## 4 Results
### Mixing Performance Analysis  
After validating the DEM model with experimental results, it was used to analyse the mixing quality as a function of mass fraction components and mixing time. To assess the homogeneity of the powder multicomponent mixture, a mixing index based on the pooled variance of the whole system according to the study developed in [25] was used, whose criteria is expressed in the following equation:  
\[M=\frac{\sigma_{0}^{2}-\sigma^{2}}{\sigma_{0}^{2}-\sigma_{r}^{2}} \tag{10}\]  
where \(M\) is the mixing index, \(\sigma^{2}\) is the unbiased sample variance of the concentration in a multicomponent mixture, \(\sigma_{r}^{2}\) is the variance for multicomponent mixtures in the com  
Figure 6: Mixing test. (**a**) Initial filling of the components in the cup; (**b**) mixture of the components after the mixing process.  
Figure 7: Powder pile from (**a**) experimental test and (**b**) DEM model.  
\begin{table}
\begin{tabular}{c c} \hline
**Repose Angle** & **Value** \\ \hline Numerical simulation & 31.98\({}^{\circ}\) \\ Experimental test & 32.00\({}^{\circ}\) \\ Relative error & 0.063\% \\ \hline \end{tabular}
\end{table}
Table 5: Comparison of repose angle between numerical simulation and experimental test.  
pletely mixed state, and \(\sigma_{0}^{2}\) is the variance for multicomponent mixtures in the completely segregated state. All these variances are calculated by DEM simulation for a defined input dataset. The \(M\) index can take values between 0, completely segregated, and 1, completely mixed.  
Every design point was simulated using the DEM model for a total mixing time of 60 s. The evolution of the mixture in the cup from the initial to the final instant of the process for the first design point is shown in Figure 8, where each represented instant matches with the end of a cycle of the mixer. A uniform mixing of particles is observed at 60 s from a visual point of view.  
As time was defined as one of the input parameters, the M index output was obtained from DEM simulation every 5 s up to 60 s for all the design points, shown in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Design**} & \multicolumn{8}{c}{**Time (s)**} \\ \cline{2-13}  & **5** & **10** & **15** & **20** & **25** & **30** & **35** & **40** & **45** & **50** & **55** & **60** \\ \hline
1 & 0.776 & 0.870 & 0.885 & 0.918 & 0.968 & 0.976 & 0.979 & 0.988 & 0.989 & 0.992 & 0.991 & 0.992 \\
2 & 0.780 & 0.892 & 0.885 & 0.953 & 0.970 & 0.985 & 0.978 & 0.989 & 0.989 & 0.987 & 0.989 & 0.991 \\
3 & 0.788 & 0.871 & 0.879 & 0.916 & 0.968 & 0.975 & 0.976 & 0.984 & 0.988 & 0.989 & 0.991 & 0.989 \\
4 & 0.716 & 0.868 & 0.857 & 0.943 & 0.965 & 0.976 & 0.967 & 0.987 & 0.987 & 0.985 & 0.987 & 0.991 \\
5 & 0.853 & 0.909 & 0.929 & 0.946 & 0.973 & 0.978 & 0.980 & 0.981 & 0.976 & 0.986 & 0.983 & 0.986 \\
6 & 0.833 & 0.912 & 0.912 & 0.964 & 0.975 & 0.977 & 0.981 & 0.985 & 0.983 & 0.984 & 0.984 & 0.992 \\
7 & 0.843 & 0.905 & 0.925 & 0.952 & 0.972 & 0.975 & 0.979 & 0.986 & 0.979 & 0.988 & 0.984 & 0.986 \\
8 & 0.829 & 0.911 & 0.909 & 0.960 & 0.971 & 0.980 & 0.977 & 0.985 & 0.979 & 0.986 & 0.985 & 0.992 \\
9 & 0.721 & 0.854 & 0.886 & 0.914 & 0.964 & 0.974 & 0.970 & 0.983 & 0.982 & 0.984 & 0.985 & 0.988 \\
10 & 0.695 & 0.862 & 0.876 & 0.934 & 0.941 & 0.970 & 0.973 & 0.981 & 0.982 & 0.988 & 0.983 & 0.990 \\ \hline \hline \end{tabular}
\end{table}
Table 6: \(M\) index output for each design point at \(t=\) 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 s.  
Figure 8: Simulation of design point 1 at different instants.  
Figure 9 shows the variation of the \(M\) index with respect to time and the component mass fraction of surfactant, carbonate, speckle, and nonionic, where it was observed that the value of \(M\) index increases with time until it stabilizes at a value close to 1 in all cases analysed.