# A Study of Precision Current Efficiency Curve Measurement with a Casing-Type Anode
## 3 Model and Numerical Analysis  
To analyze the effectiveness of the proposed measurement method, the study conducted a 2D comparison simulation of the conventional and the proposed methods of current efficiency measurement. In the numerical analysis, an alloy of 304 SS was selected as the target material. The current efficiency \(\eta\omega_{304 SS}\) of 304 SS used in both the conventional and proposed method simulations was obtained from our previous studies [16]. The equilibrium processing current in the two methods was recorded. Using Equations (4) and (5), the measured value of current efficiency was calculated. Figure 5 shows a schematic view of the models of the two measurement methods. In the analysis, the dissolution behavior of the workpiece was assumed to be primarily determined by the electric field. The following assumptions were made [24; 25; 26]:  
1. The electrolyte flow within the inter-electrode gap is assumed to be ideal and the conductivity \(\kappa\) of the electrolyte is assumed to be uniform. 2. The electrodes are defined as equipotential surfaces. 3. Joule heat, bubbles, and the anode products are negligible, and the current distribution in the inter-electrode gap is determined solely by Ohmic effects. According to classical electric field theory, the distribution of electric potential \(\varphi\) can be obtained by solving [PERSON]'s equation:  
\[\frac{\partial^{2}\varphi}{\partial x^{2}}+\frac{\partial^{2}\varphi}{\partial y ^{2}}=0, \tag{6}\]  
where \(x\) is the abscissa and \(y\) is the ordinate. Figure 4: The electric field distribution of the measurement with a casing-type anode. The evolution of the anode boundary is affected by both the feed and dissolution activities. On the basis of [PERSON]'s law and [PERSON]'s law, the evolution rate of the anode boundary can be described as:  
\[v_{ax}=\eta\omega_{304 SS}\kappa\frac{\partial\varphi}{\partial x} \tag{7}\]  
\[v_{ay}=\eta\omega_{304 SS}\kappa\frac{\partial\varphi}{\partial y}-v, \tag{8}\]  
where \(\eta\) is the current efficiency, \(\omega_{304 SS}\) is the volume electrochemical equivalent of 304 SS, and \(v\) is the feed rate of the workpiece. The steady processing current \(I_{made}\) of the anode workpiece in the conventional method and the current \(I_{mandri}\) of th mandril in the proposed method were recorded. According to Equations (4) and (5), the measured value of current efficiency was obtained. The transient simulations of current efficiency measurement were carried out using COMSOL Multiphysics, which uses the finite element method as the numerical method. All analyses were calculated for 2500 s which is long enough that the machining processes enter the equilibrium state. In this state, the processing currents no longer change. In order to make the distribution of the electric field in the gap uniform, the gap \(G\), between the mandril and the casing, should be as small as possible. In this study, 0.1 mm was selected as the size of \(G\), and nothing was filled in the gap. The thickness \(T\) of the casing was set at 2.5 mm so that the edge effect was kept away from the mandril. The simulated conditions are listed in Table 1. The relationship between \(\eta\omega\) and \(I\) is listed in Table 2.  
Figures 6 and 7 show the current and the size of the inter-electrode gap evolutions, respectively. Since the initial size of the inter-electrode gap was the same, the initial processing current under different feed rates was also the same. In the beginning, both processing methods were in a non-equilibrium state. The processing current and the gap changed rapidly at first, and then tended to be stable. Finally, the values of the processing current and the size of the inter-electrode gap did not change any more. This means that the dissolution rate of the anode was the same as the feed rate of the cathode tool.