## 4 Discussion  
The results of our study demonstrate that our extended lumped-parameter thermal model can be effectively calibrated, offering a significant improvement over the corrected-slope method, which is widely used to measure the heating power of magnetic nanoparticles. This enhancement is particularly noticeable at low heating power values, where the influence of heat transfer from the excitation coil becomes a significant factor. By comparing our refined model to the traditional corrected-slope method, we observed a reduction in measurement errors, particularly at lower heating power values. The error analysis presented in Table 2 shows that for input power values around 25 mW, the relative error decreased from 16.09% to 2.36%, while for higher power values around 190 mW, the error reduction was more moderate, improving from 2.14% to 1.58%. This suggests that the refinements introduced in our model are particularly significant when the effects of external heat transfer cannot be neglected. Calibration using the global optimization toolbox provides a solution that minimizes error in the middle range of the calibrated values, while the error at the range boundaries remains relatively higher. In our model, identifying the source of heating power determination errors related to the heat flow generated by the excitation coil allowed us to eliminate large errors at low power values, thereby increasing measurement accuracy and leading to a more symmetric error distribution at both ends of the calibrated range. Among the unmodeled sources affecting the remaining error, the most significant is the excessive heat transfer associated with factors such as the influence of ambient temperature on the sample. This is particularly relevant in the upper part of the calorimeter, where the sample is introduced into the system, and which is not covered by the water-cooling jacket. Addressing this issue would require considering the inhomogeneity of the calorimetric properties of the sample's surroundings and potentially incorporating the thermal properties of the vessel walls containing the measured liquid. Another source of error we must consider is the accuracy of the measurement probes, which is specified by the manufacturer's calibration note at 0.1 \({}^{\circ}\)C. Additionally, errors arise from the process of setting and determining the input heating power, where the primary sources of uncertainty include the tolerance of the resistor and the precision and stability of the power supply voltage. However, rather than relying solely on the nominal tolerance of the resistor, we directly measured its resistance value using a precise Agilent 4263B LCR Meter (Santa Clara, CA, USA), which offers a basic accuracy of 0.1%. This  
Figure 7: Errors in the estimation of heating power values using different non-adiabatic calorimeter models: R5C2—novel calorimeter model, R1C1—model corresponding to corrected-slope method. approach allowed us to minimize uncertainty in power determination. It should be noted that the calculated error values refer to a system where heating power is generated by a concentrated heating element (SMD resistor) submerged in liquid. However, in the case of magnetic nanoparticle heating, energy dissipation is not specifically localized in space but rather more uniformly distributed throughout the sample volume. Previous studies [20; 47] have repeatedly demonstrated that the temperature distribution within the sample is not perfectly homogeneous, mainly due to the uniformity of the magnetic field throughout the entire sample volume. This results in an additional source of error related to the positioning of the measurement probe. However, this type of error is not directly related to the calorimeter's construction or a process affecting its non-adiabaticity and overall accuracy. Instead, it stems from calibration limitations caused by the lack of an available reference heating power source in the form of a nanoparticle solution.