## 2 Materials and Methods
### Mathematical Model  
The equations of the mathematical model derived in this section are based on three assumptions:  
* The heat exchanged by the sample with the environment consists of two fluxes: heat transferred to the air by convection and the energy of ultrasound absorbed by the sample. The heat exchange between the sample and the PTFE mesh is negligibly small. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & \multicolumn{3}{c}{**Dimensions**} \\ \hline  & **Length** & **Width** & **Height** \\  & **(mm)** & **(mm)** & **(mm)** \\ \hline
**Cube** & 30 & 30 & 30 \\
**Cuboid** & 120 & 30 & 40 \\ \hline  & **Diameter** & **Height** & \\  & **(mm)** & **(mm)** & \\ \hline
**Cylinder** & 30 & 30 & \\
**Small cylinder** & 20 & 20 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the particular sample. Figure 1: Scheme of the research stand: (1) Ultrasound generator; (2) vibrating plate; (3) thermocouples; (4) datalogger; (5) sample; (6) air. * The convective heat flux between a sample and the surrounding air is described by [PERSON]'s law, i.e., it is proportional to the difference in the temperature of the sample surface and the temperature of the surrounding air. * The resistance to heat conduction inside the sample is much lower than the convective resistance to heat transport to the outside of the sample. According to the first law of thermodynamics, the decrease in internal energy \(dU\) is equal to the heat transferred to the environment \(dQ\):  
\[dU=dQ \tag{1}\]  
In the case of a metal sample, the internal heat flux density \(\overrightarrow{q}\) is proportional to the temperature gradient ([PERSON]'s law):  
\[\overrightarrow{q}=-\lambda\text{ grad }T \tag{2}\]  
where \(\lambda\) is the thermal conductivity [W/m\(\cdot\)K]. On the second side, in the air the heat flux density \(\overrightarrow{q}\) is proportional to the difference in the temperature of the sample surface \(T_{s}\) and the surrounding air temperature \(T_{a}\) (Newton's law):  
\[\overrightarrow{q}=-h(T_{s}-T_{a}) \tag{3}\]  
where \(h\) is the heat transfer coefficient [W/m\({}^{2}\cdot\)K] [41]. Comparing the Equations (2) and (3) gives the boundary condition for heat exchange:  
\[\lambda\text{ grad }T=h(T_{s}-T_{a}) \tag{4}\]  
This condition is described by dimensionless Biot's number:  
\[\text{Bi}=\frac{hl}{\lambda} \tag{5}\]  
which expresses the ratio of the resistance to heat conduction inside the sample to the resistance to heat transport to the outside of the sample. In this equation, \(l\) means the characteristic dimension of the sample [m]. If the resistance to heat conduction inside the sample is much lower than the convective resistance to heat transport to the outside of the sample (the assumption that the Biot's number is small), then the temperature gradient tends to zero. It follows that temperature \(T\) is homogeneous in the entire sample volume and is only a function of time. In this case, the inner energy \(U\) is the linear function of the internal temperature \(T\):  
\[dU=mcdT=\rho cVdT \tag{6}\]  
where \(m\) is the sample mass [kg], \(c\) is specific heat [kJ/kg\(\cdot\)K], \(\rho\) is density [kg/m\({}^{3}\)], and \(V\) is volume [m\({}^{3}\)]. The heat exchanged by the sample with the environment consists of two fluxes: heat transferred to the air by convection \(\dot{Q}_{cv}\) and the energy of ultrasound absorbed by the sample \(\dot{Q}_{abs}\):  
\[dQ=\left[\dot{Q}_{cv}+\dot{Q}_{abs}\right]dt \tag{7}\]  
According to [PERSON]'s law (3), it is assumed that the convective heat flux \(\dot{Q}_{cv}\) is proportional to the difference between the sample temperature \(T\) and the surrounding air temperature \(T_{a}\):  
\[\dot{Q}_{cv}=\overrightarrow{q}A=-hA(T-T_{a}) \tag{8}\]  
where \(h\) means the heat transfer coefficient [W/m\({}^{2}\cdot\)K] and \(A\) is the sample surface [m\({}^{2}\)] [41].