Treatment of the thermal–hydraulic uncertainties in the pressurized thermal shock analysis

Nuclear Engineering and Design 237 (2007) 143–152

Treatment of the thermal–hydraulic uncertainties in the pressurized thermal shock analysis Changheui Jang ∗ Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea Received 7 December 2005; received in revised form 19 May 2006; accepted 26 May 2006

Abstract The pressurized thermal shock (PTS) analysis is a quantitative analysis to calculate the vessel failure probability of the embrittled reactor pressure vessel. The PTS analysis consists of three major parts, such as the probabilistic safety analysis (PSA), the thermal–hydraulic analysis (T/H), and the probabilistic fracture mechanics (PFM) analysis. Because each analysis involves many parameters and assumptions associated with the uncertainties, it is important to identify and incorporate them into the analysis. Though the PSA and PFM analysis can be easily treated statistically, the thermal–hydraulic analysis results are very difficult to be treated statistically. Instead, sensitivity analyses of the thermal–hydraulic inputs were performed to understand the significance of the variation in the thermal–hydraulic inputs to the PFM analysis. In this study, the existing PFM code was modified to incorporate the uncertainties in the thermal–hydraulic inputs for the PFM analysis. The effects of the uncertainties in the thermal–hydraulic inputs for the vessel failure probabilities were evaluated using the modified code. The results showed the effects of uncertainties in the thermal–hydraulic inputs on the vessel failure probabilities are not significant for the ranges of the transient types. Even for the larger uncertainties, the effects on the vessel failure probabilities are small. Also, the effects of the thermal–hydraulic uncertainties vary depending on the transient characteristics such that the effects are greatest for the pressure dominant transient. Within the transient, the relative increases in the failure probabilities are greatest for the circumferentially oriented semi-elliptical flaws. It was found that the results of the sensitivity analysis using one standard deviation are conservative enough to bound the analysis results considering the uncertainties in the thermal–hydraulic inputs. © 2006 Elsevier B.V. All rights reserved.

1. Introduction During the operation of a pressurized water reactor, a certain type of transients could induce rapid cooldown of the reactor pressure vessel (RPV) with relatively high or increasing system pressure. This type of transients was named as the pressurized thermal shock (PTS) to distinguish it from the conventional definition of the thermal shock which only considered rapid cooling of the RPV (USNRC, 1982). For the quantitative evaluation of the vessel failure risk associated with PTS, the probabilistic fracture mechanics (PFM) analysis technique has been widely used

Abbreviations: MSLB, main steam line break; PFM, probabilistic fracture mechanics; PSA, probabilistic safety analysis; PTS, pressurized thermal shock; RPV, reactor pressure vessel; SGTR, steam generator tube rupture; SBLOCA, small break loss of coolant accident; T/H, thermal–hydraulic; USNRC, United States Nuclear Regulatory Commission ∗ Tel.: +82 42 869 3824; fax: +82 42 869 3810. E-mail address: [email protected]. 0029-5493/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2006.05.007

(Simonen et al., 1986; Dickson, 1994; Jang et al., 2001a). The PFM technique basically checks whether hypothetical flaws on the wall propagate through the vessel wall by comparing the applied stress intensity factor (crack driving force) with the fracture toughness (materials resistance to fracture) during the PTS events. The PTS analysis is a quantitative analysis to calculate the vessel failure probability of the embrittled reactor pressure vessel. As shown in Fig. 1, the PTS analysis consists of three major parts, such as probabilistic safety analysis (PSA), thermal–hydraulic analysis (T/H), and probabilistic fracture mechanics (PFM) analysis. In PSA part, the PTS initiating events are identified and event-trees are constructed by carefully analyzing the plant specific data. Next, the event frequencies of the sequences, or P(E) are quantified by PSA technique. The PTS significant transient sequences are classified and grouped based on the similarity in the thermal–hydraulic nature and the frequency of the sequence. For the selected transient sequences, the T/H analyses

