Physico–statistical approach taking defects into account and using binomial distribution - Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM) - page 105

of cells is determined by the formula [4.26].
Suppose the i i-th element of a structure is divided into i cells. Cells
are arranged in jj groups with L j i being the number of cells in each group.
The probability of nucleation of a defect in any cell P i jj (τ l ) at time τ l is
determined for each group.
To determine the number of nucleated defects in the jj -th group of cells,
it is assumed that the nucleation of defects in the group is described by a
binomial distribution, i.e. the cells are independent and P i jj (τ l ) is the same
for all cells of one group

i k i k i n k −

P kn C P ( / ) = [ ( )] [1- ( )] , t P t

[4.27]

jj n jj 1 jj 1

i
jj Pkn t is the probability of nucleation of defects in the jj -th
group of the i i-th element at time τ l ; n =

( / ) ( )

where

l

i
j L is the number of cells in the jj- th

k
C is the binomial coefficient, defined as

group,

n !

[4.28]

C

k
n

= .

( )

knk ! − !

Based on the properties of the binomial distribution [4.27], the
mathematical expectation of nucleated defects in the jj i-th group of the i -th
element at time τ is equal to

i i i

v LP ( t ×t )= ( ).

[4.29]

jj 1 jj jj 1

where v i jj (τ l ) is the mathematical expectation of the number of nucleated
defects in the jj i-th group of i i-th element at time τ l .
Then for the i i-th element the mathematical expectation of the number
of nucleated defects at time τ is defined as follows:

jj

∑

v

i

()= t v

i

( . t

[4.30]

n 1 jj 1

ii = 1

￿ ￿ ￿ ￿ ￿

where v i n (τ l ) is the expected number of nucleated defects in the i i-th element
at time τ l .
In this case, the mathematical expectation of the number of nucleated
defects at time interval Δτ l is

i i i

v ( Dt ) = v ( t × t ) v ( ).

[4.31]

n 1 n 1 ni − 1

Given that the nucleated defects are equally likely to be found in any
cell of a group of cells, it may be assumed that defects nucleate both far
away from each other and also in the neighbouring cells. Also, if a defect
formed in a cell away from other cells, the size of the nucleated defect
corresponds to the sensitivity of inspection of metal, that is, for example,
in accordance with R ef. 6 5

a = 0.54 Fl ; = 5 0.05. a

[4.32]

0.05 sens 0.05