The statistical fuel model, FRP

Nuclear Engineering and Design 56 (1980) 25-34 © North-Holland Publishing Company

THE STATISTICAL FUEL MODEL, FRP * Ib MISFELDT

Ris~ National Laboratory, Roskilde, Denmark Received 17 March 1978

The reliability of nuclear fuel for LWR reactors is generally good, but there are still some failures caused by pellet-cladding interactions (stress corrosion). Only a very small fraction of the fuel pins in operation fail, and the nature of the failures indicates a statistical problem; therefore a statistical fuel model was developed. The statistical methods are either Monte Carlo simulation or a Taylor approximation. The statistical program utilizes a deterministic fuel performance code with a stress corrosion failure criterion, verified against experimental data. The failure probability for a power ramp was evaluated for different fuel designs and ramp conditions. It is shown that the failure probability can be quite large, even when a deterministic simulation (based on the average values for design and material data) is far from the failure criterion.

1. Introduction

time) than the existing Danish fuel performance code WAFER [1 ] because the statistical methods considered require a large number of simulations for each case. The model was developed and verified independent of the statistical model. Based on available out-of-reactor stress corrosion experiments, a correlation for time to failure (time to crack penetration) was postulated. Under the assumption that the corrosive environment in irradiated fuel is comparable to that in the iodine stress corrosion experiments, a cumulative damage index for stress corrosion is calculated by means of FFRS.

It is generallyrecognized that structural safety should be expressed in probabilistic rather than in deterministic terms. Failure should be interpreted with respect to some predefined limits, a failure criterion. For a fuel rod the failure criterion could be cracking of the cladding, bowing o f the rod, or just exceeding a design limit such as the maximum allowed clad strain. Therefore, depending on the failure criterion under consideration, the concept of failure probability is applicable to both the safety and the performance of structures. A statistical fuel model, FRP (fuel reliability predictor), was developed which utilizes the deterministic fuel performance model FFRS (fast fuel rod simulator) for the calculation of statistical information on parameters important to fuel performance. The statistical methods utilized are either Monte Carlo simulations or an approximation to the moments for the parameters considered based on a Taylor expansion. The deterministic fuel performance code FFRS has to be orders o f magnitude faster (in computer

2.1. Stationary models for the regions

* This paper is a revised or up-dated version of the paper actually presented at the IAEA Specialists' Meeting on Fuel Element Performance Computer Modelling.

The cladding is treated as an axisymmetric, hollow, thin cylinder with a pressure difference between the

2. The deterministic fuel rod simulator , FFRS Only a slice (disc) of the fuel is treated in the model, except in the case of fission gas release and internal pressure where an approximation to the whole rode is used. The slice is divided into three regions: cladding, gap and fuel. The fuel is subdivided by a bridging annulus into a rigid, totally cracked zone and a perfectly plastic zone.

25

26

I. Misfeldt /Statistical fuel model, FRP Given: power, outer cladding inner pressure,

\x(~//

swelling,

temp., outer and

densification,

bridge radius,

burn-up, fluence in cladding, ductivity, etc.

Fig. 1. Cracked pellet.

choose: gap conductivity, pressure,

outside and the inside, and a superimposed axisymmetrical contact pressure acting on the inside. Elastic, plastic and thermal strains are considered. The fuel thermal expansion is calculated from the assumption of a plastic core [2]. Strains and temperature distribution in the fuel are axisymmetrical in the model. The outer, rigid zone is assumed to be totally cracked (only compressive stresses), the thermal expansion is therefore calculated as that of a rigid bar. The material in the plastic zone is allowed to expand freely and is assumed to be stress-free, except for hydrostatic pressure. A rigid annulus, the bridge, forms the boundary between the rigid and the plastic fuel zones. The position of the bridge, together with the temperature distribution in the fuel, determines the thermal expansion of the cracked pellet. The creep deformation in the fuel changes the position of the bridge. This change in position depends on the creep strain at the bridging annulus, and the total crack opening angle, see fig. 1.

