2014 IR&D Annual Report

Integrity Management of Nuclear Power Plant Components Subjected to Localized Corrosion Using Time-Dependent Probabilistic Model, 20–R8267

Principal Investigators
Pavan K. Shukla
Osvaldo Pensado

Inclusive Dates: 11/14/11 – 02/14/14

Figure 1.  Example of scatter plot of the 0.95 quantile pit depth versus time and contour plot exhibiting uncertainty
Figure 1. Example of scatter plot of the 0.95 quantile pit depth versus time and contour plot exhibiting uncertainty in the empirical parameter κ′ of the power law d = κ′ tν.

Background — Nuclear power plant operators are required to periodically inspect components by visual and volumetric examinations to maintain component integrity and ensure safety. As nuclear power plants age, more frequent inspections are expected to ensure component integrity. A framework to define inspection schedules, based on risk considerations, is needed to keep the cost of inspections constrained without compromising safety. The objective of the project is to develop an adaptive-predictive probabilistic model to forecast localized corrosion induced pit population and pit depth distributions. The forecast algorithm is intended to support integrity management programs, including integrity management for nuclear power plant components and dry storage cask systems for spent nuclear fuel. Forecasts of the evolution of corrosion damage and its uncertainty can inform maintenance and inspection program schedules and provide for savings or trigger actions when risk of failure or the extent of damage is not acceptable.

A model was developed in this project to forecast localized corrosion-induced damage of metallic components based on damage measured at a given time. For example, if a component exhibits pitting corrosion in an environment, the model will be used to estimate the distribution of pit depths as a function of time. The model accounts for previous inspection data, randomness of pit propagation, and pit growth rate as a function of time. The model could be used to estimate probability of component failure due to pitting corrosion and calibrate inspection schedules so that detection of corrosion sites occurs before failure.

Localized corrosion such as pitting commonly exhibits random scatter in the measurable parameters such as corrosion rate, maximum pit depth and time to perforation. The scatter in measurable parameters is hypothesized to arise from metal surface heterogeneities and variations in the corrosive environment over time. This randomness appears to be an inherent and unavoidable characteristic of pitting corrosion. Thus, stochastic models may be better suited to describe pitting corrosion than deterministic ones. This is especially true if the results of pitting corrosion modeling are to be used as inputs to integrity and risk management models, which should account for uncertainty in the decision making.

Figure 2.  Coupons with salt before exposure and one coupon after 8-day exposure
Figure 2. Coupons with salt before exposure and one coupon after 8-day exposure to humidity of 30 g/m3 at 50 °C.

Because measuring pit populations and their depths is time consuming and expensive for a large engineered system, experimental techniques have limited resolution, and the interest has always been in defining failure of a system, only the deepest pits are traditionally studied. There are no general methods to propagate the pit population and depth distributions in time and predict the evolution of corrosion damage. Markov chain models that are tested using Monte Carlo methods and updated with actual data using Bayes theorem have been used elsewhere to study the propagation of corrosion damage. We considered that such an approach has potential and flexibility to define an adaptive method incorporating inspection data to forecast the evolution of the system. Consequently, this method was investigated in this project.

Approach — Model development of the forecast technique consisted of three principal tasks: statistical model development, experiments, and data analysis and integration. In the statistical model, probabilities of transition between discrete states that satisfy Kolmogorov's forward equations for a pure birth process were used to forecast the evolution of depths of a population of pits. The discrete state of a pit is defined as a pit falling in a range of depths (e.g., a pit is in State 1 if its depth falls between 0 and 10 µm; State 2 if the depth is between 10 and 20 µm, and so forth). Thus, pit growth is conceptualized as a pit that transitions from one state to the next. In theory, parameters to define the transition rate between states can be obtained by measuring the average or a quantile (e.g., 95 percent quantile) pit depth as a function of time (see Figure 1). This macroscopic feature (e.g., average pit depth) can be used to quantify microscopic propagation rates.

