PSI - Issue 60

P.A. Jadhav et al. / Procedia Structural Integrity 60 (2024) 631–654 P. A. Jadhav et.al. / Structural Integrity Procedia 00 (2019) 000 – 000 Mathematically, the probability of failure can be written in terms of the marginal density functions ( ) of the stochastic variables , as in eq. 14. All of material properties involved in this study are treated as independent variables for a simplified analysis.     0 1 i n f i i i i g x P f x dx      (14) Thus the calculation of probability of failure essentially reduces to the calculation of the integral given by equation 3. A classical Monte Carlo method gives the most accurate results but is computationally very expensive. It may not work for low probability numbers (<10 -6 ). Hence, suitable sampling techniques are employed to arrive at the low probability numbers. Probability of failure is estimated by radial stratification. In multi-dimensional Gaussian space, squared radius is the sum of the squared distances from the mean for each coordinate. The squared radius follows a Chi-squared distribution with degrees of freedom equal to the number of dimensions. In radial stratification scheme multi dimensional Gaussian space is divided into n strata by hyper-spheres. If and −1 is the radius of th and ( −1) th hyper-sphere respectively, probability mass for the stratum is probability mass between these hyper-spheres and is equal to 2 ( 2 )− 2 ( 2−1 ) , where, 2 is the chi-square distribution function with n degrees of freedom. Radius of the i th hyper-sphere is chosen such that its probability mass is equal to (1-10 -i ). So that probability mass of i th stratum is (1-10 -(i) ) - (1-10 -(i-1) ) which is equal to 0.9×10 -(i-1) . Number of samples simulated in each stratum in this work is 10 5 if failure does not take in that stratum. However if failure is observed in that stratum number of samples in each stratum is increased to 10 7 adaptively. 3. Data used in the analysis The input data used for the study is listed in this section. The PT is made of Zr-2.5Nb alloy. It has an internal diameter of 82.6 mm and a thickness of 3.4 mm. The dimensions of the PT changes during service due to irradiation creep. The diameter increases and the thickness decreases as the pressure tube experiences neutron flux. However, as observed by Oh et. al. (Oh, et al., 2012) this does not have any significant effect on the probability estimates. Thus, geometrical changes over the lifetime of the PT is considered in this study. The thickness and the internal diameter is treated as a deterministic variable. The analysis is reported for sustained hot conditions. The transient condition is not considered in this analysis. Zirconium has a strong affinity for hydrogen isotopes, but it exhibits low solubility. When the reactor is in operation, heavy water and zirconium undergo a corrosion reaction, resulting in the formation of an oxide and deuterium. This deuterium (D) is absorbed into the metal matrix, adding to the low concentrations of protium (P) from the manufacturing process. The concentrations are quoted as equivalent hydrogen concentrations (Heq) where Heq = [P] + 1/2[D]. The pressure tube continues to absorb deuterium from the corrosion reaction during service. The concentration of absorbed deuterium varies along the length of the pressure tube increasing from the inlet to the outlet end. In the region of the rolled joint, the deuterium absorption is enhanced due the interaction of pressure tube with the stainless steel end fitting. Sawatzky and Ells (2000) have shown the multiple pathways through which the hydrogen isotopes accumulate in the rolled joint area. The maximum allowable initial concentration of protium in a finished pressure tube is restricted to 5 weight parts per million (ppm) for the Zr-2.5Nb material Sinha, Sinha and Madhusoodanan (2008), Rodgers et. al. (2016). The pickup rate of deuterium in the main body of the pressure tube is around 2 ppm per hot operating year Rodgers et. al. (2016). In the rolled joint area the pickup can go as high as 5 ppm per hot operating year Rodgers et. al. (2016), Langille, Coleman, & McRae (2021). In the present analysis, the probability estimates are obtained for deterministic values of Heq. The value of Heq is parametrically varied to simulate the results for effective hot operating years. These values are listed in Table 1.

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