Skip to main content
SpringerLink
Log in

An Equivalent Spherical Particle System to Describe Characteristics of Flow in a Dense Packing of Non-spherical Particles

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

Effective characteristics, such as the effective particle diameter and hydraulic radius, are usually used to determine the pressure loss associated with single-phase flow through an isotropic, dense non-spherical granular packing. These quantities can be considered as those of an equivalent spherical particle system. This paper reviews different methods to select effective particle diameters taking into account the effect of particle shape. A new method is proposed to determine an equivalent spherical particle system with both the hydraulic radius and the pressure loss equivalent to those in the actual non-spherical particle packing. This equivalent system can be used to determine the characteristics of flow in the actual system. The analyses provide theoretical justification for adoption of proper effective particle diameters and porosity as well as the dependency of Ergun–Kozeny constant on the shape factor of particles. It is also shown theoretically that the critical Reynolds number for Darcian flow depends on the porosity and the shape factor of particles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Abbreviations

a :

Empirical constant

b :

Empirical constant

c :

Circularity of the projected surface, defined as the ratio of the perimeter of a sphere with equal projected surface area to the perimeter of the projected surface of the actual particle

A :

Ergun–Kozeny constant for viscous flow

\( A_{\text{eq}} \) :

Ergun–Kozeny constant for viscous flow in the equivalent system of spherical particles

\( \bar{A}_{\text{eq}} \) :

Surface area of an equivalent spherical particle with diameter \( \bar{D}_{\text{eq}} = \bar{D}_{32} = \psi_{\text{s}} \bar{D}_{\text{v}} \), m2

\( \bar{A}_{\text{eV}} \) :

Surface area of a sphere having the same volume as an irregular particle, m2

\( A_{\text{P}} \) :

Surface area of a particle, m2

B :

Ergun–Kozeny constant for Newtonian flow

\( C_{D} \) :

Drag coefficient

\( C_{\text{Deq}} \) :

Drag coefficient of an equivalent spherical particle or the equivalent spherical particle system

\( C_{\text{Dv}}^{{}} \) :

Drag coefficient in viscous flow regime

\( C_{\text{Dn}}^{{}} \) :

Drag coefficient in Newtonian flow regime

\( d_{50} \) :

The mean grain size, m

\( d_{\text{eq}} \) :

Equivalent diameter, m

\( d_{p} \) :

Particle diameter or the diameter of an equivalent-volume sphere, m

\( d_{x} \) :

Particle size corresponding to x% passing on the particle size distribution curves, m

\( \bar{D}_{21} \) :

Length average particle size, m

\( \bar{D}_{31}^{{}} \) :

The volume-to-length mean diameter, m

\( \bar{D}_{32}^{{}} \) :

Volume-to-surface-area mean diameter, m

\( \bar{D}_{A} \) :

Projected surface equivalent sphere diameter, \( \bar{D}_{A} = \sqrt {4 \times {\text{Projected Area/}}\pi } \), m

\( \bar{D}_{eff} \) :

Effective diameter, m

\( \bar{D}_{\text{eq}} \) :

Equivalent diameter \( \bar{D}_{\text{eq}} = \psi_{\text{s}} \bar{D}_{\text{v}} \), m

\( D_{h} \) :

Mean pore diameter or average hydraulic diameter, m

\( \bar{D}_{p} \) :

Characteristic particle diameter, m

\( \bar{D}_{pore} \) :

Characteristic pore size, m

\( \bar{D}_{\text{v}} \) :

Diameter of an equivalent-volume sphere, \( \bar{D}_{\text{v}} = (6V_{\text{P}} /\pi )^{1/3} \), m

F D :

Drag force on a single particle in unhindered flow, N

F DS :

Drag force on a particle in a cluster of particles, N

g :

Acceleration of gravity, m/s2

Ga:

Fluid-particle Galileo number

J :

Hydraulic gradient defined as \( J = \nabla p/(\rho_{f} g) \)

\( k_{KC} \) :

Kozeny constant

\( l_{p} \) :

Microscopic characteristic length, m

N :

Number of particles in a granular assembly or RVE

\( N_{\text{P}} \) :

Number of particles in a unit volume of the granular bed

\( q_{i} \) :

Particle size distribution parameters

\( R_{\text{CDeq}} \) :

Ratio between the drag coefficients of a granular particle system and its equivalent spherical particle system \( R_{\text{CDeq}} = C_{\text{D}} /C_{\text{Deq}} \)

\( R_{\text{CDeqV}} \) :

Ratio between the drag coefficients of an irregular particle and an equivalent-volume spherical particle \( R_{\text{CDeqV}} = C_{\text{D}} /C_{\text{DeqV}} \)

\( \text{Re} \) :

Reynolds number

\( \text{Re}_{\text{a}} \) :

Reynolds number for angular particles

\( \text{Re}_{h} \) :

Pore Reynolds number, \( \text{Re}_{h} = \rho_{f} UR_{h} /\mu = \text{Re}_{\text{S}} /6 \)

\( \text{Re}_{\text{int}} \) :

Interstitial Reynolds number, \( \text{Re}_{\text{int}} = \frac{{\rho_{f} Ud_{\text{p}} }}{\mu } = \frac{{\rho_{f} v_{\text{s}} d_{\text{p}} }}{\mu \varepsilon } \)

\( \text{Re}_{p} \) :

Particle Reynolds number \( \text{Re}_{\text{p}} = \rho v_{s} d_{\text{p}} /\mu \)

\( {\text{Re}}_{\text{r}} \) :

Reynolds number for rounded particles

\( \text{Re}_{\text{S}} \) :

Blake/Ergun Reynolds number

\( \text{Re}_{\kappa } \) :

Reynolds number with the characteristic length being a function of permeability \( \kappa \)

\( \text{Re}_{\tau } \) :

Reynolds number defined as \( \text{Re}_{\tau } = \rho v_{s} /(\mu \bar{\tau }) \) with the characteristic length \( \bar{\tau }^{ - 1} = 2d_{50} \)

\( R_{\text{hv}} \) :

Hydraulic radius of the pore space in a spherical particle system defined as \( R_{\text{hv}} = \varepsilon d_{p} /[6(1 - \varepsilon )] \). For an equivalent spherical particle system, \( \bar{D}_{\text{eq}} = \bar{D}_{32} = \psi_{\text{s}} \bar{D}_{\text{v}} \), is used as the particle diameter, m

\( S_{pore}^{{}} \) :

Specific surface area based on the volume of pores, \( S_{\text{pore}} = A_{\text{p}} /V_{\text{pore}} \), m2

\( S_{V}^{{}} \) :

Specific surface area of a particle, \( S_{\text{v}} = A_{\text{p}} /V_{\text{p}} = \bar{D}_{32} /6 \), m2

U :

Magnitude of interstitial velocity defined as \( {\mathbf{U}} = {\mathbf{v}}_{s} /\varepsilon \), m/s

\( v_{s} \) :

Magnitude of superficial velocity \( {\mathbf{v}}_{s} \), m/s

\( V_{\text{eq}} \) :

Volume of an equivalent spherical particle with diameter \( \bar{D}_{\text{eq}} = \bar{D}_{32} = \psi_{\text{s}} \bar{D}_{\text{v}} \), m3

\( V_{\text{P}} \) :

Volume of a particle, m3

\( \beta \) :

Non-Darcian coefficient that is usually referred to as the “inertial parameter”

\( \delta \) :

Shape-dependent factor to determine the specific surface area of a particle

\( \varepsilon \) :

Bulk porosity

\( \varepsilon_{\text{eq}} \) :

Porosity of an equivalent particle system

\( \kappa \) :

Velocity-dependent permeability, m2

\( \kappa_{0} \) :

Permeability in the viscous flow regime, m2

\( \mu \) :

The dynamic viscosity of the fluid (\( \mu = 1.002 \times 10^{ - 3} Pa \cdot s \) for water at 20 °C),

\( \rho_{f} \) :

Density of fluid, kg/m3

\( \tau \) :

Tortuosity factor

\( \tau_{\text{eq}}^{{}} \) :

Tortuosity of an equivalent particle system

\( \tau_{np} \) :

Shear stress on the surface of a spherical particle induced by flowing fluid, kPa

\( \psi_{\text{s}} \) :

Shape factor or Wadell’s sphericity, which is defined as the ratio of the surface area of an equivalent-volume sphere to that of the particle

References

  • Ahuja, L.R., Cassel, D.K., Bruce, R.R., Barnes, B.B.: Evaluation of spatial distribution of hydraulic conductivity using effective porosity data. Soil Sci. 148(6), 404–411 (1989)

    Article  Google Scholar 

  • Allen, K.G., von Backström, T.W., Kröger, D.G.: Packed bed pressure drop dependence on particle shape, size distribution, packing arrangement and roughness. Powder Tech. 246, 590–600 (2013)

    Article  Google Scholar 

  • Andersson, A. C., Andersson, O., Gustafson, G.: Brunnar. Undersøking dimensjonering—borrning—drift. Rapport R42, Stockholm: Byggforskningsrådet (1984)

  • Bagheri, G., Bonadonna, C.: On the drag of freely falling non-spherical particles. Powder Tech. 301, 526–544 (2016)

    Article  Google Scholar 

  • Barree, R.D., Conway, M.W.: Beyond beta factors: a complete model for Darcy, Forchheimer, and Trans-Forchheimer flow in porous media. In: SPE Annual Technical Conference and Exhibition (SPE 89325) (2004)

  • Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)

    Google Scholar 

  • Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic, Dordrecht (1990)

    Book  Google Scholar 

  • Berg, C.F.: Permeability description by characteristic length, tortuosity, constriction and porosity. Transp. Porous Med. 103(3), 381–400 (2014)

    Article  Google Scholar 

  • Beyer, W.: Hydrogeologische Untersuchungen bei der Ablagerung von Wasserschadstoffen. Zeitschrift fuer Angewandte Geologie 12(11), 599–606 (1966)

    Google Scholar 

  • Blott, S.J., Pye, K.: Particle shape: a review and new methods of characterization and classification. Sedimentology 55, 31–63 (2008)

    Google Scholar 

  • Burcharth, H.F., Andersen, O.K.: On the one-dimensional steady and unsteady porous flow equations. Coast. Eng. 24(3-4), 233–257 (1995)

    Article  Google Scholar 

  • Carman, P.C.: Fluid flow through a granular bed. Trans. Inst. Chem. Eng. 15, 150–156 (1937)

    Google Scholar 

  • Chandra, D., Elsworth, D., VanEssendelft, D.: Mechanical and transport characteristics of coal-biomass mixtures for advanced IGCC systems. Energy Fuels 26(9), 5729–5739 (2012)

    Article  Google Scholar 

  • Cho, G.-C., Dodds, J., Santamarina, J.C.: Particle shape effects on packing density, stiffness, and strength: natural and crushed sands. J. Geotech. Geoenviron. Eng. 132(5), 591–602 (2006)

    Article  Google Scholar 

  • Clavier, R., Chikhi, N., Fichot, F., Quintard, M.: Experimental investigation on single-phase pressure losses in nuclear debris beds: identification of flow regimes and effective diameter. Nucl. Eng. Des. 292, 222–236 (2015)

    Article  Google Scholar 

  • Clennell, M.B.: Tortuosity: a guide through the maze. In: Lovetl, M. A. & Harvey, P. K. (eds), Developments in Petrophysics, Geological Society Special Publication No. 122, pp. 299–344 (1997)

  • Clift, R., Grace, J.R., Weber, M.E.: Bubbles, Drops, and Particles. Academic Press, New York (1978)

    Google Scholar 

  • Comiti, J., Sabiri, N.E., Montillet, A.: Experimental characterization of flow regimes in various porous media—III: limit of Darcy’s or creeping flow regime for Newtonian and purely viscous non-Newtonian fluids. Chem. Eng. Sci. 55(15), 3057–3061 (2000)

    Article  Google Scholar 

  • Di Felice, R.: Hydrodynamics of liquid fluidisation. Chem. Eng. Sci. 50(8), 1213–1245 (1995)

    Article  Google Scholar 

  • Dias, R.P., Fernandes, C.S., Teixeira, J.A., Mota, M., Yelshin, A.: Permeability analysis in bisized porous media: wall effect between particles of different size. J. Hydrol. 349, 470–474 (2008)

    Article  Google Scholar 

  • Dias, R., Teixeira, J.A., Mota, M., Yelshin, A.: Tortuosity variation in a low density binary particulate bed. Sep. Purif. Technol. 51(2), 180–184 (2006)

    Article  Google Scholar 

  • Dudgeon, C.R.: Relationship between porosity and permeability of coarse granular materials. In: Proceedings of 3rd Australasian Conference on Hydraulics and Fluid Mech., Sydney, 25–29 (1968)

  • Dullien, F.A.L.: Porous Media: Fluid Transport and Pore Structure, 2nd edn. Academic Press, San Diego (1992)

    Google Scholar 

  • Dwivedi, P.N., Upadhyay, S.N.: Particle-fluid mass transfer in fixed and fluidized beds. Ind. Eng. Chem. Process Des. Dev. 16(2), 157–165 (1977)

    Article  Google Scholar 

  • Ergun, S.: Fluid flow through packed columns. Chem. Engng. Prog. 48(2), 89–94 (1952)

    Google Scholar 

  • Forchheimer, P.: Hydraulik, 3rd edn. Teubner, Leipzig (1930)

    Google Scholar 

  • Ganser, G.: A rational approach to drag prediction of spherical and nonspherical particles. Powder Tech. 77, 143–152 (1993)

    Article  Google Scholar 

  • Garcia, X., Akanji, L.T., Blunt, M.J., Matthai, S.K., Latham, J.P.: Numerical study of the effects of particle shape and polydispersity on permeability. Phys. Rev. E 80(021304), 1–9 (2009)

    Google Scholar 

  • Gibilaro, L.G.: Fluidization-Dynamics: The Formulation and Applications of a Predictive Theory for the Fluidized State. Butterworth-Heinemann, Oxford (2001)

    Google Scholar 

  • Gibilaro, L.G., Di Felice, R., Foscolo, P.U., Waldram, S.P.: Fluidization quality: a criterion for the indeterminate stability. Chem. Engng J. 37, 25–33 (1986)

    Article  Google Scholar 

  • Gibilaro, L.G., Di Felice, R., Waldram, S.P., Foscolo, P.U.: Generalized friction factor and drag coefficient correlations for fluid–particle interactions. Chem. Eng. Sci. 40, 1817–1823 (1985)

    Article  Google Scholar 

  • Glicksman, L.R., Hyre, M.R., Farrell, P.A.: Dynamic similarity in fluidization. Int. J. Multiphase Flow 20(Suppl), 331–386 (1994)

    Article  Google Scholar 

  • Göğüş, M., İpekçi, O.N., Kökpinar, M.A.: Effect of particle shape on fall velocity of angular particles. J. Hydraul. Eng. 127(10), 860–869 (2001)

    Article  Google Scholar 

  • Göktepe, A.B., Sezer, A.: Effect of particle shape on density and permeability of sands. Proc. Inst. Civ. Eng. Geotech. Eng. 163(GE6), 307–320 (2010)

    Article  Google Scholar 

  • Green, L., Duwez, P.: Fluid flow through porous metals. J. Appl. Mech. 18(1), 39–45 (1951)

    Google Scholar 

  • Guo, P.: Dependency of tortuosity and permeability of porous media on directional distribution of pore voids. Transp. Porous Med. 95(2), 285–303 (2012)

    Article  Google Scholar 

  • Guo, P.: Lower and upper bounds for hydraulic tortuosity of porous materials. Transp. Porous Med. 109(3), 659–671 (2015)

    Article  Google Scholar 

  • Hassanizadeh, S.M., Gray, W.G.: High velocity flow in porous media. Transp. Porous Med. 2(6), 521–531 (1987)

    Article  Google Scholar 

  • Hayati, A.N., Ahmadi, M.M., Mohammadi, S.: How particle shape affects the flow through granular materials. Phys. Rev. E 85(036310), 1–4 (2009)

    Google Scholar 

  • Haider, A., Levenspiel, O.: Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Tech. 58(1), 63–70 (1989)

    Article  Google Scholar 

  • Huang, H., Ayoub, J.: Applicability of the Forchheimer equation for non-Darcy flow in porous media. SPE J. 13(1), 112–122 (2008)

    Article  Google Scholar 

  • Janoo, V.: Quantification of shape, angularity, and surface texture of base course materials. Special Report 98-1, US Army Corps of Engineers, Cold Regions Research & Engineering Laboratory (1998)

  • Kasenow, M.: Determination of Hydraulic Conductivity from Grain Size Analysis. Water Resources Publications, LLC, Highlands Ranch, Colorado (2002)

    Google Scholar 

  • Kececioglu, I., Jiang, Y.: Flow through porous media of packed spheres saturated with water. J. Fluids Eng. 116(1), 164–170 (1994)

    Article  Google Scholar 

  • Kenney, T.C., Lau, D., Ofoegbu, G.I.: Permeability of compacted granular materials. Can. Geotech. J. 21(4), 726–729 (1984)

    Article  Google Scholar 

  • Koponen, A., Kataja, M., Timonen, J.: Permeability and effective porosity of porous media. Phys. Rev. E 56(3), 3321–3325 (1997)

    Article  Google Scholar 

  • Kovacs, G.: Seepage through saturated and unsaturated layers. Hydrol. Sci. J. 16, 27–40 (1971)

    Google Scholar 

  • Kozeny, J.: Ueber kapillare Leitung des Wassers im Boden. Sitzungsber Akad. Wiss. Wien, 136(2a), 271–306 (1927)

  • Kunii, D., Levenspiel, O.: Fluidization Engineering, 2nd edn. Butterworth-Heinemann, Boston (1991)

    Google Scholar 

  • Leith, D.: Drag on non-spherical objects. Aerosol Sci. Technol. 6(2), 153–161 (1987)

    Article  Google Scholar 

  • Li, L., Ma, W.: Experimental study on the effective particle diameter of a packed bed with non-spherical particles. Transp. Porous Med. 89, 35–48 (2011)

    Article  Google Scholar 

  • Loth, E.: Drag of non-spherical solid particles of regular and irregular shape. Powder Tech. 182, 342–353 (2008)

    Article  Google Scholar 

  • Loth, E., O’Brien, T., Syamlal, M., Cantero, M.: Effective diameter for group motion of polydisperse particle mixtures. Powder Tech. 142, 209–218 (2004)

    Article  Google Scholar 

  • Macdonald, I.F., El-Sayed, M.S., Mow, K., Dullien, F.A.L.: Flow through porous media-the Ergun equation revisited. Ind. Eng. Chem. Fundam. 18(3), 199–208 (1979)

    Article  Google Scholar 

  • MacDonald, M.J., Chu, C.-F., Guilloit, P.P., Ng, K.M.: A generalized Blake-Kozeny equation for multisized spherical particles. AIChE J. 37(1), 1583–1588 (1991)

    Article  Google Scholar 

  • McCune, L.K., Wilhelm, R.H.: Mass and momentum transfer in a solid-liquid system. Ind. Eng. Chem. 41(6), 1124–1134 (1949)

    Article  Google Scholar 

  • Mugele, R.A., Evans, H.D.: Droplet size distributions in sprays. Ind. Eng. Chem. 43(6), 1317–1324 (1951)

    Article  Google Scholar 

  • Nemeca, D., Levecb, J.: Flow through packed bed reactors: 1. Single-phase flow. Chem. Eng. Sci. 60(24), 6947–6957 (2005)

    Article  Google Scholar 

  • Niven, R.K.: Physical insight into the Ergun and Wen & Yu equations for fluid flow in packed and fuidised beds. Chem. Eng. Sci. 57(3), 527–534 (2002)

    Article  Google Scholar 

  • Oseen, C.W.: Über die Stokes’sche formel, und über eine verwandte Aufgabe in der Hydrodynamik. Arkiv för matematik, astronomi och fysik 6(29), 1–20 (1910)

    Google Scholar 

  • Pravedny, G.H.: Design and selection of grain-size composition of filter beds for the transition zones of large dams. Energiia, Moscow (1966)

    Google Scholar 

  • Rose, H.E.: On the resistance coefficient-Reynolds number relationship for fluid flow through a bed of granular material. Proc. Inst. Mech. Eng. 153(1), 154–168 (1945)

    Article  Google Scholar 

  • Rubinstein, G., Derksen, J., Sundaresan, S.: Lattice Boltzmann simulations of low-Reynolds-number flow past fluidized spheres: effect of stokes number on drag force. J. Fluid Mech. 788, 576–601 (2016)

    Article  Google Scholar 

  • Rumer, R.R., Drinker, P.A.: Resistance to laminar flow through porous media. J. Hydraulics Div. 92(HY5), 155–164 (1966)

    Google Scholar 

  • Rumpf, H., Gupte, A.R.: Einflüsse der porosität und korngrößenverteilung im widerstandsgesetz der porenströmung. Chem. Ing. Tec. 43, 367–375 (1971)

    Article  Google Scholar 

  • Schaap, M.G., Lebron, I.: Using microscope observations of thin sections to estimate soil permeability with the Kozeny–Carman equation. J. Hydrol. 251, 186–201 (2001)

    Article  Google Scholar 

  • Shepherd, R.G.: Correlations of permeability and grain size. Gr. Water 27(5), 633–638 (1989)

    Article  Google Scholar 

  • Singh, R., Saini, R.S., Saini, J.P.: Nusselt number and friction factor correlations for packed bed solar energy storage systems having large sized elements of different shapes. Sol. Energy 80, 760–771 (2006)

    Article  Google Scholar 

  • Sprouse, K.M., Schuman, M.D.: Dense-phase feeding of pulverized coal in uniform plug flow. AIChE J. 29(6), 1000–1007 (1983)

