Using probabilistic methods for solving the problem of ensuring leak tightness of heat exchanger tubes of nuclear power plant steam generators - Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM) - page 291

noted that in Ref. 132 the number of actual cracks was determined without
consideration of repeated inspections using only function P d.d .
The actual number of cracks is defined as the aggregate number of
cracks found in ISI and the number of cracks missed due to imperfect
ISI. The method of calculating the amount of actual cracks is based on
determining the probability of detecting defects:

Number of detected defects N

Pa ( )=

det

;

[9.9]

d.d

Actual number of defects N

act

N

[9.10]

N =

det

.
()

act

Pa

d.d.

The number of actual cracks is calculated using equation [9.10] 132 .
According to the authors of Ref. 132, this equation cannot be used for
periodic ISI inspection because it does not account for multiple inspections.
To count the number of actual cracks using the data from periodic ISI,
it is important to consider both the probability of detection of defects
and of non-detection of defects. The probability of detecting a crack in
n inspections is calculated by the probability of non-detection of cracks..
The probability of detecting a defect x times in n inspections is
calculated using the binomial function:

n

 n ! x nx − x nx −

P x () = pq pq = ,

[9.11]



d.d.

x

xn x !( − )



Here, p is the probability of detection in a single inspection, and q (= 1 - p )
is the probability of missing the defect.
The probability of detecting a crack at least once in n inspections is:

( )

n

Px

d.d. (

−=−− 1) 1 1 p .

[9.12]

￿ ￿ ￿ ￿ ￿

To apply equation [9.12], we must know the exact history of crack
growth and the number of inspections for each crack. However, these
parameters are unknown.
Since each crack may be subject to inspection several times, the actual
probability of detection is different from P d.d . Let the actual probability of
detection be the effective probability. The number of actual cracks is then
expressed as:

N

N = .

()

det

[9.13]

act

Pa

d.d.ef

To know the history of crack growth and the number of inspections for
each crack, we use the statistical modelling approach.