## 3 Local Heat Flux Estimation Procedure  
The heat flux locally exchanged at the fluid-wall interface was estimated by using the acquired IR temperature maps as inputs for the IHCP resolution approach. For the data processing, each FPPHP branch in the adiabatic section was considered as a plane-symmetric geometry (Figure 3a) due to the same boundary conditions on both outer surfaces of the device (free convection coupled with radiation). In the present analysis, the only channel walls having thickness \(\mathrm{s}=\mathrm{x}_{\mathrm{ext}}-\mathrm{x}_{\mathrm{int}}=0.5\) mm were considered (red dashed lines in Figure 3a) to better understand the heat transfer phenomena involving internal interactions between the working fluid and the device wall. For such a thin wall element (Figure 3b), the transient energy balance equation without heat generation reads as:  
\[k\
abla^{2}T=\rho c_{p}\frac{\partial T}{\partial t} \tag{1}\]  
where \(k\), \(\rho\) and \(c_{p}\) are the solid wall thermal conductivity, density and specific heat, respectively. The boundary conditions for Equation (1), applied on the inner and outer surfaces of the channel, are, respectively, defined by Equations (2) and (3). \[-k\left.\frac{\partial T}{\partial x}\right|_{x=x_{int}}=q \tag{2}\]  
\[k\frac{\partial T}{\partial x}\right|_{x=x_{ext}}=-h_{env}(T-T_{env}) \tag{3}\]  
Here, \(T_{env}\) is the room temperature and \(h_{env}\) is the overall heat transfer coefficient between the channel wall and the surrounding environment by both natural convection and radiation. The thermal conductivity \(\mathrm{k}\) of copper was certified equal to 401 W/mK at 20 \({}^{\circ}\)C and the heat transfer coefficient \(h_{env}\) was considered equal to 10 W/m\({}^{2}\)K. The temperature at \(x=x_{int}\) was assumed to be equal to that at \(x=x_{ext}\) (thin-wall approximation), thus simplifying the three-dimensional solid domain as a bidimensional one, where the temperature is considered to vary only along the y and z coordinates, as well as over time. \begin{table}
\begin{tabular}{l l} \hline \hline _Wavelength_ & 2.5-5.1 \(\mathrm{\SIUnitSymbolMicro m}\) \\ \hline _Focal length_ & 27 \(\mathrm{mm}\) \\ \hline _F-number_ & 3 \\ \hline _Cooler_ & Close-cycle (rotary) Stirling cooler \\ \hline _Field of view_ & \(20^{\circ}\times 16^{\circ}\) \\ \hline _Temperature range_ & \(-20\,^{\circ}\)C to 3000 \({}^{\circ}\)C \\ \hline _Distance from the framed portion during the tests_ & 50 \(\mathrm{cm}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Camera settings. Figure 3: Cross-sectional view of a generic FPPHP channel within the adiabatic section (**a**) and three-dimensional generic wall element (**b**). By considering the infinitesimal wall section of Figure 3b with dimensions \(\Delta y\) and \(\Delta z\), Equation (1) assumes the following formulation:  
\[\rho c_{p}\frac{\partial T}{\partial t}s\Delta y\Delta z=\left(Q_{y}-Q_{y+\Delta y }\right)+\left(Q_{z}-Q_{z+\Delta z}\right)+Q_{int}-Q_{ext} \tag{4}\]  
The power terms in Equation (4) are expressed as:  
\[\left(Q_{y}-Q_{y+\Delta y}\right)=\frac{k\delta^{2}T}{\partial y^{2}}s\Delta y \Delta z \tag{5}\]  
\[\left(Q_{z}-Q_{z+\Delta z}\right)=\frac{k\delta^{2}T}{\partial z^{2}}s\Delta y \Delta z \tag{6}\]  
\[Q_{int}=q\Delta y\Delta z \tag{7}\]  
\[Q_{ext}=h_{env}(T-T_{env})\Delta y\Delta z \tag{8}\]  
When Equations (5)-(8) are substituted in Equation (4), the wall-to-fluid heat flux q reads as:  
\[q=s\left(\rho c_{p}\frac{\partial T}{\partial t}-k\frac{\partial^{2}T}{ \partial y^{2}}-k\frac{\partial^{2}T}{\partial z^{2}}\right)+h_{env}(T-T_{env}) \tag{9}\]  
During the experiments, \(T_{env}\) varied from 18\({}^{\circ}\) C to 21 \({}^{\circ}\)C. For the cases under investigation, the term \(h_{env}(T-T_{env})\) was negligible with respect to the other heat transfer terms appearing in Equation (9), thus proving that the value assumed by the overall heat transfer coefficient \(h_{env}\) does not play a significant role in the present wall-to-fluid heat flux evaluation.