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Nomenclature a

crack depth normalized to the thickness of the vessel c crack length normalized to the thickness of the vessel Cn coefficients in stress polynomial Gn influence coefficients IRTNDT initial reference temperature-nil ductility transition K stress intensity factors Kapp applied stress intensity factor at the crack tip KIC static fracture toughness KIC,m mean static fracture toughness crack arrest fracture toughness KIR KIR,m mean arrest fracture toughness Kt , Kp , and Kr stress intensity factors calculated from the thermal stress, the pressure stress, and the residual stress, respectively mT number of standard deviation for temperature, σ T mP number of standard deviation for pressure, σ P P(E) sequence frequency P(F/E) conditional vessel failure probability Pmean system pressure from the best-estimate T/H analysis result Q shape factor, 1 + 1.454(a/c)1.65 RTNDT reference temperature-nil ductility transition RTNDT increase in reference temperature-nil ductility transition t time tP min time at which the system pressure is the lowest tT min time at which the coolant temperature is the lowest Tmean coolant temperature from the best-estimate T/H analysis result W thickness of the vessel Greek symbols σ stress acting on crack line σP standard deviation for pressure σT standard deviation for temperature

are performed using the transient analysis codes to calculate the time histories for vessel internal pressures, downcomer coolant temperatures, and coolant heat transfer coefficients. If thermal stratification within the cold leg is suspected, more detailed mixing analyses are needed to obtain the localized temperature near the RPV wall in the downcomer region. The next step is the PFM analysis. As shown in Fig. 1, system pressures, downcomer coolant temperatures near the RPV wall, and heat transfer coefficients obtained from the T/H and mixing analyses are provided as inputs to the PFM analysis. The specific vessel data such as, physical material properties, geometry, surveillance capsule data, etc. are also needed for the analysis. Through the PFM analysis, the conditional vessel failure probability, P(F/E) for each PTS significant transient sequence

Fig. 1. Overall flow of the PTS analysis.

is calculated. The vessel failure probability at the event of the specific PTS sequence is calculated by multiplying P(F/E) and the sequence frequency, P(E). Finally, the overall vessel failure probability, or the PTS risk is determined by simply adding the vessel failure probability of all transient sequences analyzed. Because all three major parts of the analysis involve many parameters and assumptions associated with the uncertainties, it is important to identify and incorporate them in the PTS analysis. In fact, the USNRCs guideline for the PTS analysis requires that the uncertainties of the analysis should be identified and incorporated in the PTS risk calculation (USNRC, 1987). In the PSA part, the frequencies of the specific sequences are statistically calculated to provide the best estimate frequency and the associated uncertainty distribution. In the PFM codes, the embrittlement related parameters, such as chemistry, fluence level, fracture toughness, etc. are statistically treated considering the associated uncertainties (Dickson, 1994; Jang et al., 2001a). On the other hand, applied stress intensity factors are calculated from the T/H inputs which are provided as the bestestimate values. Because the T/H analysis results of the PTS transients depend on many variables and assumptions, it also should be treated statistically. However, the T/H inputs are very difficult to be treated as statistical variables. In one part, it is not easy to quantify the associated uncertainties in the T/H results (USNRC, 2005). Another part is due to the fact that once the T/H inputs are changed to consider the uncertainties, the deterministic analysis has to be repeated. This is practically not possible for many PFM codes in which the deterministic analyses are separately performed and the results are stored in separate files to be used in the probabilistic analysis later. Therefore, there has

