## 2 Materials and Methods
### Experimental Material Selection  
Two characteristic scales (\(\lambda_{m}\) and\(D_{p}\)) [23; 24], are used to measure the propagation of electromagnetic waves in the sample and the selection of heating materials.  
\[\left\{\begin{array}{l}\lambda_{m}=\frac{\sqrt{2}\varepsilon}{\sqrt{M+ \varepsilon_{r}\times f}}\\ D_{p}=\frac{\sqrt{2}\varepsilon}{\sqrt{M-\varepsilon_{r}\times 2\pi f}}\end{array}\right. \tag{5}\]  
where \(M=\sqrt{\left(\varepsilon_{r}\right)^{2}+Q_{e}{}^{2}}\), \(f\) and \(c\) denote the microwave frequency and the speed of light, respectively. The parameter \(\lambda_{m}\in(0,\frac{\varepsilon}{f})\) determines the number of waves formed inside the material under microwave heating. The penetration depth, denoted by \(D_{p}\), refers to the distance microwaves can travel through a material before being absorbed completely. Microwave absorbing materials such as SiC, carbon, and water can easily convert microwave energy into heat, their penetration depth \(D_{p}\) is typically in the order of centimeters.  
Let \(f_{p}=\lambda_{m}/2\pi D_{p}\), \(f_{w}=\lambda_{m}/\lambda_{0}\)[24], where \(\lambda_{0}=c/f\), \(\lambda_{m}\) and \(D_{p}\) have undergone de-unitization and normalization. Inspired by reference [24], materials can be categorized into four groups using parameters \(f_{p}\) and \(f_{w}\), as illustrated in Table 1.  
Two representative materials were selected for the experiment: Potato was chosen for the first group (\(f_{p}\leq 0.2\), \(f_{w}\leq 0.2\)), and SiC was chosen for the third group(\(f_{p}>0.2\), \(f_{w}\leq 0.2\)). Potatoes are commonly used in the food processing industry and are a staple in daily life, in contrast, the SiC is primarily utilized in high-temperature materials research. Table 2 summarizes the relevant properties of the two materials.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Group** & **Material Type** \\ \hline \(f_{p}\leq 0.2\), \(f_{w}\leq 0.2\) & Carrot, Apple, Potato, Radish, Beef, Lamb \\ \(f_{p}\leq 0.2\), \(f_{w}>0.2\) & Bread, Pizza, Alumina, Zirconia, PET \\ \(f_{p}>0.2\), \(f_{w}\leq 0.2\) & White pudding, Ham, Tomato sauce, SiC \\ \(f_{p}>0.2\), \(f_{w}>0.2\) & Almonds, Walnut and peanut butter \\ \hline \hline \end{tabular}
\end{table}
Table 1: Groups of materials are categorized based on the magnitudes of \(f_{p}\) and \(f_{w}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material Properties** & **SiC a** & **Copper b,c** & **Potato a** & **Air a,b,c** \\ \hline Relative permittivity (real part) & \(\varepsilon\) (\(T\)) & 1 & \(\varepsilon^{potato}(T)\) & 1 \\ Loss tangent & \(\tan\delta\) (\(T\)) & - & - & - \\ Electrical conductivity [S/m] & - & \(5.998\times 10^{7}\) & 0 & 0 \\ Thermal conductivity [W/(m K)] & \(K\) (\(T\)) & 0.648 & 0.648 & 0 \\ Density(kg\(\cdot\)m\({}^{-3}\)) & 3100 & 8700 & 1050 & - \\ Heat capacity at constant pressure [J/(kg K)] & \(C\) (\(T\)) & 3600 & 3640 & - \\ \hline \hline \end{tabular}  
* Reference [25], b Reference [2], c COMSOL built-in material library.  
\end{table}
Table 2: Material parameters of the finite element analysis numerical calculation model.  
Where  
\[SiC\left\{\begin{array}{l}\varepsilon(T)=-2\times 10^{-4}T^{2}+0.4503T+124.26\\ K(T)=8\times 10^{-5}T^{2}-0.325T+326.69\\ C(T)=4\times 10^{-7}T^{3}-17\times 10^{-4}T^{2}+2.3729T+115.43\\ \tan\delta(T)=5\times 10^{-10}T^{3}+9\times 10^{7}T^{2}+6\times 10^{-4}T+0.2801 \end{array}\right.\]  
\[\varepsilon^{potato}(T)=-6.4\times 10^{-3}T^{2}+2\times 10^{-1}T+56.8\]