CrossMark

Available online at www.sciencedirect.com

ScienceDirect

Procedía Engineering 130 (2015) 1554 - 1563

Procedía Engineering

www.elsevier.com/loeate/procedia

14th International Conference on Pressure Vessel Technology

Case Study on the Relationship between Safety Margin and Conditional Probability for the Integrity Analysis of Reactor Pressure Vessel under Pressurized Thermal Shock

Y.B. Lia, Z.L. Gao3'*, Y.B. Leia

aZhejiang University of Technology, Institute of Process Equipment and Control Engineering, Hangzhou, Zhejiang 310032, China

Abstract

Both probabilistic and deterministic analysis methods are used in structural integrity assessment of the reactor pressure vessel (RPV) under pressurized thermal shock (PTS). In the deterministic analysis, the assessment of integrity according to the linear elastic fracture mechanics principle is carried out by comparing the stress intensity factor (SIF) Ki with the fracture toughness Kic which is characterized by a reference transition temperature Tref. Then the safety margin ATm can be determined. In the probabilistic analysis, the conditional probability on the probabilistic fracture mechanics analysis can be obtained by Monte Carlo simulation, where some input parameters are regarded as random variables, such as fracture toughness, crack size, neutron fluence, etc. Then, the relationship between the conditional probability and the temperature margin is analyzed for some typical PTS transients. From the analysis, a good correlation between the conditional probability and the temperature margin is obtained. ©2015 The Authors.PublishedbyElsevierLtd. Thisis an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICPVT-14

Keywords: Safety margin; Conditional probability; Probabilistic fracture mechanics; Pressurized thermal shock; Reactor pressure vessel.

1. Introduction

The reactor pressure vessel (RPV), as one of the most important safety barriers of pressurized water reactors, is exposed to neutron irradiation, which results in embrittlement of the RPV steel. One potential challenge to the

* Corresponding author. E-mail address: zlgao@zjut.edu.cn

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICPVT-14

doi: 10.1016/j.proeng.2015.12.324

structural integrity of the RPV in a pressurized water reactor is posed by pressurized thermal shock (PTS). Both probabilistic and deterministic analysis methods are used in structural integrity assessment of the RPV under PTS [!]•

In the deterministic analysis [2], the assessment of integrity according to the linear elastic fracture mechanics principle is carried out by comparing the stress intensity factor (SIF) K with the fracture toughness K\c. The SIFs along the crack tip can be evaluated for the analyzed PTS transients. The fracture toughness curves with crack tip temperature Kc-T are usually fixed with a reference transition temperature Tref. In the ASME approach, the reference transition temperature is defined as RTndt, reference temperature for nil ductility transition [3]. Then, a maximum allowable transition temperature RT^dt for the PTS transient corresponds to the fracture toughness curve shifted horizontally up to the point where it intersects relative to the maximum value of Ki during the PTS event. The difference between RT^ and RTndt of RPV material determines the safety margin ATm , which should larger than or equal to zero.

The conditional probability on the probabilistic fracture mechanics analysis is obtained by Monte Carlo simulation, where some input parameters are regarded as random variables, such as fracture toughness, crack size, neutron fluence, etc [4-7]. Comparisons between the conditional probability and the temperature margin are made for some typical PTS transients. From the analysis, a good correlation between the conditional probability and the temperature margin is obtained.

2. Analysis Approaches

2.1. Deterministic approach

For the assessment of RPV integrity, the deterministic approach is usually based on linear elastic fracture mechanics (LEFM). The fracture toughness value taking into account radiation embrittlement, KIc, is compared with the applied SIF, Kj, at the tip of the flaws. The crack initiation criterion can then be written as:

K > KIC (1)

then the safety margin of fracture toughness is the difference between KIc and Kj.