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been limited attempt to analytically consider the uncertainties of T/H inputs in the PTS analysis. Instead, sensitivity analyses of T/H inputs were performed to understand the significance of the variation in the T/H inputs (Jeong et al., 2001; USNRC, 2005). In this paper, the existing PFM code was modified to incorporate the T/H uncertainties into the PFM analysis to calculate the PTS risk. Then, the effects of the T/H uncertainties on the calculated vessel failure probabilities were investigated. 2. Methodology and analysis 2.1. VINTIN code An advanced probabilistic fracture mechanics code, VINTIN (Vessel INTegrity analysis—INner flaws) was developed for the quantitative risk assessment of the RPV at the events of the pressurized thermal shock (Jang et al., 2001a). Since the development, VINTIN has been continuously modified to expand the applicability of the code and improve the accuracy of the stress intensity factor and the vessel failure probability calculation (Jang et al., 2001b; Jang et al., 2003; Jhung et al., 2005). The schematics of the PFM analysis method are shown in Fig. 2. As shown in the figure, the analysis consists of two parts, such as the deterministic analysis and the probabilistic analysis. In the deterministic analysis part, the temperature profiles and the resulting thermal stress along the thickness of the reactor pressure vessel are calculated from the given thermal–hydraulic conditions. The distribution of stresses from other sources like pressure and residual stresses are separately calculated. The stress intensity factor, K from each stress components is calculated by Eq. (1), known as the Raju–Newman method (Raju and Newman, 1982)

145

using the appropriate influence coefficients for the flaw shapes and orientations (Jang et al., 2001a): 3 G n Cn a n √ πaW K = n=0√ Q for σ = C0 + C1 a + C2 a2 + C3 a3

(1)

where, Gn is the influence coefficients corresponding to the term Cn an in stress polynomial. Cn are the coefficients in stress polynomial; a the crack depth normalized to the thickness of the vessel; W the thickness of the vessel; σ the stress acting on crack line; Q = 1 + 1.454(a/c)1.65 ; c is the crack length normalized to the thickness of the vessel. Then, the stress intensity factor components calculated for the various stress components are added to be the total applied stress intensity factor, Kapp at the crack tip: Kapp = Kt + Kp + Kr

(2)

where Kt , Kp , and Kr are the stress intensity factors caused by the thermal stress, pressure stress, and residual stress, respectively. This method can be readily applied to calculate the applied stress intensity factors in the base metal of the reactor pressure vessel. However, because of the lower thermal conductivity of stainless steels, the temperature profile in the cladding considerably deviates from that in base metal made of low alloy steels. Also, the thermal stress profiles in the cladding region deviates from that in the base metal which can be fitted as smooth third order polynomials. Special treatment scheme was developed to handle such steep deviations in the cladding region (Jang et al., 2003). In the probabilistic analysis part, variety of statistical parameters such as flaw size, neutron fluence, copper and nickel

Fig. 2. Schematics of the probabilistic fracture mechanics analysis.

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contents, and the reference temperature-nil ductility transition (RTNDT ) are simulated for each hypothetical reactor pressure vessel. From the temperature profile and the RTNDT , the mean static fracture toughness KIC,m and the mean arrest fracture toughness, KIR,m at the tip of the flaws are calculated using the equation derived from the lower-bound fracture toughness (ASME, 1995): √ KIC,m = 52.22 + 32.62 exp[0.036(T − RTNDT )], MPa m (3) KIR,m = 36.72 + 17.11 exp[0.0261(T − RTNDT )], MPa

to initiate and grow a certain distance. Then, at the new flaw size, new values of RTNDT , Kapp and KIR are simulated and compared. If Kapp is smaller than KIR , the flaw is considered to be arrested. Otherwise, the flaw size is increased again and the arrest check is repeated until the end of the transient. By repeating the above analysis millions of times, a statistically significant conditional probability of the vessel failure for the specific thermal hydraulic boundary condition is determined as follows: P(F/E) =

number of vessels failed . number of vessels simulated

(5)

√

m

2.2. Modification of VINTIN code

(4) Finally, using the mean values and the associated uncertainties, the fracture toughness values like KIC and KIR are simulated to be compared with the applied stress intensity factors at the tip of the flaws, Kapp . If Kapp is larger than KIC , the flaw is assumed

As mentioned above, in VINTIN, the thermal and stress analyses, the stress intensity factor calculations are all performed analytically, and the deterministic results are directly used in the subsequent probabilistic analysis within the same program run (Jang et al., 2001a, 2003). Therefore, with a slight modification

Fig. 3. Schematic flow of the mod-VINTIN code to incorporate the T/H uncertainties.

