Optical method for measuring the volume fraction of granular media: Application to faced-centered cubic lattices of monodisperse spheres

In order to understand the dynamics of granular flows, one must have knowledge about the solid volume fraction. However, its reliable experimental estimation is still a challenging task. Here, we present the application of a stochastic-optical method (SOM) [L. Sarno et al., Granul. Matter 18, 80 (2016)] to an array of spheres arranged according to faced-centered cubic lattices, where spheres' locations are known a priori. The purpose of this study is to test the robustness of the image binarization algorithm, introduced in the SOM for the indirect estimation of the near-wall volume fraction through an optically measurable quantity, defined as two-dimensional volume fraction. A comprehensive range of volume fractions and illumination conditions are numerically and experimentally investigated. The proposed binarization algorithm is found to yield reasonably accurate estimations of the two-dimensional volume fraction with a root-mean-square error smaller than 0.03 for all investigated illumination conditions. A slightly worse performance is observed for samples with relatively low volume fractions (<0.3), where the binarization algorithm occasionally cannot identify the surface elements in the second and third layers of the regular lattice.

Figure 1

Sketch of the experimental setup for the SOM method. The viewing direction is normal to $\mathrm{\Delta }$, while $\zeta$ is the angle between viewing and lighting directions.

Figure 2

Centered FCC disperse lattice of equal spheres ($\mathbf{\nabla }$-type orientation) with $\lambda =1.33$.

Figure 3

(a) Relationships ($c_{2\mathrm{D}},c_{3\mathrm{D}}$) for FCC deterministic lattices ($5d\times 5d$ IWs). The averages of $c_{2\mathrm{D}}$ are reported together with their minimum and maximum values, obtained by shifting the IW in a range $\pm 1.25d$ along $x$ and $z$ directions. (b) Standard deviations ${\sigma }_{c_{2\mathrm{D}}}\left(c_{3\mathrm{D}}\right)$. Inset: averaged standard deviations, $\overline{{\sigma }_{c_{2\mathrm{D}}}}$.

Figure 4

Binarized images of S-virt-7 ($c_{3\mathrm{D}}=0.222$) under different illumination conditions. White pixels correspond to visible and illuminated surface elements, concurring to $c_{2\mathrm{D}}$ calculation. (a) POS-1; (b) POS-2; (c) POS-3; (d) POS-4.

Figure 5

Picture of the sample S-4 ($c_{3\mathrm{D}}=0.374$) taken under the illumination condition POS-1.

Figure 6

Comparisons between experimental and theoretical values of $\overline{{\sigma }_{c_{2\mathrm{D}}}}$ for various angles of incidence of light, ζ.

Figure 7

Comparison between the experimental data ($c_{2\mathrm{D}},c_{3\mathrm{D}}$), obtained by image binarization ($D_N=1.5d, p=1.0d$), and the corresponding numerical data: (a) POS-1; (b) POS-2; (c) POS-3; (d) POS-4.

Figure 8

Comparison between the experimental data ($c_{2\mathrm{D}},c_{3\mathrm{D}}$), obtained by image binarization ($D_N=1.5d, p=1.5d$), and the corresponding numerical data: (a) POS-1; (b) POS-2; (c) POS-3; (d) POS-4.

Figure 9

POS-1: comparisons among binarized images (IW $5d\times 5d$), obtained with $D_N=1.5d$ and different values of the MAF ($p=0.5d, 1.0d, 1.5d$), and the binary images, obtained numerically.

Figure 10

POS-4: comparisons among binarized images (IW $5d\times 5d$), obtained with $D_N=1.5d$ and different values of the MAF ($p=0.5d, 1.0d, 1.5d$), and the binary images, obtained numerically.