Transactions of the 17th International Conference on Structural Mechanics in Reactor Technology (SMiRT 17) Prague, Czech Republic, August 17 –22, 2003
Paper # D04-3
Ageing Evaluation Model of Nuclear Reactors Structural Elements
Antanas Ziliukas, Audrius Jutas, Vitalis Leisis
Kaunas University of Technology, Lithuania
ABSTRACT
In this article the estimation of non-failure probability by random faults on the structural elements of nuclear reactors is presented. Ageing is certainly a significant factor in determining the limits of nuclear plant lifetime or life extensions. Usually the non-failure probability rates failure intensity, which is characteristic for structural elements ageing in nuclear reactors. In practice the reliability is increased incorrectly because not all failures are fixed and cumulated. Therefore, the methodology with using the fine parameter of the failures flow is described. The comparison of non-failure probability and failures flow is carried out. The calculation of these parameters in the practical example is shown too.
KEY WORDS: ageing, degradation, material, damage, non-failure probability, failures flow, reliability, embrittlement, creep; fatigue; corrosion, cracking, nuclear, power plant, service.
INTRODUCTION
Ageing is defined as the continuous time dependent degradation on materials due to normal service conditions, which include normal operation and transient conditions, postulated accident and past accident conditions are excluded [1], [2] [3]. The austenitic stainless steels for reactor pressure vessel internals and reactor pressure vessel cladding is exploited usually. An intensity of ageing process is going on at low speed in these steels. Prevention of any negative impact on safety and reliability is the primary target of plant management.
The main concern ageing is: a) changes of physical properties; b) irradiation embrittlement; c) thermal embrittlement; d) creep; e) fatigue; f) corrosion, including corrosion erosion and corrosion assisted cracking; g) wear (e.g. fretting) and wear assisted cracking (e.g. fretting fatigue). From here the ageing represents the cumulative changes over time that may occur within a component or structure owing to one or more of these factors and is clear that ageing is a complex process that begins as soon a component or structure is produced and continues throughout its service life. Ageing is certainly a significant factor in determining the limits of nuclear plant lifetime or life extensions. All materials in a nuclear power plant can undergo ageing and can to incur losses, partially or totally. The increase of failure intensity is the ageing indication of structural elements [4], [5]. Usually the non-failure probability during the evaluation of the faults that occur in the constructions is expressed:
1) by the reliability function which is calculated only for the increasing failure intensity [6];
2) by the reliability function in the case when high failure intensity is at the initial stage and failure flow increases at the end of operating period, i.e. the average increase of failure intensity is observed.
Thus, we meet non-exponential case of average increase of failure intensity, which is characteristic for structural elements ageing in nuclear reactors, also. The intensity of ageing depends strongly on both the service conditions and the sensitivity of materials to those conditions.
AGEING EVALUATION MODEL FOR STRUCTURAL ELEMENTS
In the case when structural elements that are damaged because of their ageing, the parameter of failure flow ω
( )
t is taken into consideration but not the parameter of failure intensity λ( )
t . The parameter of failure flow equals to the average number of faults per time unit [4], [7]( ) ( )
( )
,t T ,..., 2 , 1 i ; t N
t n t
i
t i r
i ∆ = ∆
∆ = ω = ∆ ω =
ω (1)
where ni
( )
∆t is the number of elements damaged at the time interval of[
i∆t,( )
i+1∆t]
; N is the number of elements at the beginning of operating period; T is resource, r i is number of intervals.It is important to define the parameters λ
( )
t and ϖ( )
t before making the organizational decision to increase the reliability of constructions.( )
( )
, t k t n t i N 1 j ij i i∑
= ∆ = ω = ω (2)where ni
( )
∆t is the number of element faults occurring during the operating period of[
i∆t,( )
i+1∆t]
; N is the number ofioperating constructions during the defined period; t is the output of j element during the period ij
[
i∆t,( )
i+1∆t]
; k is thenumber of elements in the construction.
To simplify the calculations of failure flow parameter ω
( )
t the obtained Ni and t values are grouped into initial andijfinal ones on the basis of which the average value is obtained later. Hence the average error is
, 1 n t t t j P j I j j − − = δ (3)
where I j
t and P j
t are output values given at the first and the last column of j number; n is the number of columns inj
which j number is indicated.
