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On the Permeability of Fractal Tube Bundles

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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,–

References

  • Bhattacharya A., Calmidi V.V., Mahajan R.L.: Thermophysical properties of high porosity metal foams. Int. J. Heat Mass Transf. 45, 1017–1031 (2002)

    Article  Google Scholar 

  • Boomsma K., Poulikakos D.: On the effective thermal conductivity of a three-dimensionally structured fluid-saturated metal foam. Int. J. Heat Mass Transf. 44, 827–836 (2001)

    Article  Google Scholar 

  • Boomsma K., Poulikakos D., Ventikos Y.: Simulations of flow through open cell metal foams using an idealized periodic cell structure. Int. J. Heat Fluid Flow 24, 825–834 (2003a)

    Article  Google Scholar 

  • Boomsma K., Poulikakos D., Zwick F.: Metal foams as compact high performance heat exchangers. Mech. Mater. 35, 1161–1176 (2003b)

    Article  Google Scholar 

  • Boyd D.W.: The sequence of radii of the apollonian packing. Math. Comput. 39, 249–254 (1982)

    Article  Google Scholar 

  • Costa A.: Permeability–porosity relationship: a reexamination of the Kozeny–Carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett. 33, L02318 (2006)

    Article  Google Scholar 

  • Deville S., Saiz E., Tomsia A.P.: Freeze casting of hydroxyapatite scaffolds for bone tissue engineering. Biomaterials 27, 5480–5489 (2006)

    Article  Google Scholar 

  • Donatti D.A., Vollet D.R., Mesquita A.I., Silva T.F.: Mass fractal characteristics of silica sonogels as determined by small-angle X-ray scattering and nitrogen adsorption. Phys. Rev. B. 71, 014203 (2005)

    Article  Google Scholar 

  • Fritton S.P., Weinbaum S.: Fluid and solute transport in bone: flow-induced mechanotransduction. Ann. Rev. Fluid Mech. 41, 347–374 (2009)

    Article  Google Scholar 

  • Fuchs, E., Sanden, K.: Prime number and local to global conjectures in Apollonian circle packings (2009) (in press)

  • Gimenez D., Perfect E., Rawls W.J., Pachepsky Ya.: Fractal models for predicting soil hydraulic properties: a review. Eng. Geol. 48, 161–183 (1997)

    Article  Google Scholar 

  • Gurav, J.L., Jung, I., Park H., Kang, E., Nadargi, D.: Silica aerogel: synthesis and applications, J. Nanomater. Article ID 409310 (2010)

  • Henderson N., Brettas J.C., Sacco W.F.: A three-parameter Kozeny–Carman generalized equation for fractal porous media. Chem. Eng. Sci. 65, 4432–4442 (2010)

    Article  Google Scholar 

  • Hirata E., Uo M., Nodasaka Y., Takita H., Ushijima N., Akasaka T., Watari F., Yokoyama A.: 3D collagen scaffolds coated with multiwalled carbon nanotubes: initial cell attachment to internal surface. J. Biomed. Mater. Res. B Appl. Biomater. 93, B2 (2010)

    Google Scholar 

  • Jacquenot, G.: 2D Apollonian gasket with four identical circles http://www.mathworks.com/matlabcentral/fileexchange/15958-2d-apollonian-gasket-with-four-identical-circles (2007)

  • Jacquin C.G., Adler P.M.: Fractal porous media II: geometry of porous geological structures. Transp. Porous Media 2, 571–596 (1987)

    Article  Google Scholar 

  • Lapidus, L., Pearse, E., Winter S.: Pointwise tube formula for fractal sprays and 2 self-similar tilings with arbitrary generators. arXiv:1006.3807v1 [math.MG] (2010)

  • Manna S.S., Herrmann H.J.: Precise determination of the fractal dimensions of Apollonian packing and space-filling bearings. J. Phys. A Math. Gen. 24, L481–L490 (1991)

    Article  Google Scholar 

  • Neimark A.V., Ruetsch S., Kornev K.G., Ravikovitch P.I.: Hierarchical pore structure and wetting properties of single-wall carbon nanotube fibers. Nanoletters 3, 419–423 (2003)

    Article  Google Scholar 

  • O’Brien F.J., Harley B.A., Waller M.A., Yannas I.V., Gibson L.J., Prendergast P.J.: The effect of pore size on permeability and cell attachment in collagen scaffolds for tissue engineering. Technol. Health Care 15, 3–17 (2007)

    Google Scholar 

  • Perfect E., Gentry R.W., Sukop M.C., Lawson J.E.: Multifractal Sierpinski carpets: theory and application to upscaling effective saturated hydraulic conductivity. Geoderma 134, 240–252 (2006)

    Article  Google Scholar 

  • Perfect E., Sukop M.C.: Models relating solute dispersion to pore space geometry in saturated media: a review. Physical and chemical Processes of water and solute transport/retention in soil. Soil Science Society of America: Special publication no. 56, 77–146 (2001)

    Google Scholar 

  • Perrier E., Bird N., Rieu M.: Generalizing the fractal model of soil structure, the pore-solid fractal approach. Fractals in Soil Science. Elsevier Science B.V, Amsterdam (2000)

    Google Scholar 

  • Pierre A.C., Pajonk G.M.: Chemistry of aerogels and their Applications. Chem. Rev. 102, 4243–4265 (2002)

    Article  Google Scholar 

  • Predoi-Racila M., Stroeb M.C., Croletb J.M.: Human cortical bone: the SiNuPrOs model. Part II - a multi-scale study of permeability. Comput. Methods Biomech. Biomed. Eng. 13, 81–89 (2010)

    Article  Google Scholar 

  • Radin C.: The pinwheel tilings of the plane. Ann. Math. 139, 661–702 (1994)

    Article  Google Scholar 

  • Rodriguez E., Giacomelli F., Vazquez A.: Permeability–porosity relationship in RTM for different fiberglass and natural reinforcements. J. Compos. Mater. 38, 259–268 (2004)

    Article  Google Scholar 

  • Shah R.K.: Laminar flow forced convection in ducts. Academic Press, London (1978)

    Google Scholar 

  • Thormann A., Teuscher N., Pfannmçller M., Rothe U., Heilmann A.: Nanoporous aluminum oxide membranes for filtration and biofunctionalization. Small 3, 1032–1040 (2007)

    Article  Google Scholar 

  • West G.B., Brown J.H., Enquist B.J.: A general model for the origin of allometric scaling laws in biology. Science 276, 122–126 (1997)

    Article  Google Scholar 

  • Wolfram S.: Geometry of binomial coefficients http://www.stephen29.com/publications/articles/ca/84-geometry/1/text.html (1984)

  • Yu B., Liu W.: Fractal analysis of permeabilities for porous media. Am. Inst. Chem. Eng. J. 50, 46–57 (2004)

    Article  Google Scholar 

  • Zinovik, I., Poulikakos, D.: On permeability of fractal tube bundles. http://arxiv.org/abs/1109.2302; arXiv:1109.2302 [physics.flu-dyn] (2012)

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  1. Laboratory of Thermodynamics in Emerging Technologies, Department of Mechanical and Process Engineering, ETH Zurich, Zurich, 8092, Switzerland

    I. Zinovik & D. Poulikakos

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  1. I. Zinovik
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  2. D. Poulikakos
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Correspondence to I. Zinovik.

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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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