Percolation of a fine particle in static granular beds

We study the percolation of a fine spherical particle under gravity in static randomly packed large-particle beds with different packing densities $\varphi$ and large to fine particle size ratios $R$ ranging from 4 to 7.5 using discrete element method simulations. The particle size ratio at the geometrical trapping threshold, defined by three touching large particles, $R_t=\sqrt{3}/(2-\sqrt{3})=6.464$, divides percolation behavior into passing and trapping regimes. However, the mean percolation velocity and diffusion of untrapped fine particles, which depend on both $R$ and $\varphi$, are similar in both regimes and can be collapsed over a range of $R$ and $\varphi$ with the appropriate scaling. An empirical relationship for the local percolation velocity based on the local pore throat to fine particle size ratio and packing density is obtained, which is valid for the full range of size ratio and packing density we study. Similarly, in the trapping regime, the probability for a fine particle to reach a given depth is well described by a simple statistical model. Finally, the percolation velocity and fine particle diffusion are found to decrease with increasing restitution coefficient.

Figure 1

Left: Perspective view of computational box and two representative fine-particle trajectories (sampled at 5 ms intervals) in the free sifting regime ($R=6.5$) for a static large-particle bed with $\varphi =0.577$. Color map indicates dimensionless time, $t/\sqrt{d_l/g}$, during the trajectory. Upper right: Top view of the trajectory of one particle. Bottom right: Pore throat with a fine particle (red) at the geometrical trapping threshold limit.

Figure 2

Positions of ${10}^4$ noninteracting fine particles in the $xz$ plane at different times for packing density $\varphi =0.639$ in (a) the trapping regime ($R=5$), and (b) the passing regime ($R=6.5$). Black dashed outlines are the fully periodic boundaries of the computational domain in the $xz$ plane; large particles are omitted for clarity. Red curves indicate the fine particle number distributions versus depth, and blue dashed curves in (b) are fits to a normal distribution.

Figure 3

Passing regime: Nondimensional mean vertical displacement versus nondimensional time interval for fine particles with $R=7$ at various $\varphi$.

Figure 4

Trapping regime: Conditionally averaged nondimensional mean vertical displacement versus nondimensional time interval with respect to trapping time for fine particles with $R=5$ at different $\varphi$. Symbols are defined in Fig 3. The three regions show where fine particles percolate at a constant velocity (green), decelerate (yellow), and stop (red). Stopping distance, $s_d$, is the vertical distance over which fine particles slow to a stop.

Figure 5

(a) Nondimensional percolation velocity $v_p/\sqrt{g{d}_l}$ vs. packing density $\varphi$ for various $R$ across the passing and trapping regimes for restitution coefficient $e=0.8$. The dashed curve approximates the boundary between the passing (above) and trapping (below) regimes. (b) Scaling the non-dimensional percolation velocity by $\sqrt{R}$ removes the $R$ dependence. (c) Scaling the non-dimensional percolation velocity using both $R$ and $\varphi$ makes it nearly constant. Data points for $R=7$ and $\varphi =0.577$ at different simulation conditions (black $\times$) are offset slightly to the right to make them more visible. In (b) and (c), the dashed line is a linear fit to the data.

Figure 6

Nondimensional mean squared displacement (MSD) of fine particles versus time interval $\mathrm{\Delta }t$ in the passing regime ($R=7$): (a) projected to the $xy$ plane (horizontal) and (b) in the $z$ direction (vertical). Results for various $\varphi$ are shown, and symbols are defined in Fig. 3.

Figure 7

Diffusion coefficient in the (a)–(c) $xy$ plane projection (horizontal) and (d)–(f) $z$ direction (vertical) scaled by (a), (d) large particle diameter, $\sqrt{g{d}_l^3}$; (b), (e) $\sqrt{g{d}_l^3{R}^3}$; and (c), (f) expressed as Péclet numbers versus packing density for various $R$ in the passing and trapping regimes for restitution coefficient $e=0.8$. Data symbols for $R=7$ and $\varphi =0.577$ at different simulation conditions, as indicated in (a), are offset slightly to the right to make them more visible. Dashed curves in (a), (d) approximate the boundary between passing (above) and trapping (below) regimes.

Figure 8

Effects of restitution coefficient $e$ in the passing regime on (a) percolation velocity, (b) diffusion coefficients and (c) Péclet numbers. Dashed black line is linear fit of experimental results for ${\text{Pe}}_{xy}$ (small red filled circles) [20]. For (b) and (c), only results for $R=7,\varphi =0.577$ are shown for clarity.

Figure 9

Probability $P$ in the trapping regime that a fine particle reaches scaled depth $(z_0-z_f)/d_l$ for (a) $R=4$, (b) $R=5$, and (c) $R=6$ for various $\varphi$ (symbols defined in Fig. 3). Uncertainty bars plotted in (b) for $R=5, \varphi =0.577$ are due to finite fine particle number (see text). Solid lines are model predictions using Eq. (4).

Figure 10

(a) Red sphere inscribed in a tetrahedron formed from four large bed particles. (b) Four green spheres positioned at the narrowest point of each of the four pore throats of the tetrahedron indicate the maximum diameter of a passable fine particle. (c) Two-dimensional illustration of two Delaunay cells that can be merged (see text) [31, 32].

Figure 11

Two randomly packed large particle beds (purple spheres) and corresponding pore-throat spheres (color map indicates pore-throat size) obtained using the L1 method [32] for (a) $\varphi =0.577$ and (b) $\varphi =0.639$.

Figure 12

(a) Pore-throat size distributions, (b) corresponding cumulative distribution functions (CDF), and (c) passing probabilities versus $R=d_l/d_f$ for different $\varphi$. Color and corresponding $\varphi$ values are as in Fig. 3, and only data for $\varphi =0.526,0.577,0.607$, and 0.639 are shown in (a) for clarity.

Figure 13

Average vertical distance normalized by large particle diameter, $\mathrm{\Delta }{\overline{z}}_p/d_l$, between passable pore throats on the paths of all fine particles for different particle size ratios $R$ at different packing densities $\varphi$. Uncertainty bar depicting the standard deviation of the measurement is only shown for $R=4$ for clarity.

Figure 14

(a) Nondimensional local percolation velocity, $v_{p,l}/\sqrt{g{d}_l}$, versus pore throat to fine particle size ratio, $d_p/d_f$, for $R=7.5$ at various $\varphi$. Symbols are defined in Fig. 3. (b) Multiplying $v_{p,l}/\sqrt{g{d}_l}$ by $\varphi$ collapses the data. (c) Additional data for other size ratios, $4⩽R⩽7.5$, both in the trapping and passing regimes with symbols defined in Fig. 5, and the full range of packing densities (not differentiated by symbols) also collapse. Uncertainty bars indicate the standard deviation of the measurement for $\varphi =0.577$ and $R=7.5$. The white line is a linear fit through the data.

Figure 15

Nondimensional local percolation velocity, $v_{p,l}/\sqrt{g{d}_l}$, versus pore throat to fine particle size ratio, $d_p/d_f$, for three restitution coefficients with $R=7$ and $\varphi =0.577$.