However, it is difficult to measure fracture toughness directly, especially under radiation environment. Many assessment procedures use reference temperature to characterize the fracture toughness. For example, the reference nil-ductility temperature is used in ASME Section XI, which can be evaluated by Charpy test. In ASME Section XI, Klc is the lower bound fracture toughness value at the assessment temperature and can be evaluated from the following equation:

KIc = 36.5 + 22.783exp[0.036(TT - RTndt )] (2)

Where, T is the metal temperature at crack tip and RTndt is the reference nil-ductility temperature with the

effects of neutron irradiation. Both T and RTndt are in unit of °C, KIc is in MPaVm , with an upper shelf fracture

toughness of 195 MPaVm •

The vessel material transition temperatures can be evaluated as:

RTndt = RTndt(u) + M + ARTndt (3)

M = 2yl &U + al

ARTndt = (CF) f(0-28-0-10106 f) (5)

where, RTndt(U) is the reference temperature for a reactor vessel material in the pre-service or unirradiated condition, evaluated according to the procedures in the ASME Code NB-2331 [8]. M is the margin to be added to account for uncertainties in the values of RTndt(U) , copper and nickel contents, fluence and the calculation procedures,

evaluated with the standard deviation, a^, for RTndt(U) and crA for ARTndt . ARTndt is the mean value of the transition temperature shift, or the change in RTndt , due to irradiation. CF (°C) is the chemistry factor, which is a function of copper and nickel contents, f is the best estimate neutron fluence, in units of 1019 n/cm2 (E >=1 MeV), at the clad-base metal interface on the inside surface of the vessel at the location where the material in question receives the highest fluence for the period of service in question. RT^y must be determined in a PTS analysis for each vessel beltline material using the EOL fluence for that material.

According to the IAEA good practice guidance, the vessel's maximum allowable transition temperatures RT^dt can be evaluated by following the procedure shown in Fig. 1. For each individual analyzed PTS sequence, the material behavior in terms of allowable SIF, and the crack loading path in terms of SIF are considered as a function of temperature and should be presented in a SIF respectively allowable SIF vs. temperature diagram.

Therefore, it is convenient for taking the place of fracture toughness by reference temperature, to determine the safety margin. The difference between the maximum allowable transition temperature of the vessel and the vessel material's transition temperature determines the safety margin ATm , as

ATm = RT^T - RTndt (6)

The value of this safety margin should be larger than or equal to zero depending on the national regulatory requirements and considering the reliability of individual input data, such as material properties and effectiveness of NDE.

0 50 100 150 200 250 300

Temperature [°C]

Fig.l. Scheme ofthe approach for establishing the allowable indexing parameter for fracture toughness, on the basis ofa) the tangent point and b) the 90% or 1.0 maximum KI intersection point between the KIC and crack driving force (KI) curves.

2.2. Probabilistic approach

In contrast to the deterministic analysis, the probabilistic approach takes into account the scatter and uncertainties of material properties, loading conditions, and defect distribution in an appropriate manner. Thereby it is assumed

that the quantities entering a failure criterion are given in terms of probability distributions. Then, the failure probability can be calculated by some reliability analyses methods.

Usually, Monte Carlo simulation is used to calculate the failure probability based on probabilistic fracture mechanics. The calculation process is shown in Fig. 2. LFEM analysis is performed by stepping through discrete transient time steps to examine the temporal relationship between the SIF and the fracture initiation toughness KIc at the crack tip. For a specified PTS transient, total # RPV trials are simulated with consideration of the uncertainty in material and neutron fluence. Every RPV trail may contain a number of cracks. The SIF should be evaluated for the zth RPV trial and jth crack, at the transient time step r, and denoted by KY (r)^j^. For the comparison with

deterministic analysis, it is assumed that there is only one crack in every RPV trial. But the crack size is different in every RPV trial.

Fig. 2. Schematic ofan assessment process with probabilistic approach.

In general, the statistical distribution of fracture toughness is considered to be Wei-bull distribution [9]. The failure probability under a given SIF can be obtained from its cumulative distribution function, as follows:

KIH aKi

K i («")(, )> aKi

where, the parameters of the distribution are a function of AT : aKu = 11.9727 + 25.734exp(0.00414AT)

bKic = 16.2169 + 46.845 exp(0.02232AT) (8)

c^ = 2.03025 + 0.4983 exp(0.0243AT)

Where, AT = T(r)- RTndt in °C.