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Table 1 Thermo-mechanical properties of the RPV Material property

Carbon steel (base and weld)

Stainless steel cladding

Thermal conductivity (W/m ◦ C) Specific heat (kJ/kg ◦ C) Density (kg/m3 ) Modulus of elasticity (GPa) Thermal expansion coefficient (m/m ◦ C) Poisson’s ratio

40.897 0.5091 7809.5 182.85 13.29E−6 0.3

17.238 0.5259 7851.0 185.26 17.12E−6 0.3

of the code, the uncertainty of T/H inputs can be incorporated in the analysis. The schematics of VINTIN with modifications, or mod-VINTIN, are shown in Fig. 3. In the figure, modifications for treating T/H uncertainties are shown in dashed lines. The overall flow of the analysis are: (1) the new temperature and pressure vs. time can be simulated with given mean temperature and pressure and the associated uncertainties, then (2) the deterministic analysis is performed using the simulated T/H conditions, and subsequently and (3) the probabilistic analysis is performed to find the vessel failure probability using the deterministic analysis results for the simulated input conditions. The steps (1)–(3) are repeated for sufficient number of times to give statistically significant results with reasonable analysis time.

Fig. 4. Temperature and pressure variation during the PTS-like transients. (a) Temperature; (b) pressure.

2.3. Vessels and transients The reactor pressure vessel and the typical PTS transients assumed in this study are selected from those of the previous plant-specific PTS evaluation (Jeong et al., 2001). The reactor pressure vessel is a typical Westinghouse two-loop plant with the dimension of radius of 1676 mm, thickness of 165 mm, and clad thickness of 4.76 mm. The thermo-physical properties of the base metal and the stainless steel cladding are shown in Table 1. The mean values and the uncertainties of the radiation embrittlement related parameters are summarized in Table 2. The assumed flaw conditions are summarized as follows: • flaw orientation: axial and circumferential flaws; • flaw shape and location: infinite and 1/6 semi-elliptical surface breaking flaws; • flaw size distribution: Marshall flaw size distribution. Table 2 Irradiation embrittlement related properties of the RPV: means and standard deviations Parameters

Mean value

S.D.

Copper content (w/o) Nickel content (w/o) Initial RTNDT (◦ C) RTNDT shift equation Fluence at RPV inner surface Errors in KIC reference curve Errors in KIR reference curve Flux attenuation (m)

0.29 0.68 −32.8 ◦ C Reg. guide 1.99 rev. 2 3.00 × 1019 /cm2 ASME derived mean ASME derived mean 9.45

0.07 0.05 0 15.6 ◦ C 0.16 0.15 0.1

The three typical PTS-like transients such as the MSLBtype, the SGTR-type, and the SBLOCA-type are selected for the assessment of the effects of uncertainties in the T/H inputs. The variations of temperature and pressure during the transients are plotted in Fig. 4. The MSLB-type and the SGTRtype transients are characterized as re-pressurization during the transient with moderate final coolant temperature. The SBLOCA-type transient results in the lowest final coolant temperature but the relatively low pressure. From temperature and pressure variations shown in the figures, it is presumed that the pressure effect is dominant in the MSLB-type transient and the temperature effect is dominant in the SBLOCA-type transient. 2.4. Uncertainties in the T/H Inputs It is not a simple task to quantify the uncertainties associated with the best-estimate T/H analysis results. For the purpose of this study, the standard deviation for pressure (σ P ) was rather arbitrarily chosen as 0.345 MPa (50 psi), that for temperature (σ T ) as 8.33 ◦ C (15 ◦ F) after analyzing on the recent assessment results (USNRC, 2005). The way the uncertainties in the T/H inputs are treated in the PFM analysis are schematically shown in Fig. 5 along with the best-estimate values. As shown in the figure, the uncertainty factors, mT and mP within ±3 were simulated and multiplied with the uncertainties in temperature and pressure at the time of the minimum temperature or pressure. Then temperatures and pressures at other times are calculated

