# 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

Jay Fisher

Gary Burkhardt

Inclusive Dates: 11/14/11 – Current

**Background** — The objective of this project is to develop an
adaptive-predictive probabilistic model to forecast localized-corrosion-induced
pit population and pit depth distributions. Nuclear power plant (NPP) operators
are required to periodically inspect components by visual and volumetric
examinations to maintain integrity and ensure safety. As nuclear power
plants age, however, more frequent inspections are expected to ensure component
integrity. A framework to define an inspection schedule based on risk
considerations is needed to keep the cost of inspection constrained without
compromising safety. SwRI is developing a model to forecast
localized-corrosion-induced damage of NPP 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 and an initial state. The model is expected to account for
previous inspection data, randomness of pit generation and 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-induced degradation occurs
before failure.

**Approach** — Model development consists of the following
tasks: statistical model, experiments, and data analysis and integration.
Probabilities of transition between discrete states that satisfy Kolmogorov's
forward equations for a pure birth process can be used to describe 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 µm and 100 µm; it is in state 2 if the depth is between 100 µm and 200
µm; and so on). Thus, pit growth is conceptualized as a pit that transitions
from a state to the next. Parameters to define the transition rate between
states can be obtained by measuring the average pit depth as a function of time.
Experiments are being conducted with coupons and tubes made of stainless steel
316. The coupons and tubes have been placed in a tub filled with synthetic sea
water. Sea water is used in some NPPs as coolant in the open-cycle cooling water system. The tubes will be inspected using a eddy current technique for pit
population and their depths. One-by-one, coupons will be removed from the tub at defined intervals. When a coupon is removed
from the tub, it will be inspected using laser profilometry. The eddy current
method will provide a coarse measurement of pit population and pit depth
distribution in tubes, representative of results that would be achieved during
in-service inspections of NPP components, whereas the laser profilometer will
provide more accurate measurements. The data collected from coupons will be used
to estimate pit depth versus time. Both coupon and tube data will be used to
estimate statistical model parameters, such as transition rates, separately for
coupons and tubes. The model will be used to forecast the next pit depth
distribution. At the next inspection, the model will be updated with the
collected data and the next forecast will be performed.

**Accomplishments ** — A numerical solution for the system
of Kolmogorov's forward equations for a pure birth process was implemented,
with non-homogeneous birth rate or transition rate to develop the model.
Several dependencies for the transition rate were investigated to find a
functional dependence that yields a growth rate for the average pith depth
that is consistent with a time dependence empirically observed. The common
growth law is of the form *d* = κ(*t* + *t**ο*)*ν*, where
*d* is a reference pith depth,
*t* is the time, *t**ο* is a reference time, and κ and ν are empirical constants.
In the numerical solution, the empirical parameter ν is provided as an input
to define state transition rates, and the parameter κ is used to map
computer time to a physical time. Given an initial distribution of pit
depths, the numerical solution can be used to forecast the distribution at a
later time, preserving the relationship *d* = κ(*t* + *t**ο*)*ν*, where *d* is
interpreted as the average pit depth. The experimental work has been initiated, and the data from the experiments will be fed in the model to test the adaptive-predictive forecasting approach. Specifically, later stages of the project will be devoted to track pit population and their depth with time using the model, i.e., forecast the pit population and depth distribution, inspect experimental data and compare with the model forecast, and if necessary, update the model to make next forecast. This will be repeated till a satisfactory match is found between the model forecast and experimental data.