# Numerical Investigation of a Novel Grinding Device
## 3 Results
### Effect of Grinding Bead Size  
The balance between the size of precursor and grinding beads is a critical aspect of bead milling processes. Indeed, a relatively large inter-bead space in a close packing configuration can dramatically reduce the milling efficiency [10]. In order to reproduce different bed configurations, further simulations were carried out by varying the size and number of grinding beads. In particular, three other systems with fixed total solid volume fraction were considered:  
* Bi-disperse, 4 Ag beads of 3 mm and 135 ZrO2 beads of 2 mm diameter;
* Tri-disperse, 4 Ag beads of 3 mm and 55 ZrO2 beads of 2 mm and 10 ZrO2 beads of 4 mm diameter;
* Autogenous, 44 Ag beads of 3 mm diameter. The comparison among these new systems and the reference in terms of impact kinetic energy, impact angle, and collision frequency is shown in Figure 6, where, for the sake of simplicity, only the median values are reported. Figure 5: Density histogram and contour plot of the joint probability density function for impact angle and impact kinetic energy at (**a**,**d**) 300 rpm, (**b**,**e**) 600 rpm, and (**c**,**f**) 900 rpm. The left column refers to collisions between all particles in the system, while the right column refers to those involving at least one Ag particle. On the edges of each map, the marginal probability density functions and the corresponding quartiles are also shown (the median in black and the first and third quartiles in red). Furthermore, the joint probability density functions of the bi- and tri-disperse systems are shown in Figure 7, while the PDFs of the autogenous system are not reported since they are very similar to the standard case for the overall collisions (left-hand column of Figure 5). The impact angle was found to be almost independent of the filling configuration, with the greatest variation at low \(\Omega\) (see, e.g., Figure 7a,d corresponding to Figure 6b at 300 rpm). The detected median value of 30\({}^{\circ}\) in the experimental operative regime is then characteristic of the geometry under investigation. The smaller and more numerous grinding beads of the bi-disperse system induce a reduction in the impact kinetic energy and an increase in the impact frequency (Figure 6). The joint distribution is slightly more spread towards higher \(\theta_{\text{impact}}\) values (Figure 7). The tri-disperse system, with fewer small and some large particles, partially recovers in terms of \(k\epsilon_{r}\), but \(v_{\text{c}}\) is reduced almost to the standard configuration value. The shape of the PDF, which is more similar to the standard case than the bi-disperse one, is in agreement with this observation. Assuming that a certain level of impact energy must be reached in order to induce abrasion and produce NPs, neither the bi- nor the tri-disperse systems seems to be optimal. Indeed, the same energy level of the standard system can be achieved by increasing the stirring speed, but this determines a reduction in the impact frequency to approximately the same value of the mono-disperse system, thus nullifying the benefits. Figure 6: Comparison of the investigated grinding systems in terms of (**a,d**) impact kinetic energy, (**b,e**) impact angle, and (**c,f**) collision frequency as a function of the stirring velocity. The left column refers to collisions between all particles in the system, while the right column refers to those involving at least one Ag particle. Figure 7: Density histogram and contour plot of the joint probability density function for impact angle and impact kinetic energy at (**a,d**) 300 rpm, (**b,e**) 600 rpm, and (**c,f**) 900 rpm for the bi-disperse (left column) and tri-disperse (right column) system. On the edges of each map, the marginal probability density functions and the corresponding quartiles are also shown (the median in black and the first and third quartiles in red). A different discussion applies to the autogenous system.