## 3 Results and Discussion  
### Bulk Density  
The results of loose density (\(\rho_{l}\)), tapped density (\(\rho_{t}\)), and theoretical density are summarized in Table 3. The results show that the bulk densities of solid particles depend somewhat on the particulate size distribution, as well as on the compaction. The loose bulk density of CB with the small particle size (0.3 mm) is slightly lower (around 0.9% lower) than the tapped bulk density of the 0.5 mm beads, indicating that the larger and nearly spherical 0.5 mm beads pack slightly better under gravity alone. In contrast, the tapped bulk density of the 0.3 mm CBs is noticeably greater (by around 6.0%), indicating that the smaller CBs may be slightly less uniform in size and shape and that the surface friction between the smaller beads may deter close packing under gravity alone. In all cases, the porosity of the tapped CB beads (41% and 46%) is reasonably close to but not less than the expected porosity of randomly packed spheres (around 36%). This behavior is expected for the nearly spherical shape of these artificial particles. Generally, for all particulate materials, the results showed that the tapped bulk density is at least somewhat larger than the loose density for solid particulates of a small size. This is attributed to the rearrangement of particles after disturbance thereby decreasing the inter-particle voids.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material Type** & **Theoretical Density \(\rho_{th}\)** & **Loose Density \(\rho_{l}\)** & **Tapped Density** & **Porosity [38; 42]** \\  & **[kg/m\({}^{3}\)]** & **[kg/m\({}^{3}\)]** & **[kg/m\({}^{3}\)]** & **\(\left(\mathbf{1}-\frac{\rho_{b}}{\rho_{th}}\right)\times\mathbf{100}\)** \\ \hline RWS & 2756 \(\pm\) 28.7 & 1571 \(\pm\) 8.8 & 1653 \(\pm\) 37 & 43 & Loose \\  & & & & 40 & Tapped \\ \hline RRS & 2771 \(\pm\) 29 & 1574 \(\pm\) 8.8 & 1679 \(\pm\) 38 & 43 & Loose \\  & & & & 39 & Tapped \\ \hline SOS & 3260 \(\pm\) 29.7 & 1873 \(\pm\) 9.3 & 2155 \(\pm\) 37 & 43 & Loose \\  & & & & 34 & Tapped \\ \hline USOS & 1913 \(\pm\) 29.2 & 1445 \(\pm\) 8.6 & 1572 \(\pm\) 36 & 24 & Loose \\  & & & & 18 & Tapped \\ \hline CB–0.3 mm & 3321 \(\pm\) 36.6 & 1796 \(\pm\) 10.2 & 1944 \(\pm\) 43 & 46 & Loose \\  & & & & 41 & Tapped \\ \hline CB–0.5 mm & 3321 \(\pm\) 42.3 & 1813 \(\pm\) 10.3 & 1828 \(\pm\) 44 & 45 & Loose \\  & & & & 45 & Tapped \\ \hline \hline \end{tabular}
\end{table}
Table 3: Measurement results of loose and tapped density of particulate materials at ambient temperature.  
Comparing the measured theoretical density results in Table 3 with those in Table 1, which are adapted from [16; 17; 18; 19; 20; 21; 22; 23], it is clear that for RWS and RRS, the theoretical densities are slightly higher than that of pure SiO\({}_{2}\). This is likely due to the small quantities of heavier impurities (like iron oxides) in the natural particulate samples. The density of SOS is almost identical to that of pure olivine [17], indicating that it has minimal impurities. However, the density of USOS is much smaller. That is due to a large proportion of impurities in the sample. The loose density measured by [53] for typical sand is 1450 kg/m\({}^{3}\), whereas that for RWS is 1571 kg/m\({}^{3}\). The higher loose density for RWS is due to its 98% SiO\({}_{2}\) content, whereas that in [53] is 91 to 97%. The other reason for the deviation is that [53] used relatively bigger sizes of particles up to 465 um. The porosity measured in [38] for CB is almost the same as that measured in this study.