## 4 Results and Discussion
### Comparison Results of the Uncertainty Analysis and Deterministic Models  
To investigate the dispersions of the model outputs of the maximum temperature and thermal stress inside the receiver tube wall under multi-source uncertainties, an uncertainty propagation evaluation was performed through MC simulations using 5000,  
Figure 8: Temperature distribution of the tube. (**a**) Whole tube wall (z = 1:60); (**b**) Cross-wall of the tube (z = 0 m).  
Figure 9: Thermal stress distribution of the tube. (**a**) Whole tube wall (z = 1:60); (**b**) Cross-wall of the tube (z = 0 m).  
10,000, and 15,000 samples. The results showed that there was no appreciable difference in the cumulative frequency of the model output when the sample size exceeded 10,000. Therefore, 10,000 samples were used to conduct the uncertainty propagation analysis.  
The frequency and cumulative frequency of the maximum temperature and thermal stress inside the tube wall under uncertainty are, respectively, shown in Figures 10 and 11. Figure 10 demonstrates that the lower and upper bounds of the maximum temperature inside the tube wall under uncertainty were, respectively, 847 K and 895 K, with an expectation of 871 K and a standard deviation of 8 K. The deterministic result of the maximum temperature inside the tube wall (871 K) is also depicted in Figure 10, and the corresponding cumulative frequency was 50%. This demonstrates that there was a 50% probability that the maximum temperature of the tube wall under uncertainty would be more than the deterministic value of 871 K, and the maximum deviation of the uncertainty analysis result from the deterministic value was 24 K, which may cause the overheating of the receiver. As a result, to prevent receiver failure caused by overheating, the dispersion of the maximum temperature inside the tube wall under uncertainty should be considered in the design process of the receiver.  
Figure 11 illustrates the lower and upper maximum thermal stresses inside the tube wall under uncertainty, which were, respectively, 173 MPa to 245 MPa, with an expectation of 204 MPa and a standard deviation of 12 MPa. The results demonstrated that there was a 50% probability of achieving maximum thermal stress exceeding 200 MPa and a 25% probability of achieving maximum thermal stress exceeding 208 MPa. The deterministic result of the maximum thermal stress inside the tube wall is also plotted in Figure 11, which shows a 40% probability of exceeding the deterministic result of the maximum thermal stress inside the tube wall (204 MPa), and the maximum deviation of uncertainty analysis result from the deterministic value was 41 MPa, which may cause fatigue damage in the receiver. It is worth noting that although the probability distributions of the uncertain input parameters in this paper were all symmetrical, the deterministic result of the maximum thermal stress was not at the cumulative frequency of 50%. This can be attributed to nonlinear models and responses. All in all, considering that the model input parameters generally have uncertainties, the adaptability of the deterministic model has to be questionable. Thus, to  
Figure 10: Probability distribution of the maximum temperature inside the tube wall under uncertainty.  
\(p_{6}\) was the least sensitive to the maximum thermal stress of the tube wall, with a total sensitivity index \(S_{\text{T}}\) and contribution rate of nearly 0.