3.1 Introduction
3.2 Empirical Data for Different Materials and Microstructure Types
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Cooper et al. [6], comparison of various τ-types specified below, LSCF cathode
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τdir_pore_centroid, (image analysis/Avizo Fire, voxel based)
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τindir _diff_sim. (sim conductivity, ‘StarCCM+’/Laplace s., Fourier’s law, mesh based)
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τindir _diff_sim. (sim bulk diffusion, ‘AvizoXlab’/Laplace solver, Fick’s law, voxel based)
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τindir _diff_sim, (simbulk diffusion, ‘TauFactor’/Laplace solver, Fick’s law, voxel based)
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τindir _diff_sim. (simulation of bulk diffusion, (in-house)/random walk, voxel based)
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Tjaden et al. [8], comparison of tortuosity types, porous YSZ support layer:
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τdir_FMM, (image analysis/in-house (Matlab)/FIB-SEM)
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τdir_FMM, (image analysis/in-house (Matlab)/X-ray nano CT)
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τindir_ therm, (sim. of thermal cond., ‘StarCCM+’/Laplace, Fourier, mesh based/FIB-SEM)
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τindir_therm, (sim. of thermal cond., ‘StarCCM+’/Laplace, Fourier, mesh based/XnCT)
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τindir_diff, (diffusion cell experiment/Gas diffusion at 30 and 100 °C/Fick’s law)
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Pecho et al. [12], supplementary material
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Holzer et al. [19], 2 tortuosity types, porous Zr-oxide used as Diaphragm in pH sensor:
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τdir_geodesic, (image analysis/in-house)
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τindir_ele, (simulation of electrical conduction, GeoDict/Laplace solver/Ohm’s law)
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Holzer et al. [29], 2 tortuosity types, PEM GDL dry IP and TP, compression series:
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τdir_geodesic, (image analysis, in-house) var. thickness from in-situ μ-CT compression experiment
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(29b)
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τindir_ele (simulation of electrical conduction, GeoDict/Laplace solver/Ohm’s law)
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Holzer et al. [30], 2 tortuosity types, PEM GDL wet IP and TP, μ-CT imbibition experiment:
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(30a)
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τdir_geodesic, (image analysis/in-house, from dynamic XCT-imbibition experiment)
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(30b)
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τindir_ele, (simulation of electrical conduction, GeoDict/Laplace solver/Ohm’s law)
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Cooper et al. [32], LI-Battery (LiFePO4):
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τdir_pore_centroid, (image analysis/Avizo Fire, voxel based)
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(32b)
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τindir_therm, (sim. thermal cond., ‘StarCCM+’/Laplace solver Fourier’s law, mesh based)
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Saomoto et al. [55], mixed tortuosity types, monosized 2D ellipsoids, aspect ratios 1- 5
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τmixed_hydr_Streamline, (sim. of flow with Comsol/Image Analysis (IA) of 2D vel. field)
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(55b)
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τmixed_ele_Streamline, (simulation of el. conduction with Comsol/IA of 2D velocity field)
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τmixed_hydr_Vav, (simulation of flow with Comsol/IA of 2D velocity field)
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τmixed_ele_Vav, (simulation of el. conduction with Comsol/IA of 2D velocity field)
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3.3 Empirical Data for Different Tortuosity Types
3.4 Direct Comparison of Tortuosity Types Based on Selected Data Sets
3.4.1 Example 1: Indirect Versus Direct Pore Centroid Tortuosity
3.4.2 Example 2: Indirect Versus Direct Medial Axis Tortuosity
3.4.3 Example 3: Indirect Versus Direct Geodesic Tortuosity
3.4.4 Example 4: Indirect Versus Medial Axis Versus Geodesic Tortuosity
3.4.5 Example 5: Direct Medial Axis Versus Direct Geodesic Tortuosity
3.4.6 Example 6: Mixed Streamline Versus Mixed Volume Averaged Tortuosity
3.5 Relative Order of Tortuosity Types
3.5.1 Summary of Empirical Data: Global Pattern of Tortuosity Types
3.5.2 Interpretation of Different Tortuosity Categories
3.5.2.1 Direct and Mixed Tortuosities
3.5.2.2 Indirect Tortuosities
3.6 Tortuosity–Porosity Relationships in Literature
3.6.1 Mathematical Expressions for τ–ε Relationships and Their Limitations
Nr.
