# Optimizing the Shape of the Spinning Electrode for Needleless Coaxial Electrospinning
## 2 Design and Simulations
### Mathematical Model of the Optimized Design  
The optimized electrode design, as shown in Figure 6, was created. The metal wire was extended to its ends beyond the edges of the electrode body, and discharge eliminators (spheres) were placed on them. It was calculated that reducing the curvature due to the large radius of the spheres would eliminate high electric field values at the edges of the electrode. Through FEM, an analysis of the electric field intensity distribution near the electrode was performed for the original design, both with the wire alone and for the new concept for two types of eliminator material (HDPE and metal). The spheres' size at the electrode's edges was optimized to achieve electric field homogeneity along the spinning area's entire length. The values mentioned above were observed, and the diameter of the spheres was searched for, where the Eedge value was equal to the Emid value. The maximum value on the spheres' surface was also monitored and marked as Esphere. Simulations were performed for plastic spheres with diameters of 10 mm, 20 mm, 30 mm, 40 mm, 50 mm and 60 mm. The electric field intensity values from the focused locations are in Table 1. As this is a symmetrical model, the results for both  
Figure 5: Electric field distribution on the wire electrode of the original design. Figure 6: The conceptual design of the new electrode. sides are identical and presented in Table 1, as well as in the text for only one side. The E\({}_{\text{edge}}\) and E\({}_{\text{mid}}\) values are shown in the graph; see Figure 7. The calculated values show that the optimal diameter of the plastic spheres is 40 mm. In this case, the E\({}_{\text{mid}}\) and E\({}_{\text{edge}}\) values ratio is closest to 1. The optimal value of the sphere diameter can also be reached from the graph in Figure 7. The optimal value corresponds to the intersection of the curve for E\({}_{\text{mid}}\) and the curve for E\({}_{\text{edge}}\). Figure 8 shows the electric field distribution near the electrode and plastic spheres with a diameter of 40 mm. In the second case, the electric field distribution on the electrode's surface was analyzed in an arrangement with metal spheres at the ends of the electrode. The magnitude of the electric field was obtained again from the three main points (sphere, amid and edge). The diameters of the metal spheres, including the values of the electric field intensities, are shown in Table 2. The graphical representation of the results is shown in Figure 9. In the graph, we can find the intersection of the E\({}_{\text{mid}}\) and E\({}_{\text{edge}}\) dependence curves, showing the optimal diameter of the spheres to be 7 mm and the graphic demonstration of electric field distribution for metal spheres; see Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline
**Ball Diameter** & \multicolumn{4}{c}{**Electric Field [V/mm]**} & \\ \cline{2-5}
**[mm]** & **E\({}_{\text{mid}}\)** & **E\({}_{\text{edge}}\)** & **E\({}_{\text{ball}}\)** & **E\({}_{\text{mid}}\)/E\({}_{\text{edge}}\)** \\ \hline no ball & 7809 & 8986 & x & 0.869 \\ \hline
10 & 6765 & 7285 & 1652 & 0.929 \\ \hline
20 & 6694 & 7020 & 789 & 0.954 \\ \hline
30 & 6643 & 6793 & 523 & 0.978 \\ \hline
40 & 6601 & 6576 & 385 & 1.004 \\ \hline
50 & 6554 & 6389 & 311 & 1.026 \\ \hline
60 & 6508 & 6168 & 267 & 1.055 \\ \hline \end{tabular}
\end{table} 
Table 1: Electric field values for electrodes with plastic spheres. Figure 7: Electric field dependence on the plastic spheres’ diameter: the red line is the value of the edge position (E\({}_{\text{edge}}\)); the blue line is the value of the middle position (E\({}_{\text{mid}}\)). Figure 8: The intensity distribution on the electrode with the optimized diameter of the plastic spheres.