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on the vessel failure probability. Finally, the sensitivity analysis results of temperature and pressure were presented and compared with those of the mod-VINTIN that directly incorporated the T/H inputs in the PFM analysis. As shown in Fig. 6, for the sensitivity analysis, the maximum errors of one standard deviation were assumed at the times of minimum temperature and pressure, then temperatures and pressures at other times are calculated using following equations: T (t) = Tmean (t) − σT

t , tmin

T (t) = Tmean (t) − σT , P(t) = Pmean (t) + σP Fig. 5. Schematics of the simulation of the temperature and pressure variation considering uncertainties.

t > tmin t

tmin

P(t) = Pmean (t) + σP ,

t ≤ tmin ;

,

(8)

t ≤ tmin ;

t > tmin

(9)

as follows: T (t) = Tmean (t) + mT σT

t tT min

T (t) = Tmean (t) + mT σT , P(t) = Pmean (t) + mP σP

,

t > tT min

t , tP min

P(t) = Pmean (t) + mP σP ,

3. Results and discussion

t ≤ tT min ; (6)

t ≤ tP min ;

t > tP min

(7)

In this approach, it is assumed that the maximum deviation from the best-estimate curve happens at times of the lowest temperature and pressure, respectively. 2.5. Analysis To verify the accuracy of the modified VINTIN (or, modVINTIN), the calculation results of the mod-VINTIN assuming zero T/H uncertainties were compared with those of the original VINTIN code. The maximum number of simulations in VINTIN was set as 10 million. To get the same number of simulation, 3163 T/H input simulations and 3163 vessel simulations were used in the mod-VINTIN code. After the verification, the effects of the uncertainties in the T/H inputs were evaluated using the mod-VINTIN code at three different fast neutron fluence levels, such as 3.711, 4.876, and 5.852 × 1019 /cm2 . Then the uncertainty in temperature was increased up to 27.8 ◦ C (50 ◦ F) to assess the effects of the level of uncertainty

3.1. Verification of VINTIN modification The vessel failure probabilities were calculated using the original VINTIN and the mod-VINTIN codes to verify the modification. Some of the comparison results are summarized in Table 3. It was found that the differences between two codes were within ±4% for all the analysis conditions considered in this paper. Considering the wide spectrum of transient types and fluence levels, flaw shapes and orientations, such level of differences was considered as an excellent agreement. Therefore, the mod-VINTIN code was proved to be as accurate and consistent enough as the original VINTIN code to be used in the subsequent analyses. It is generally believed that at least a million simulations are recommended to give statistically significant vessel failure probabilities for the PTS analysis. However, the actual number of simulation to give reasonably accurate failure probability depends on the failure probability and the analysis condition. For example, more than 10 million simulations were needed to determine the failure probabilities when the vessel failure probabilities were not high enough (Jeong et al., 2001). Because the number of the vessel simulation for each thermal–hydraulic input simulation was substantially reduced to 3163 when total number of simulation is 10 million, it is necessary to check whether the number of simulation used in the study is enough to

Fig. 6. Schematics of the temperature and pressure variation for the sensitivity analysis.

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Table 3 Comparison of the calculated vessel failure probabilities Fluence (1019 /cm2 )

MSLB Pfail

VINTIN (10 million simulation) 3.711 1.71E−02 4.876 2.83E−02 5.852 3.63E−02

SGTR Pini.

Pfail

SBLOCA Pini.

Pfail

Pini.

