# A Study of Precision Current Efficiency Curve Measurement with a Casing-Type Anode
## 3 Model and Numerical Analysis
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Feed rate \(v\) & 0.1/0.2/0.3/0.4/0.6/0.8/1.0 mm/min \\ Electrolyte conductivity \(\kappa\) & 15 5/m \\ Voltage between electrodes \(U\) & 10 V \\ Workpiece & 304 stainless steel (304 SS) \\ Initial inter-electrode gap & 1 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation conditions. Figure 5: Schematics of models: (**a**) conventional constant voltage method; (**b**) casing-type anode measurement. Figures 8 and 9 show the simulated contours in the workpiece obtained by the conventional and proposed methods, respectively. The profile of the workpiece was smooth and flat in the central region at the end of the workpiece in both methods. However, there were distinct machined fillets at the edge of the workpiece, especially at a low feed rate. Compared to the conventional method, the fillets of the proposed method can be ignored. The edge effect was also effectively suppressed by using the proposed method. \begin{table}
\begin{tabular}{c c} \hline \hline \(i\) **(A-cm\({}^{-2}\))** & \(\eta\omega\) **(mm\({}^{3}\)-Amin\({}^{-1}\))** \\ \hline
0 & 0 \\
12.28 & 0.814 \\
19.61 & 1.019 \\
27.43 & 1.093 \\
35.37 & 1.130 \\
50.68 & 1.183 \\
66.74 & 1.198 \\
83.19 & 1.202 \\ \(\geq\)83.19 & 1.202 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(\eta\omega\)-\(I\) of 304 SS. Figure 6: Diagrams of the current evolutions. Figure 7: Diagrams of the inter-electrode gap evolutions. Figure 10 shows the current density distribution of the conventional and proposed methods with a feed rate of 0.1 mm/min. The current distribution in the central region was consistent with the ideal uniform current density in both of the methods used. However, the current distributions at the edge and the sidewall were different from the ideal current. The current distributions at the fillets and sidewall changed dramatically. The current on the edges was smaller than the ideal current because the surface of the workpiece was far away from the cathode tool at the edge. The current distribution at the sidewall decreased gradually as it moved away from the end and dropped to 0 at the edge of the fixture. The current distribution at the sidewall caused the measured current to be larger, and this was not consistent with the hypothesis of a uniform electric field. In comparison to the ideal current, the influenced area at the end face and the sidewall was 2.160 mm and 3.883 mm in the conventional method, while it was only 0.223 mm and 0.253 mm in the proposed method. This means that the edge effect and sidewall corrosion were significantly inhibited. The error between the processing current and the ideal current can be obtained by comparing the processing current of the two processes with the ideal current. As shown in Figure 11, the maximum error of the conventional method was 11.0% when the feed rate was 0.1 mm/min. The error of the conventional method had a tendency to decrease continuously as the feed rate increased. When the feed rate was 1 mm/min, the conventional method had the minimum error at 2.0%. The error between the processing current of the proposed method and the ideal current was almost negligible, differing from the conventional method. In the proposed method, the maximum error was only 0.2% when  
Figure 8: Contours of the workpiece in the conventional method. Figure 9: Contours of the workpiece in the proposed method. the feed rate was 0.1 mm/min. The current measured by the proposed method was closer to the ideal current value when compared to the conventional method, which means that the proposed method can obtain a more accurate current efficiency. Figure 11: The error between the processing current and the ideal current. Figure 10: The current density distribution on the contour of the workpiece (\(v\) = 0.1 mm/min). From calculations using Equations (3) and (4), the current efficiency was obtained.