# Comparative Study to Evaluate Mixing Efficiency of Very Fine Particles
## 4 Numerical Analysis Models
### Sampling Zone Definition for LMI Calculation  
To quantitatively assess mixing uniformity using the LMI, the internal domain of each mixing system was discretized into uniformly sized cubic cells. This discretization enabled the measurement of local particle compositions, which is essential for statistical variance calculations in LMI. We uniformly divided the internal volume of each mixer into regularized cubic cells, ensuring equal spatial resolution. The cell dimensions were determined based on each mixer's geometry to balance resolution with computational feasibility.  
As mentioned earlier, only cells containing more than 10 particles during the sampling period were considered in the final variance calculation. This threshold removed sparsely populated zones, which could introduce statistical noise and distort the LMI values. Figure 11 shows representative examples of spatially subdivided mixing geometries. All cells in these figures are of equal size and represent the initial sampling zones prior to any filtering. The described spatial partitioning and filtering approach was consistently applied across all five mixing processes to ensure comparability and statistical reliability when evaluating mixing efficiency. Figure 12 presents the filtered sampling zones, highlighting only those regions that satisfy the particle-count threshold. This filtering strategy ensured that the LMI would reflect actual mixing behavior rather than being influenced by sparse or underpopulated regions.  
Figure 11: Uniform cell-based spatial division of mixing domain: (**a**) tumbling V mixer; (**b**) ribbon-blade mixer; (**c**) paddle-blade mixer; (**d**) vertical-blade mixer; (**e**) conical-screw mixer.  
This zone partitioning and filtering approach was consistently applied across all five mixing processes to maintain comparability and statistical reliability in the LMI evaluation.  
To ensure meaningful statistical evaluation under varying CGM scaling conditions, the bin size used for spatial discretization was adjusted accordingly. In cases where the particle size increased due to coarse graining, using smaller bins led to many cells having insufficient particle counts, which would potentially compromise the reliability of the LMI  
Figure 12: Sampling zones satisfying particle-count threshold (>10 particles per cell): (**a**) tumbling V mixer; (**b**) ribbon-blade mixer; (**c**) paddle-blade mixer; (**d**) vertical-blade mixer; (**e**) conical-screw mixer.  
calculations. Therefore, the bin size was scaled to ensure that most cells would contain more than 10 particles, thereby satisfying the threshold for inclusion in the LMI computation. Additionally, although the bin sizes varied across cases, the total particle mass involved in the LMI evaluation was consistently maintained to ensure a fair comparison. While a formal sensitivity analysis was not conducted, this sampling strategy was implemented to minimize numerical artifacts and ensure robust LMI-based mixing assessments.  
The influence of the sampling resolution and bin size on the LMI accuracy is acknowledged to be an important factor and will be considered in future investigations.