Mechanical trapping of particles in granular media

Mechanical trapping of fine particles in the pores of granular materials is an essential mechanism in a wide variety of natural and industrial filtration processes. The progress of invading particles is primarily limited by the network of pore throats and connected pathways encountered by the particles during their motion through the porous medium. Trapping of invading particles is limited to a depth defined by the size, shape, and distribution of the invading particles with respect to the size, shape, and distribution of the host porous matrix. Therefore, the trapping process, in principle, can be used to obtain information about geometrical properties, such as pore throat and particle size, of the underlying host matrix. A numerical framework is developed to simulate the mechanical trapping of fine particles in porous granular media with prescribed host particle size, shape, and distribution. The trapping of invading particles is systematically modeled in host packings with different host particle distributions: monodisperse, bidisperse, and polydisperse distributions of host particle sizes. Our simulation results show quantitatively and qualitatively to what extent trapping behavior is different in the generated monodisperse, bidisperse, and polydisperse packings of spherical particles. Depending on host particle size and distribution, the information about extreme estimates of minimal pore throat sizes of the connected pathways in the underlying host matrix can be inferred from trapping features, such as the fraction of trapped particles as a function of invading particle size. The presence of connected pathways with minimum and maximum of minimal pore throat diameters can be directly obtained from trapping features. This limited information about the extreme estimates of pore throat sizes of the connected pathways in the host granular media inferred from our numerical simulations is consistent with simple geometrical estimates of extreme value of pore and throat sizes of the densest structural arrangements of spherical particles and geometrical Delaunay tessellation analysis of the pore space of host granular media. Our results suggest simple relations between the host particle size and trapping features. These relationships can be potentially used to describe both the dynamics of the mechanical trapping process and the geometrical properties of the host granular media.

Figure 1

The two essential schematics of mechanical trapping mechanisms: (a) internal filter cake, when particles are trapped inside the porous media; (b) external filter cake, when particles are trapped at the surface above porous media. Note the particle size dependence and difference in deposit morphology (modified after McDowel et al. [2]).

Figure 2

2D schematic of the numerical procedures to simulate mechanical trapping and percolation of fine invading particles in granular medium. First, the host matrix is generated by pouring host particles into a cylindrical container under the influence of gravity. Second, the fine particles invade host matrix using a similar numerical procedure. Both spherical and nonspherical particle shapes can be implemented in this methodology.

Figure 3

Four circles are mutually tangent at six distinct points. One solution of Descartes' theorem, Eq. (2), is the diameter of inscribed sphere (a), and other solution is diameter of circumscribed sphere (b).

Figure 4

Visualization of the Delaunay tessellation of the pore space of a random three-dimensional loose monodisperse packing of spherical particles (a) into a set of nonoverlapping tetrahedral structures (b).

Figure 5

The distribution of the largest inscribed sphere with the diameter $d_p$ that can be placed inside of each formed nonoverlapping tetrahedral structures whose vertices are the centers of four neighboring spheres from the Delaunay tessellation analysis in (a) monodisperse, (b) bidisperse with $D_1/D_2=2$, (c) $D_1/D_2=3$ and varying volume percent of the smallest particle size, (d) polydisperse with normal distribution of diameter of host particle and varying standard deviation ratio, $\gamma =\sigma /D_{50}, 0.13\le \gamma \le 0.33$, (e) polydisperse with log-normal distribution of diameter of host particle and varying standard deviation ratio, $\gamma =\sigma /D_{50}, 0.255\le \gamma \le 0.691$, (f) polydisperse with beta distribution of diameter of host particle with fixed $\alpha =2.0$ and varying $\beta$ parameter from 2.0 to 5.0.

Figure 6

(a) Fraction of trapped particles out of 200 invading into the monodisperse packing of 2000 identical frictionless spherical particles (open circles are ten realizations, solid squares are average of ten realizations) is compared with Ghidaglia et al.'s [11] lab data (solid yellow diamonds); (b) ${\lambda }_m$, normalized median penetration depth of invading particles into host matrix, compared with Ghidaglia et al.'s [11] lab data.

Figure 7

(a) Fraction of trapped particles out of 200 invading into the monodisperse packing of 2000 identical spherical particles with ratio $\zeta =D_{\mathrm{container}}/H_{\mathrm{container}}$ between container diameter $D_{\mathrm{container}}$ and height $H_{\mathrm{container}}$ of 0.5, 1, 3.5, and 12; (b) ${\lambda }_m$, normalized median penetration depth of invading particles into host matrix for different container diameters to height ratios $\zeta$, compared with Ghidaglia et al.'s [11] lab data.

Figure 8

(a) Fraction of trapped particles out of 50, 200, 250, 500, 1000, 2000, and 4000 invading into the monodisperse packing of 2000 identical spherical particles; (b) number of trapped particles as a function of number of entering particles for size ratios $d_{\mathrm{invading}}/D_{\mathrm{host}}$ of 0.15, 0.16, 0.17, and 0.18.

Figure 9

(a) Host and invading particles with spherical and nonspherical shape. Nonspherical shape is generated using coherent noise (Perlin noise) modification of the spherical particle surface. Note that the spherical and nonspherical particles have the same volume, and the ratio between smallest diameter of nonspherical particle and diameter of spherical particle is 0.84; (b) normalized median penetration depth of invading particles into medium for spherical and nonspherical shapes.

Figure 10

Fraction of trapped particles out of 200 invading into the bidisperse packing of 2000 spherical host particles with (a) $D_1/D_2=2$, (b) $D_1/D_2=3$ and varying volume percent of the smallest particle size between $20\%$ and $100\%$ (monodisperse).

Figure 11

(a) Normal distribution of the host particle diameter normalized by $D_{50}$ (50th percentile) diameter of host particle and varying standard deviation ratio, $\gamma =\sigma /D_{50}, 0.13\le \gamma \le 0.33$; (b) fraction of trapped particles out of 200 invading into the polydisperse packing of 2000 spherical particles with normal distribution of host particle sizes.

Figure 12

(a) Beta distribution of the host particle sizes normalized by $D_{50}$ (50th percentile) diameter of host particle with fixed $\alpha =2.0$ and varying $\beta$ parameter from 2.0 to 5.0; (a) fraction of trapped particles out of 200 invading into the polydisperse packing of 2000 spherical particles with beta distribution of host particle sizes.

Figure 13

(a) Log-normal distribution of the host particle diameter by $D_{50}$ (50th percentile) diameter of host particle and varying standard deviation ratio, $\gamma =\sigma /D_{50}, 0.255\le \gamma \le 0.691$; (b) fraction of trapped particles out of 200 invading into the polydisperse packing of 2000 spherical particles with log-normal distribution of host particle sizes.

Figure 14

Relation between host particle size at 50th percentile and invading particle size at $50\%$ of trapped particles in monodisperse, bidisperse, and polydisperse packing of spherical particles.

Figure 15

Relation between $d_{\mathrm{IQRN}}$ of host particle size and $d_{\mathrm{IQRN}}$ of invading particle size in monodisperse and polydisperse packing of spherical particles.