## 3 Results and Discussion
Figure 11c,d shows, respectively, the concentration contour in the vertical mid-section plane of the pipe and the cross-section, demonstrating that the accumulation occurs towards the outside wall, in the longitudinal mid-section plane of the pipe. Another interesting aspect is that the position or location of the area of greatest accumulation within the elbow, although it occurs in the middle plane, is not completely defined in which deviation (\(\varphi\)) of the elbow it will occur. This is clear when comparing, in Figure 10, the C\({}_{\mathrm{max}}\) zone of the cases C13d1, C12d1, C11d1, since there is no important difference in the location of the maximum concentration between one and the other (\(90^{\circ}\), \(87^{\circ}\) and \(0.25\)D of the bend, respectively), even though the Dean vortex formation point is near the inlet of the elbow for the case with the highest curvature ratio (C13d1) and practically at the exit for the lowest curvature ratio (C11d1). Similarly, if we compare the cases according to Re, no specific trend is noticeable regarding the concentration. However, when comparing equivalent cases, but using different particle sizes, for instance C13d1 and C13d2, it could be observed that, for larger particle size (dp), the accumulation point was delayed. On the other hand, this would not explain why the particles are focused on the mid-section plane and towards the outside wall of the pipe. A possible explanation could be that there is an accumulation mechanism related to the interaction between the secondary flow (vortices) and the inertial responsiveness of the particles to be centrifuged by it. Figure 10: Maximum distribution of concentration in the medial plane of the elbow. Of the cases C11d1, C12d1 (reference case) and C13d1. C\({}_{\rm max}\) is considered as a benchmark in our study in addition to the Stokes numbers for the prediction of the concentration or accumulation of particles within an elbow. Table 6 summarizes the C\({}_{\rm max}\) of the 12 solved cases and Table 7 summarizes the values of the ad hoc formulated Stokes numbers together with the relative concentration (C), where the relative concentration is obtained by dimensioning the C\({}_{\rm max}\) of the selected case by the lowest C\({}_{\rm max}\) recorded among the solved cases, being C = C\({}_{\rm max}\)/C\({}_{\rm max}\)_C13d2. Additional information on this topic is presented in Appendix B, including the values of the maximum velocity (Table A1) and maximum tangential velocity (Table A2) obtained and used for the calculation of the Stokes numbers. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Codification** & **C11d1** & **C12d1** & **C13d1** & **C21d1** & **C22d1** & **C23d1** & **C11d2** & **C12d2** & **C13d2** & **C21d2** & **C22d2** & **C23d2** \\ \hline C\({}_{\rm max}\) [kg/m\({}^{3}\)] & 19.4 & 45.8 & 201.5 & 19 & 30 & 39 & 18.9 & 22.1 & 16.8 & 18.6 & 20.5 & 20.7 \\ \hline \end{tabular}
\end{table} 
Table 6: Cases studied and their corresponding Maximum concentration within the elbow. Figure 11: Results of particle concentration. Reference Case C12d1. (**a**) Instant positioning of the X-Z plane particles (**b**) Instant positioning of the particles in 3D view (**c**) Concentration contour in the X-Z plane of the mid-section of the pipe (**d**) Concentration contour in the Z-Y plane of the maximum concentration cross section. Snapshots are taken for a statistically steady flow.