## 2 Materials and Methods
### Equivalent Thermo-Electrical Circuit  
The approximate electrical equivalent circuit of the thermal system with lumped elements is shown in Figure 2. The current source \(i_{\mathrm{m}}(t)\) represents the controlled heat flow source in the form of the SMD 0408 16 k\(\Omega\) resistor immersed in the water sample. The input thermal power was a rectangular pulse function corresponding to the resistive power dissipated by the heating resistor, with the adjustable amplitude regulated for each of the \(m\)-th measurements. The capacitor \(C_{1}\) described the heat capacity of the 2 mL water sample and could be roughly estimated according to Formula (1) as 8.368 J/(g \({}^{\circ}\)C). The heat capacity of the calorimeter's airspace was introduced into the model by \(C_{2}\). \(R_{1}\) characterized the thermal resistance between the water sample and the environment, while \(R_{2}\) corresponded to the thermal resistance between the water sample and the airspace inside the calorimeter. \(R_{3}\) represented the thermal resistance between the calorimeter's airspace and the environment. In our project, we used Litz wire for the construction of the induction coil, in contrast to other authors who used copper tube windings cooled by forced water flow inside the tube [46]. The use of Litz wire was particularly advantageous due to the need to reduce the coil's resistance by mitigating the skin effect at high RF frequencies. The system was cooled using a water jacket with an automatically cooled continuous water circuit, which absorbed heat from the heating carcass of the induction coil while simultaneously maintaining a constant reference temperature in the immediate vicinity of the heated sample. Despite using water cooling, we observed that a small portion of the temperature rise in the sample resulted from the inward heat flow generated by the resistive energy dissipated in the excitation coil. This effect was modeled by adding an external temperature source connected to the calorimeter's airspace and the sample via thermal resistances \(R_{4}\) and \(R_{5}\), respectively. The transient temperature curve registered by the probe placed inside the excitation coil's carcass served as an input signal \(u_{m}(t)\). A direct current was applied to the coil during the calibration experiments to generate resistive heat. Using an alternating magnetic field in the calibration measurements would have caused the uncontrolled heating of the resistor and wires due to eddy currents, as reported in [42; 43]. This problem was eliminated in our experiments by measuring the temperature with nonmagnetic fiber optics sensors and the application of direct current in the excitation coil during the calibration procedure. While DC could not heat the nanoparticles, it allowed us to control the thermal power produced by the excitation coil. Figure 2: Diagram of the calorimetric measurement setup. Using a refrigated circulator, we set the desired controlled temperature to 25 \({}^{\circ}\)C, similar to the environment temperature measured during our experiments. This allowed us to assume that the uniform ground potential for the whole model would be 25 \({}^{\circ}\)C.  
The range of power values selected for calibration was based on the observed temperature increase in the initially tested nanoparticle solutions (OceanNanotech SPA-25, San Diego, CA, USA). With the magnetic fields used, in the order of several mH, a maximum temperature rise of about 10 \({}^{\circ}\)C was observed in the nanoparticle solution during the 3600 s measurement, which approximately corresponds to around 200 mW in a lossless, adiabatic system. To balance the calibration time with the accuracy required in determining the model parameters, five different heating resistor power values (0 mW, 14.1 mW, 56.4 mW, 127.0 mW, 225.7 mW) were selected. These corresponded to the voltage values set on the generator (15 V, 30 V, 45 V, 60 V), with 60 V being the highest available setting on the voltage power supply.