The initial t and final 0,j t output values after selecting the average value of error can be calculated as:g,j
(
)
2(
n 1)
t t t ; 1 n 2 t t t t j 1 j p j p j j , g j 1 j p j 1 j j , 0 − − + ≤ − − − ≥ (4)
The algorithm of calculation is:
( )
( )
[
]
( )
[
]
[
( )
]
( )
[
]
[
( )
]
( )
∆ ≤ ∆ + ≤ = ∆ + ∆ ≥ ∆ + ∆ ∈ − ∆ ≤ ∆ + ∆ ∈ ∆ + ∆ ∈ − ≤ ∆ + ∆ ∈ ∆ ≤ ∆ − ≤ ∆ + ≤ ∆ ≤ ∆ = t i t and t 1 i t when , 0 t ; t 1 i , t i t and t 1 i , t i t when , t t i t ; t 1 i , t i t and t 1 i , t i t when , t t t ; t 1 i , t i t and t i t when , t i t t ; t 1 i t and t i t when , t t j , g j , 0 ij j , g j , 0 j , 0 ij j , g j , 0 j , 0 j , g ij j , g j , 0 j , g ij j , g j , 0 ij (5)Seeking to obtain the reliable evaluation of ωj the length of interval t∆ must be chosen in respect to the number
( )
tni ∆ .
The number of intervals must not exceed the value of
t T
i r
max = ∆ , where T is interrepair recourse, which can reachr
even several thousands of hours.
The parameter λt can also be calculated on the basis of operating failure reports but only if the number ni
( )
∆tincorporates not all the reports about elements which failed in the period of
[
i∆t,( )
i+1∆t]
, but only those which were damaged for the first time. Still it is hard to define the moment of the first fault for the elements that were renewed after the failure and has operated further. Besides, not all the elements are exploited equally and are observed not at the same time. Hereafter given equation for the failure intensity calculation allows to avoid these problems [4]:( )
( )
, t t n t i N 1 j ij i i∑
= ∆ = λ = λ (6)where ni
( )
∆t is the number of first faults during the period[
i∆t,( )
i+1∆t]
; N is the number of objects in this period; i t is'ij( )
+ −
− =
∆t 1 i time the to up moment t
at happened failure
first the when , i∆ t
; ∆t time the to up object j in failures no were there when , t
t i∆ time the to up object th j in happened element
the of failure first the when , 0 t
j j
ij '
ij (7)
In order to ascertain that the failure of the element is the first, the number of the first failures during this period must be defined, '
ij
t must be measured precisely and must be observed from the beginning of the operating period. Consequently the length of t∆ interval calculated on the basis of Eq. (6) must be significant and the function λ
( )
t =λi will be stepped.Shortcomings of the presented methodology: 1) repeated data of failure observation are not used;
2) in order to determine λ
( )
t values long-term data must be used; 3) hard to define the moment of λ( )
t increase;4) λi increase must be defined on the basis of stepped function.
In practice failure intensity can be calculated by Eq.(6) on the basis of failure flow parameter ω
( )
t .Let us suppose that every element after its failure is renewed and goes on operating as a new one. So it allows on the basis of ω
( )
t defining the output density distribution between the failures f( )
t .( ) ( )
= +∫
(
−) ( )
ωω t
0
dx x x t f t f
t (8)
Failure intensity λ
( )
t on the basis of classical expression is calculated:( ) ( )
( )
t , Ft f t =
λ (9)
where f
( )
t is the density of distribution; F( )
t is non-failure probability. Then it is possible to write:( )
( )
( )
∫
− =
λ t
0
dx x f 1
t f
t (10)
In order to calculate Eq. (8) big stepped integral equations must be solved. Practically it is impossible to obtain the precise solution.