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τ type
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\({\tau }^{2} { }\left( {{or \tau }!} \right) = { f}\left( {\varepsilon } \right)\)
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Material type or microstructure type
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x = ..
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Reference(s)
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---|---|---|---|---|---|
1
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Geom. model
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\(\tau_{geometric} = \frac{1}{2}\left[ {1\frac{1}{2}\sqrt {1 - \varepsilon } + \frac{{\sqrt {\left( {1 - \sqrt {1 - \varepsilon } } \right)^{2} + \left( {1 - \varepsilon } \right)/4} }}{{1 - \sqrt {1 - \varepsilon } }}} \right]\)
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Geometric 2D model of square particles
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Yu and Li [81]
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3,
2
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Geom. model
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\(\tau_{geometric} = 1.23\frac{{\left( {1 - \varepsilon } \right)^{4/3} }}{{ x^{2} \varepsilon }}\)
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Packed particles →
Packed spheres →
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0.75
1
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Lanfrey et al. [82]
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4
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Geom. model
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\(\tau_{geometric} = \left( {\frac{19}{{18}}} \right)^{{\ln \left( \varepsilon \right)/\ln \left( {8/9} \right)}}\)
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Pore fractal model
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Li and Yu [83]
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5,
6,
7
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Hydr
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\(\tau_{hydr}^{2} = 1 - x\ln \left( \varepsilon \right)\)
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Spheres + fibers →
Plates + flakes →
High aspect ratio particles →
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0.5
1
3
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Pech and Renaud, cited in Comiti and Renaud [84]
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8
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Hydr
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\(\tau_{hydr}^{2} = \varepsilon^{ - x}\)
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Binary mixture of spheres
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0.4
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Mota et al. [85]
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9
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Hydr
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\(\tau_{hydr}^{2} = \frac{\varepsilon }{{1 - \left( {1 - \varepsilon } \right)^{2/3} }}\)
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Isotropic granular material
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Du Plessis et al. [86]
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10
11
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Hydr
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\(\tau_{hydr}^{2} = \sqrt {\frac{2\varepsilon }{{3\left[ {1 - x\left( {1 - \varepsilon } \right)^{2/3} } \right]}} + \frac{1}{3}}\)
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Monosized spheres:
Cubic packing →
Tetrahedral packing →
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1.209
1.108
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Ahmadi [87]
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12
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Hydr
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\(\tau_{mix\_hyd\_Vav} = 1 + 0.8 \left( {1 - \varepsilon } \right)\)
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Mono-sized solid rectangles in 2D
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Koponen et al. [88]
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13
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Electr
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\(\tau_{ele}^{2} = 1 + 0.5\left( {1 - \varepsilon } \right)\)
\(\tau_{ele}^{2} = \frac{{\left( {3 - \varepsilon } \right)}}{2}\)
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Dilute suspension of spheres
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Maxwell, 1873 [89]
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14,
15
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Electr
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\(\tau_{ele}^{2} = \varepsilon^{1 - x}\)
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Cementation Exponents:
For rocks →
For sediments →
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1.1
2
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Archie, 1942 [79]
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16
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Electr
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\({\varvec{\tau}}_{{{\varvec{ele}}}}^{2} = {\varvec{\varepsilon}}^{{1 - {\varvec{x}}}}\), identical with Bruggeman (see Nr 27)
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Packing of poly-dispersed spheres
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1.5
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Archie, 1942 [79]
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17,
18,
19,
20,
21,
22
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Electr
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\(\tau_{ele}^{2} = 1 - x\ln \left( \varepsilon \right)\)
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Spherical particles →
Monosized spheres →
Cubic particles →
Cylinders →
Overlapping spheres →
Monosized spheres →
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0.41
0.49
0.63
1.00
0.5
0.5
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Pech and Renaud, cited in Comiti and Renaud [84]
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23
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Diff
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\(\tau_{diff}^{2} = \varepsilon^{ - x}\)
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Mono-sized spheres
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0.4
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24
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Diff
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\(\tau_{diff}^{2} = \frac{\varepsilon }{{1 - \left( {1 - \varepsilon } \right)^{1/3} }}\)
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Catalyst
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Beeckman [91]
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25
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Diff
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\(\tau_{diff}^{2} = \varepsilon + x\left( {1 - \varepsilon } \right)\)
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Sand, silt, sediments
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2
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Iversen and Jørgensen [92]
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26
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Non spec