h ~=m

gap, and contact

g' and P' cp

calculate: temperature'distribution (depends on h~), stress distribution (depends on P~p)'v elastic, thermal and permanent new bridge radius

strains,

t

calculate:

gap and contact pressure, g and Pcp' from the calculated strai~s, assuming additional elastic strain to avoid negative gap

(g, Pcp) ~ (g', P c p )

< (g' PcP ) = (g '' P cp

temperature,

stress and strain distri-

butions found

Fig. 2. Solution of the stationary equations.

(1)

0 b = PcpRfs/Rb,

where Pep is the contact pressure, Rfs the fuel surface radius, and R b the bridge radius. The U02 creep, e, at the bridge is found from the bridge temperature and oe by the U02 creep equation. The area which the material from Rb to Rb + dR will occupy as a result of creep is Act = e dR 27rRb.

gas con-

(2)

creep of the bridge dR = e47rw/R b.

(4)

The connection between fuel and cladding is the gap. The gap conductance is modelled according to Ross and Stoute [3] with modifications taking account of the eccentricity.

The crack area between R b and Rb + dR may be approximated by

2.2. The stationary solution for the fuel rod

Acr k = 7r(dR) 2 w/27r.

The common stationary solution for the regions is found by simultaneous solution of the equations for the regions with some given boundary and initial

(3)

Equating these areas [eqs. (2) and (3)] yields the

L Misfeldt / Statistical fuel model, FRP

conditions for the fuel rod. The boundary and initial conditions are, for example, outer cladding temperature, heat load, pressure (outer and inner), cold geometry and material conditions. The solution is found iteratively as shown schematically in fig. 2.

27

is found for a fixed time step, and the creep deformation is therefore treated as a time-independent plastic deformation. The simulation of a realistic irradiation case requires consideration of time-dependent boundary conditions as well as changes in the materials with time. During a period with constant heat load, the most important changes result from creep and changes in the material conditions such as swelling, densification and fission gas release.

2.3. The time-dependent quasi-stationary model The stationary model as outlined in fig. 2 includes creep in fuel and cladding; but the solution

[choose the length of I Ithe next time step

based

on

last

temperature

bution,

caIcuiate:

fission

gas

release,

densification,

I

steady

swelling, conductivity,etc.

t

power

t

Ir p°wer eduction

ramp

power

calculate: temperature, stress and strain distributions, bridge radius = solution of the stationary equations

gas

distri-

assuming strains

eiastic

only,

calculate: temperature, stress and strain distributions, new bridge radius by cracking

assuming elastic strains only, calculate: temperature, stress a n d strain distributions

t

,t

Calculate: temperature, stress and strain distributions, bridge radius = solution of the stationary equations

F ~ . 3. The time~ependent quasistationary model FFRS.

L Misfeldt /Statistical fuel model. FRP

28

The solution is obtained b y an incremental theory; the temperature distribution from the last time-step is used in the evaluation of swelling and fission gas release during the time-step considered, b u t stress, strain and temperature distributions are found for the time-step shown in fig. 3. Power ramps are divided into "small" ramps. In each "small" ramp the bridge is moved a fraction o f the pellet radius towards the centre and then allowed to creep back as far as the creep rate and the time allow. Hence the bridge position is fixed b y a balance between ramp rate and creep rate at the bridging annulus. During a fall in heat load the bridge position is assumed unchanged in the model. The power level (in a new ramp) at which the thermal expansion again starts moving the bridge towards the centre, thus opening the cracks, is decreased with burn-up from the level before the fall in power to the actual power.

2.4. Verification o f FFRS The fuel model was verified in a number o f irradiation experiments, including the EPRI benchmark cases [4]. The results generally agreed well with the experimental results (for the EPRI benchmark cases, as good as any o f the compared codes). A few results are listed in table 1 together with the experimental values.

