# Comparative Study to Evaluate Mixing Efficiency of Very Fine Particles
## 5 Numerical Analysis Results
### Mixing Efficiency Analysis of Tumbling V Mixer with CGM Scaling  
We analyzed the impact of various CGM scaling factors on the mixing efficiency in the tumbling V mixer, based on the DEM simulation results. Simulations were performed under six CGM conditions (CGM = 1, 2, 3, 4, 5, and 7). For each case, the temporal variation in the LMI and the evolution of the particle distributions were evaluated. Figure 13 illustrates the temporal evolution of the LMI values across all CGM cases. While the absolute LMI values differed among the CGM levels, the general trends were consistent. Notably, the differences in the LMI values between the scaling levels tended to diminish over time, which indicates that the mixing behavior converged as the simulation progressed. Although the simulations were concluded before complete mixing was achieved, we anticipate that the deviations between the CGM cases would continue to diminish if the simulations were extended beyond 30 s.  
Figure 14 displays the particle distributions at multiple time points for each CGM condition. Across all CGM levels, the macroscopic mixing patterns appeared visually consistent, which indicates that CGM scaling does not significantly alter the overall mixing behavior. However, simulations with finer CGM levels (e.g., CGM = 1) provided smoother and more detailed representations of the mixing interface, akin to high-resolution imaging. As the CGM factor increased, the granularity of the particle representation became coarser, slightly reducing visual clarity, although the overall mixing characteristics remained unchanged. Table 3 presents the statistical metrics of the LMI (average, maximum, and minimum) during the final second of the simulation (29-30 s), along with the total number of particles, the simulation time, and the estimated error percentages. The results demonstrate that increasing the CGM factor substantially reduced the computational time by more than two orders of magnitude, while maintaining the LMI error within an acceptable range. We expect that prolonged simulation would reduce the error even further, which highlights the practical advantages of CGM for large-scale simulations. In Table 3, the error is defined as the absolute difference between the average LMI value of each CGM case and the non-CGM  
case (CGM = 1), expressed as a percentage. This definition is used consistently throughout the following subsections. For the tumbling V mixer, the LMI error exceeded 10% in most CGM cases, with the highest observed deviations surpassing 16%. Even the lowest-error case (CGM = 7) showed a non-negligible deviation of 8.56%. These elevated errors are attributed to the fact that the system had not yet reached a sufficiently mixed state by the end of the simulation, as reflected in the relatively low final LMI of approximately 0.300 in the non-CGM case. To investigate this trend further, we conducted additional comparisons at earlier time intervals (i.e., 25-26 s and 27-28 s). The results indicate a progressive decrease in the error over time, suggesting that the high errors observed at 29-30 s are likely transient effects of incomplete mixing. These results are summarized in Table 4, which presents the LMI values and the corresponding error percentages across multiple-time windows and CGM levels.