Physico–statistical models based on the residual defectiveness of structural materials - Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM) - page 173

The constants A, n, α are determined by solving a system of three
equations for these constants:
The first equation is derived for a point with the coordinates ( a = 1 mm,
N det = 20) in Fig. 5.43:

20 = A · 1 - n {1 - exp [-α (1-0.6)]};

The second equation is obtained for a point with the coordinates ( a =
5mm, N det = 4) in Fig. 5.43:

4 = A · 5 - n {1-exp [-α(5-0.6)]};

The third equation is derived for a point with the coordinates ( a = 13mm,
N det = 0.66) in Fig. 5.43:

0.66 = A · 13 - n {1- exp [-α (13-0.6)]}.

For the third equation the value N det = 0.66 was obtained by averaging
the number of identified defects in the range from 11 to 13 mm, which
was equal to 2/3, where 2 is the number of identified defects, 3 is the
number of intervals.
Finally, the system of equations has the form:

20 = A · [1- exp (-0.4 α)]
4 = A · 5 - n [1- exp (-4.4 α)]
0.66 = A · 13 - n [1-exp (-12.4 α)].

The system of equations with respect to A, n , α produced the following
results:

A = 1000 mm, n = 2.56, α = 0.05 mm -1 .

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Substituting the constants A, n , α in the corresponding equations gives:
• equation for the initial defectiveness:

N in = 1000 a -2.56 (curve 2 in Fig. 5.44);

• equation for the probability of detecting a defect:

P dd = 1 - exp [-0.05 ( a- 0.6)];

• equation for residual defectiveness (curve 3 in Fig. 5.44):

N res (χ) = N in (χ) - N det (χ).

The following equations are solved: