## 2 Microwave Chamber's Geometry  
For the purpose of the experiment, a model of an exemplary cavity was developed: a cylindrical chamber with a cylindrical waveguide joined at the top of a chamber, as is displayed in Figure 2. The model is described with four basic dimensions: the waveguide radius, \(r\); the waveguide's length, \(l\); the chamber's radius, \(R\); and the chamber's length, \(L\)  
Figure 1: The detailed mechanism of heat generation in microwave heating: dipole rotation [11].  
It was assumed that the device would operate in a resonant frequency of 2.45 GHz, as the most commonly used microwave range in ovens operates at frequencies of about 2.45 GHz [33]. Resonant frequency and field distribution shape is dependent on wave type and chamber dimensions and is related to the possibility of standing wave generation. The resonant frequency \(f_{nmp}\) of the cylindrical chamber for the transverse electric (TE) mode, where the electric field is transverse to the \(Z\) axis [34], is described as [35]:  
\[f_{mnp}=\frac{c}{2}\sqrt{\left(\frac{p^{\prime}_{mn}}{\pi R}\right)^{2}+\left( \frac{p}{L}\right)^{2}}, \tag{1}\]  
where \(p^{\prime}_{mn}\) is the \(n\)-th root of derivative of \(m\)-th order Bessel function, \(p\) is an axial mode number, \(R\) is chamber's radius and \(L\) is the chamber length.  
The chamber length \(L\), as well as the waveguide length \(l\), was an integer multiple of the wavelength \(\lambda\): \(l=2\times\lambda\cong 2\times 120=240\) mm.  
The waveguide radius, \(r\), was estimated using an equation describing cut-off angular frequency \(\omega_{c}\) and phase constant \(\beta\)[36]:  
\[\omega_{c}=\frac{p^{\prime}_{mn}}{r}, \tag{2}\]  
\[\beta=\sqrt{\mu\epsilon(\omega^{2}-\omega_{c}^{2})}, \tag{3}\]  
where \(p^{\prime}_{mn}\) is the \(n\)-th root of derivative of \(m\)-th order Bessel function, \(r\) is the waveguide radius and \(\omega=2\pi f\) is the chamber's angular resonant frequency. The condition used for deriving the waveguide radius is defined by a phase constant (3), which must be a positive value. Fulfilling the condition (3) guarantees lossless wave propagation. The biggest waveguide radius \(r\) fulfilling the condition (3) was selected as \(r=80\) mm. Using Equation (1), the chamber radius \(R\) can be calculated as \(R=75.5\) mm.  
In order to generate multiple wave modes, the length \(L\) and the radius \(R\) of the chamber should be an integer multiple of the previously calculated dimensions: \(L=2\times 240=480\) mm and \(R=2\times 75.5=151\) mm.  
On the base of Equations (1)-(3), the resonance should occur for the chamber radius \(R\) equal to 151 mm. However, it should be highlighted that the analytical model given by Equations (1)-(3) enables only a rough estimation of the size of the microwave resonant chamber. Moreover, this estimation is possible only for a simple shape of such a chamber,  
Figure 2: Geometry of the microwave chamber.  
whereas for technical purposes, much more sophisticated shapes are often required. As a result, the dimensions of the microwave chamber necessary for the resonance can be estimated only on the basis of the FEM model. Moreover, achieved results can be verified only experimentally, after the manufacturing of the chamber.  
Estimations made for device dimensions and dimensional tolerances based on finite elements modelling significantly reduces the cost of development and practical validation of the operability of microwave resonant chambers for industrial applications. In addition, even in simplified cases, the achieved results significantly vary from the results of Equations (1)-(3). These differences are caused by, e.g., the significant size of the microwave waveguide and strongly influence the operation of real devices. In the next section, the technical method of validation of the FEM mesh granularity value is proposed, which can be used for any shape of microwave resonant chamber.