Dispersion of particles by spontaneous interparticle percolation through unconsolidated porous media

We have performed extensive experimental and numerical studies of spontaneous percolation of small beads through an unconsolidated porous media made with large glass beads. In this paper, an experimental setup and a fast “discrete element method” algorithm are presented to deal with large numbers of particles during our interparticle percolation phenomenon studies. In all the experimental and numerical analyses, the size ratio between the moving beads and the stable packing was chosen larger than the geometrical trapping threshold: ${\xi }_c={\left(\frac{2}{\sqrt{3}}-1\right)}^{-1}=6.464\dots$. We measure the longitudinal and transverse dispersion coefficients versus the height of the porous medium or the number of falling small beads. The influence of bead properties such as density, diameter, or restitution coefficients was investigated by using either steel or glass beads. The individual description of these effects and their explanations were made possible by confrontation and coupling between experimental and numerical results. Indeed, with our numerical model, individual analysis of the effects of these mechanical or geometrical parameters were made possible and carried out.

Figure 1

Schematic description of the experimental setup. The small bead dispenser is placed at $h$ from the top of the porous media of an height $H$. The collecting box is put on the scale.

Figure 2

Illustration of contact between two soft overlapping spheres $i$ and $j$ in the MD approach. Notations used in this paper are also mentioned.

Figure 3

(Color online) Distribution function of particle positions at the exit of the porous medium for a launch of 10 000 glass particles of mass $m_s=2.1\times {10}^{-3}\text{g}$ and with a porous medium of height $H/D=8.125$.

Figure 4

Normalized mass measurements delivered by the piezoelectric scale. A fit with Eq. (11), and determined with unweighted least-squares method, is also represented with continuous lines. $N$ corresponds to the number of falling steel particles and $N_p$ is the rescaled value, which was defined in Sec. 2. Here $H/D=10$.

Figure 5

Mean transit time dependency on $H/D$ for $N=500$, $N=10000$, and $N=70000$. Results obtained with steel percolating particles of $d=1\text{mm}$ inside glass beads of diameter $D=16\text{mm}$. Linear fits are also represented.

Figure 6

Variance of the particle distribution function in the transverse direction at the output of the porous medium versus $H$ for different numbers of falling particles. In these experiments, we used small steel beads (a) and glass ones (b) of diameter $d=1\text{mm}$ inside packings of glass beads of diameter $D=16\text{mm}$ (i.e., size ratio $D/d=16$). Unweighted linear least-squares regressions are also represented.

Figure 7

Evolution of transverse dispersion coefficient $D_{\perp }$ versus $N_p$. The experiments were performed with small steel (●) and glass (◻) beads of diameter $d=1\text{mm}$ through a porous medium made with glass bead of diameter $D=16\text{mm}$ (size ratio $D/d=16$). We have also represented unweighted linear least-squares regressions.

Figure 8

Dispersion coefficients in the flow direction determined with Eq. (11), versus the parameter $N_p$ for three different packing heights $H$. In these experiments, we use small steel beads of diameter $d=1\text{mm}$ inside a packing of glass beads of diameter $D=16\text{mm}$ (size ratio $D/d=16$). In order to improve visual quality of the figure, all error bars are not displayed.

Figure 9

(Color online) Snapshots of flow of 20 000 particles through a porous medium. Note that for convenience only half system is represented.

Figure 10

Time evolutions of $⟨{\left(\Delta z\right)}^2⟩$ and $⟨{\left(\Delta r\right)}^2⟩$ versus time obtained by the MD model of soft spheres, with $N=8000$, $D/d=10$, $D=20\text{mm}$, $e_1=0.84$, $e_2=0.99$, and $H/D=14$. Linear fits obtained with unweighted least-squares method are represented. The linear evolution of $⟨{\left(\Delta z\right)}^2⟩$ is observed between 0.2 and 1 s as shown by the straight line. After 1 s the decrease in the moving particles amount explains the divergence which is only an artifact of the calculus method. The same behavior is observed on evolution of $⟨{\left(\Delta r\right)}^2⟩$ but divergence starts to be visible after 1.5 s.

Figure 11

Evolution of dispersion coefficients versus $N_p$ for a size ratio $D/d=10$. These results have been obtained with $H/D=14$, $D=20\text{mm}$, $e_1=0.84$, and $e_2=0.99$. Error bars are also represented.

Figure 12

Evolution of dispersion coefficients versus $N_p$ for a size ratio $D/d=16$. Error bars are shown for accuracy analysis. Results are obtained with $H/D=14$, $D=20\text{mm}$, $e_1=0.84$, and $e_2=0.99$.

Figure 13

Evolution of $D_{\perp }$ versus $D/d$ for different values of $N$. The results are obtained with $H/D=14$, $D=20\text{mm}$, $e_1=0.84$, and $e_2=0.99$. To improve the readability, only significant error bars are displayed.

Figure 14

Evolution of $D_{\parallel }$ versus $D/d$ for different values of $N$. Results have been obtained by the MD model of soft spheres with $H/D=14$, $D=20\text{mm}$, $e_1=0.84$, and $e_2=0.99$.

Figure 15

Evolution of (a) $D_{\perp }$ and (b) $D_{\parallel }$ with $e_1$ for $N=1000$, $N=2000$, and $N=5000$. The results are obtained with $H/D=14$, $D/d=10$, $D=20\text{mm}$, and $e_2=0.99$. Unweighted linear least-squares regressions are also represented.

Figure 16

Evolution of (a) $D_{\perp }$ and (b) $D_{\parallel }$ versus $e_2$ for $N=2000$, $D/d=10$, $D=20\text{mm}$, $H/D=14$, and $e_1=0.84$. An unweighted linear least-squares regression is represented in case (a).

Figure 17

Evolution of $D_{\parallel }$ and $D_{\perp }$ versus density ${\rho }_s$ of the small particles for $N=2000$, $D/d=10$, $D=20\text{mm}$, $H/D=14$, $e_2=0.99$, and $e_1=0.84$.