Method for calculating stress corrosion cracking failure probability and confidence interval of pressurized water reactor primary circuit component

Abstract

The application provides a method for calculating stress corrosion cracking failure probability and confidence interval of a pressurized water reactor primary circuit component, which can quantitatively predict the stress corrosion cracking failure probability of the structural material of a pressurized water reactor primary circuit device, provides a reference for the maintenance of the device, and has a wide application prospect; meanwhile, the method has a simple process, high calculation efficiency, is suitable for engineering application, and is beneficial to improving the safety and economy of the pressurized water reactor nuclear power plant.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of pressurized water reactor structure reliability analysis, and particularly relates to a pressurized water reactor one-loop component stress corrosion cracking failure probability and confidence interval calculation method. BACKGROUND

[0002] Stress corrosion cracking refers to a damage mode that may be caused by a metal in a corrosive medium under the condition of bearing tensile stress. Stress corrosion cracking leads to material fracture, which is called stress corrosion cracking. Stress corrosion of a metal is a complex process, and once a corrosion crack in a metal structure is formed, it is difficult to control, the safe operation of equipment cannot be guaranteed, and the integrity and working performance of the structure are seriously threatened, so stress corrosion cracking is an important content in metal failure evaluation. Stress corrosion cracking failure is an important failure mode of metal materials, and probability analysis of stress corrosion cracking failure is also an important part of metal structure risk evaluation.

[0003] The equipment of the pressurized water reactor one-loop is long-term in a coolant environment of high temperature and high pressure, and has the possibility of causing stress corrosion cracking failure, which seriously threatens the safety of the nuclear power plant. At present, the nuclear power plant sets up redundant safety facilities to cope with the influence of stress corrosion cracking failure for safety consideration, which reduces the economy of the nuclear power plant. It is particularly important to establish a pressurized water reactor one-loop component stress corrosion cracking failure probability and confidence interval calculation method to calculate the stress corrosion cracking failure probability of the equipment, and then replace the equipment before failure to improve the safety and economy of the nuclear power plant. SUMMARY

[0004] The purpose of the present application is to provide a pressurized water reactor one-loop component stress corrosion cracking failure probability and confidence interval calculation method. The method can quantitatively predict the stress corrosion cracking failure probability of the structural material of the pressurized water reactor one-loop equipment. Thus, the method provides a reference for the maintenance of the pressurized water reactor one-loop equipment of the nuclear power plant, and is conducive to improving the safety and economy of the pressurized water reactor nuclear power plant.

[0005] The purpose of the present application is achieved by comprising the following steps:

[0006] Step 1, setting the number of sample points, considering that the material properties and other parameters of the pressurized water reactor one-loop component have uncertainty, the failure probability needs to be calculated by a random sampling method, and only in the case of a certain number of sample points can the result be true and reliable;

[0007] Step 2, according to the actual situation of the component, determining the stress, temperature borne by the pressurized water reactor one-loop component, and the yield strength, fracture toughness, thickness, crack initial size, crack aspect ratio and other parameters of the structural material of the component;

[0008] Step 3, on the basis of step 2, the above parameters are divided into uncertain parameters subject to certain distribution rules and deterministic parameters with fixed values, for uncertain parameters, random extraction is performed according to their distribution rules;

[0009] Step 4, the stress intensity factor K I The calculation formula and the empirical correlation of crack propagation rate are obtained, and the parameters obtained in step 3 are substituted into the differential equation of crack depth and crack propagation rate, and the relationship between crack depth and time can be obtained by numerical method;

[0010] Step 5, on the basis of step 4, the crack depth at different times is substituted into the stress intensity factor calculation formula to obtain the relationship between stress intensity factor and time;

[0011] Step 6, on the basis of step 5, the fracture failure probability at different times is calculated according to the criterion of stress corrosion fracture failure and the distribution rule of material fracture toughness;

[0012] Step 7, extract a group of input parameters, complete one calculation as a sample, judge whether the sample number reaches the pre-set number, if not, return to step 2;