So, the best way is to approximate real function λ
( )
t by a certain model (Fig. 1).λ (t)
t
λ
λ λ
max
0 p
p
t a b T
Fig. 1 Simplified model of failure intensity variations
In the interval 0<t<a the function λ
( )
t can be described according to A. Weibull when m=0.5, i.e.( )
1 0.5 ptt 2 1 t = − −
In the other intervals λ
( )
t is obtained, respectively:( )
(
)
> −
λ + λ
≤ ≤ λ
= λ
b t when , b t
; b t a when , t
1 0
0 (12)
With the application of this model the changes of failure flow corresponds to the density of output distribution between the failures:
( )
( )(
)
( )(
)
> − λ + λ
≤ ≤ λ
= ∫
= +
λ − λ + λ − λ
−
λ −
b t when
, e
b t
; b t a when , e
e t f
2 2 1 1 0 0
b t 2 t b 1
0 t 0 dt t
(13)
The function of failure flow ω
( )
t due to the theory of recovery is:( ) ( ) (
= + −τ) ( )
ωτ τωt f t
∫
t dt 0
(14)
Marking the functions f
( )
t and ω( )
t by Laplas distribution we will write them in the form of f( )
s and ω( )
s .Hence, the Eq. (8), when
∫
f(
t−τ) ( )
ωτdτ→f( ) ( )
sωs can be written as:( )
( )
( )
s f 1s f s
− =
ω (15)
From the Eq. (12) we obtain:
( )
( ) s 2 L s ,s s e
s s
f
1 0 1 0
b s 0
0 0
λ + λ λ
π − + λ × +
+ λ
λ
= −λ+ (16)
where ∫
π =
∞ −
a 2
2 x 2/2 e xdx a
e 2 1
L .
Then from the Eq. (15) we will get:
( )
(
)
( ) ( )
λ + λ λ
π + λ − + λ −
−
−
+ λ −
= ω
1 0 1 0
0 sb1 s 2 L s 1 0
e 1
s s 1
s (17)
It is impossible to obtain all values of ω
( )
t from the last equation, only the marginal ones. i.e. ω( )
t →sω( )
s : t→0,∞ →
s ; ω
( )
t →sω( )
s : t→∞, s→0. Thus from the Eq. (17) we get:( )
t =λ0ω ; (18)
( )
λ + λ λ
π λ − −
λ = ∞
ω −λ
1 0 1 0 b 0
s L 2 1 e
1 0 (19)
On the basis of the theorem of recovery theory
( )
t = 1∞ ,ω
where ∞
g
T is time duration up to one failure.
When the density distribution corresponds to the Eq. (13) and taking into account that the duration of observations is infinite, after integration we get:
( )
λ λ λ
π λ − +
× λ =
=∞ −λ
∞
∫
1 0 1 0 b 0
0
g L
2 1 e 1 1 t tf
T 0 (21)
The comparison of Eq. (19) and Eq. (21) shows that they are identical.
Thus Eq. (18) and Eq. (20) relates three unknown parameters λ0, b , λ1. The left side of these equations can be
obtained calculating ωi and deriving the average from the 2 or 3 first values of ω
( )
0 and 2 or 3 last values of ω( )
∞ , e.g. ifb is known other two parameters λ0 ,and λ1 can be defined, e.g. taking b it is easy to analyse possible cases ti∆ , when max
i ,..., 2 , 1
i= , because imax is not significant. In order to find λ1, only one non-linear Eq. (19) must be solved because 0
λ isdetermined only by Eq. (18).
After the solutions (λ0, λ1, b ) are obtained they can be grouped into the sets, one of which is λ1>0 and other is
( )
t =const.λ It is possible to answer the question if the constructional element is ageing taking into consideration two hypotheses:
0
H : λ1
( )
t >0, i.e. the best of the solutions (λ0, λ1, b )when λ1>0;1
H : λ
( )
t =const, i.e. the best of the solutions (λ0, λ1, b )when λ1≤0.Suitable hypothesis about failure correspondence to the operating period can be evaluated on the basis of χ2 criterion.
Quality statistics will be written as:
( )
, ff fˆ nmax
i 1
i i
2 i i
2
∑
=
− =
χ (22)
where fi is the theoretical density of non-failure probability, which is calculated according to Eq. (13), when t=i∆t; fˆ isi
experimental density of non-failure probability. Experimental value of fˆ is obtained from Eq. (2): i
∆ ϖ +
∆ ϖ − ∆ ϖ − ϖ =
∆ ϖ +
∆ ϖ − ∆ ϖ − ϖ =
∆ ϖ +
∆ ϖ − ϖ =
ω =
− − t 1
t f t f fˆ
... ... ... ... ...
