# Effect of Plate Curvature on Heat Source Distribution in Induction Line Heating for Plate Forming
## 2 Model and Method
### Electromagnetic  
In the induction heating process, the alternating current in the coil generates an alternating magnetic field. The alternating magnetic field generates an induced electromotive force in the workpiece and then generates an induced eddy current. When the induced eddy current flows in the workpiece, it heats up due to overcoming the internal resistance of the conductor itself, to achieve the purpose of heating the workpiece. We obtained the eddy current distribution by solving the Maxwell equations [16], see Equations (1)-(4):  
\[\Delta\times\stackrel{{\rightarrow}}{{H}}=\stackrel{{ \rightarrow}}{{J}}+\frac{\partial\stackrel{{ \rightarrow}}{{D}}}{\partial t} \tag{1}\]  
\[\Delta\times\stackrel{{\rightarrow}}{{E}}=-\frac{\partial \stackrel{{\rightarrow}}{{B}}}{\partial t} \tag{2}\]  
\[\Delta\cdot\stackrel{{\rightarrow}}{{B}}=0 \tag{3}\]  
\[\Delta\cdot\stackrel{{\rightarrow}}{{D}}=\rho_{v} \tag{4}\]  
where \(\stackrel{{\rightarrow}}{{H}}\) is the magnetic field strength (Wb), \(\stackrel{{\rightarrow}}{{J}}\) is the conduction current density (A/m\({}^{2}\)), \(\frac{\partial\stackrel{{\rightarrow}}{{D}}}{\partial t}\) is the displacement current density, \(t\) represents time (s), \(\stackrel{{\rightarrow}}{{E}}\) is the electric field (V/m), \(\stackrel{{\rightarrow}}{{B}}\) is the magnetic field (T), \(\stackrel{{\rightarrow}}{{D}}\) is the electric flux density (C/m\({}^{2}\)), and \(\rho_{v}\) is the electric charge density. We calculated heat generation using Joule's Law, Equation (5):  
\[q=\frac{\left|\stackrel{{\rightarrow}}{{J}}\right|^{2}}{\gamma} \tag{5}\]  
where q is the Joule heat generation density (J/m\({}^{3}\)), \(\stackrel{{\rightarrow}}{{J}}\) is the current density (A/m\({}^{2}\)), and \(\gamma\) refers to the electric conductivity (S\(\cdot\) m). Due to the skin effect in the process of induction heating, the induced eddy current and Joule heat generated on the workpiece usually concentrates on the workpiece surface [17], so the heat source model can be simplified as the surface heat source, and the simplified model was verified [12; 14]. The skin effect depth can be described in Equation (6):  
\[\delta=503\sqrt{\frac{1}{f\mu_{r}\gamma}} \tag{6}\]  
Figure 1: Main strategy for the calculation. where \(\delta\) is the skin effect depth (mm), \(f\) is the current frequency (Hz), and \(\mu_{r}\) is the relative magnetic permeability. We obtained the surface heat source by calculating the integral of the Joule heat generation in the thickness direction, Equation (7):  
\[q_{surf}=\int_{z}q\mathrm{d}z \tag{7}\]  
where \(q_{surf}\) is the surface heat source distribution (J/m\({}^{2}\)). We used the commercial software ANSYS 15.0 [18] to perform the electromagnetic calculation. Figure 2 shows the main dimensions of the inductor coil and the cross section. Figure 2a shows the shape of the coil, of which the width is 220 mm, and the length is 200 mm. The cross section of the coil is 8 mm \(\times\) 20 mm rectangle. The workpiece, cooling water, coil, core, and air field were modeled. In this paper, it was considered that the workpiece was made of carbon steel, the coil was made of copper, and the material of the core was ferrite. The material properties refer to [PERSON]'s [11] data. The whole model, loads, and mesh division are shown in Figure 3 (the air field is excluded). The 3-D 20-Node electromagnetic solid element (SOLID236) was used for the whole model. The mesh on the workpiece and cooling water were generated with brick shape elements. Meshes on the other parts were generated with the tetrahedral-shaped elements, and the pyramid-shaped elements were used for transition. The whole model had 1,380,763 elements and 2,142,703 nodes in total. Due to the skin effect in induction heating, fined elements were used in the top surface of the workpiece. The frequency of the inductor was 12.7 kHz. Figure 3: Finite element method (FEM) model for the electromagnetic calculation and the mesh of the air gap.