atw - International Journal for Nuclear Power | 11/12.2019

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atw Vol. 64 (2019) | Issue 11/12 ı November/December RESEARCH AND INNOVATION 530 redistributed, leading to the first cladding temperature peak, as shown in Figure 8. The cladding temperature passes from the steady state value (around 617 K) to 955 K. The result of the core voiding is a sudden drop of the water collapsed level in the reactor vessel, as shown in Figure 9, under the Bottom of Active Fuel (BAF). When the pressure in the primary system is lower than the accumulator initial pressure, the accumulator check valves open and water is discharged in the primary system; this happens around 11 s after the transient initiation. The water initially injected in the cold legs by the accumulators bypasses the vessel lower plenum through the upper down comer region and it is directed to the break without penetrating the core. After the first fast depressurization phase, the primary pressure continues to reduce at a lower rate, equalizing the containment pressure (around 0.45 MPa after around 40 s) and ending the blowdown phase. Water is injected in the primary system by the accumulators and the low pressure safety injection system start to inject water after around 30 s from the break initiation. The refill phase starts around 40 s, when the emergency core coolant water reaches the vessel lower plenum and the collapsed level in the vessel starts to rise. During this phase the core is mainly uncovered and heat is not removed from the fuel rods, with the exception of a small amount of heat removed by thermal radiation and natural convection of the steam present in the core. For this reason during the refill period the cladding temperature increases (Figure 8) due to the quasi adiabatic heating of fission product decay. When, around 50 s from the LOCA initiation, the water level reaches the core bottom (Figure 9) the refill period ends and the reflood phase starts. Water collapsed level rises quickly up to around 65 s (time of end of accumulators injection), and it continues at a lower rate due to the LPIS. In this phase the net core flow rate is positive, even if very small and with many oscillations. The water entering the core is heated up, starts to boil and entrains water droplets that help the cooling of the hottest parts of the core. With the rising of the water level, the cooling is increased and the cladding temperature starts to decrease. The complete rewetting of the cladding surface caused by the rising water level produces a strong temperature drop (core quenching). This happens around 125 s after the beginning of the LOCA. Analyzing the dispersion of the results, the primary pressure (Figure 6) presents an almost negligible dispersion during the blowdown phase and the predictions of the 59 runs are very similar; after the blowdown the dispersion band width is always lower than 0.1 MPa with a final average value of 0.475 MPa. As regards the core mass flow rate (Figure 7), cladding temperature (Figure 8) and vessel water collapsed level (Figure 9), the results dispersion is very limited during the blowdown phase, while it is more noticeable in the refill phase, especially for the vessel water collapsed level, and it is higher during the reflood phase. In particular, the refill initial time shows a dispersion band width of 12 s (33 – 45 s); during the reflood the collapsed level dispersion band width is around 1 m and the Top of Active Fuel (TAF) is reached in a time band of 33 s (126 – 159 s). The cladding temperature has, at the first peak, a low dispersion band width of 14 K and the peak has almost the same timing for all runs; instead, the dispersion band width is higher for the second peak (50 K) and with a time dispersion band width of 10 s. The hot rod cladding quenching time is also affected by a dispersion band width of 20 s (115 – 135 s). | Tab. 6. Primary system pressure predicted by TRACE code. | Tab. 7. Core mass flow rate predicted by TRACE code. | Tab. 8. Cladding temperature of the hot rod predicted by TRACE code. | Tab. 9. Reactor vessel water collapsed level predicted by TRACE code. Research and Innovation Evaluation of a Double-Ended Guillotine LBLOCA Transient in a Generic Three-Loops PWR-900 with TRACE Code Coupled with DAKOTA Uncertainty Analysis ı Andrea Bersano and Fulvio Mascari
