## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \(Stk_{\text{carved}}\) & \(Stk_{t_{e}}\) & \(Stk_{w_{e}}\) & \(Stk_{w_{e}}\) & **C = C\({}_{\rm max}\)/C\({}_{\rm max}\),C13d2** \\ \hline C11d1 & 0.11 & 0.45 & 0.29 & 0.55 & 1.15 \\ C12d1 & 0.92 & 1.80 & 1.14 & 0.86 & 2.73 \\ C13d1 & 1.87 & 3.60 & 13.23 & 1.46 & 11.99 \\ C21d1 & 0.01 & 0.16 & 0.10 & 0.20 & 1.13 \\ C22d1 & 0.10 & 0.64 & 0.40 & 0.34 & 1.79 \\ C23d1 & 0.21 & 1.27 & 0.81 & 0.51 & 2.32 \\ C11d2 & 0.04 & 0.05 & 0.03 & 0.06 & 1.12 \\ C12d2 & 0.33 & 0.20 & 0.13 & 0.10 & 1.38 \\ C13d2 & 0.68 & 0.40 & 0.25 & 0.16 & 1.00 \\ C21d2 & 0.005 & 0.02 & 0.01 & 0.02 & 1.11 \\ C22d2 & 0.04 & 0.07 & 0.04 & 0.04 & 1.22 \\ C23d2 & 0.08 & 0.14 & 0.09 & 0.06 & 1.23 \\ \hline \end{tabular}
\end{table} 
Table 7: Cases studied and their corresponding Stokes numbers. The inertial responsiveness of each particle size is critical in describing how the particles will be concentrated. Therefore, the results are grouped and compared for d1 and d2. Figures 12 and 13 present a comparison of the different Stokes numbers for each case in addition to the value of the relative concentration. Overall, the four formulations of Stokes number mirror the accumulation process, with the four Stk showing the same overall trend as the concentration number. Figure 12 shows that, in cases with particle size 50 \(\upmu\)m (d1), the Stokes numbers qualitatively show a correlation between the increase of the four Stokes numbers and the relative concentration (i.e., the higher the value of the four Stokes numbers, the higher the underlying maximum concentration). Three scenarios can be identified: First, if  
Figure 12: Bar graph of the Stokes numbers and trend line of relative concentrations between cases, for particle sizes d1 = 50 \(\upmu\)m.  
Figure 13: Bar graph of the Stokes numbers and trend line of relative concentrations between cases, for particle sizes d2 = 150 \(\upmu\)m.  
Finally, regarding the different Stokes numbers \((Stk_{\text{\emph{curved}}},\ Stk_{\text{\emph{tr}}},\ Stk_{\text{\emph{\omega}}_{ \text{\emph{c}}}},\ Stk_{\text{\emph{\omega}}_{\text{\emph{v}}}})\), it was observed that simultaneous values greater than one were linked to a noticeable increase of the maximum concentration inside the pipe bend. Likewise, for simultaneous values of Stokes numbers less than one, a minimum of relative concentration is present. In addition, there is a correlation between the Stokes value with intermediate levels of concentration as Stokes numbers increase for values of \(Stk_{\text{\emph{curved}}},\ Stk_{\text{\emph{\omega}}_{\text{\emph{c}}}},\ Stk_{\text{\emph{\omega}}_{\text{\emph{v}}}}\) close to the unity and \(Stk_{\text{\emph{tr}}}>1\), the latter being the one that apparently presents the greatest correlation force. Conceptualization, S.H.L. and M.J.T. ; formal analysis, S.H.L. ; methodology, S.H.L. and M.J.T. ; resources, M.J.T. ; software, S.H.L. ; supervision, M.J.T. ; validation, S.H.L. ; visualization, S.H.L. ; writing--original draft, S.H.L. ; writing--review and editing, S.H.L., M.J.T., R.G.S. and J.Z.V. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The authors declare no conflict of interest. Figure 18: 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. Figure 14: Results of particle concentration. Case C21d1. (**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.