The failure probability that described as above, in fact, is a conditional probability under the condition that the transient occurred. Therefore, it is defined that cpi(r^ is the conditional probability of initiation at the crack tip at

time t, and cpi{r\.^ = Pr{Kx (r)^-^ > KIc} . In this work, the frequency of the transient is ignored. The conditional probability of initiation is focused in the work. For this transient, the maximum value of cpiir)^ is regarded as the conditional probability of initiation of the transient, which can be written as:

3. Analysis Models

3.1. RPV and Flaw Models

The reactor vessel considered in this analysis is a typical PWR with an inner surface radius of 1994 mm, a base metal thickness of 200 mm and a cladding thickness of 7.5 mm. The postulated crack is an axial semielliptical surface breaking crack. For the deterministic analysis, the crack depth is considered to be 19.5mm, and the length to bell7 mm. The material properties are adopted from IAEA report-

3.2. PTS transient

The scenario describes PTS regime for a reactor pressure vessel of the 3 loop typical French PWR. The PTS scenario with repressurization is selected, which was used for PROSIR benchmark. Time variations of primary pressurep, coolant temperature T and convective heat-transfer coefficient A are drawn in Fig. 3 and 4.

3.3. Random variables

cpi d )=max w^X)

Then, a mean of CPI can be obtained from the N RPV trials.

CPI m =£ CPI (i)

The main input parameters taken into account as stochastic variables are the flaw size, the neutron fluence, the chemical element contents of steel which is relevant to material embrittlement (copper and nickel), the initial

reference transition temperature RTndt(u) , the material fracture toughness KIc. These parameters can be found in Table 2. From the temperature profile and RTndt , the fracture toughness, KIc, at the tip of the crack is represented by a normal distribution using the equation derived from the lower-bound fracture toughness in ASME.

0 3000 6000 9000 12000 15000

Time [s]

Fig. 3. Primary pressure and coolant temperature.

6000 9000

Time [s]

Fig. 4. Heat-transfer coefficient.

Table 1. Variables and distributions used in PFM analysis.

Distributions Average Standard deviation

Crack depth (mm) Normal 19.5 1.0

R1NDT(LI) ( C) Normal -20 10

Cu (wt%) Normal 0.06 0.006

Ni (wt%) Normal 0.85 0.085

Fluence (1019n/cm2) Normal 1.0-7.0 10% ofaverage

4. Analysis Results 4.1. Temperature field

The variation of temperature through the RPV wall thickness at different time is shown in Fig. 5. It can be seen that the temperature difference through thickness at 3600s arrives about 81 °C, which will cause thermal stress.

0 0.2 0.4 0.6 0.8 1

Normalized thickness x/(t+tc)

Fig. 5. Variation oftemperature through the RPV wall thickness at i=3600s and i=7200s.

4.2. Stress field

Normalized thickness x/(t+tc)

Fig. 6. Variation ofhoop stress through the RPV wall thickness at i=3600s and i=7200s.

The variation of hoop stress through the RPV wall thickness at different time is shown in Fig.6. At 3600s, the internal pressure is low, and the stress is mainly composed of thermal stress. However, at 7200s, the temperature difference is low, and the stress is mainly composed of primary stress with re-pressure.

4.3. Deterministic analysis

The average values of the input parameters are used to perform deterministic analysis. For the fracture toughness, ASME low boundary is considered. The SIF and fracture toughness with temperature are shown in Fig. 7. It can be

seen that the intersection point between the Kic and Ki curves is located at about 70 °C (7200s). The maximum allowabletransitiontemperatures RT^Jdt isabout58.59 °C.

The vessel material transition temperatures under different fluence can be evaluated following equations 3-5. The results are shown in Table 2.

60 90 120 150 180 210 240 270 300 Temperature [ °C ]

Fig. 7. SIF and fracture toughness with temperature.

Table 2. Temperature margins (°C).

