Solid State Phenomena Vol. 120 (2007) pp. 37-42 online at http://www.scientific.net (2007) Trans Tech Publications, Switzerland Failure Probability Estimation of Pressure Tube Using Failure Assessment Diagram Joon-Seong Lee 1,a, Sang-Log Kwak 1,b, and Chang-Ryul Pyo 1,c 1 Div. of Mechanical System Design Engineering, Kyonggi University San 94-6, Iui-dong, Yeongtong-gu, Suwon, Gyonggi-do 443-760, Korea 2 Safety Technology Research Team, Korea Railroad Research Institute 360-1, Woulam-dong, Uiwang, Gyeonggi-do 437-757, Korea 3 Dept. of Mechanical System, Induk Institute of Technology San 76, Wolgae-dong, Nowon-gu, Seoul 139-0749, Korea a jslee1@kyonggi.ac.kr, b slkwak@krri.re.kr, c crpyo@induk.ac.kr Keywords: Failure Probability, Monte Carlo Simulation, Failure Assessment Diagram, Pressure Tube, Delayed Hydride Cracking, Probability Density Function Abstract. Pressure tubes are major component of nuclear reactor, but only selected samples are periodically examined due to numerous numbers of tubes. Pressure tube material gradually pick up deuterium, as such are susceptible to a crack initiation and propagation process called delayed hydride cracking (DHC), which is the characteristic of pressure tube integrity evaluation. If cracks are not detected, such a cracking mechanism could lead to unstable rupture of the pressure tube. Up to this time, integrity evaluations are performed using conventional deterministic approaches. So it is expected that the results obtained are too conservative to perform a rational evaluation of lifetime. In this respect, a probabilistic safety assessment method is more appropriate for the assessment of overall pressure tube safety. This paper describes failure probability estimation of the pressure tubes using probabilistic fracture mechanics. Failure assessment diagram (FAD) of pressure tube material is proposed and applied in the probabilistic analysis. In all the analyses, failure probabilities are calculated using the Monte Carlo simulation. As a result of analysis, failure probabilities for various conditions are calculated, and examined application of FAD and LBB concept. Introduction There are four CANDU type reactors in Korea. Since the first commercial operation of a CANDU type reactor, 19 years have been passed. In the CANDU reactors, cold-worked Zr-2.5Nb pressure tubes are used to contain fuel bundles and heavy water coolant. Shape of pressure tube is a thin-walled pipe with 6400mm(length) 104mm(inner diameter) 4.2mm(thickness). At normal operating conditions, the pressure tubes are subject to internal pressure of 10MPa and to temperatures ranging from 260 C at the inlet to 310 C at the outlet. Since several incidents of leaking in the pressure tube have been experienced, significant efforts have been made to improve the pressure tube integrity design, material, and fabrication upgrades during the last decade. As a result of research, the fitness for service guideline (FFSG) provides the flaw acceptance criteria for pressure tubes. FFSG is developed under the basis of ASME Code Sec. XI [1], FFSG [2] uses conventional deterministic fracture mechanics approaches. Due to the uncertainty in inspection data, conservative data are used for the integrity evaluation. For example, lower bound in fracture toughness and upper bound in stress data, crack growth rate are used. It results in, therefore, difficulty in estimating lifetime. The 53 tubes at CANDU type reactors in Korea are inspected as of 2001, which is far beyond the code requirement [3]. Statistical analysis of inspection results by Park [4] shows that about 45% of pressure tubes have defects whatever the significance. Considering the inspection coverage is limited to 15% of pressure tubes, it is necessary to develop a tool for evaluating pressure tube integrity accounting uninspected portion. The All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 218.159.88.87-18/04/07,02:47:45)
