## 3 Results  
The RC parameters of the proposed lumped model and the model derived from the corrected-slope function were estimated based on the MATLAB Global Optimisation toolbox and measured temperature curves. All the fitted parameters for both models are listed in Table 1. The fitted model response and the corresponding temperature curves measured for the same exemplary input power are presented in Figure 5. It is noteworthy that model C1R1 does not account for the heating effect had by the excitation coil on the sample, which, for the case of power losses in the excitation coil of around 60 W, resulted in an increase in the temperature of the sample by approximately 0.35 \({}^{\circ}\)C (Figure 5(a2)). From all the fitted values, the value of \(C_{1}\) could be easily compared to the theoretical thermal capacity of the 2 mL water sample, which, according to Formula (1), equals 8.368 J/(g \({}^{\circ}\)C) and differs by up to 12% in each model compared to the theoretical value. This discrepancy could result from the imprecise placement of the thermometer in the center of the sample, as mentioned by several authors before [16; 47]. Our calibration protocol could be performed each time a new nanoparticle sample is placed, along with the positioning of all the fiber-optic thermometer probes. After calibration is completed and all probes are fixed in place within the measurement setup, the power resistor is removed from the nanoparticle solution while the positions of the probes remain unchanged, which partially eliminates the positioning error. Therefore, even though the probes' positions might vary slightly with each measurement, this positioning error is partially compensated during the calibration process, where slightly different capacitance values are fitted instead of using the exact values calculated from the formula. This approach also enables the fitting of capacitance values for a variety of solutions, not limited to water-based ones, but also including more complex solvent mixtures. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Element** & **Symbol** & **C2R5** & **C1R1** \\ \hline Thermal capacity of the water sample & \(C_{1}\) & \(7.60\)\(\frac{1}{\text{g}^{\circ}\text{C}}\) & \(9.32\)\(\frac{1}{\text{g}^{\circ}\text{C}}\) \\ Thermal resistance between the water sample and the environment & \(R_{1}\) & \(44.25\)\(\frac{1}{\text{g}^{\circ}\text{C}}\) & \(44.80\)\(\frac{1}{\text{W}}\) \\ Thermal resistance between the water sample and the airspace & \(R_{2}\) & \(1.39\)\(\frac{1}{\text{mW}}\) & - \\ Thermal capacity of the calorimeter airspace & \(C_{3}\) & \(0.05\)\(\frac{1}{\text{g}^{\circ}\text{C}}\) & - \\ Thermal resistance between the calorimeter’s airspace and the environment & \(R_{3}\) & \(1.23\)\(\frac{1}{\text{mW}}\) & - \\ Thermal resistance between the external temperature source and the sample & \(R_{4}\) & \(171.17\)\(\frac{1}{\text{mW}}\) & - \\ Thermal resistance between the external temperature source and the & \(R_{5}\) & \(1.50\)\(\frac{1}{\text{mW}}\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the model. The polyamide test tube was precisely filled with deionized water using the automatic pipette. However, it should be noted that although the test tubes were closed tightly, two tiny holes in the cup for the fiber-optic probe and the heating resistor were only partially isolated, allowing for the evaporation of a small amount of water, especially during the long 48 h calibration measurements. Both models were validated for five selected power values that were generated by the heating resistor and a single constant value of thermal power that was generated by the excitation coil. All of the excitations were different from those used for the model identification. The input power values were estimated based on the recorded heating curves and the equivalent lumped thermo-electric model, with the fitted parameter values assigned to the corresponding elements.