atw Vol. 64 (2019) | Issue 11/12 ı November/December | Tab. 10. Pearson’s simple regression correlation coefficients. 4.2 Uncertainty analysis response correlations 4.2.1 Description of regression correlations in DAKOTA As a result of the uncertainty analysis, DAKOTA [7, 8, 20] computes four response correlation coefficients: simple, partial, simple rank and partial rank. The simple coefficient is related to the actual input and output data. The simple coefficient r between an input variable x and an output variable y, in n samples, is computed using the Pearson’s correlation. It is a measure of the degree of linear correlation between the two variables and its value is comprised between -1 and 1. If r0 the correlation is positive (an increment of x leads to an increment of y). The simple correlation coefficient between x and y is obtained by dividing the covariance of the two variables by the product of their standard deviations [27]: (2) The partial correlation coefficient is computed similarly to simple one but taking into accounts the effects of the other variables. This is useful, for example, if there is a strong correlation between two inputs; in this way the correlation of the second input on the output may be adjusted after having considered the correlation between the first input and the output [7]. Rank correlation coefficients use the ranked data instead of the actual ones. Ranks are obtained by ordering the data in ascending order, and are more convenient to be used when inputs and outputs are characterized by sensible difference in magnitude; it is possible to understand if the input sample with the lower rank is associated to the output with the lower rank and so on [7, 20]. To compute the rank correlation, DAKOTA uses the Spearman’s rank correlation that is similar to Pearson’s one but with the ranked data instead of the actual values. If two variables are monotonically related, without repetitions, the Spearman coefficient is -1 or +1 (depending if the function is monotonically decreasing or increasing), since the ranked values are used. Moreover, Spearman’s correlation is less sensitive to possible outlier values of the variables than Pearson’s one. 4.2.2 Results of response correlation coefficients for the cold leg LBLOCA transient The time dependent computation of response correlation coefficients has been performed extracting the FOM value at different selected instant of the transient evolution. Figure 10 shows the Pearson’s simple correlation coefficient for the six input uncertain parameters from the LOCA beginning to the complete core quenching (after 130 s). Figure 11 shows the Spearman’s rank correlation coefficient for the same parameters. On both graphs the values 0.2 and 0.5 (and -0.2 and -0.5) have been highlighted as measure of the correlation between the input parameter and the FOM. As indicated in the [20], for the Spearman coefficient, if the coefficient is higher than 0.5 (or lower than -0.5) the correlation is significant, if it is between 0.2 and 0.5 (or -0.2 and -0.5) the correlation is moderate, otherwise it is low [20]. In | Tab. 11. Spearman’s simple rank regression correlation coefficients. the following analysis the same threshold values have been adopted also for the Pearson coefficient. In this application, the two response correlations show similar trends; the main advantage of this time dependent analysis is the possibility to have a measure and characterize the correlation of the different input parameters on the uncertainty of the selected FOM in all the phases of the transient. In the blowdown phase (0-40 s) the effect of the initial core power on the hot rod cladding temperature is positive due to the heat stored in the solid structures. Both Pearson’s and Spearman’s coefficients are higher than 0.8 for this parameter so the correlation is significant and almost monotonically linear. In the remaining part of the transient both coefficients are close to 0.2 so the corre lation is much weaker than in the initial phase. After around 20 s from the LOCA initiation the accumulators initial pressure has a negative Pearson’s and Spearman’s coefficients around -0.3; therefore the lower is the accumulator pressure the higher is the cladding temperature since the accumulator starts to inject water later. In the refill phase (40-50 s), also the SIS characteristic has both response correlation coefficients around -0.3; in fact, with a lower injection flow rate the hot rod cladding temperature is higher. In the reflood phase (after 50 s) also the initial containment pressure has negative Pearson’s and Spearman’s coefficients around -0.35. This is due to the fact that with a lower pressure in the containment a greater amount of coolant water is expelled through the break and the cladding temperature is higher. The uncertainty in the accumulators and SIS temperature have a low correlation with the FOM. RESEARCH AND INNOVATION 531 Research and Innovation Evaluation of a Double-Ended Guillotine LBLOCA Transient in a Generic Three-Loops PWR-900 with TRACE Code Coupled with DAKOTA Uncertainty Analysis ı Andrea Bersano and Fulvio Mascari
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atw - International Journal for Nuclear Power | 11/12.2019

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