38 Safety and Structural Integrity 2006 probabilistic approach would be a suitable way for the evaluation in taking into account the uncertainties associated with important integrity parameters, such as in-service inspection sampling sizes, flaws distribution, initial hydrogen concentration, and material properties [5]. For realistic analysis failure criteria, which is more suitable for probabilistic analysis, is proposed based on failure assessment diagram (FAD). In this paper, the evaluation of failure probabilities in pressure tubes considering delayed hydride cracking (DHC) mechanism was investigated using Monte Carlo simulation [6]. Also, Probabilistic integrity evaluation code for CANDU pressure tube was developed. To determine the failure of pressure tube, FAD is applied. Deterministic Integrity Assessment Since Monte Carlo simulation is iterative technique repeating deterministic analysis, deterministic integrity assessment is the basis of probabilistic analysis. Many references [13] are available for details of deterministic analysis on CANDU pressure tubes. The characteristic of pressure tubes integrity assessment is cracking mechanism. Pressure tube cracking mechanism is governed by delayed hydride cracking (DHC) whereas other structures are governed by fatigue or corrosion. While the fatigue crack growth is in the order of 10-8 m/cycle, the crack growth by DHC is in the order of 10-6 m/cycle [7] in pressure tube. Therefore, integrity evaluation is focused on the contribution of DHC in this study. Flaw initiation and growth. According to inspection results analysis by Park [4], axial crack is dominant in pressure tube because the metallurgical structure of pressure tube is favorable for axial cracking rather than circumferential direction and hoop stress is higher than axial stress by two times. In general, the axial defects found in pressure tubes are associated with debris or scratch in axial direction. Once a sharp notch is made, the stress concentrates at the tip. The hydride formation and its associated crack are very susceptible to high stress location. When a hydrogen concentration is higher than terminal solid solubility (TSS) and a stress intensity factor is above a threshold value, hydrides are predicated and crack growth by DHC occurred [8]. The delayed hydride cracking velocity (DHCV) was independent of the stress intensity factor once the threshold was exceeded. Since the velocity is a function of temperature, the cumulative crack growth was determined by numerically integrating the velocity over the cooldown cycle. In the deterministic analysis 95% upper bound of DHC velocity of test data are used, where as actual distribution of DHC velocity are used in the probabilistic analysis. Axial crack velocity is used to estimate flaw growth along the tube length. From Eq. (1), the DHC growth can be integrated numerically by sectioning the cooldown transient curve. a= DHCV t j=1 j N (1) where DHCV is the DHC velocity in m/s, a is the crack extension in m, t j is time period for each transient loading condition in seconds and N is number of cooldown cycles expected during the evaluation period. In the probabilistic analysis DHC velocity are assumed to be probabilistic variables so that mean value and standard deviation are used. In the analytical procedure by FFSG, DHC propagation is possible if the hydrogen concentration at the flaw equals or exceeds the terminal solid solubility for hydride in dissolving (TSSD). However, in the present probabilistic analysis, hydrogen concentrations with terminal solid solubility for hydride precipitation (TSSP) temperature are used. Failure Criteria. Pressure tube material has enough ductility at installation because it was made by cold-work. During operation it loses ductility due to hydride precipitation and irradiation embrittlement. Current failure criteria used in FFSG are unstable fracture and plastic collapse for the consideration of material behavior change. However, some test data cannot be explained by these
Solid State Phenomena Vol. 120 39 failure criteria. More exact failure criteria are required because no safety margin is applied in probabilistic analysis. FAD can explain these data more exactly. For the application of FAD [9], failure parameters such as stress intensity factor (K I ), fracture toughness (K IC ), plastic collapse stress ( σ c ) and applied stress ( σ a ) are used. In this study the FAD is used for the consideration of material behavior change. Stress intensity factor and fracture toughness. Raju-Newman equation [10] is applicable up to radius/thickness ratio(r/t) 10 but the pressure tube has R/t ratio between 12 and 14. The stress intensity factor data were, therefore, derived by performing 3D finite element models for various surface crack shapes considering R/t ratio. A total of 40 analyses were performed for the ratio of crack depth to length from 0.2 to 1.0 and crack depth to wall thickness from 0.2 to 0.8 [11,12]. The loading condition was set to 10.4 MPa of internal pressure considering normal operating condition. One of the commercial finite element codes, ABAQUS [13] was used for the analysis. Unstable fracture is to occur when stress intensity factor exceeds the fracture toughness value. In the deterministic analysis, lower-bound value of fracture toughness and safety margin are used for the integrity evaluation. However, in the probabilistic analysis, probability density function (PDF) [11] of the fracture toughness should be used, which is fitted to be log-normal distribution and the coefficient of variance is assumed to be 0.2. Determination of failure. The global plastic collapse stress is determined using Zahoor s solution [14] whereas local collapse is determined using Kiefner solution [15]. Failure is assume to occur when crack depth to thickness ratio is greater than 0.8 [11]. LBB (Leak-Before-Break) condition cannot be applied simultaneously in the deterministic analysis. The LBB analysis means that when an unlikely leak through-wall DHC crack occurs, throughout the period of time needed for leak detection, confirmation, location and finally placing the reactor in a cold depressurized state, the growing crack length is always less than the critical crack length (CCL) in order to assure that the probability of an unstable pressure tube rupture is extremely low. Even with through-wall crack that produces leak, the failure of pressure tube does not occur immediately. Under the local collapse condition, plastic collapse is assumed to occur when crack depth is greater than 80% of wall thickness [12]. Therefore, the LBB can be considered at global collapse condition. As shown in Fig. 1 that is FAD, straight line (box) is current failure criteria, whereas FAL is failure criteria used in this study. FAL is determined by recent SINTAP [9] results. Current criteria cannot explain region B in Fig. 1. In the FAD, two data sets are displayed: one is derived from actual tensile strength and fracture toughness, and the other from mean value of tensile strength and fracture toughness. Failure assessment line (FAL) used in FAD is shown in Eqs. (2) ~ (4). K 1/ 2 r, FAL = r r 2 L r 1 + 2 K = K / K Lr I =σ / σ a c IC 6 [ 0.3+ 0.7 exp( 0.6L )] (2) (3) (4)
40 Safety and Structural Integrity 2006 Fig. 1 Failure assessment line versus burst test data Fig. 2 Flow of PFM analysis Probabilistic integrity assessment Among various probabilistic approaches, Monte Carlo (MC) method is used in this study to evaluate failure probabilities as a function of operation time. Fig. 2 illustrates the flow chart of this code. Some probabilistic variables are selected according to an employed analysis model. Probabilistic variables include initial crack size, material properties and so on. Crack growth simulations by DHC are performed by deterministic fracture mechanics considering ISI and transients data. Finally, failure probabilities are calculated as a function of operation time. In MC simulation the convergence was obtained when number of iterations equal to 10 6. In order to decrease the random number effect, five independent simulations are carried and an average value is used as a result. To check the failure criteria effect, some case studies were carried out. Probabilistic density function (PDF) for each probabilistic variable assumed in this study was derived from ISI and testing data. Only axial surface crack grown by DHC is considered in this study because the fatigue crack growth is negligible comparing with that of DHC. Details of probabilistic variables [11] are shown in Table 1 and sensitivity analyses are explained in the following. Table 1 Details of probabilistic variables Probabilistic variables PDF type Mean S.T.D. Min. Max. value value Aspect ratio (a/c) Exponential 0.12 N.A. 0.1 1.0 Depth ratio (a/t) Log-normal 0.10 0.08 0.01 0.5 Fracture toughness (K IC ) [MPa m] Log-normal 67.0 12.0 20.0 120.0 Coeff. of radial crack extension ( 10-2 ) [m/s] Log-normal 5.30 0.58 2.0 14.0 Coeff. of transverse crack extension ( 10-2 ) [m/s] Log-normal 2.40 0.48 1.0 5.5 Initial hydrogen [ppm] Normal 8.30 2.65 5.0 15.5 Flow stress [MPa] Normal 1063.3 55.4 600 1400 Effect of failure criteria. To verify the proposed FAD in this study, four analyses are performed. The Zahoor s equation [14] is used to calculate the global plastic collapse stress whereas Carter s equation [16] is used to local collapse. Leak and failure probabilities are shown in Figs. 3 and 4.
