## 4 Conclusions  
This paper introduces an analytical model to predict the temperature profile in a planar moving induction heating process. The power density of induced eddy current q\({}_{\rm v}\) in the heated workpiece can be obtained based on Maxwell equations, which govern the general induction heating. Then the transient temperature evolution will be calculated using the mathematical equations defined as a rectangle moving heating source with the uniform power P\({}_{\rm L}\). In addition, several finite element simulations and experiments are conducted to validate the analytical calculation model, and the following conclusions can be achieved:  
(1) The temperature profile calculated by the analytical model introduced in this work is similar to the finite element simulation results. By comparing with finite element simulation and the corresponding validation experiments, the maximum temperature in the workpiece is verified while the velocity is 5 mm/s and 8 mm/s, respectively. It is noticed that the maximum temperature by analytical calculation is higher than the simulation results and the experimental results, due to the fact that the simulation model has taken into consideration of the surface heat transfer effect while the analytical model ignores it. It is presented that the relative errors between the analytical maximum temperature and the corresponding experimental data are lower than 25%, and there is a low error level when the temperature is higher than 200, which is matched with the fact that the convective heat transfer coefficient decreases with the temperature rises in reality. (2) The higher input current intensity \(I\) of the coil, the higher temperature is acquired, which has a good agreement with the finite element simulation results and the experiment results. Similarly, the heated temperature rises as the frequency \(f\) increases at the same level of \(I\). It is also proved that the moving velocity \(v\) influences the temperature in the planar induction heating process, thus it is essentially to treat \(v\) as an important parameter in the temperature precisely prediction. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Test** & **\(I\)/A** & \(f\)/kHz & **Velocity/**ms\({}^{-1}\)** & **Analytical** & **FE** & **Experimental** & **FE Simulation** & **Analytical** \\
**Simulation/\({}^{\circ}\)C** & **Simulation/\({}^{\circ}\)C** & **Results/\({}^{\circ}\)C** & **Error/\%** & **Error/\%** \\ \hline
1 & 485 & 26.7 & 5 & 100.10 & 85.79 & 80.91 & 6.53 & 23.7 \\
2 & 430 & 37.6 & 5 & 227.20 & 213.28 & 195.7 & 8.98 & 16.1 \\
3 & 767 & 26.7 & 5 & 284.97 & 272.32 & 258.54 & 5.33 & 10.2 \\
4 & 890 & 37.6 & 5 & 357.18 & 346.43 & 331.27 & 4.58 & 7.82 \\
5 & 485 & 26.7 & 8 & 97.94 & 78.92 & 73.72 & 7.06 & 24.7 \\
6 & 430 & 37.6 & 8 & 221.39 & 201.08 & 189.96 & 5.85 & 16.5 \\
7 & 767 & 26.7 & 8 & 277.50 & 258.98 & 240.22 & 7.81 & 15.5 \\
8 & 890 & 37.6 & 8 & 347.64 & 330.34 & 317.23 & 4.13 & 9.59 \\
9 & 430 & 37.6 & 3 & 240.93 & 220.62 & 210.71 & 4.7 & 14.3 \\
10 & 430 & 37.6 & 10 & 187.65 & 169.98 & 155.57 & 9.26 & 20.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum temperature and the relative errors with experimental results. Figure 17: Maximum temperature validation in the surface of the workpiece (\(I\) = 430 A,\(f\) = 37.6 kHz, Test 9: \(v\) = 3 mm/s; Test 2: \(v\) = 5 mm/s; Test 6: \(v\) = 8 mm/s; Test 10: \(v\) = 10 mm/s). (3) The computation efficiency of the analytical model in this work is higher than the finite element simulation model. Taking into consideration of the indispensable simplification during the analytical modeling and the difficulties in precisely calculation and controlling of the induction coil during the planar moving induction experiment system, it is believed that the established analytical model in this work can be used to predict the temperature evolution in the heated workpiece by the planar moving induction heating.