As part of this project, conditions were selected to induce pitting in 304 stainless steel (SS). An experimental system was designed to monitor pit propagation in time. Small coupons of 304 SS with salt deposits were exposed to controlled temperature and humidity. This environment caused pitting readily after a few hours of exposure (see Figure 2). The 304 SS coupons were systematically removed from the humidity chamber for pitting corrosion evaluation and inspected using laser profilometry. The data collected from coupons were used to estimate pit depth versus time.

Figure 2.  Coupons with salt before exposure and one coupon after 8-day exposure
Figure 3. Example of data post-processing approach to identify pitting corrosion features: (a) photograph of metallic sample with pitting corrosion; (b) density plot generated with laser profilometry data; (c) filtered data keeping only pitting corrosion features and removing surface roughness features; (d) identified deepest pit; (e) contour plot of depths of deepest pit, (f) three-dimensional surface plot of the deepest pit; and (g) micrograph of deepest pit.

As part of the data analysis and integration step, the model was used iteratively to forecast the pit depth distribution at the next inspection or detection time. At the next inspection or detection time, the model parameters are updated with the collected data and a Bayesian update algorithm, and the next forecast is performed. A forecast sequence spanning 200 days and longer term forecasts were performed to demonstrate the implementation and use of the forecast model and associated techniques.

Accomplishments — A technique was developed to efficiently identify corrosion pits and measure pit depths using profilometry data. This technique is more efficient than known alternatives because it does not need to manually focus on one pit at a time. The technique automates the process of focusing on individual pits and extracting data for contour plots and two- and three dimensional plots of individual pit depths, as well as counting pixel clusters potentially associated with pits over a given detection area (see Figure 3). The technique allows for quantifying features of pitting corrosion such as the area and volume of the metal affected by pitting. An invention disclosure on the profilometry data filtering technique was submitted recently.

Based on Markov chain concepts and transition probabilities, a differential model was developed to forecast the evolution of corrosion damage, expressed as a distribution of pit depths. The only inputs to the model are empirical parameters, κ′ and ν, of the power law d = κ′ tν (d = pit depth, t = time) that can be determined from profilometry measurements or alternative techniques. It is demonstrated that the power law d = k′ tν directly arises from the propagation algorithm for a given class of transition probabilities or transition rates. The model also can be used to describe damage due to other corrosion modes that exhibit power law propagation.

A forecast algorithm for distribution of pit depths was developed that accounts for uncertainty in those distributions (see Figure 4). The uncertainty forecast employed a Bayesian update algorithm to combine forecasts with measured distributions and correct distributions of the empirical parameter κ′. Figure 4 graphically shows how the parameter κ′ controls the speed of the estimated pit-depth. In the forecast sequence, the κ′ posterior distribution is shifted to the right (i.e., the κ′ magnitude is increased) when the forecasts underestimate the pit penetration depths with respect to measurements, and the posterior distribution is shifted to the left when the forecast depths overestimate the measurements. In other words, the κ′ distribution is the mechanism to slow or accelerate forecasts based on comparisons to depth measurements. The κ′ distributions summarize information on the time propagation and spatial variability of the pits. A κ′ distribution can be interpreted to carry information of the previous forecast cycles. The algorithm is a method to combine and average measurements taken at different times, and extrapolate in time distributions of depths based on previous trends. As part of the project, longer-term forecasts were executed to demonstrate how the method would be applied to estimate the lifetime of components subject to pitting corrosion.

Figure 4.  200-day forecast sequence.
Figure 4.  200-day forecast sequence.
Figure 4.  200-day forecast sequence.
Figure 4.  200-day forecast sequence.
Figure 4. 200-day forecast sequence. The forecast is on plots on the left, gray curves. The red curve on the left is the measured cumulative distribution of pit depths. The plots on the right show the prior cumulative distribution function (CDF) for the rate constant parameter k′, the posterior CDF computed with the forecast outputs and the measured pit depth distribution, and the weighted average (dashed line curve) to be used as prior in the next forecast cycle.
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