## 4 Validation of the Estimation Procedure  
A dedicated validation of the estimation method by means of synthetic temperature data has been performed. The validation of the inverse procedure for the wall-to-fluid heat flux evaluation is based on four different steps [44]:  
1. _Direct problem implementation._ The heat transfer by conduction through the studied solid domain is simulated on a CFD environment by providing a known heat flux distribution \(q_{exact}\) as a boundary condition at the wall-fluid interfaces;
2. _Simulation of experimental data acquisition._ The resulting synthetic temperature distribution is randomly spoiled with Gaussian noise levels \(\sigma\);
3. _Inverse problem implementation._ The causal factor (imposed heat flux) that produces the given set of synthetic, experimental observations (noisy temperature distribution on the outer wall surface) is estimated. To this aim, the three-dimensional noisy temperature, being a function of the two space coordinates and time, is filtered by means of Equation (19) and used as input for Equation (9), thus attempting to evaluate back the given wall-to-fluid heat flux \(q_{exact}\). In particular, since the considered inverse problem concerning transient heat conduction in the bidimensional domain is intrinsically ill-posed from a mathematical standpoint (due to the presence of noise in the input temperature data which undermine the stability of the solution), Equation (19) is employed as a regularization method by filtering out the undesired noise;
4. _Evaluation of the estimation error._\(q_{exact}\) is finally compared with the heat flux evaluated in step (3), \(q_{restord}\), thus assessing the goodness of the estimation procedure for the given heat flux distribution and noise levels. Step (1) was performed through the Comsol Multiphysics(r) software (version 5.6, COMSOL Inc., Stockholm, Sweden) by the finite elements method. More than 150,000 elements were found to guarantee the grid-independency of the model. To ensure the reliability of the method validation, the \(q_{exact}\) given in step (1) has to be as similar as possible to the thermofluidic interactions occurring in the real device. Hence, according to the expected physical behavior in experimental cases [3; 18], the convective heat flux is assumed to exhibit a sinusoidal shape described by time-space variations, representative of the working fluid oscillations through the adiabatic section:  
\[q_{exact,A}=Amp\text{-}cos(2\pi t\text{-}f-\varphi)+\frac{Amp}{2}-\frac{z}{H} \cdot Amp \tag{20}\]  
where \(H\) is the length of the considered portion of the adiabatic section, \(f\) is the oscillation frequency of the sinusoidal signal and \(z\) is the axial coordinate. Here, variations over the \(z\) coordinate are expressed by the term \(\frac{z}{H}\cdot Amp\), while variations over y are introduced by means of the phase \(\varphi\). Adjacent channels are assumed to exhibit a phase shift equal to \(\pi\). In fact, in the real FPPHP device, when hot fluid flows from the evaporator to the condenser in one channel, cold fluid is expected to flow back to the evaporator in the adjacent branches. This may result, for fixed time instants, in positive wall-to-fluid heat flux peaks in one channel (hot fluid warming up the device wall), and negative wall-to-fluid heat flux peaks (device wall cooled down by cold fluid) in the adjacent branches. In addition, the wall-to-fluid heat flux is expected to vary between the thin walls and side walls surfaces A and B, respectively, highlighted in Figure 4 for a generic channel, thus leading to the following definition of \(q_{exact,B}\):  
\[q_{exact,B}=2q_{exact,A} \tag{21}\]  
Specifically, the heat flux exchanged over surface B is believed to present greater heat flux amplitudes due to the higher thermal inertia of the side walls; a validation of this assumption by preliminary experimental data will be provided at the end of the present Section.