1.72E−02 2.84E−02 3.64E−02

7.42E−02 9.88E−02 1.12E−01

7.45E−02 9.90E−02 1.13E−01

2.45E−02 3.43E−02 4.14E−02

2.90E−02 3.83E−02 4.49E−02

Mod-VINTIN (10 million simulation with zero uncertainties) 3.711 1.69E−02 1.70E−02 4.876 2.72E−02 2.73E−02 5.852 3.70E−02 3.71E−02

7.59E−02 9.83E−02 1.13E−01

7.62E−02 9.85E−02 1.13E−01

2.46E−02 3.44E−02 4.29E−02

2.91E−02 3.84E−02 4.64E−02

0.62% 0.50% 3.69%

0.39% 0.37% 3.44%

(Pmod-VINTIN − PVINTIN )/PVINTIN 3.711 −1.16% 4.876 −4.00% 5.852 1.98%

−1.15% −3.93% 1.99%

2.29% −0.44% 0.50%

2.22% −0.43% 0.51%

Axially oriented 1/6 semi-elliptical flaws were assumed. Note: Pmod-VINTIN = probabilities calculated using the modified VINTIN code; PVINTIN = probabilities calculated using the VINTIN code.

give reasonably accurate failure probabilities. The convergence of the results were calculated and plotted in Fig. 7 for total simulation number of 1 million (1000 T/H simulation × 1000 vessel simulation), 10 million (3163 T/H simulation × 3163 vessel simulation), and 100 million (10,000 T/H simulation × 10,000 vessel simulation). As shown in the figure, the failure probabilities converge fairly rapidly especially for the axially oriented flaws where the failure probabilities are in the order of 10−2 . Even the result of 1 million simulations was close enough to that of 100 million simulations. On the other hand, the calculated

failure probability was not fully converged even at 10 million simulations for the circumferentially oriented flaws where the failure probabilities were in the order of 10−4 . In this case, 100 million simulations would provide fully converged failure probabilities but with much greater analysis time. Based on the results above, it was considered that 10 million simulations would give reasonably accurate failure probabilities with reasonable analysis time for wide ranges of transients and analysis conditions. Therefore, 10 million simulations were used in the analyses afterward. 3.2. Impacts of the thermal–hydraulic uncertainties on the failure probabilities

Fig. 7. Comparison of the failure probabilities for different number of total simulations. (a) For axial flaws; (b) for circumferential flaws.

The calculated vessel failure probabilities with and without the T/H uncertainties are shown in Fig. 8. As shown in the figures, the effects of incorporating the T/H uncertainties are not significant for the typical PTS transients considered in the study. The increases in the failure probabilities are relatively large for the MSLB-type transient, which is a pressure dominant transient with rapid re-pressurization. On the other hand, the increases in the failure probabilities are minimal for the SBLOCA-type transient, which is a temperature dominant transient with rapid de-pressurization. For the pressure dominant transients, the fractional increases are greater when the failure probabilities are lower, such as when the circumferentially oriented semi-elliptical flaws are assumed. In all, by considering T/H uncertainties, the factor of increase in failure probability is less than 2. Considering that the transients treated in the study are among the most severe PTS transients that contribute significant fraction of the total PTS risk, it can be said that the overall PTS risk would be less than twice of that using the best-estimate T/H inputs. To evaluate the effects of the magnitude of the uncertainties, the larger uncertainties in temperatures were assumed for the MSLB-type transient in which the effects of T/H uncertainties were the largest. While maintaining the uncertainty in pres-

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Fig. 9. Effects of the larger uncertainties in temperature on the vessel failure probabilities of the MSLB-type transient. (a) For infinite flaws; (b) for semielliptical flaws.

summarized in Fig. 10. If the uncertainties in temperature and pressure increase, the increase in the failure probabilities become greater for all cases evaluated. In this case, increases in the failure probabilities for the circumferentially oriented semi-elliptical flaws are 5–8 times. This again emphasizes that the orientation and shape of the assumed flaws are important parameters in evaluating the effects of the T/H uncertainties. 3.3. Comparison with sensitivity analysis Fig. 8. Effects of the uncertainties in T/H inputs on the vessel failure probabilities. (a) MSLB-type transient; (b) SGTR-type transient; (c) PTS-type transient.