[0013] Step 8, observe whether the mean value of failure probability tends to be stable, if not, the sample number is insufficient and return to step 1 to increase the sample point number;

[0014] Step 9, draw the failure probability frequency histogram at each time, and observe the distribution form of the failure probability at each time;

[0015] Step 10, on the basis of step 9, transform the failure probability at each time through mathematical method, so that the transformed failure probability frequency distribution histogram is in normal distribution form;

[0016] Step 11, on the basis of step 10, calculate the confidence interval of the transformed failure probability;

[0017] Step 12, on the basis of step 11, convert the confidence interval of the transformed failure probability into the failure probability confidence interval before transformation through mathematical method.

[0018] Further, in step 4, the stress intensity factor K I The calculation formula is:

[0019]

[0020] In the formula, R p0.2 is the yield strength of the material, MPa;

[0021] σ is the tensile stress, MPa;

[0022] a is crack depth, m;

[0023] b is material thickness, m;

[0024] c is crack width, a / c is called aspect ratio;

[0025] The empirical correlation of crack propagation rate is:

[0026]

[0027] In the formula, C and n are coefficients related to environment and material.

[0028] Further, in step 6, the criterion of stress corrosion cracking failure is the stress intensity factor criterion according to the linear elastic fracture mechanics theory, according to the characteristics of the stress corrosion crack being an I-type crack, when the fracture toughness K IC <Stress intensity factor K I is greater than K

[0029]

[0030] In the formula, f(x) is the fracture toughness probability density function.

[0031] Compared with the prior art, the beneficial effects of the present application are: the present application proposes a calculation method of stress corrosion cracking failure probability and its confidence interval of pressurized water reactor components under the condition of coolant, which can quantitatively predict the stress corrosion cracking failure probability of the structural materials of the pressurized water reactor primary circuit equipment, provide a reference for the maintenance of the equipment, and has a wide application prospect; at the same time, the method process is simple, has high calculation efficiency, is suitable for engineering application, and is beneficial to improve the safety and economy of the pressurized water reactor nuclear power plant. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 is a calculation flowchart;

[0033] Figure 2 is an equivalent schematic diagram of a crack;

[0034] Figure 3 is a schematic diagram of failure probability and sample number;

[0035] Figure 4 is a schematic diagram of failure probability frequency distribution histogram at a certain time;

[0036] Figure 5 is a schematic diagram of failure probability frequency distribution histogram after deformation at a certain time;

[0037] Figure 6 is a result schematic diagram. DETAILED DESCRIPTION

[0038] The present application will be described in detail below with reference to the accompanying drawings and embodiments. The specific examples are the stress corrosion cracking failure probability of a certain device structure material in the primary circuit of a pressurized water reactor under coolant conditions.

[0039] Figure 1 To calculate the flowchart, as shown in Figure 1 The stress corrosion cracking failure probability calculation method of the primary circuit component of the pressurized water reactor according to the present application is as follows:

[0040] Step 1, set the number of sample points. Considering the uncertainty of material properties and other parameters, the failure probability needs to be calculated by random sampling method. Only in a certain number of sample points, the result can be true and reliable. In the specific embodiment, the number of sample points is set to N.

[0041] Step 2, according to the actual situation of the calculated component, determine the stress σ, temperature t, yield strength R p0.2 , fracture toughness K IC , thickness B, crack initial size a0, crack aspect ratio a / c and other parameters of the component structure material. Considering the uncertainty of the above parameters, the parameter distribution type, distribution range, mean, variance and other parameters need to be determined according to the actual situation. For the parameters that cannot determine the distribution type, they are processed according to the uniform distribution. Step 3, on the basis of step 2, the above parameters are extracted according to their distribution law to obtain the sample set of input parameters σ i , t i , (R p0.2 ) i , B i , (a0) i , (a / c) i , where i is the label of the sample set, 1≤i≤N.

