PSI - Issue 60

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A. Kumar et al. / Procedia Structural Integrity 60 (2024) 541–552 Akshay Kumar/ StructuralIntegrity Procedia 00 (2023) 000 – 000

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physical properties nearby Zircaloy-4 material. Thus, in this work the values given for Zircaloy-4 material for a temperature range of 300-1100 K is taken for the probabilistic analysis. The mean value of emissivity is taken as 0.9 with standard deviation of 0.045 and Normal distribution is assumed for this variable. The uncertainty in the stress exponent value and the activation energy for the Nortons’ creep law is not available in literature. Based on the reported deterministic data, the mean value of the stress exponent value is taken as 1.8 with a cov of 1% and 5%. It is assumed to follow a lognormal distribution. The mean of ⁄ is taken as 29000 with a cov of 1%. It is assumed to follow a lognormal distribution. 5. Methodology for uncertainty analysis The validated model is used to demonstrate the uncertainty analysis. The methodology presented here can be applied to the real reactor scenario. The uncertainty propagation is accomplished by two methods, i.e., Wilks’ Method and Monte Carlo simulation 5.1 Wilks’ Method Wilks’ method (Wallis 2003, William and Wallis 2004, Rastogi 2009) is a popular method employed in the field of thermal hydraulics to make decisions using models which have uncertain input parameters. The Wilks’ method helps in estimating an output value which is greater than percentile with a confidence of 95%. The Wilks’ formula for estimating the number of trials required to arrive at this output value is given by eq, 11. = (1 − ) ( ) (11) Where and are percentile of results covered in the Monte-Carlo simulation and the confidence level respectively. For example, for 90 percentile data with 95% confidence level, the corresponding values of and are 0.9 and 0.95 respectively. For an output value which is greater than 95 percentile, which is estimated with a confidence of 95% (95-95 value), 59 trials would be required. Thus 59 sets of values are generated randomly for each of the uncertain variables defined in Section 4 based on the probability density function. For each set thus obtained, deterministic model is used to predict the contact temperature and the contact time. The maximum value of these trials gives the contact temperature and the contact time which is greater than the 95 percentile value and this value is obtained with a confidence of 95%. It is to be noted that this value thus will lie in between 95 percentile and 100 percentile and will not be the exact 95 percentile value. The advantage of this method is that the estimation is done with limited trials and the confidence on the result is also obtained. This method is very useful when the simulation is computationally expensive. 5.2 Monte Carlo Method If the combination of the computational resources and the time complexity of the finite element model permit a large number of simulations, Monte Carlo method can be employed. The output obtained from this estimation procedure is used to generate the appropriate statistics of interest. This statistics is used for making the decisions. 6. Results of uncertainty analysis The simulation was performed using 59 trials for Wilks’ method and 60 trials (for demonstration of methodology) for Monte Carlo method. The results from two different cases are reported here with the coefficient of variation (cov) of 1% and 5% on stress exponent and activation energy. The results are listed in Table 2.