sure at 0.345 MPa (50 psi), that of temperature was increased to 16.7 ◦ C (30 ◦ F) and 27.8 ◦ C (50 ◦ F). The results are summarized in Fig. 9. As expected, the conditional vessel failure probabilities increased as the uncertainties in temperature increased. As shown in Fig. 9a, when the infinite axial flaws are assumed, the factors of increase in the failure probabilities are less than 2 even with the quite large uncertainty of 27.8 ◦ C. On the other hand, the factors of increase in the failure probabilities are the most significant when the semi-elliptical circumferential flaws are assumed (Fig. 9b), which result in the lowest failure probabilities. The effects of the larger uncertainties in both pressure and temperature were analyzed for the MSLB-type transient and

As mentioned above, because of the difficulties in treating the uncertainties in the T/H results, sensitivity analysis results have been indirectly used to assess the impact of the uncertainties (Jeong et al., 2001; USNRC, 2005). The uncertainty analysis results of the study are compared with the sensitivity analysis and shown in Fig. 11. It is clear that the overall impact of the T/H uncertainties is far less than the sensitivity analysis of temperature which assumes one standard deviation greater temperature than the best-estimate values during the transient. Also, the sensitivity of temperature is always greater than that of pressure. 3.4. Implication to PTS analysis In the plant-specific PTS analysis guideline, the detailed procedures and methods for the PTS integrity analysis are described.

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151

Fig. 10. Effects of the larger uncertainties in both temperature and pressure on the vessel failure probabilities of the MSLB-type transient.

In that, it is required that the major sources of the uncertainty should be identified, and the magnitude of the uncertainty should be estimated (USNRC, 1987). The three major parts of the PTS analysis are the PSA, T/H, and PFM analysis. For both the PSA and PFM analysis, the results are provided as the best-estimate values with the associated uncertainty ranges and distributions. However, the uncertainties in the T/H analysis are hard to be

Fig. 11. Comparison of sensitivity analysis of T/H inputs with incorporating T/H uncertainties (for MSLB-type transients). (a) Axial semi-elliptical flaws; (b) circumferential semi-elliptical flaws.

determined and treated in most of the analysis. In the recent PTS re-evaluation study, the uncertainties of the T/H results were semi-quantitatively assessed using the sensitivity analysis results (USNRC, 2005). In that study, the lower bound, nominal, and upper bound values of the key PTS contributing parameters were used to assess their impact on the sensitivity indicators like final coolant temperature. However, the sensitivity indicators may not be generally applied because the fracture mechanics results are non-linear with respect to the wide ranges of the transient types and the vessel geometries. Therefore, the best way to treat the T/H uncertainties is to combine them with the PFM code instead of comparing number of sensitivity analysis cases. The fully statistical approach developed in this study is compared with the previous approach in Fig. 12. In the previous approach, the T/H input uncertainties are treated as a nonstatistical manner by using the lower bound, nominal, and upper bound values of temperatures and pressures in the sensitivity analysis. The sensitivity analysis can provide only the upper and the lower bound of the vessel failure probability which would be several orders of magnitude larger or smaller than the nominal failure probabilities calculated using the best-estimate T/H inputs. Thus the sensitivity analysis results cannot be used to understand the statistical characteristics of the vessel failure probability caused by the uncertainties in T/H inputs. On the other hand, in the newly developed approach, the T/H input uncertainties are statistically treated and directly incorporated in the PFM analysis. As results, the calculated conditional vessel probabilities contain the effects of T/H input uncertainties as well as those of other fracture mechanics parameters. Therefore, if the uncertainties in the T/H results can be expressed with the best-estimate values with uncertainty distributions, the every components of the PTS analysis become truly statistical analysis. In this study, the uncertainties in temperature and pressure were considered as normal distributions. Without any detailed analysis on the uncertainty distributions, the distributions were

