Abstract
The permeability of a porous medium is strongly affected by its local geometry and connectivity, the size distribution of the solid inclusions, and the pores available for flow. Since direct measurements of the permeability are time consuming and require experiments that are not always possible, the reliable theoretical assessment of the permeability based on the medium structural characteristics alone is of importance. When the porosity approaches unity, the permeability–porosity relationships represented by the Kozeny–Carman equations and Archie’s law predict that permeability tends to infinity and thus they yield unrealistic results if specific area of the porous media does not tend to zero. The aim of this article is the evaluation of the relationships between porosity and permeability for a set of fractal models with porosity approaching unity and a finite permeability. It is shown that the tube bundles generated by finite iterations of the corresponding geometric fractals can be used to model porous media where the permeability–porosity relationships are derived analytically. Several examples of the tube bundles are constructed, and the relevance of the derived permeability–porosity relationships is discussed in connection with the permeability measurements of highly porous metal foams reported in the literature.
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Abbreviations
- L :
-
Length of tube bundle, m
- ΔP :
-
Pressure drop across tube bundle, Pa
- d :
-
Tube diameter, m
- q :
-
Tube flow rate, m3/s
- μ :
-
Dynamic viscosity, Pa s
- β :
-
Tube flow form factor, –
- α :
-
=(β / μ)(ΔP/L), m−1s−1
- n :
-
Number density of distribution of diameters d of bundle tubes m−1
- D :
-
Fractal dimension, power coefficient of power law function, –
- τ :
-
= 1 + D, –
- C :
-
Constant pre-factor of the power law function, m(τ -1)
- N :
-
Cumulative number distribution, –
- \({\varphi}\) :
-
Porosity, –
- Q :
-
Total flow rate of tube bundle, m3/s
- m k :
-
k-moment of distribution density function, n, mk
- A :
-
Cross sectional area of tube bundle, m2
- K D :
-
\({=\frac{\mu}{{\Delta}P/L}\frac{Q}{A};}\) Darcy permeability, m2
- N k :
-
Number of tubes that are added into the bundle at iteration “k”, –
- x :
-
= N k+1/N k ; Pattern (tube number) scaling factor, –
- a k :
-
Area of cross section of tube which is added at iteration k, m2
- y :
-
= (a k /a k+1)1/2; Length scaling factor, –
- f :
-
Subscript index denoting parameters related to fractal, –
- K f :
-
\({=\frac{K_{\rm Df}}{\beta A};}\) Dimensionless fractal permeability, –
- K :
-
\({=\frac{K_D}{\beta A};}\) Dimensionless permeability of prefractal bundle, –
- p,r:
-
Fitting parameters of the Archie’s law, m2,–
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Zinovik, I., Poulikakos, D. On the Permeability of Fractal Tube Bundles. Transp Porous Med 94, 747–757 (2012). https://doi.org/10.1007/s11242-012-0022-0
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DOI: https://doi.org/10.1007/s11242-012-0022-0