Fluence 1.0 3.0 5.0 7.0

RTndt 16.62 22.87 25.36 26.76

ATm 41.97 35.73 33.23 31.84

1.0E-01

1.0E-03

^1.0E-05

J 1.0E-07 "V o

§ 1.0E-09 •a

1.0E-11 1.0E-13 1.0E-15

iViViViVi'i

lTnmmVn TnnVn

0 1800 3600 5400 7200 9000 10800 12600 14400 Time [s]

Fig. 8. Conditional probability of the transient with time.

4.4. Probabilistic analysis

The failure probability under different fluence can be evaluated as above. The results are shown in Table 3.

Table 3. Failure probability under different fluence. Fluence 1.0 3.0 5.0 7.0

P( 3.03x10« 5.02x10« 6.82xl0-8 7.87x10'

According to the probabilistic approach, the failure probability can be calculated with these random variables. For the parameters used in deterministic analysis expect the fracture toughness, the failure probability with time cpi(t) is shown in Fig. 8. Then the conditional probability of initiation of the transient is the maximum value of 3.24* 10"3. After 1000 RPV trails, the mean of conditional probability can be evaluated about 6.82* 10"8.

4.5. Discussions

Figure 9 shows the relationship between the conditional probability and temperature margin. It can be seen that there is a linear correlation between the conditional probability and temperature margin in semilog coordinate. Therefore, if the temperature margin is calculated by deterministic approach, the conditional probability can be obtained with the linear correlation.

30 32 34 36 38 40 42

Temperature Margin [°C]

Fig. 9. Temperature margin and conditional probability.

5. Conclusions

Two approaches for RPV integrity analysis under PTS are introduced. The safety margin based on reference temperature is proposed, as a result of the deterministic approach. Meanwhile, the conditional probability can be calculated by Monte Carlo method with probabilistic approach. Then a case study is performed to build a relationship between the conditional probability and temperature margin. The result shows that there is a linear correlation between the conditional probability and temperature margin in semilog coordinate. Therefore, if the temperature margin is calculated by deterministic approach, the conditional probability can be obtained with the linear correlation. It is suggested that several PTS transients should be performed to build a precise relationship for the engineering application.

Acknowledgements

The research work was supported by the Zhejiang Public Welfare Technology Application Research Project (No. 2014C23001).

References

[1] Qian Guian, Niffenegger Markus, 2014. Deterministic and probabilistic analysis of a reactor pressure vessel subjected to pressurized thermal shocks. Nuclear Engineering and Design, 273, 381-395.

[2] IAEA, 2010. Pressurized Thermal Shock in Nuclear Power Plants: Good Practices for Assessment. Technical Report IAEA-TECDOC-1627, IAEA, VIENNA.

[3] ASME Boiler and Pressure Vessel Code, 2013. Section XI Rules for Inservice Inspection of Nuclear Power Plant Components. ASME, New York.

[4] Dickson T, Malik S, 2001. An updated probabilistic fracture mechanics methodology for application to pressurized thermal shock. International Journal ofPressure Vessels and Piping, 78, 155-163.

[5] Gao Zengliang, Li Yuebing, Lei Yuebao, 2013. A comparison between probabilistic and deterministic fracture mechanics assessments of the structural integrity of a reactor pressure vessel subjected to a pressurized thermal shock transient. ASME 2013 Pressure Vessels & Piping Conference, PVP2013-97569.

[6] Qian G, Niffenegger M, 2013. Procedure, method and computer codes for probabilistic assessment of reactor pressure vessels subjected to pressurized thermal shocks. Nuclear Engineering and Design, 258, 35-50.

[7] Pistora Vladislav, Posta Miroslav, Lauerova Dana, 2014. Probabilistic assessment of pressurised thermal shocks. Nuclear Engineering and Design, 269, 165-170.

[8] ASME Boiler and Pressure Vessel Code, 2013. Section III, Rules for Construction of Nuclear Facility Components, Division 1 -Subsection NB, Class 1 Components. ASME, New York.

[9] Williams P T, Bowman K O, Bass B R, Dickson T L. Weibull statistical models of KIc/KIa fracture toughness databases for pressure vessel steels with an application to pressurized thermal shock assessments of nuclear reactor pressure vessels. International Journal of Pressure Vessels and Piping, 78, 165-178.