## 3 Methods  
### Wall-to-Fluid Heat Flux Estimation  
The procedure for the estimation of the heat flux exchanged between the FPPHP thin walls and the working fluid is based on the post-processing of the thermographic data by the inverse heat conduction problem (IHCP) resolution approach, coupled with a three-dimensional Gaussian filter for the regularization of the raw space-time temperature maps. Such a procedure is carried out within the adiabatic section of the device, and it is thoroughly explained in [29], together with its error estimation. In Figure 3, the areas of interest for the wall-to-fluid heat flux estimation, i.e., where the FPPHP channels present thin walls, are highlighted in red.  
The resulting wall-to-fluid heat flux distributions \(q\) are bi-dimensional maps referred to each FPPHP channel over 0.04 m (length of the section of interest) and 20 s (duration of each MWIR acquisition). In particular, by considering the energy balance equation in transient  
Figure 3: Three-dimensional sketch of a FPPHP portion within the adiabatic section; in red, areas of interest for the wall-to-fluid heat flux evaluation.  
Figure 2: Investigated device orientations with respect to the gravity field.  
conditions at the thin wall element and by rearranging all the terms, the wall-to-fluid heat flux finally assumes the following formulation [29]:  
\[q=s\bigg{(}\rho c_{p}\frac{\partial T}{\partial t}-k\frac{\partial^{2}T}{ \partial y^{2}}-k\frac{\partial^{2}T}{\partial z^{2}}\bigg{)}+h_{env}(T-T_{env}) \tag{1}\]  
where \(s\) is the wall thickness, \(\rho\), \(c_{p}\) and \(k\) are density, specific heat and conductivity of copper, respectively, \(h_{env}\) is the outer heat transfer coefficient by free convection and radiation to the environment, and \(T_{env}\) is the environmental temperature. To obtain consistent values of heat flux \(q\), the input temperature is filtered by following the steps below [29]:  
* The three-dimensional, raw space-time temperature map is converted to the frequency domain by the discrete fast Fourier transform (DFFT);
* The transfer function for the Gaussian filter is applied to the frequency image of the input temperature map. A first attempt cut-off frequency is initially guessed;
* The temperature map is converted back to the space-time domain by applying the inverse DFFT;
* The residual between the raw input temperature map and the filtered temperature is computed;
* Steps 2-4 are iteratively replicated by varying the cut-off frequency until the evaluated residual equals the standard deviation of the raw temperature data, \(\sigma\) = 0.04 K (i.e., fulfilment of the [PERSON]'s discrepancy principle [31]). The goodness of the estimation procedure is therefore assessed by means of the following steps [29]:
* The direct problem is implemented in a COMSOL Multiphysics(r) environment. The solid domain of the device within the adiabatic section is simulated by using, as a boundary condition at the inner wall-fluid interface, a given space-time heat flux distribution that reflects the one exchanged in the real device. Equations for three-dimensional transient conduction are solved by the in-built solver;
* The synthetic temperature distribution resulting from the simulation is then spoiled with a noise level equal to the standard deviation of the raw data, and processed by means of the filtering technique;
* The wall-to-fluid heat flux is estimated back by means of Equation (1), i.e., leading to the inverse problem solution. The estimated wall-to-fluid heat flux is therefore compared with the one given as a boundary condition in Step 1, allowing the evaluation of the estimation error for the procedure (quantified in [29], equal to about 15% of the heat flux given in Step 1).