0.6!

~

--~---

~

~-

-

T

o Experimentel m i d - p e l l e t stretn 7 x FFRS ,/ /

0.5

i

o.~.

I

~o.1! i

0.2

i J

O.O

4

-0.1

&

~

0

100

-02

4 , 200

, 300 Heet

- ~ /.,00 Iood.

, 500

, 600

, 700

--

w/cm

Fig. 4. Measured and calculated tangential strain for the X-264 pin.

As an illustration of the modelling o f a p e l l e t - c l a d ding contact situation during a power ramp, the tangential strain (calculated and experimental) is shown in fig. 4 for the Canadian X-264 irradiation experiment [4]. In fig. 5 the heat load and the calculated hot gap for a commercial PWR rod (case D o f the EPRI benchmark cases [4]) is shown. The fuel was unstable (high and fast densification), which caused a large gap between fuel and cladding, high temperatures and fast swelling that then rapidly closed the gap.

Table 1 Comparison between experimental PIE data and the values calculated by FFRS Pin no.

M20-1B Pa29-4 M2-2C AG17-2 AG17-3 HCD X-260 X-264 ELP-9 PWR rod

EOL average strain

Max. temperature

Released fission gas

Exp. (%)

Calc. (%)

Exp. (°C)

Calc. (°C)

Exp. (%)

Calc. (%)

-0.35 0 -0.15 0.2-0.3 0.28 0.36 -0.59

-0.29 -0.25 -0.36 -0.16 -0.17 0.16 0.21 0.16 -0.71

1950 1900 1850 1232 a) 2015 2166 2200 1650

2100 2260 2070 1970 1960 1268 a) 1970 2140 2230 1860

40 47 37 3 (2.3) 23 12.7

35 61 49 21 22 22 19 5

a) At 4710 MWd/MTU and 400 W/cm.

J 800

Ref.

[5 ] [6 ] [6] [7] [7] [4] [4 ] [4] [4] [4]

I. Misfeldt /Statistical fuel model, FRP 0.12

the form

300

OlO

P,

~

"--

29

tvsc

250

(environment) X f2 (stress, material condition)

= fl

X f3 (temperature) 0.08 E E ~. 0.06

200 E

was derived. In the stress corrosion tests the dominating environmental factor is the iodine concentration which is perhaps influenced by the presence of catalyzing components (air, iron, etc.). The in-reactor fuel rod environment is almost unknown, but the occurrence of stress corrosion failures indicates a similar environment with respect to stress corrosion. As a first approximation the corrosivity is assumed to be proportional to the fission gas pressure in the plenum until a saturation pressure is reached, f2 and fa are assumed identical in-reactor and out-of-reactor but, of course, fluence hardening changes the material conditions. Fig. 6 shows the proposed correlation for I"2. The environment is assumed to be at the saturation limit 0rl = 1). The temperature dependence is

150

8

o

"~ oo~

I00

-Jr-

-r o,o !

0

i

0

,

,

i

i

I

i

4000 8000 Time at Power. Hours

1200

Fig. 5. Heat load and calculated hot gap for the PWR rod.

3. Failure criterion The failure mode which is today considered the most important for LWR fuel during normal operation and for minor accidents, is stress corrosion (S-C). From the available data (all out-of-reactor) a correlation for the time to failure for stress corrosion of 1.2

-

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(5)

f 3 ( T ) = 10 (360-T)/40 ,

Tin °C .

(6)

The experimental values from a recent investiga-

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1.0 _.~..

360 "C-•

Stress-relieved:

400 "C - V

360 C - , ;

400 £;-,

o9 ~

08 ""

•

0.7

".

"'"

~I-ee

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"'-

m N

approximahon to flcrN) "

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~t,=

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dev. band

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0 Z

O.3 0.2 0.5 0.05

•

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5

0.1

0.5

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10

so

1oo

1

5

10

Time to failure

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i

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soo

1ooo

1o0oo at 36o l:

50

100

1000 at 400"C

(hours)

Fig. 6. The dependence of time to failure for stress corrosion on the normalized stress.

