3.2. 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 °C. The flow rate from the pressurizer is 0.05 m/s, and the temperature of the fluid in the pressurizer is 330 °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 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 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].
where
is the length of turbulent flow penetration, Ω
0 is the entry swirl,
D is the inner diameter of the surge line,
U is the average velocity,
DR is the inner diameter of the hot leg,
Usrg is the average flow velocity of the surge line,
ξ is taken as 6 for the horizontal pipe section, and
β is taken as 1.4 by experience. The term “±1” in Equation (5) accounts for the uncertainty in model predictions that is ±1 diameter corresponding, to a 95% confidence limit. The calculated turbulent penetration length ranges from 1095–1665 mm (about 10
Dsrg), which is similar to the CFD result.
When the thermal stratification phenomenon exists, the fluid temperature difference in different locations is very large. In dividing the thermal zone of the thermal stratification region, some experts divide the thermal stratification region into two thermal zones, and the temperature of the upper zone is taken as the temperature of the fluid from the pressurizer, and the temperature of the lower zone is taken as the temperature of the fluid from the hot leg.
However, this kind of division of thermal zones is too simple to describe the temperature field of thermal stratification, and the error in the temperature field will cause large errors in the calculation of thermal stress. Therefore, a much more accurate description of the thermal zone is needed. Referring to the thermal stratification monitoring work for the surge line of the Paks Nuclear Power Plant [
9], seven thermocouples were installed on the outer wall of the surge line. In order to accurately describe the fluid temperature in the thermal stratification region, all temperatures at the seven measuring points were used to divide the thermal stratification region into six thermal zones, which were named zone 4–9. The division of the thermal zone of the surge line is shown in
Figure 6.
During the operation of a nuclear power plant, the working conditions are very complicated. The stratification state and thermal hydraulic parameters in the surge line are different under different working conditions. Based on the power plant operation experience, seven flow cases were set according to the presence or absence of thermal stratification, the flow rates of the hot leg and the outlet of the pressurizer. Among the seven flow cases, case 3 is a start-up condition, and the thermal stratification phenomenon is most obvious under this flow case. Cases 6 and 7 are accident conditions, which correspond to the power off of the main pump and the low flow rate in the hot leg. Generally, such an accident occurs when the hot leg flow rate drops below one-tenth of the rated rate
Qn. The judgments and description of every flow cases is shown in
Table 1.
When determining the wall–fluid convective heat transfer coefficient of each thermal zone, the fluid temperature and flow rate should be considered. When the flow rate is low, it is natural convective heat transfer controlled. When the flow rate is high enough, it is forced convective heat transfer controlled. In the turbulent penetration region, spiral convective heat transfer should be more suitable. So, convective heat transfer coefficients should be determined carefully with suitable empirical equations and thermal hydraulic parameters [
15].
In this paper, the start-up condition (case 3) is taken as an example, and the calculation results of the convective heat transfer coefficient of each hot zone under this condition is shown in
Table 2.
3.3. The Establishment of Thermal Stress Green’s Function
3.3.1. Material Properties
Thermal stress Green’s function is generally obtained by applying a certain step temperature on each thermal zone. However, the physical parameters of metal materials are greatly affected by temperature. Koo and Kwon [
16] confirmed that the temperature characteristics of materials have a significant influence on the maximum peak stress, and the temperature dependence of material properties affects the maximum stress range for fatigue evaluation; they used the weighting function method to make the Green’s function method more accurate. Therefore, it is necessary to consider this effect when using the Green’s function method to monitor the fatigue damage of the actual operating conditions of nuclear power plants. Therefore the temperature-dependent material properties were defined in the current study. The surge line and the connected hot leg are Z3CND20.09M and Z2CND17.12, respectively, and the material properties were taken from RCC-M specifications, as shown in
Figure 7. The Poisson’s ratio is taken as a constant of 0.3, and the density is taken as a constant of 7930 kg/m
3, and the fluid physical parameters change with temperature, as shown in
Figure 8.
3.3.2. Selection of Fatigue Evaluation Sections and Paths
FRAMATOME’s research on the surge line of a 900 MW pressurized water reactor nuclear power plant revealed that the elbow of the vertical pipe section is the position most prone to thermal fatigue [
10]. In addition, large thermal stress is likely to occur at the butt-weld of the hot leg nozzle. So, these two sections were set as monitoring sections, as shown in
Figure 9, and named Section A and Section B, respectively.
In order to determine the analysis path, 36 paths are set every 10° in the monitoring sections, as shown in
Figure 10, and named Path 0 to Path 350.
In monitoring Section A, the maximum stress occurs in Path 60, and in Section B, the maximum stress occurs in Path 270, as shown in
Figure 11. So, Path A-60 was set as the evaluation path of Section A, and Path B-270 was set as the evaluation path of Section B.
3.3.3. 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 °C as the zero-thermal stress temperature of the structure and a uniform temperature of 20 °C in all thermal zones as the beginning of the transient thermal analysis. A step temperature of 320 °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 temperature 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.
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
Sint 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
Gij, 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
G0,ij, and the transient-state value is
Gij −
G0,ij, where
i and
j have the same meaning as the subscript of Green’s function, and
g0,ij and
gij represent the steady-state and transient values of Green’s function after normalization, respectively
The equation for calculating the thermal stress of the evaluation path using Green’s function is as follows.
φi(
t) is the fluid temperature of thermal zone
i, and
Tref is the fluid temperature at time 0,
n is the number of thermal zones, and
S is the thermal stress of the evaluation path.
3.3.4. Verification of the Green’s Function
In this paper, a working condition was designed in order to verify the accuracy of the Green’s function. The temperature change in each thermal zone with time in the designed condition is shown in
Figure 14.
We used the established Green’s function of each thermal zone to perform convolution calculation with the temperature–time change of each thermal zone, and added the stress calculation results for each thermal zone to obtain the stress change with time under this working condition. The stress calculation result of Green’s function method was compared with the stress result calculated by finite element calculation software, and the results are shown in
Figure 15. Comparing the calculation result of Green’s function method with the calculation result of finite element method, the error of the maximum stress value was 3.451%.
The difference in the step temperature load applied to the thermal zone will cause fluctuations in the results after the normalization of the Green’s function. In order to reduce the relative error of the comparison between the Green’s function and the finite element calculation result, it is necessary to adjust the step temperature load applied to the different thermal zones. The step temperature was set to 100 °C, 200 °C, 300 °C, 400 °C, 500 °C and 1000 °C, respectively, to find the optimal one.
The
Sint curves of the Green’s function method and the finite element result at different step temperatures are compared in
Figure 16. The relative errors of the maximum stress between the Green’s function method calculation result and the finite element calculation result at different step temperatures are shown in
Figure 17.
It can be seen from
Figure 17 that step temperatures above 300 °C yield small relative error. A step temperature of 400 °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 pressurizer, 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 pressurizer. So, we focused on the determination of the fluid temperature for thermal zones 4–9 when stratification occurs.