## 2 Materials and Methods  
### Mathematical Modeling and Implementation  
A 2D computational model of an air-based density separator with a length (\(L\)) of 6.6 m and a height (\(H\)) of 2.5 m is presented in Figure 1. The vibrating bed is 4 m in length (\(l\)) and 2.5 m in height (\(h\)). The total angular rotation of the vibrating bed is 8\({}^{\circ}\) from the initial position A (\(\theta\) = 0\({}^{\circ}\)) to the final position B (\(\theta\) = 8\({}^{\circ}\)). This angular rotation improves the recycling process by increasing the distribution of separated plastic waste from the vibrating bed to the trash outlet. A small guide is also placed near the air inlet at an angle of 60\({}^{\circ}\) for directing the heavier PET materials, which fails with the air stream to the vibrating bed, to trash outlet. The airflow in the density separator was turbulent, with a high Reynolds number of 60,100 at an inlet air velocity of 35 m/s. Turbulence was modeled by the Reynolds averaged Navier-Stokes (RANS) equations, because the description of the flow at all points in time and space is not practical. The RANS equations contain the weight of all turbulent flow vortices and denote the equations for average flow magnitudes. This representation is valuable because of its modification of every quantity in space and time once turbulence is attained by the fluid. The governing equations of RANS are as follows:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+u_{j}\left(\frac{\partial u_{i}}{\partial x _{j}}\right)=-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\frac{\partial }{\partial x_{j}}\left[v\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right)\right]+\frac{\partial}{\partial x_{j}} \left(-u_{i}^{\prime}u_{j}^{\prime}\right) \tag{2}\]  
in which \(\rho\), \(v\) represent the density (kg/m\({}^{3}\)) and kinematic viscosity (Pa-s) of air, \(u\) is the average velocity (m/s), \(u^{\prime}\) is the velocity fluctuation (m/s), and \(p\) denotes pressure (Pa). The \(k\)-\(\epsilon\) turbulence model was used to treat turbulent flow with high Reynolds number in the density separator. This turbulence model is suitable for complex flow patterns, as the model produces reasonable and stable results with less computational time [28]. The governing equations of turbulent kinetic energy (\(k\)) and dissipation rate (\(\epsilon\)) for the \(k\)-\(\epsilon\) turbulence model are as follows:  
\[\rho\frac{\partial k}{\partial t}+\rho u_{j}\frac{\partial k}{\partial x_{j}} =\tau_{ij}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial}{\partial x_{ j}}\left[\left(\mu+\frac{\mu_{i}}{\sigma_{k}}\right)\frac{\partial k}{ \partial x_{j}}\right]-\rho\epsilon \tag{3}\]  
Figure 1: Computational model set-up of the air-based density separator system. \[\frac{\partial\epsilon}{\partial t}+\rho u_{j}\frac{\partial\epsilon}{\partial x_{j} }=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{\epsilon }}\right)\frac{\partial\epsilon}{\partial x_{j}}\right]+C_{\epsilon 1}\frac{\epsilon}{k}\tau_{ij}\frac{\partial u_{i}}{ \partial x_{j}}-C_{\epsilon 2}\rho\frac{\epsilon^{2}}{k} \tag{4}\]  
in which \(\sigma_{k}\) and \(\sigma_{\epsilon}\) are the turbulent Prandtl number for kinetic energy and the dissipation rate, respectively, and \(C_{\epsilon 1}\) and \(C_{\epsilon 2}\) are the first and second experimental model constants for the dissipation rate, respectively. In the present study, we assigned the following values to the above-mentioned constants: \(\sigma_{k}\) = 1.0, \(\sigma_{\epsilon}\) = 1.3, \(C_{\epsilon 1}\) = 1.44, \(C_{\epsilon 2}\) = 1.92, and \(C_{\mu}\) = 0.09.