## 2 Physical Modelling and Numerical Methodology
### Study Cases  
In this paper, the four ad hoc formulated Stokes numbers were calculated using numerical methods, and their relationship with the maximum concentration formed in a 90\({}^{\circ}\) bend, for different configurations was analyzed. A turbulent air flow with a dilute solid phase of mono-dispersed particles (mass load 42%, \(\alpha_{p}<10^{-3}\)) was modeled. Two flow conditions \(\mathrm{Re1}=\) 61,508 (V\({}_{1}=\) 8.7 [m/s]) and \(\mathrm{Re2}=\) 173,972 (V\({}_{2}=\) 24.6 [m/s]), two particle sizes \(d_{p1}=\) 50 [\(\mathrm{\upmu m}\)] and \(d_{p2}=\) 150 [\(\mathrm{\upmu m}\)] and three different radii of curvature \(\mathrm{Rc}/\mathrm{R}=\) 1, \(\mathrm{Rc}/\mathrm{R}=\) 4 and \(\mathrm{Rc}/\mathrm{R}=\) 8 were considered, totaling twelve study cases identified according to the code shown in Figure 7. Until now, the Dean number, a dimensionless number that determines flow structures at elbows, had not been introduced. It is defined: \(\mathrm{De}=\mathrm{Re}\sqrt{\mathrm{D}/2\mathrm{Rc}}\). The Dean number mainly marks the transition from laminar to turbulent flow, determining the internal structure that the secondary flow will manifest, although there is no universal solution since the parameter depends largely on the curvature ratio. According to the literature, a completely turbulent flow can be considered for De \(>400\) in circular pipes, that is, without vortices that present undulations, twists and eventually fusion and division of pairs. The cases developed, as can be seen in Figure 7, are high, so it can be considered a completely turbulent flow. Although it is important to take into account the Dean number in order to know with which flow one is working, in the present work it was not used as a direct parameter to measure the dependence of the Stokes numbers with the concentration because, as it depends so much on the Reynolds number as in the curvature relationship. For equal Dean numbers (De) it does not show a clear correlation because the weighting of the Reynolds effect and the radius of curvature is not the same for the concentration of particles.  
The presented Eulerian-Lagrangian approach was used, where the RNG \(k\)-\(\epsilon\) model was selected as the method for solving the fluid phase and the ANSYS Fluent discrete phase module for Lagrangian tracking of the particles. Only the effect of the drag force of the flow (air) and the action of gravity were considered on the particles. The interaction between phases was modeled using a bidirectional coupling and fully elastic collisions were considered. According to the diagram of [PERSON] et al. [13], the volume fraction is delimited between \(10^{-6}<\alpha_{p}<10^{-3}\).  
The geometrical models are shown in Figure 8. Table 4 summarizes the construction parameters.  
Figure 8: Geometrical models generated in ANSYS SpaceClaim. (**a**) Rc/R = 1; (**b**) Rc/R = 4; (**c**) Rc/R = 8.  
Figure 7: Coding of study case.  
The domain was discretized using a structured mesh of hexahedral elements whose nodes were equally spaced in azimuthal and axial directions. In the radial direction and in general, the resolution and quality values of the Mesh 2 used in the validation of the transport phase and discrete phase were used (Sections 2.2.1 and 2.2.2). Table 5 shows the number of elements used in each case and their characteristic dimensions, and Figure 9 shows an example of the type of mesh constructed.