## 2 Physical Modelling and Numerical Methodology
### Relationship between Stokes Numbers and Particle Accumulation at a 90\({}^{\circ}\) Elbow  
The Stokes number describes the response of a particle to a specific flow condition by determining how inertial the particle behaves (i.e., whether it behaves as a flow tracer (\(Stk<1\)) or not (\(Stk>1\))). In its classical form, the Stokes number is defined as the relaxation time of the particle \"\(t_{p}\)\" over the characteristic time of the flow \"\(t_{f}\)\" (Equation (8)). \[Stk=\frac{t_{p}}{t_{f}} \tag{8}\]  
However, for the analysis of particle accumulation, the Stokes number can be inverted, as the Stokes number is used to indicate an accumulation of particles (\(Stk>1\)). The accumulation) occurs according to the inertial capacity of each particle to migrate between streamlines (as non-tracer particles) and finally be deposited in a certain area. The latter depends on the accumulation mechanism, which will be defined by the physics of the problem. The relevant physical and geometrical effects in a bend pipe were previously investigated [11], and possible characteristic lengths, velocities and/or times of the flow were  
Figure 6: Longitudinal flow velocity profile in z\({}^{\prime}\)/D = \(-1\), \(\varphi\) = 0\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\) and 90\({}^{\circ}\). RNG \(k\)-\(\varepsilon\) model. Experimental data of [PERSON] et al. (Re = 60,000) [10]. identified according to the physical problem, leading to four different formulations of the Stokes number based on different time scales of fluid:  
1. _Curved Stokes Number_ (\(Stk_{curved}\)), this number quantifies the effect given by the geometry, where the ratio of radii (\(\mathrm{Rc}/\mathrm{R}\)) and velocities (\(V_{mean}/V_{max}\)) are inversely proportional [7]. Given this relationship, the effect of geometry can be quantified by the radius of curvature and the maximum velocity generated within the bend \(t_{1}=\mathrm{R_{c}}/V_{max}\). \[Stk_{curved}=\frac{t_{p}}{t_{1}}=\frac{\rho_{p}d_{p}^{2}\ V_{max}}{18\mu_{g} \ \mathrm{R_{c}}}\] (9)
2. _Stokes Number of particle transit through the vorticial zone (\(Stk_{t}\)),_ as the accumulation is directly related to the residence time of a particle in the bend, we will have a characteristic time \(t_{2}=L_{s}/V_{mean}\) representing the average time it would take for a fluid particle to travel a distance equal to the arc length of the bend \"\(L_{s}\)\". The shorter the average flow time the longer the particle residence time, favoring particle accumulation. \[Stk_{t}=\frac{t_{2}}{t_{p}}=\frac{18\mu_{g}\ L_{s}}{\rho_{p}d_{p}^{2}\ V_{ mean}}\] (10)
3. _Primary Centrifugal Stokes Number (\(Stk_{\omega_{c}}\)),_ quantifies the effect of the change of direction, where the particles are affected in their equilibrium position by the centrifugal force caused by the curvature. Then, it will be represented by the response of the particles to the angular velocity (\(\omega_{c}=V_{mean}/Rc\)) measured with respect to the center of curvature that the particle is affected (\(t_{3}=1/\ \omega_{c}\)). \[Stk_{\omega_{c}}=\frac{t_{3}}{t_{p}}=\frac{18\mu_{g}}{\rho_{p}d_{p}^{2}\ \omega_{c}}=\frac{18\mu_{g}\ \mathrm{R_{c}}}{\rho_{p}d_{p}^{2}\ V_{ mean}}\] (11)  
The greater the centrifugal force, the greater the \(\omega_{c}\), causing the increase of centrifuged particles to the external face of the elbow, generating greater accumulation in this zone. 4. _Secondary Centrifugal Stokes Number (\(Stk_{\omega_{c}}\)),_ aims to quantify the susceptibility of the particle to be centrifuged out of the Dean vortices (secondary flow) towards the central plane and zones of lower vorticity. Therefore, it will be determined by the relative angular velocity that the particles are subjected to with respect to the rotating center of the vortex (\(\omega_{v}\)).