30

L MisfeMt / Statistical fuel model, FRP

tion of stress corrosion [8] are marked on the figure; the different test temperatures are normalized by means off3, as indicated on the time-scale. Stress corrosion damage is assumed to be cumulative and the damage during a time-step of length A t i is

ASCDi =

At/. •

(7)

tFSC,i The accumulated damage at time tn is

tn

n

SCO(tn) = ~ ASCD i

or

i=1

:

dt

to tvsc(t)

,

(8)

where tFSC, i or tysc(t) is the time to failure under the conditions existing in time-step i, or at the time t. Very often the permanent tensile strain is used as a failure criterion for fast power ramps. The criterion can be based on the experimental data of Scott [9], where the mean value for uniform strain is 0.21%, and the standard deviation is 0.04%. The total elongation was, on average, 1.7% with a standard deviation of 0.35%. It should be noted that numerous metallographic examinations of fuel rods show that the dominating failure mode is not overstrain (O-S).

assuming uncorrelated variables Xi, the expressions for the mean value and the variance are n

mean (Yi) = Fi(X~ + i ~ 0 2 F i 0 ~ ) 2/=1 3X 2 var (X/),

var (Yi) = ~ ~ : o A / ]=1

var (X/)

(9)

(10)

where Fi(X) is the state variable Yi as calculated by FFRS, and Fi(X ) is the state variable calculated by FFRS, when all design and material parameters are at their mean values. The partial derivatives are evaluated numerically. If the variables Xi are correlated, the expression for the mean and the variance contains an additional term which includes the covariance between all Xi values. One major advantage of this approch is the separately calculated contributions to the variance. They give useful information on the relative importance of the variables Xi. However, since the variance for some of the parameters is large, the accuracy of the approximation can be quite poor.

4.2. Monte Carlo simulation 4. The statistical model, FRP Two methods, Monte Carlo simulation and Taylor approximation, are employed in FRP for the estimation of the distribution functions for the parameters describing the fuel rod state. Typically, only the distributions for some specified parameters are evaluated such as the end-of-life values (strain, released fission gas, damage index for stress corrosion, etc.).

Monto Carlo simulation consists of computing a number of values of F(X), each based on a new random sample from X. If the number of samples is large, the calculated values of F(Xi) form a good approximation to F(X'). Using this method the pdf (probability density function) for the state variables can be approximated arbitrarily well, regardless of the form of the pdfs for the Xi values, but no information is obtained regarding the sensitivity of F(X) to the individual Xi values.

4.1. Taylor approximation For specified irradiation conditions (power, outer cladding temperature, etc. as a function of time), the fuel state parameters Y are functions of the stochastic variables (design and material parameters) X. For a random sample of X, the values of Y are found by a deterministic calculation with FFRS. The moments about the mean can, for the state variable Yi, be approximated by a Taylor expansion. Retaining the terms up to the second order and

5. Applications From numerous ramp experiments it is obvious that the distribution of the experimental results is important. The profit from these experiments can be considerably increased if probabilistic considerations are introduced into the planning of the experiments, as well as into the evaluation of the experimental results.

L MisfeMt / Statistical fuel model, FRP Since the reliability is the logical basis for design comparisons, propabilistic methods should be included when comparing different designs. For a certain "reference power history" the reliability as well as the influence from design and material parameters can be calculated. These calculations give a quantitative measure for the design differences and a measure for the importance of the tolerances specified for the fuel. Very often quite serious restrictions are imposed on reactor operation by the fuel supplier (under the terms of the guarantee), especially in connection with control rod movements (BWR), refuelling, and power increase after longer periods of operation at reduced power. These limits are normally based on conservative extrapolations of ramp experiments together with deterministic fuel simulations. However, as the failures are clearly of a statistical nature, it is very difficult to decide on reasonable restrictions.