; t 1
t f t f fˆ
; t 1
t f fˆ
; fˆ
0 1 i 1 i 0 i i i
0 1 1 0 2 2 2
0 0 1 1 1
0 0
(23)
The Eq. (23) are obtained from Eq. (8) after the transformation of integral into the sums
∑
= −ν ν
∆ ω +
=
ω i
0 y i i
i f f t (24)
The following is obtained if i=0,1,2,... values are taken instead of i:
∆ ω + ∆ ω + ∆ ω + = ω
∆ ω + ∆ ω + = ω
= ω
... ... ... ... ... ... ...
; t f t f t f f
; t f t f f
; f
2 0 1 0 0 2 2 2
1 0 0 1 1 1
0 0
The statistics can be calculated for all three parameters (λ0, λ1, b ). Those three for which Eq. (22) statistics is
minimal will be the best for the failure evaluation, e.g. those three for which λ1>0 will meet H hypothesis. 0
Example of Methodology Application
Hereafter given calculations to illustrate how the sharp increase of failure intensity can be noticed on the basis of failure intensity flow parameters. Let us assume that pipe of circulating contour before the fault was noticed was working for Tg=50000 h. Inter-maintenance resource Tr =30000 h. The failure intensity increases sharply after the operating
period of b=15000 h and increases twice at the end, i.e. λmax =2λ0 and λ0=1⋅10−5 1/h. Putting these values into
Eq. (18) and Eq. (19) we get:
( )
( )
1(
1 0.85(
1 0.38)
)
1.7 10 = − − − ≈
ω ∞ ω
If b=2400 h and λmax =4λ0, the following is obtained ω
( ) ( )
∞ ω0 =2.5. Thus the parameter of failure flow ωI step behind the failure intensityλI, i.e. the later ageing fault manifests, the less is the parameter of failure flowω.Therefore, the methodology of ageing to investigate non-failure probability will be writing:
1) The number of hours up o the fault T is defined and compared with the inter-maintenance resourceg T . r
2) Tg<Tr, the first fault is singled from all.
3) If Tg >
(
5...10)
Tr, failure intensity within the limits of this resource must be constant.4) If Tg <
(
5...10)
Tr, experimental values of failure flow ωI are calculated in accordance with the Eq. (1) for the cases... 2 , 1 , 0
= (imax-1).
5) Experimental values of non-failure function fˆ are calculated using Eq. (23) and on the basis of i ωI values for the cases
(
i 1)
...2 , 1 , 0
i= max − .
6) The assumption is made that b=ν∆t and λν
0 and λν1 parameters are found from Eq. (18) and Eq. (19), i.e. one set of
three parameters (λ0, λI, b ), which describe the failure intensity (the number of sets will be i) is obtained.
7) Inserting the obtained values of λν
0, λν1 and b into Eq. (13) the density of non-failure probability v fiν is calculated. The
number of such sets will be (v=0,1,2...,imax).
8) According to Eq. (22) the equivalent of every set of three parameters (λν
0, λν1, b ) will be defined for the existing dataν
obtained earlier (fˆ and i fiν).
9) That set λ0, λI, b of the solution the number of which obtained by Eq. (22) is minimal will be accepted as the main,
e.g. if λ1>0 the hypothesis of ageing existence after duration b will be accepted. The obtained conclusions are checked in accordance with the confidence degree p of x
2
χ distribution tables.
CONCLUSIONS
The ageing evaluation model of nuclear power structural elements with using parameters of the durability is presented. The finest characteristic – the parameter of failure flow ω
( )
t can be put into estimation of the failure. The phenomenon and dependence between the parameter of failure flow ω( )
t and the failure intensity λ( )
t of structural elements is described.REFERENCES
1. Chung M., Thermal Ageing of Decommissioned Reactor Cast Stainless Steel Components and Methodology for Life Prediction, ASME PVP, Vol.171, N.J., July 1989.
2. Getman A., “Primary circuit metal state with require to NPP life times extension”, Pressure Vessel Technics, Trans. ASME, February 1993, pp. 85-90.
3. Mallinson L., Ageing Studies and Lifetime Extension of Materials, AWE Plc., Alder-maston, UK., 2001. 4. Ziliukas A., Mechanics important on safety, Technologija, Kaunas, LT, 2001.
5. Vilram N. Shah, Phillip E. and MacDonald., Ageing and life extension of major light water reactor components. -Amsterdam - London - New York - Tokyo, 1993.
6. Ziliukas A., Diagnostic and reliability of aircraft, Technologija, Kaunas, LT, 1998.