[0042] Step 4, the crack propagation empirical correlation formula selected in the specific embodiment is (Terachi T, Yamada T, Miyamoto T, et al. SCC growth behaviors of austenitic stainless steels in simulated PWR primary water[J]. Journal of Nuclear Materials, 2012, 426(1-3): 59-70.):

[0043]

[0044] In the formula, R is a constant, 8.314; T is temperature, K.

[0045] The stress corrosion crack is conservatively simplified to a semi-elliptical crack, as Figure 2 shown. The calculation formula for its stress intensity factor is shown in Equation (1). By联立the stress intensity factor K I calculation formula (see Equation 1) and the empirical correlation formula for crack growth rate, a differential equation for crack depth and crack growth rate is obtained:

[0046]

[0047] Substitute the input parameter sample set obtained in Step 3 into Equation (5), and solve the differential equation by numerical method to obtain the crack depth a i (τ) at N different times τ.

[0048] Step 5, substitute the crack depth a i (τ) at N different times τ obtained in Step 4 into Equation (1) to obtain the stress intensity factor (K I ) i (τ) at different times τ.

[0049] Step 6, if the probability density function f(x) of the fracture toughness K IC is known, substitute the stress intensity factor (K I ) i (τ) at different times τ into Equation (3) to obtain the failure probability at different times. For the case where the probability density function g(x) of the J-integral critical value J IC of the material is known, but the probability density function of the fracture toughness K IC is not known, it is necessary to convert J IC to K IC . The conversion relationship between J IC and K<00000​​​​​​​​​​​​​​​​​​​​​​​

[0055] Step 8, observe whether the failure probability calculation result tends to be stable, if not, return to step 1, in a specific example, the failure probability mean under different sample sizes is calculated, and a schematic diagram of the relationship between the failure probability and the sample size is shown as Figure 3 .

[0056] Step 9, plot the failure probability at each time as a failure probability frequency histogram, and observe the distribution form of the failure probability at each time, in a specific embodiment, a schematic diagram of the frequency distribution histogram at a certain time is shown as Figure 4 , according to relevant experience, the frequency of failure probability usually presents a negative skew distribution.

[0057] Step 10, the purpose of this step is to convert the skew distribution into a normal distribution by a transformation formula, so as to calculate the confidence interval under a certain confidence. The transformation formula is:

[0058] g i (τ)=(-f i (τ)+m) a (8)

[0059] In the formula, i is the number of failure probability, i = 1 ~ N, N is the sample size; g i (τ) is the i th transformed failure probability at τ time; a is the transformation index, a = 0 ~ 1, the value is determined by using an optimization algorithm, so that the P value of K-S test of the transformed distribution is maximum; f i (τ) is the i th failure probability before transformation at τ time; m is the maximum value of all failure probabilities before transformation, that is, the maximum value of N failure probabilities before transformation.

[0060] After transformation is completed, N transformed failure probabilities will be obtained. In this specific embodiment, a schematic diagram of the frequency distribution histogram at a certain time after transformation is shown as Figure 5 . After K-S test, the P value is greater than 0.05, so the transformed failure probability distribution is considered to present a normal distribution.

[0061] Step 11, the purpose of this step is to calculate the mean and variance of the failure probability g(τ) at τ time, and then calculate the transformed confidence interval. On the basis of step 10, the mean μ(τ) and variance var(τ) of the failure probability distribution at τ time after transformation are obtained, and the mean μ(τ) and variance var(τ) are respectively:

[0062]

[0063] In the formula, g i (τ) is the i th transformed failure probability at τ time; N is the set sample size.

[0064]

[0065] wherein μ(τ) is the mean value of the failure probability distribution at time τ; var(τ) is the variance of the failure probability distribution at time τ; u

[0066] The distribution g(τ) of N failure probabilities at time τ obeys a normal distribution with mean μ(τ) and variance var(τ), and the upper limit L upper (τ) and the lower limit L lower (τ) of the confidence interval of the normal distribution are:

[0067]

[0068] wherein μ(τ) is the mean value of the failure probability distribution at time τ; var(τ) is the variance of the failure probability distribution at time τ; u α / 2 is the upper α / 2 quantile of the standard normal distribution, which can be obtained by consulting the standard normal distribution table.