Table 2 Design data and reference power history for the fuel pins Specification

BWR 8× 8

BWR 7 X 7

Inner cladding diameter (mm) Cladding thickness (mm) Diametral gap (mm) Density (% of theoretical) Grain size (~tm) Densification Fill gas/pressure (atm) Heat load, 0-8000 h (W/cm) Heat load, 800016 000 h (W/cm) Heat load after the ramp (W/cm) Fast flux at max. power (n/cm 2) Burn-up at the ramp, % FIMA Burn-up at the ramp, MWd/tUO2

(10.8, 0.015) a)

(12.42, 0.017) a)

(0.864, 0.022) a)

(0.94, 0.024) a)

(0.228, 0.023) a) (94.4, 0.66). a)

(0.30, 0.03) a) (94.4, 0.66) a)

31

5.1. Failure probability for a fuel rod subject to a po wet in crease As an illustrative example, the failure probability for two standard BWR designs was evaluated. The design data are given in table 2. The reference power history corresponds to one cycle (8000 h) in the peak power position, one cycle at very low power (control rod inserted), and finally a power ramp to 25% higher than the former peak power. Irradiation is continued at this power until 4000 h after the ramp start. Power ramps of this size could be the consequence of the withdrawal of a control rod near the peak power position after the control rod had been inserted for a long period, or it could result from the moving of an element from the outer regions to the inner region o f the reactor. A few data for the reference power history are given in table 2. Fig. 7 shows the calculated failure probability as a function o f the ramp time (the time to reach 25% overpower from the reduced power). The probability for the average strain to exceed the uniform limit (021%) is also shown for the fastest ramp. It is meaningless to calculate the O - S (overstrain) failure probability for the slow ramps. The failure probability as a function of the overpower in the ramp is shown in fig. 8. The ramp time is one hour.

100 ~

,

,

,

~

~"--0-5,7 x 7 (25, 5.0) a) stable 1 He

(25, 5.0) a) stable 1 He

440

556

150

193

550

707

1.5 X 1014

1.5 X 1014

2.5

2.5

20 000

20 000

l

_/ ."=_3F

~ Burn-up:2.5%F'MA~'~

°I0~. 550W/cm, 8x8 n I Ramp from193(556)to

0.01

0.1 I 10 100 Ramp time. Hours

000

Fig. 7. Failureprobabilityfor the 7 X 7 and the 8 X 8 B W R

a) Mean, standard deviation

designs as a function of the ramp rate.

32

L Misfeldt / Statistical fuel model, FRP 100 .....

•

....~

--

,

max.stress,8x8 "

--~---~ %F

o - S.joC___o.__--o

30Le 30

J

g2s

i -= 10

S-C

7x7

e

i

E~

I

~

=~20

~

Ramp

time:l

I

c

c

-

~

0

tO

~

I

~

C

0

--

o-°-×

I

D"/

1

c

oo

0

o.

fuel

:

£

@

Hour

/ 0.~

~5

' 10

' 15

2tO

25

[-I

o

Overpower,%

Fig. 8. Failure probability for the 7 X 7 and the 8 X 8 BWR designs as a function of the overpower in the ramp.

T h e values for t h e stress c o r r o s i o n d a m a g e i n d e x a n d t h e p e r m a n e n t tensile strain as c a l c u l a t e d b y a d e t e r m i n i s t i c c a l c u l a t i o n w i t h F F R S are listed in t a b l e 3. It is seen t h a t a l m o s t n o i n f o r m a t i o n a b o u t t h e i n f l u e n c e o f t h e r a m p t i m e is o b t a i n e d f r o m t h e determ i n i s t i c calculations. Stress c o r r o s i o n d a m a g e is negligibly small for b o t h designs, regardless o f t h e r a m p rate, a n d if o v e r s t r a i n is used as t h e failure c r i t e r i o n , t h e 7 X 7 design will fail w i t h a very h i g h

Fig. 9. Contribution to the variance on the maximum stress for the 8 × 8 BWR design.

p r o b a b i l i t y a n d t h e 8 X 8 design will b e far f r o m t h e d a n g e r o u s region regardless o f t h e r a m p rate.

