# A Fast and Accurate Method for Computing the Microwave Heating of Moving Objects
## 2 Mathematical Model
### Multiphysics Simulation of a 3-D Model  
For validating our method, we take a 3-D model like the article of [PERSON] et al. [16], which includes four parts: a cavity, a glass turntable, a rectangular waveguide, and a potato sample (50 mm \(\times\) 60 mm \(\times\) 15 mm), as shown in Figure 2. The initial temperatures of the air, rotating turntable, and the moving objects inside the oven are all 20 \({}^{\circ}\)C. The potato, whose dielectric properties are considered as temperature-dependent, is centered at the glass turntable. The waveguide filled with air works in the TE\({}_{10}\) mode, providing a power of 300 W and operating at 2.45 GHz. The input parameters of the geometric model are given in Table 1[9, 16, 17, 18, 19]. Figure 1: Overall flowchart of the proposed method. It is crucial to choose an appropriate number of mesh elements for numerical simulation because this affects not only the calculation accuracy but also the calculation time. Therefore, a mesh convergence study for 3-D electric field calculation is performed by the proposed method and the ALE method to determine the reasonable amount of mesh elements. Figure 3 shows the normalized power absorption (NPA) [20] as a function of the total number of elements in the mesh. It can be clearly seen that the simulation results are mesh-independent after the number of elements reaches 100,000. Accordingly, the total elements consisting of 138,438 and 126,480 quadratic tetrahedrons are used in the proposed method and the ALE method, respectively. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Property** & **Domains** & **Value** & **Source** \\ \hline Relative permittivity & Air & 1 & [18] \\  & Potato & \((-6.4\times 10^{-3}\text{T}^{2}+0.2\text{T}+56.8)\) & [9] \\  & & \(-\text{j}(-1\times 10^{-4}\text{T}^{2}-0.108\text{T}+16.1)\) & [16] \\  & Glass & 4 & [16] \\ Conductivity (S/m) & Air & 0 & [18] \\  & Potato & 0 & [18] \\  & Glass & 0 & [16] \\ Thermal conductivity (W/m-K) & Potato & \(5.998\times 10^{7}\) & [17] \\ Density (kg/m\({}^{3}\)) & Potato & 1050 & [17] \\ Specific heat capacity (J/kg-K) & Potato & 3630 & [17] \\ Heat transfer coefficient & Potato & 10 & [19] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input parameters used in the simulations. Figure 3: Normalized power absorption (NPA) as a function of the total number of elements for the proposed method and the arbitrary Lagrangian–Eulerian (ALE) method. Figure 2: Schematic of the microwave oven system used for modeling (all sizes in millimeters). As a discrete simulation, the continuous rotation of the turntable and the heated food should be divided into many discrete steps and the discrete angle plays an important role in obtaining the temperature distribution with good accuracy. To study the effects of rotational angle on simulation,  
The average temperatures of the potato at the 24 th second under different speeds and different rotation angles are calculated, as shown in Figure 4. This shows that when the rotational speed is faster than 5 rpm, a rotation angle of 30\({}^{\circ}\) per time step is sufficient to obtain accurate results. However, when the speed is lower than 5 rpm, a smaller angle should be selected to ensure the accuracy of the simulation results. Therefore, 30\({}^{\circ}\) rotational degrees per step is suggested to obtain reasonable results by the proposed method in the next study, i.e., the oven is assumed to be at 12 discrete positions for each rotation cycle. In order to evaluate the accuracy and efficiency of our method, we simulate the same 3-D model and compare our results with those obtained by the ALE method [16], which has been proved to be more efficient and accurate than the conventional discrete method for simulating the microwave heating of moving objects and verified by experiments. Moreover, we also study the effects of rotational speeds on the microwave heating process. In addition, the effects of food deformation, moisture migration, and phase change are ignored because the heating time is very short.