[0069] Step 12, on the basis of step 11, the confidence interval of the failure probability at time τ after deformation is transformed by mathematical means to obtain the confidence interval of the failure probability before deformation, and the conversion formula is:

[0070]

[0071] wherein L upper (τ) is the upper limit of the failure probability confidence interval under the normal distribution at time τ; L lower (τ) is the lower limit of the failure probability confidence interval under the normal distribution at time τ; L upper (τ) is the upper limit of the failure probability confidence interval under the original distribution at time τ; L lower (τ) is the lower limit of the failure probability confidence interval under the original distribution at time τ; m is the maximum failure probability of the original distribution, which is consistent with m in formula (8); a is the deformation index, which is consistent with a in formula (8).

[0072] In specific embodiments, the failure probabilities and confidence intervals of the component at different times are as shown in Table 1. Figure 6

[0073] The above description is merely preferred embodiments of the present application, and is not intended to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.​

Claims

1. A method for calculating the probability and confidence interval of stress corrosion fracture failure of primary circuit components in a pressurized water reactor, characterized in that, The steps are as follows: Step 1: Set the number of sample points based on the material properties of the primary loop components of the pressurized water reactor; Step 2: Determine the stress, temperature, yield strength, fracture toughness, thickness, initial crack size, and crack aspect ratio of the primary circuit components of the pressurized water reactor. Step 3: Based on Step 2, the above parameters are divided into uncertain parameters that follow a certain distribution law and deterministic parameters that are constant. For uncertain parameters, they are randomly selected according to their distribution law. Step 4: Combine stress intensity factor K I The calculation formulas for crack propagation rate and crack depth and crack propagation rate are obtained. The parameters obtained in step 3 are substituted into the differential equation for crack depth and crack propagation rate. The relationship between crack depth and time is obtained by numerical method. Step 5: Based on Step 4, substitute the crack depth at different times into the stress intensity factor calculation formula to obtain the relationship between stress intensity factor and time; Step 6: Based on Step 5, calculate the probability of fracture failure at different times according to the criteria for stress corrosion cracking failure and the distribution law of material fracture toughness. Step 7: Extract a set of input parameters. One calculation is considered one sample. Determine if the number of samples has reached the preset number. If not, return to step 2. Step 8: Observe whether the mean failure probability tends to stabilize. If it does not tend to stabilize, the number of samples is insufficient and return to step 1 to increase the number of sample points. Step 9: Plot the failure probability at each time point as a failure probability frequency histogram and observe the distribution of the failure probability at each time point; Step 10: Based on step 9, transform the failure probability at each time step using mathematical methods so that the transformed failure probability frequency distribution histogram is in the form of a normal distribution. Step 11: Based on Step 10, calculate the confidence interval of the failure probability after deformation; Step 12: Based on Step 11, convert the failure probability confidence interval after deformation into the failure probability confidence interval before deformation using mathematical methods; In step 4, the stress intensity factor K I The calculation formula is: In the formula, R p0.2 Where σ is the material yield strength (MPa), σ is the tensile stress (MPa), a is the crack depth (m), B is the material thickness (m), and c is the crack width (a / c is called the aspect ratio). The crack propagation rate is calculated using the following formula: In the formula, C and n are coefficients that are related to the environment and materials.

2. The method for calculating the probability and confidence interval of stress corrosion fracture failure of primary circuit components in a pressurized water reactor according to claim 1, characterized in that, In step 6, the criterion for stress corrosion cracking failure is based on the stress intensity factor criterion of linear elastic fracture mechanics theory. Given that stress corrosion cracks are type I cracks, when the fracture toughness K... IC Stress intensity factor K I When fracture failure occurs, the formula for calculating the probability of fracture failure is: In the formula, f(x) is the fracture toughness probability density function.

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