Solid State Phenomena Vol. 120 41 Fig. 3 Leak probability for various plastic collapse condition Fig. 4 Failure probability for various plastic collapse conditions Effect of dimensional change. Physical deformations of pressure tubes, such as diametral increase, elongation, wall thickness decrease and so on, are produced as operation time elapses due to thermal creep, irradiation creep and irradiation growth. The design allowance for the diametral expansion of pressure tubes is conservatively limited to 5% of the initial value. The analysis of inspection data [4] shows that the dimensional changes in diameter and wall thickness are linearly dependent with operation time. The maximum change rates are obtained from ISI data with 0.03mm/year in thickness reduction and 0.11mm/year in radius increase. ISI data shows that there is strong correlation between thickness decrease and diametral increase as shown in Fig. 5. Dimensions used in these analyses are shown in Table 2. As shown in this table, three different cases were considered in the following. Case 1: No dimensional change with thickness is 4.3 mm, inner diameter is 104.0 mm Case 2: Average dimensional change with 0.02 mm decrease in thickness and 0.16 mm increase in diameter Case 3: Maximum dimensional change with 0.03 mm decrease in thickness and 0.22 mm increase in diameter Results for the various conditions are shown in Fig. 6. Fig. 5 Correlation between diametral expansion and thickness decrease Fig. 6 Comparison of leak probability change as dimensional change Analysis Results. The results are represented by failure or leak probabilities of pressure tube as a function of time, which are shown in Figs. 3~6. In general, a maximum allowable failure probability is provided for reactor pressure vessel and nuclear piping. But it is not for CANDU pressure tube. It is assumed to be 10-5 for the comparison of analyses results. Under these assumption failure criteria has some effect on failure probability as shown in Figs. 3 and 4. For all cases failure or leak probabilities
42 Safety and Structural Integrity 2006 are less than 10-5 during design life. But five-year difference was observed as wall penetration criteria change as shown in Fig. 3. When crack depth reaches to 80% of wall thickness, the leak probability between the local collapse and the global collapse, which is used by failure criteria, has no difference almost. In case crack depth reaches to 100% of thickness, however, analysis result of two failure criteria shows some measure difference. The leak probability is reached in allowable failure probability after 38 and 43 years, respectively. Fig. 6 shows an effect of dimensional change. As dimensional change condition varies 20-year life difference is observed. Dimensional change has the highest effect on failure probability as shown in figure. It means that pressure tubes which have high dimensional change must be inspected in detail during ISI. In all case integrities are assured within design life. But integrity is not assured in extended period, when maximum dimensional change is occurred. Conclusions A probabilistic fracture mechanics code which could evaluate the integrity of CANDU pressure tube using MC simulation was developed. Analyses about failure probability are performed using probabilistic variables taken from ISI data. As a result of the research, the effects of TSS and failure criteria based on FAD were investigated. It is known that the governing failure mode was leakage or plastic collapse in all cases analyzed in this study, whereas unstable fracture was governing failure mode in deterministic analysis. Also, sensitivity analyses were conducted for evaluating CANDU pressure tube integrity. For more realistic failure estimation FAD are applied instead of current failure criteria that explains test data more exactly. Among the integrity parameters selected for the probabilistic failure analysis, the dimensional changes have high sensitivity for failure probability. References [1] ASME: B&PV Code Sec. XI (1996) [2] J.R. Hopkins, E.G. Price, R.A. Holt and H.W. Wong, Proc. of ASME PVP, Vol. 360 (1998) [3] CSA, CAN/CSA N285.4 (1994) [4] W.Y. Park, S.S. Kang and B.S. Han, Nuclear Engineering Design 212 (2002), pp. 41-48 [5] J.M. Bloom, ASME PV&P Vol. 92 (1984), pp. 1-19 [6] R.Y. Rubinstein, Simulation and Monte Carlo Method, John Wiley & Sons. Inc. (1981) [7] G.D. Moan, C.E. Coleman and E.G. Price, J. of PV&P, Vol. 43 (1990), pp. 1-21 [8] Z.L. Pan, I.G. Ritche and M.P. Puls, J. of Nuclear Materials, Vol. 228 (1996), pp. 227-237 [9] O. J. Ruiz, S.F. Gutierrez and M.A. Gonzalez, Cantabria Univ., Report/SINTAP/UC/05 (1997) [10] I.S. Raju and J.C. Newman, Engineering and Fracture Mechanics, Vol. 15 (1981), pp. 185-192 [11] S.L. Kwak, J.S. Lee, Y.J. Kim and W.Y. Park, Int. J. of Modern Physics 17 (2003), pp. 1587 [12] S.L. Kwak, J.B. Wang, Y.W. Park and J.S. Lee, Trans. of KSME, Vol. 28 (2004), pp. 289 [13] Hibbitt, Karlsson and Sorensen, ABAQUS User s manual, Ver. 5.8 (1999) [14] A. Zahoor, Ductile Fracture Handbook, Novetech Corp. & EPRI, Vol. I, II, III (1991) [15] J.F. Kiefner, W.A. Maxey, R.J. Eiber and A.R. Duffy, ASME STP 536 (1973), pp. 461 [16] A.J. Carter, Nuclear Electric Report TD/SID/REP/0190 (1992)