# 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
### Division of Thermal Zones and Working Conditions  
In order to divide thermal zones, computational fluid dynamics (CFD) simulation was performed for a start-up condition of the surge line with transient analysis by using Ansys Fluent. The flow rate in the hot leg is 15.5 m/s, and the temperature of the fluid in the hot leg is 290 \({}^{\circ}\)C. The flow rate from the pressurizer is 0.05 m/s, and the temperature of the fluid in the pressurizer is 330 \({}^{\circ}\)C. After the calculation result reaches stable, the thermal stratification is very significant in the horizontal pipe section, as shown in Figure 4. The turbulent penetration phenomenon exists in the hot leg end. In this region, the streamline  
Figure 1: The 3-D model of the surge line. Figure 3: The finite element model of the surge line and the constraints. (**a**) Ends of the hot leg. (**b**) The pressurizer end. Figure 2: The finite element model of the surge line. (**a**) Ends of the hot leg. (**b**) The pressurizer end. presents swirling flow, and there is no significant difference between the temperature of the top and bottom fluids. Figure 5 shows the temperature of the top and bottom fluid at different positions of the straight pipe section near the hot leg. When the distance from the inlet of the branch pipe is less than 1200 mm, the temperature difference is small, which indicates swirling flow with penetration. When the distance is more than 1200 mm, the temperature difference quickly gets larger, which indicates that the turbulent penetration phenomenon disappears. So the length of the turbulent penetration is about 1200 mm. As for the vertical pipe section, there is little difference in the temperature of the fluid in the pipe. According to the CFD simulation results, the entire flow field can be divided into the following parts: the hot leg, the turbulent penetration region, the thermal stratification region, and the vertical pipe region. As the fluid temperature in the hot leg and vertical pipe stays almost constant, one thermal zone can be assigned for each region, namely, zone 1 for hot leg and zone 10 for the vertical pipe section. The turbulent penetration region and the nozzle are divided into  
Figure 4: The streamline of the turbulent penetration region. Figure 5: The temperature of the top and bottom fluid changes with the distance from the hot leg. two zones, named zone 2 and 3. They generally have the same heat transfer boundary, but thermal zone 2 (the upper half) and thermal zone 3 (the lower half) can employ different heat transfer boundaries for large surge flow rates, in which turbulent penetration may be suppressed. The turbulent penetration length can also be calculated by Equation (5), and the empirical Equations (6) and (7) are based on the test data [14]. \[\frac{\overline{x}_{m}}{D}=\frac{L_{\Omega}}{D}\left[\left(\frac{\Omega_{0}D}{ 2 Ul}\right)^{1/\beta}-1\right]\pm 1 \tag{5}\]  
\[\frac{\Omega_{0}D}{2 Ul}=0.63\bigg{(}\frac{D}{D_{R}}\bigg{)} \tag{6}\]  
\[\frac{L_{\Omega}}{D}=3.2 \tag{7}\]  
where \(\overline{x}_{m}\) is the length of turbulent flow penetration, \(\Omega_{0}\) is the entry swirl, \(D\) is the inner diameter of the surge line, \(Ul\) is the average velocity, \(D_{R}\) is the inner diameter of the hot leg, \(U_{sg}\) is the average flow velocity of the surge line, \(\xi\) is taken as 6 for the horizontal pipe section, and \(\beta\) is taken as 1.4 by experience. The term \"\(\pm 1\)\" in Equation (5) accounts for the uncertainty in model predictions that is \(\pm 1\) diameter corresponding, to a 95% confidence limit. The calculated turbulent penetration length ranges from 1095-1665 mm (about \(10D_{sg}\)), which is similar to the CFD result. When the thermal stratification phenomenon exists, the fluid temperature difference in different locations is very large.