## 4 Validation of the Estimation Procedure
The capability of the inverse procedure of effectively evaluating the wall-to-fluid heat flux was quantified by adopting different amplitudes and oscillation frequencies of \(q_{exact}\). In particular, \(Amp\) was considered equal to 2000, 4000 and 8000 W/m\({}^{2}\)K, while f was considered equal to 0.5, 0.8 and 1.2 Hz. From the direct problem simulation, the maximum residual between the inner and outer wall temperatures, i.e., for \(T(x=x_{int})\) and \(T(x=x_{ext})\) according to Figure 2b, was found to be less than 0.02 K. Since such a value was lower than the noise level evaluated in Section 3 for the present experimental set-up (0.04 K), the thin-wall approximation was consequently verified. For step (2) of the validation procedure, three different noise levels \(\sigma\) were adopted, namely 0.01, 0.04 and 0.08 K. To provide references for the restoration procedure of the  
Figure 4: Two surfaces at which \(q_{exact}\) is defined for the generic device channel. A: thin wall; B: side wall. given exact wall-to-fluid heat flux distributions, the validation data referred to two different \(q_{exact}\) cases for a single channel (\(\varphi=0\)) are shown in Figure 5 by considering \(\sigma=0.04\) K and the axial coordinate \(z=0.02\) m.  
In Figure 5a-c, the noisy temperature \(T_{noise}\), resulting from step (2), and the filtered temperature \(T_{filtered}\), resulting from step (3), are reported for \(Amp=2000\) W/m\({}^{2}\),\(f=1.2\) Hz and \(Amp=8000\) W/m\({}^{2}\),\(f=0.5\) Hz, respectively. In Figure 5b-d, the exact and restored heat fluxes are additionally shown for the same test cases. In Figure 5a, the lower amplitude of the considered \(q_{exact}\) results in lower temperature variations and lower signal-to-noise ratio, leading to a hard restoration of the exact heat flux peaks (Figure 5b) [33]. On the contrary, the heat flux of Figure 5d is restored with slightly greater accuracy thanks to the higher signal-to-noise ratio of \(T_{noise}\) (Figure 5c), even though some variations from the exact value are still appreciable due to residual noise in the \(T_{filtered}\) signal. Figure 5: Noisy and filtered temperature signals (\(\sigma=0.04\) K, z = 0.02 m) for \(Amp=2000\) W/m\({}^{2}\), \(f=1.2\) Hz (**a**), and \(Amp=8000\) W/m\({}^{2}\), \(f=0.5\) Hz (**b**), and corresponding exact and restored heat fluxes (**c**,**d**), respectively. The fixed axial coordinate \(z=0.02\) m is considered. Moreover, the goodness of the assumption of Equation (21) was verified by comparing preliminary experimental data with numerical data obtained by the heat flux restoration procedure on synthetic distributions. In Figure 6, all the terms appearing in Equation (9) are plotted as a function of time for a generic PHP channel and a median axial coordinate. Specifically, \(q_{z}=-sk\frac{\partial^{2}T}{\partial z^{2}}\), \(q_{y}=-sk\frac{\partial^{2}T}{\partial y^{2}}\), \(q_{t}=spc_{p}\frac{\partial T}{\partial t}\) and \(q_{env}=h_{env}(T-T_{env})\). In other words, \(q_{z}\), \(q_{y}\) and \(q_{t}\) are the terms constituting the inverse solution, while \(q_{A}\) and \(q_{B}\) in Equation (21) are the parameters used in the direct problem for the generation of synthetic data. Specifically, the amplitude of the heat flux exchanged on the B surface may influence the terms of the heat flux exchanged on the thin-wall surface due to a redistribution of temperature by conduction in the solid domain. Hence, \(q_{z}\), \(q_{y}\) and \(q_{t}\) were considered in both the numerical and the experimental cases to better understand the relationship between \(q_{A}\) and \(q_{B}\), and to qualitatively assess the reasonability of the assumption of Equation (21). Figure 6a refers to the numerical data, while Figure 6b refers to the preliminary experimental data. From the restoration procedure on synthetic data (Figure 6a), obtained by adopting Equation (21), the \(q_{t}\) peaks assume values about half, in module, of those assumed by \(q_{y}\), while \(q_{z}\) approaches the positive peaks of \(q_{t}\).