    Article  Google Scholar 

  • Steinour, H.H.: Rate of sedimentation: suspensions of uniform-size angular particles. Ind. Eng. Chem. 36(9), 840–847 (1944)

    Article  Google Scholar 

  • Teng, H., Zhao, T.S.: An extension of Darcy’s law to non-Stokes flow in porous media. Chem. Eng. Sci. 55, 2727–2735 (2000)

    Article  Google Scholar 

  • Terzaghi, K.: Principles of soil mechanics: III—determination of permeability of clay. Eng. News Rec. 95(21), 832–836 (1925)

    Google Scholar 

  • Thompson, T.L., Clarke, N.N.: A holistic approach to particle drag prediction. Powder Tech. 67, 57–66 (1991)

    Article  Google Scholar 

  • Torskay, T., Shabro, V., Torres-Verdín, C., Salazar-Tio, R., Revil, A.: Grain shape effects on permeability, formation factor, and capillary pressure from pore-scale modeling. Trans. Porous Med. 102, 71–90 (2014)

    Article  Google Scholar 

  • Tran-Cong, S., Gay, M., Michaelides, E.E.: Drag coefficients of irregularly shaped particles. Powder Tech. 139, 21–32 (2004)

    Article  Google Scholar 

  • Tsuji, Y., Kawaguchi, T., Tanaka, T.: Discrete particle simulation of two-dimensional fluidized bed. Powder Tech. 77, 79–87 (1993)

    Article  Google Scholar 

  • Urumovic, K., Urumovic Sr., K.: The effective porosity and grain size relations in permeability functions. Hydrol. Earth Syst. Sci. Discuss. 11, 6675–6714 (2014)

    Article  Google Scholar 

  • van Der Hoef, M.A., Beetstra, R., Kuipers, J.A.M.: Lattice–Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force. J. Fluid Mech. 528, 233–254 (2005)

    Article  Google Scholar 

  • van Lopik, J.H., Snoeijers, R., van Dooren, T.C.G.W., Raoof, A., Schotting, R.J.: The effect of grain size distribution on nonlinear flow behavior in sandy porous media. Transp. Porous Med. 120, 37–66 (2017)

    Article  Google Scholar 

  • Vidal, D., Ridgway, C., Pianet, G., Schoelkopf, J., Roy, R., Bertrand, F.: Effect of particle size distribution and packing compression on fluid permeability as predicted by lattice-Boltzmann simulations. Comput. Chem. Eng. 33(1), 256–266 (2009)

    Article  Google Scholar 

  • Vuković, M., Soro, A.: Determination of hydraulic conductivity of porous media from grain-size composition. Water Resources Publications, LLC, Littleton (1992)

    Google Scholar 

  • Wadell, H.: The coefficient of resistance as a function of Reynolds number for solids of various shapes. J. Frankl. Inst. 217, 459–490 (1934)

    Article  Google Scholar 

  • Watanabe, H.: Drag coefficient and voidage function on fluid flow through granular packed beds. Int. J. Eng. Fluid Mech. 2(1), 93–108 (1989)

    Google Scholar 

  • Wen, C.Y., Yu, Y.H.: Mechanics of fluidization. Chem. Eng. Prog. Symp. Ser. 162, 100–111 (1966)

    Google Scholar 

  • Williams, E.B.: Fundamental concepts of well design. Groundwater 19(5), 527–542 (1981)

    Article  Google Scholar 

  • Yang, W.-C.: Handbook of Fluidization and Fluid-particle Systems. Marcel Dekker Inc, New York (2003)

    Book  Google Scholar 

  • Yazdchi, K., Luding, S.: Towards unified drag laws for inertial flow through fibrous materials. Chem. Eng. J. 207–208, 35–48 (2012)

    Article  Google Scholar 

  • Zeng, Z., Grigg, R.: A criterion for non-Darcy flow in porous media. Transp. Porous Med. 63, 57–69 (2006)

    Article  Google Scholar 

Download references

Acknowledgements

Funding provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, McMaster University, Hamilton, ON, L8S 4L7, Canada

    Peijun Guo, Dieter Stolle & Shannon X. Guo

Authors
  1. Peijun Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dieter Stolle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shannon X. Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peijun Guo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, P., Stolle, D. & Guo, S.X. An Equivalent Spherical Particle System to Describe Characteristics of Flow in a Dense Packing of Non-spherical Particles. Transp Porous Med 129, 253–280 (2019). https://doi.org/10.1007/s11242-019-01286-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-019-01286-y

Keywords

Search

Navigation