5.2. Influence o f the individual design and material parameters T h e c o n t r i b u t i o n t o t h e variance o f t h e m a x i m u m average t a n g e n t i a l stress a n d o f t h e p e r m a n e n t tensile strain d u r i n g t h e r a m p is s h o w n in figs. 9 a n d 10 for

'fable 3 Stress corrosion failure index and tensile strain as calculated by FFRS, when all material and design data take on their mean values Ramp time 00

0.01 0.1 1.0 10.0 100.0 1000.0 1.0 1.0 1.0 1.0 1.0

Overpower (%)

25 25 25 25 25 25 5 10 15 20 25

BWR 8 × 8

BWR 7 X 7

S-C Damage index

O-S ema x (%)

S-C Damage index

O-S ema x (%)

1.3 6.6 6.6 4.3 0 0 2.5 1.2 1.3 3.6 6.6

0.1 0.1 0.09 0.08 0.076 0.062 0.004 0.02 0.04 0.06 0.09

1 3 3.8 0.5 0 0 0 1.1 1.7 3.3 3.8

0.34 0.32 0.32 0.30 0.30 0.23 0.15 0.21 0.22 0.26 0.32

× × × x

10 - 3 10 - 3 10 - 3 10 - 3

x x x × x

10 - 4 10 - 3 10 - 3 10 - 3 10 - 3

x x × x

10 - 3 10 - 3 10 - 3 10 - 3

x × × ×

10 - 3 10 - 3 10 - 3 10 - 3

L Misfeldt /Statistical fuel model, FRP max s t r a i n , 8 x 8 fuel

%1 c25 ID

>o 0j20 ,.C

-6

U-

o

-6 E L-

5 20

~

5

Fig. 10. Contribution to the variance on the average tensile strain for the 8 × 8 BWR design.

the 8 × 8 design. The mean values and the standard deviations as calculated by the Taylor approximation are, for both stress and strain, in good agreement with the values calculated by the Monte Carlo approximation. The Taylor approximation to the stress corrosion damage index is very poor due to the very long tail of the pdf - actually it seems as if the moments do not exist for this pdf. For a fixed ramp rate the stress corrosion damage is correlated (not linearly) with the stress. As can be seen from figs. 9 and 10, the variance for stress and strain is dominated by contributions from different material parameters for stress and strain, respectively. This again leads to the conclusion that strain alone cannot be used, except under special conditions, to predict the failures caused by stress corrosion. Furthermore, it is observed that manufacturing tolerances have virtually no influence on stress or strain at this high burn-up. The large uncertainty associated with the plenum volume is primarily due to the uncertainty with respect to the free volume in the pin, and has very little to do with the manufacturing tolerances.

33

gap size, etc.) is associated with a six-standard-deviation band. A tolerance band for sample inspection (density, yield strength, etc.) is associated with a four-standard-deviation band. This follows Haugen [10]. The design tolerances are those of Rose et al. [11] and Stehle et al. [12]. The material equations are based on experimental data available in the literature. For example, the Zircaloy creep equation is that proposed by Gittus et al. [13]. All the uncertainty in the creep prediction is associated with a factor F defined as F = actual/calculated creep. The distribution of F was estimated from available in-reactor creep experiments (a total of 65 experimental values from different sources was included). F was found to be log-normally distributed with the mean value 1.2 and a standard deviation of 0.50. It should be noted that the distributions for the material data are based on literature data from various sources, and they include an uncertainty resulting from a lack of experimental information. Therefore the distributions are not typical of a single case. The reduction of the uncertainty would require experimental data for each fuel batch.