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\(\tau^{2} = 2 - \varepsilon\)
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Petersen [93]
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27
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Diff
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\(\tau_{diff}^{2} = \varepsilon^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} = \varepsilon^{1 - x}\)
for spheres identical with Archie
(See Nr 16)
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Polydisperse granular media:
Spheres →
Cylinders →
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1.5
2
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Bruggeman 1935 [94]
see also Tjaden et al. [95]
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28
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Electr
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\(\tau_{diff}^{2} = x_{1} \varepsilon^{{1 - x_{2} }}\)
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Battery electrodes and separators
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x1: 0.1–2.6
x2:1.27–5.2
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29
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Electr
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\(\tau_{ele}^{2} = \varepsilon^{ - x}\)
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Battery electrode
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0.5–2
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Ebner [39]
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30
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Diff
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\(\tau_{diff}^{2} = \varepsilon^{ - 1/3}\)
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Monodisperse granular media
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van Brakel and Heertjes [97]
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31
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Hydr
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\(\tau_{diff}^{2} = 1 - \ln \left( \varepsilon \right)\)
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Overlapping cylinders
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Tomadakis and Sotirchos [98]
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32
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Diff
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\(\tau_{diff}^{2} = 1 - \frac{\ln \left( \varepsilon \right)}{2}\)
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Overlapping spheres
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Weissberg [99]
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33
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Diff
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\(\tau_{diff} = 1 - x\ln \left( \varepsilon \right)\)
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Clays dry
and hygroscopic
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0.357
0.503
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Sun et al. [100]
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34
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Hydr
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\(\tau_{hydr}^{2} = \left( {\frac{{\left( {2 - \varepsilon } \right)}}{\varepsilon }} \right)^{2}\)
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Cation exchange resin
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Mackie [101]
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35
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Diff
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\(\tau_{diff} = \left[ {1 - x\left( {1 - \varepsilon } \right)^{ - 1} } \right]\)
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Granular media with: spheres →
Cubes →
Large and →
Small parallelepipeds →
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0.6
0.73
1.07
1.21
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Pisani [102]
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36
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Hydr
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\(\tau_{mixed\_hydr\_Vav}^{2} = x_{1} - x_{2} \ln \left( {\varepsilon^{2} } \right)\)
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Monosized spheres
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x1: 1.1842
x2: 0.6579
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Jin et al. [103]
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37
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Hydr
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\(\tau_{mixed\_hydr\_Vav}^{2} = x_{1} - x_{2} \ln \left( {\varepsilon^{2} } \right)\)
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Monosized spheres
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x1: 0.9463
x2: 0.7173
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38
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Hydr
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\(\tau_{mixed\_hydr\_Vav} = x_{1} \left( {1 - \varepsilon } \right) + x_{2}\)
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Monosized spheres
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x1: 0.8002
x2: 1.0454
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Jin et al. [103]
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39
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Hydr
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\(\tau_{mixed\_hydr\_Vav} = x_{1} \left( {1 - \varepsilon } \right) + x_{2}\)
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Monosized spheres
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x1: 0.9119
x2: 0.9340
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40
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Hydr
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\(\tau_{mixed\_hydr\_Vav} = x_{1} - x_{2} \ln \left( \varepsilon \right)\)
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Monosized spheres
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x1: 1.1133
x2: 0.4845
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Jin et al. [103]
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41
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Hydr
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\(\tau_{mixed\_hydr\_Vav} = x_{1} - x_{2} \ln \left( \varepsilon \right)\)
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Monosized spheres
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x1: 1.0104
x2: 0.5541
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42
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Hydr
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\(\tau_{mixed\_hydr\_Vav} = 1 - x_{1} \ln \left( \varepsilon \right)\)
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2D monosized spheres/cubes
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0.5/0.541
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Saomoto and Katagiri [54]
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43
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Ele
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\(\tau_{mixed\_ele\_Vav} = 1 - x_{1} \ln \left( \varepsilon \right)\)
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2D monosized spheres/cubes
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0.2/0.19
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Saomoto and Katagiri [54]
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