# Fast Calculation of Transient Thermal Stress in Surge Lines with Thermal Stratification Region
## 3 Fast Calculation of Thermal Stress Based on Green's Function Method
### The Establishment of Thermal Stress Green's Function
#### 3.3.2. Strategy to Construct Thermal Stress Green's Functions  
Thermal stress Green's functions should be worked out for every fatigue evaluation section and path with respect to the step temperature loading and convective heat transfer coefficient of every thermal zone. Thermal stress Green's functions may be shared if convective heat transfer coefficients are identical for different flow cases. Generally, thermal stress Green's functions have a huge database, covering every flow case. For example, we assumed 20 \({}^{\circ}\)C as the zero-thermal stress temperature of the structure and a uniform temperature of 20 \({}^{\circ}\)C in all thermal zones as the beginning of the transient thermal analysis. A step temperature of 320 \({}^{\circ}\)C was applied as the fluid bulk temperature only in the appointed thermal zone 4 with a specified convective heat transfer coefficient corresponding to flow case 3. Then, 6000 s was used to obtain a stable state for the tempera  
Figure 8. Material parameters of water at different temperatures. (**a**) Density. (**b**) Dynamic viscosity. Figure 9. The location of the fatigue evaluation sections. ture response, which is illustrated in Figure 12, for the outside point of the evaluation paths in the two sections. The temperature response data is the temperature Green's function of each thermal zone under each working condition. Figure 11: Stress distribution in Section A and B. (**a**) Section A. (**b**) Section B.  
Figure 10: The paths in the monitoring sections. Then, elastic thermal stress can be calculated by reading the transient thermal analysis results. The time length of Green's function is 6000 s, and the Green's functions of the evaluation paths are shown in Figure 13. It is clear that all directional stresses and \(S_{int}\) remain stable after 2000 s. This reveals that 6000 s is quite enough to obtain the stable state of the step response. So the time length of Green's functions, which is the set decay time, is much larger than the actual decay period, and this means the time length of Green's function is reasonable. The obtained stress response needs to be normalized to get Green's function datum. The Green's function of each thermal zone is denoted as \(G_{ij}\), where the subscript \(i\) varies from 1 to 10, corresponding to 10 thermal zones, and \(j\) varies from 1 to 7, corresponding to seven different flow cases. The steady-state value of each Green's function is recorded as \(G_{0,ij}\), and the transient-state value is \(G_{ij}-G_{0,ij}\), where \(i\) and \(j\) have the same meaning as the subscript of [PERSON]'s function, and \(g_{0,ij}\) and \(g_{ij}\) represent the steady-state and transient values of [PERSON]'s function after normalization, respectively  
\[g_{0,ij}=G_{0,ij}/\Delta T \tag{8}\]  
Figure 12: Temperature response of the outside point of the evaluation paths. (**a**) Section A. (**b**) Section B.  
Figure 13: Transient thermal stress (total stress and \(\text{S}_{\text{int}}\)) response of evaluation paths based on FEM. (**a**) Section A-60 (**b**) Section B-270. It can be seen from Figure 17 that step temperatures above 300 \({}^{\circ}\)C yield small relative error. A step temperature of 400 \({}^{\circ}\)C gives the smallest relative error of 3.422%. From the above analysis, it was found that the Green's functions can yield transient thermal stress in quite good agreement with that calculated directly by FEM. However, we cannot say for sure that the method provides the real transient thermal stress. The shortcoming is the assuming fluid temperatures for thermal zones especially in stratification region, which has not given any evidence to close the problem. As factory datum are generally available for hot leg fluid and outlet fluid from the pressurized, temperatures for thermal zones 1, 2, 3, and 10 may be easily determined. With large surge out or in flow case, temperatures in zone 4-9 can also be taken as the temperature of the fluid from the hot leg or outlet fluid from the pressurized.