5.4. Calculation time FRP is written in FORTRAN and implemented on a Burroughs B 6700 installation. A deterministic calculation for the reference power history in the example takes approximately 10 s CPU time. An early version of the code was implemented on a CDC 6600 installation, and a factor of approximately 10 was observed between the time used on the B6700 and the time used on the CDC 6600. Around 1000 s (100 on the CDC 6600) are used for the calculation of the Taylor approximation when all possible design and material parameters are included (38 of the parameters can be considered as stochastic variables). In most of the Monto Carlo simulations 100 deterministic calculations were used, which normally gave satisfactory accuracy.

5.3. Material and design data The standard deviation for the design data is based on the specified tolerances. A tolerance band for 100% inspection (cladding diameter and thickness,

6. Conclusion The computer program FRP was developed for the calculation of the reliability of nuclear fuel. The

34

I. Misf~,Mt / Statistical fuel model, FRP

deterministic fuel performance model, F F R S , utilized in FRP, was verified through the simulation of a large number of irradiation experiments. A failure criterion based on out-of-reactor stress corrosion experiments is utilized in FRP for the failure prediction. It is demonstrated that stress corrosion failures cannot in general be correlated with the average (or the maximum) strain. For constant ramp rate, a correlation between stress corrosion failures and maximum stress seems possible. The large uncertainty on some o f the material properties, together with the exponential dependence on stress for the time to failure, results in a p d f for the stress corrosion damage index with a very long right tail. Therefore the deterministically calculated values for the stress corrosion damage index cannot be used to predict the probability o f failure, or to compare the reliability o f different designs. The calculated mean values for strain, stress, fission gas release, etc. are close to the values calculated by a deterministic simulation, therefore these values should be used for the verification o f deterministic models. For the reference power history considered, the material parameters dominate in the contribution to the variance for stress (and strain), therefore little improvement in reliability is obtained by decreasing the fabrication tolerances. Efforts should rather be concentrated on obtaining additional information on important material properties such as creep, swelling, fission gas release, and gap conductance, thereby decreasing the uncertainty on these parameters.

Finally, it should be noted that although the deterministic codes seem unsuitable for failure prediction, the statistical model is based on a deterministic code and the accuracy of the calculations depends upon the deterministic fuel simulations.

References [1] N. Kjaer-Pedersen, in: Trans. SMIRT Conf., San Francisco, California, 15-19 August 1977 (D1/3). [2] M.J.F. Notley, A.S. Bain and J.A.L. Robertson, AECL2143 (1964). [3] A.M. Ross and R.L. Stoute, AECL-1552 (1962). [4] M.G. Andrews, H.R. Freeburn and S.R. Pati, CENPD218 (Apr. 1976). [5] P. Knudsen, Paper presented at the ANS Winter Meeting, San Francisco, Nov. 1977. [6] H. Carlsen, Paper presented at the IAEA Specialists Meeting on Fuel Element Performance Computer Modelling, 13-17 March 1978, Blackpool, U.K. [7] P. Knudsen, C. Bagger and M. Fishier, Paper presented at the ANS Topical Meeting on Water Reactor Fuel Performance, 9-11 May 1977, St. Charles, Illinois. [8] C.C. Busby, R.P. Tucker and J.E. McCauley, J. Nucl. Mater. 55 (1975) 64-82. [9] D.B. Scott, WCAP-3269-41 (1965) 68 pp. [10] E.B. Haugen, Probabilistic Approaches to Design (John Wiley & Sons, New York, 1968). [11] R. Rose, K. Lunde and S. Aas, Nucl. Eng. Des. 33 (1975) 219-229. [12] H. Stehle, H. Assmann and F. Wunderlich, Nucl. Eng. Des. 33 (1975) 230-260. [13] J.H. Gittus, D.A. Howl and H. Hughes, Nucl. Appl. Technol. 9 (1970) 40-46.