# Characterization of Deburring by Abrasive Flow Machining for AL6061
## 4 New Objective Function
### Suggestion for a New Objective Function  
In viscous fluid flow, streamlines cannot be formed along the burr edge due to the no-slip boundary condition. In reality, edge erosion and deburring can be achieved by abrasive flow, which can be understood in terms of pathlines rather than streamlines. A fine abrasive particle can nearly follow a pathline, with random micro-motion generated by the irregularity of the particle geometry. For particles following the pathlines near the burr edge, there are chances for impact and momentum transfer. Meanwhile, pathlines coincide with streamlines for steady flow.  
[PERSON]'s single particle erosion model is defined as follows [23]:  
\[Q=\frac{mV_{p}^{2}}{p\psi K}\bigg{(}\sin 2\alpha-\frac{6}{K}\sin^{2}\alpha \bigg{)},\ \ \ if\ \tan\alpha\leq\frac{K}{6} \tag{3}\]  
\[Q=\frac{mV_{p}^{2}}{p\psi K}\bigg{(}\frac{K\cos^{2}\alpha}{6}\bigg{)},\ \ \ if\ \tan\alpha\geq\frac{K}{6} \tag{4}\]  
where \(Q\) is the volume of material removed, \(m\) is the mass of the abrasive particle, \(V_{p}\) is the speed of the abrasive particle, and \(p\) is the plastic flow stress of the workpiece. \(\psi\) is the ratio of the depth of collision to the depth of cut. \(K\) is the ratio between the vertical and horizontal components of the contact force, and \(\alpha\) is the attack angle of particle velocity with respect to the contact surface of the workpiece.  
Based on the above equations, the total volume of the eroded material was proportional to the mass of a single particle and the square of the collision speed. A new objective function could thus be proposed along these lines in the following form:  
\[f_{ab,\ New}=\frac{\tau_{xy}}{|\kappa|}\cdot V^{2} \tag{5}\]  
where \(\tau_{xy}\) is the wall shear stress, and \(\kappa\) is the local curvature of the streamline near the burr edge, \(V\) is the flow speed at the observation point.  
The values of the new objective function at each flow speed, along with [PERSON]'s objective function [18], are listed in Table 7.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **3.3 m/s** & **4.5 m/s** & **5.6 m/s** & **6.7 m/s** & **7.8 m/s** \\ \hline \(\tau_{xy}\) (kPa) & 14.05 & 18.46 & 22.62 & 26.72 & 30.58 \\ \(K\) (1/mm) & 1.28 & 1.28 & 1.29 & 1.29 & 1.30 \\ \(f_{ob,\ Kim}\) (N/m) & 10.7 & 14.1 & 17.2 & 20.2 & 23.1 \\ Deburred area (mm\({}^{2}\)) & 0.046 & 0.132 & 0.280 & 0.430 & 0.763 \\ \hline \end{tabular}
\end{table}
Table 6: Values of objective function suggested by [PERSON] [18] from CFD analyses at different speeds along with the deburred area observed for the flow length of 200 km (Table 2).  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **3.3 m/s** & **4.5 m/s** & **5.6 m/s** & **6.7 m/s** & **7.8 m/s** \\ \hline \(f_{ob,\ Kim}\) (N/m) & 10.7 & 14.1 & 17.2 & 20.2 & 23.1 \\ \(f_{ob,\ Kim}\) (N/m) & 10.2 & 23.6 & 44.5 & 75.1 & 115.1 \\ \end{tabular}
\end{table}
Table 7: Values of [PERSON]’s objective function [18] and the new objective function for different flow speeds.  
For comparing the new objective function with the one suggested by [PERSON] [18], the values of the deburred area at each of the six steps for various flow speeds in Table 5 were normalized by each of the objective function values enlisted in Table 7, as shown in Figures 16 and 17.  
While the curves in Figure 16 have different slopes from the origin, the curves in Figure 17 are less divergent. The new objective function seems to be more representative of the actual deburring performance.