# Fast Calculation of Transient Thermal Stress in Surge Lines with Thermal Stratification Region
## 4 Determination of Fluid Temperatures of Thermal Zones in Stratification Region
### Determination of Fluid Temperatures of Thermal Zone 4-9  
We assumed the temperature of the outer wall at the monitoring point is \(T_{\text{O1}}\), \(T_{\text{O2}}\), \(T_{\text{O3}}\), \(T_{\text{O4}}\), \(T_{\text{O5}}\), \(T_{\text{O6}}\), \(T_{\text{O7}}\) from top to bottom in order, and the corresponding fluid temperature of each layer is \(T_{\text{f12}}\), \(T_{\text{f23}}\), \(T_{\text{f34}}\), \(T_{\text{f45}}\), \(T_{\text{F56}}\), \(T_{\text{f67}}\).  
The temperature data was selected from the stable section after 20,000 s, as shown in Figure 21, the top measured temperature decreases a little from the pressurizer side to the hot leg side, and the bottom measured temperature decreases a little from the hot leg side to the pressurizer side. This is due to the heat loss of the fluid along the flow direction. However, with the help of pipe insulation, the heat loss effect can be neglected. Thus, the difference of temperature in the axial direction is ignored, and the fluid temperature of one zone is constant.  
The top and bottom measured temperatures in each monitoring section were compared with the temperature of the outlet fluid from pressurizer \(T_{\text{hot}}\) and the temperature of the hot leg fluid \(T_{\text{cold}}\), respectively, as shown in Figure 22.  
Figure 21: Comparison of the temperature at the top and bottom monitoring points in different monitoring sections.  
Figure 22: The temperature of the fluid from the pressurizer and hot leg.  
In time span B (70,000-130,000 s) and time span C (130,000-280,000 s) before stable operation, the measured temperatures fluctuate sharply due to changes in surge flow, which strongly disturbs the fluid temperature field. During stable operation, the top and bottom measured temperatures are roughly the same as the temperature of the fluid from pressurizer and hot leg, respectively. It can be seen that at the time of stable operation, the top measured temperature is slightly lower than \(T_{hot}\), and the bottom measured temperature is slightly lower than \(T_{cold}\). Taking into account slight thermal diffusion through the pipe insulation, it is reasonable that the outside wall temperatures are slightly lower than that of the fluid temperature. So, it is reasonable to assume that during stable operation, the temperature of the top zone fluid is equal to \(T_{hot}\), and the temperature of the bottom zone is equal to \(T_{cold}\), which is  
\[T_{f1}=T_{hot} \tag{11}\]  
\[T_{f7}=T_{cold} \tag{12}\]  
Then, from the radial heat transfer Equations (13) and (14), the heat flux densities at the top and bottom can be obtained,  
\[q_{1}=\frac{T_{f1}-T_{01}}{\frac{1}{\alpha_{1}}+\frac{\delta}{ \lambda}} \tag{13}\]  
\[q_{7}=\frac{T_{f7}-T_{07}}{\frac{1}{\alpha_{7}}+\frac{\delta}{ \lambda}} \tag{14}\]  
where \(\alpha_{1}\) and \(\alpha_{7}\) are the convective heat transfer coefficient of thermal stratified flow used in the calculation of Green's functions, \(\lambda\) is the thermal conductivity of the surge line metal material, and \(\delta\) is the thickness of the surge line.  
As for other monitoring points, heat flux may be assumed with the obtained \(q_{1}\) and \(q_{7}\). Two kinds of heat flux distribution are suggested. One is uniform heat flux as the maximum of \(q_{1}\) and \(q_{7}\) as shown by Equation (15). Another is linear heat flux distribution as shown by Equation (16).  
\[q_{i}=\max(q_{1},q_{7}) \tag{15}\]  
\[q_{i}=q_{1}+\frac{q_{7}-q_{1}}{6}(i-1),i=2,3,\ldots,6 \tag{16}\]  
The obtained heat flux is used to calculate the fluid temperature at the corresponding position of the monitoring points:  
\[T_{fi}=T_{oi}+q_{i}(\frac{1}{\alpha_{1}}+\frac{\delta}{\lambda}),i=2,3,\ldots,6 \tag{17}\]  
The fluid temperature in the uppermost thermal zone is taken as \(T_{hot}\), and the fluid temperature in the lowermost thermal zone is taken as \(T_{cold}\). The fluid temperature in other fluid areas is taken as the average of the fluid temperatures corresponding to two adjacent monitoring points.