# CFD Analysis and Validation of a Foreign Material Winnowing Machine for Pepper Harvester
## 2 Materials and Methods
### Aerodynamic Analysis of Winnowing
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Pepper Fruit** & **Branch** & **Unit** \\ \hline Length (\(L_{1}\)) & 0.11 & 0.2 & m \\ Diameter (\(L_{2}\)) & 0.017 & 0.004 & m \\ Weight (\(m\)) & 0.012 & 0.004 & kg \\ Terminal velocity (\(V_{t}\)) & 13.5 & 11.5 & m/s \\ Air density (\(\rho\)) & 1.27 & 1.27 & kg/m\({}^{3}\) \\ Acceleration of gravity (\(g\)) & 9.81 & 9.81 & m/s\({}^{2}\) \\ Velocity (\(V_{air}\)) & 10, 12.5, 15, 17.5 & 10, 12.5, 15, 17.5 & m/s \\ Winnowing machine angle & 34 & 34 & \({}_{\circ}\) \\ x-axis start direction (\(S_{x0}\)) & 0 & 0 & m \\ y-axis start direction (\(S_{y0}\)) & 0.4 & 0.4 & m \\ Analysis time interval (\(t\)) & 0.1 & 0.1 & s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculation formula parameter for pepper fruit and branch in the blowing fan. the initial time (\(t\)) is 0, \((V_{xp})\) becomes 0. The acceleration is assumed to be constant at the analysis time interval of 0.1 s.  
\[F_{Dx}=m\ a_{x} \tag{3}\]  
where \(F_{Dx}=\) drag force of x-axis direction (N); and \(a_{x}=\) acceleration of x-axis direction (m/s\({}^{2}\)). \[a_{x}=\frac{F_{Dx}}{m}=\frac{1}{2}\ \frac{C_{D}\ \rho\ A_{p}\ \big{(}V_{air}\ cos\ \theta-V_{xp}\big{)}^{2}}{m} \tag{4}\]  
where \(V_{air}=\) wind velocity (m/s); and \(V_{xp}=\) x-axis direction velocity of object (m/s). The drag applied to chili pepper fruit and chili pepper branch in the y-axis direction is affected by the gravitational acceleration, which is shown in Equation (5) [40]. The acceleration applied in the y-axis direction \((a_{y})\) is shown in Equation (6) [40]. \[F_{Dy}=-m\ a_{y}+m\ g \tag{5}\]  
where \(F_{Dy}=\) drag force of y-axis direction (N); and \(a_{y}=\) acceleration of y-axis direction (m/s\({}^{2}\)). \[a_{y}=g-\frac{F_{Dy}}{m}=g-\frac{1}{2}\ \frac{C_{D}\ \rho\ A_{p}\ \big{(}V_{air}\ sin\ \theta-V_{yp}\big{)}^{2}}{m} \tag{6}\]  
where \(V_{yp}=\) y-axis direction velocity of object (m/s). Velocities of chili pepper fruit and chili pepper branch in the x-axis direction, \(V_{x}\), and in the y-axis direction, \(V_{y}\), are expressed using acceleration values \((a_{x})\) and \((a_{y})\) and calculated at the analysis time (\(t\)) of 0.1 s as shown in Equation (7) [40]. Based on the analysis time (\(t\)), the initial velocity \((V_{n-1})\) of \(V_{x}\) or \(V_{y}\) is the velocity value previously calculated. It becomes the initial value. \[V_{n}=V_{n-1}+a_{n}\ t \tag{7}\]  
where \(V_{n}=\) velocity of an object \((V_{x}\) or V\({}_{y})\) (m/s); \(V_{n-1}=\) previous value for velocity of object (m/s); \(a_{n}=\) acceleration \((a_{x}\) or \(a_{y})\) (m/s\({}^{2}\)); and \(t=\) time interval of analysis (s). The travel distance in the x-axis direction, \(S_{x}\), expressed using the acceleration in the x-axis direction and the y-axis direction \((a_{x},\ a_{y})\) and velocity \((V_{x},\ V_{y})\) values calculated at each time (\(t\)) are shown in Equation (8) [40]. The travel distance in the y-axis direction, \(S_{y}\), is shown in Equation (9) [40]. The initial travel distance \(S_{x0}\) is 0 m. \(S_{y0}\) is fixed to the fall height of 0.4 m. As to the analysis time (\(t\)), the calculation is made at an interval of 0.1 s for up to 5 s.  
\[S_{x}=S_{x0}+V_{x}\ t+\frac{1}{2}\ a_{x}\ t^{2} \tag{8}\]  
where \(S_{x}=\) travel distance of x-axis direction (m) and \(S_{x0}=\) initial travel distance of x-axis direction (0 m). \[S_{y}=S_{y0}+V_{y}\ t+\frac{1}{2}\ a_{y}\ t^{2} \tag{9}\]  
where \(S_{y}=\) travel distance of y-axis direction (m) and \(S_{y0}=\) initial travel distance of y-axis direction (0.4 m).