverse pores
- ‖:
-
as subscript, streamwise pores
- Μ :
-
as subscript, due to viscous shear
- c :
-
as subscript, critical value
- D :
-
as subscript, downstream side of solid cube
- f :
-
as subscript, fluid phase
- f s :
-
as subscript, fluid-solid interface
- U :
-
as subscript, upstream side of solid cube,
- 0:
-
as subscript, low Reynolds number limit
- ∞:
-
as subscript, high Reynolds number limit
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Prieur Du Plessis, J. Analytical quantification of coefficients in the Ergun equation for fluid friction in a packed bed. Transp Porous Med 16, 189–207 (1994). https://doi.org/10.1007/BF00617551
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DOI: https://doi.org/10.1007/BF00617551