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Fig. 12. Comparison of fully statistical analysis and partly statistical analysis.

rather arbitrarily chosen for the simplicity. In fact, the temperature and pressure distribution may not be symmetric and the standard deviation may be different for different transients. Despite of these limitations, the mod-VINTIN code used in the study can be further modified by incorporating the uncertainty distributions, if provided. 4. Conclusions • The existing PFM code was modified to incorporate the uncertainties in the T/H inputs for the PFM analysis. The effects of the uncertainties in the T/H inputs for the vessel failure probabilities were evaluated using the modified code. • The effects of incorporating the T/H uncertainties in the PFM analysis were analyzed using the typical PTS transients. • The effects of the T/H uncertainties vary depending on the transient characteristics such that the effects are greatest for the pressure dominant transient. Within the transient, the relative increases in the failure probabilities are greatest for the circumferentially oriented semi-elliptical flaws. • The effects of the uncertainties in the T/H inputs on the vessel failure probabilities are not significant even for very large uncertainties. Actually, the results of the sensitivity analysis using one standard deviation are conservative enough to bound the uncertainty analysis of the T/H inputs. • By incorporating the T/H input uncertainties in the PFM analysis, every components of the PTS analysis become the truly statistical analysis.

References ASME, 1995. Boiler and Pressure Vessel Code, Section XI. Dickson, T.L., 1994. FAVOR: A Fracture Analysis Code for Nuclear Reactor Pressure Vessels, Release 9401, ORNL/NRC/LTR/94/1. Jang, C.H., Moonn, H.R., Jeong, I.S., Hong, S.Y., 2001a. Development of the Improved Probabilistic Fracture Mechanics Analysis Code: VINTIN. In: Presented at the Proceedings of the KNS 2001 Spring Meeting, Cheju, Korea, May 23–24. Jang, C.H., Jeong, I.S., Hong, S.Y., 2001b. Treatment of stainless steel cladding in pressurized thermal shock evaluation: deterministic analysis. J. Korean Nucl. Soc. 33 (2), 132–144. Jang, C.H., Kang, S.C., Moonn, H.R., Jeong, I.S., Kim, T.R., 2003. The effects of the stainless steel cladding in pressurized thermal shock evaluation. Nucl. Eng. Des. 226, 127–140. Jeong, I.S., Jang, C.H., Park, J.H., Hong, S.Y., Jin, T.E., Yeum, H.G., 2001. Lessons learned from the plant-specific PTS integrity analysis on an embrittled reactor pressure vessel. Int. J. PVP 78, 99–109. Jhung, M.J., Choi, Y.H., Kim, H.J., Jang, C.H., 2005. Probabilistic structural integrity of reactor vessel under pressurized thermal shock, paper D06-3. In: Presented at the Proceedings of the SMiRT18, Beijing, China, August 7–12. Raju, I.S., Newman Jr., J.C., 1982. Stress–intensity factors for internal and external surface cracks in cylindrical vessels. J. Pressure Vessel Technol. 104, 293–298. Simonen, F.A., Johnson, K.I., Liebetrau, A.M., Engel, D.W., Simonen, E.P., 1986. VISA-II. A Computer Code for Predicting the Probability of Reactor Vessel Failure, NUREG/CR-4486. USNRC, 1982. Pressurized Thermal Shock (PTS), Enclosure A of SECY 82465, November. USNRC, 1987. Format and Content of Plant-Specific Pressurized Thermal Shock Safety Analysis Reports for Pressurized Water Reactors, Regulatory Guide 1.154. USNRC, 2005. Technical Basis for Revision of the Pressurized Thermal Shock (PTS) Screening Criteria in the PTS Rule (10CFR50.61), NUREG1806.