## 4 Discussion  
Modifying the internal surface of the PTC receiver tube with fins and using nanofluids as the working fluid are two popular techniques to enhance the performance of the PTC systems. The turbulence of the working fluid is increased by fins, and the conductivity is improved using nanofluids. Thereby, the heat transfer of the receiver tubes is enhanced. However, using the two techniques makes the receiver systems more complex; consequently, much more cost is required for the analysis and optimization investigations using experimental or CFD methods. Therefore, to address this problem, two ROMs based on CNNs were developed in this study.  
According to the results of Section 3.1, the first ROM could accurately and quickly predict the temperature field of the outlet cross section, including the fluid and solid domain, which could provide some interesting local temperature information for the designer. As commonly known, local high temperature could cause local thermal stress and deformation, further leading to the PTC structure failure. In our problem, the sunlight concentrates on the lowest part of the tube wall, then logically, as shown in Figure 9, the lower part of the tube shows the highest temperature. Furthermore, the larger fins heat more nanofluid, and its own temperature also rises. This may be due to the fact that in our study, the stream velocity is fixed, so a larger cross-sectional area of fins results in a smaller fluid cross-sectional area, and thus a smaller fluid flow. Therefore, there is insufficient working fluid to carry the heat away from the absorption tube. Hence, if we use the proposed temperature field predictor in the fin-shape optimization process, the local high temperature could be known and avoided.  
The second predictor is developed to estimate the heat flux of the outlet cross section, then calculate the useful heat production of the PTC receiver, which is one of the most important physical quantities for PTC absorber tubes, measuring the heat captured by the working fluid passing through the absorber tube. According to the comparison in Table 4, the size of the fin has a significant effect on the outlet heat flux, and an increased height of the fins leads to a decrease \(\mathrm{q_{out}}\) in the outlet cross section. This is probably because the increase in the height of the fin reduces the nanofluid flow cross-sectional area. And in the case of a fixed inflow velocity, the flow rate of the absorber tube is reduced, and there is not enough working fluid to transfer the heat from the tube. Furthermore, we have investigated the influence of dataset size on the performance of the proposed \(\mathrm{q_{out}}\) predictor. The predictions with all tested data sizes agree well with the reference, and the biggest dataset of 2000 data points leads to the highest accuracy.  
Benefitting from the proposed high-performance ROMs in this study, the future investigation could be the fin-shape optimization of the PTC receiver using the performance enhancement criteria and an appropriate optimization [55; 56]. Conventional optimization framework usually uses a CFD solver to estimate the observation points. Meanwhile, for a complex, highly nonlinear heat transfer system, the optimization process requires a large number of observation points, and each estimation needs to solve lots of PDEs. Therefore, the required computational time grows exponentially. Fortunately, the proposed ROMs enable fast and accurate performance prediction of the nanofluid-filled receiver tube with fins. Therefore, replacing the CFD solver with the ROMs will significantly improve the optimization efficiency.