PROBABILISTIC METHODS FOR '

- LMFBR APPLICATION

D. S. D u t t Hanford Engineering Development Laboratory

Richland, Washington ..

c \ K . H. Chen and 3. D . Stepnen General Electric Company - Advanced Reactor Systems Department

Sunnyvale, Cal i fornia A

Rockwell - Energy Systems Group Pal o A1 to, Cal i forni a

,% To Be Presented A t International 'Conference On Fast Breeder Reactor Fuel Performance

March 5-8, 1979 i n

Monterey, Ca1 i fornia

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PROBABILISTIC METHODS FOR LMFBR APPLICATION

D. S. D u t t Hanford Engineering Development Laboratory

Richland, Washington

K. H. Chen and J. D. Stephen . . .

.General .E lec t r ic . - Advanced Reactor Systems ~ e ~ a r t m e n t Sunnyvale, Cal i f orni a

91~77 . . Be D. O'Reilly 95>

Rockwe 11 Energy Systems Group . . Canoga Park, California

'. . . .

INTRODUCTION

Understanding and predicting the performance of an FBR fue l systen~ core includes dealing with uncertainties i n fabricat ion, operations, properties and modeling. The net e f fec t of the interact ions among these uncertainties can be determi ned through probabi 1 i s t i c analysis. The gain i n knowledge from quantifying the margin between what i s expected and what might occur improves the rat ionale fo r design specifications and licensing defense- Several elegant and easy-to-use methods have been developed using Monte Car10 and rnonents techniques fo r probab i 1 i st i c analysis of the sodium and cladding temperature, fue l temperature and fuel pin l ifetime i n fuel assemblies. Recent applications include a number of immediate FBR design concerns. General Electr ic Advanced Reactor Systems Division (GE-ARSD) , Hanf ord Engi neeri ng Developrnerlt Laboratory (HEDL) and Rockwell Energy Systems Group (R-ESG), sponsored by . t h e U.S. DOE-RRT, are working together t o continue development and ver i - f ica t ion of these methods. This paper i s not a s t a t i s t i c a l presentation, but rather a report on the accompl ishrnents~d s ta tus of e f f o r t s t o pro- vide analysis tools to quant i ta t ively account for the e f fec ts of uncer- t a i n t i e s i n our treatment of cr-i t icdl core design questions-

METHODS . . . .

The metkbds developed use accepted determinis t ic (mean value) per- . :. formance models as . t r ans fe r functions t o . evaluate the e f f ec t 0% input; parameter values on systcm response. These t r ans fe r ' functions are .then ' .

combined w i t h a "driver" which allows ,the e f fec t of .parameter var iat ion .

on system response to be evaluated. The two methods of evaluating the .

frequency d is t r ibut ion of the response are Monte Carlo simulation - and moments met hod.

. . Monte Carlo

The Monte Carlo approach simulates a random process by providing a s e t of randomly sampled parameters t o .the t ransfer function. The average 1 outcome of a large number of responses will then. represent t h e expected response of the physical process (Figure 1) t o the input parameters- . .. . . ,

A straightforward Monte Carlo simulation involving a l l i n p u t and response variables i n a core-wide analysis may not be prac t ica l when a very large number. of t r i a l s needs t o be car r ied . out. Biased sampling can be used t o reduce the number of t r i a l s , ye t maintain the required accu-

. . racy. Also preliminary analyses may be performed t o -identify the most c r i t i c a l variables and a search f o r " c r i t i c a l " regions will ident i fy the

' "most probable" areas through a conservative estimate, A de ta i led Monte . . Carlo analysis can then be carried out f o r the critica.'l regions of t h e

sens i t ive random quant i t ies ,

Moments Method . . . .

The moments method of probabi 1 i s t i c analysis involves the calculat ion of the moments (parameters of the probabili ty d is t r ibut ion function) of the response variable to specify i t s frequency dis t r ibut ion. For simple model transformations the moments may be e f f i c i en t ly calculated using numerical integration techniques, whyil e for more complex models a Taylor se r i e s expansion is needed. In many cases complex model transformations may be accurately approximated using only the l inear terms of a Taylor se r i e s expansion ( f i r s t order l inear approximation). Arbitrary i n p u t parameter d is t r ibut ions and smooth mode1 transformations a re allovied; however, the form of the output variable must be determined i n order to calculate probabi l i s t ic estimates (Figure 1) . This may be done by evaluating the higher moments of the response variable or by performing a Monte Car lo simulation. The probabi 1 i s t i c coding requirements can be qui te sophisticated but i t provides the advantage of a very shor t comput- ing time.

The development a c t i v i t i e s are surnmarimed i n Tab'te I * 'I'hey are directed toward obtaining s t a t i s t i c a l techniques to provide performance analyses f o r LMFBR fuels . The progression to des-ign a c t i v i t i e s awaits ver i f ica t ion .-and validation of probabi 1 i s t i c codes by comparison t o fuel performance data (current ly from EBR-I1 and from FFTF in ear ly 1980)-

TABLE I. PROBABILISTIC MODELING CODES AVAILABLE

PURPOSE CODES AVAILABLE

Thermal Hydraulics . . PACT(^)

Fuel Pin .Power P E F T ( ~ ) Capabi 1 i t y ' ' PROBMELT(~)

. . NSIEX

Fuel P i n Life P E C S ( ~ )

General Methods E R P R O P ( ~ { PO STAT ( 6

DEVELOPING SITE

G E

METHODS

Monte Carlo

Monte Carlo, . .

Monte Carlo Monte Carlo

Monte Car l o , .

Moments Met hod . .

Moments Met hod

COMPARISON OF STANDARD DESIGN METHODS WITH PROBABII-I S'TXC METHODS

Generally, fuel pin design methods comprise a conservative analysis to show t h a t "worst case" conditions have been met by a pin design. By comparison, a probabi l i s t ic design will quant i ta t ively evaluate the im- pact of fabricat ion parameter variations and model uncertai n t i e s t o determine the r e l i abi 1 i t y of the to ta l system performance. By replaci r ~ y a "worst case" analysis w i t h a probabi l is t ic one, the designer wil l be able to:

(1) Quantify the r e l i a b i l i t y associated with a spec i f ic design and assess the r i sk involved in allowing variat ion of design parameters from the i r nominal design value.

(2) Properly account fo r the interdependence of -input parameters on . the to t a l system performance.

. . ( 3 ) Relate design variables t o fabrication specif icat ions. . .

The following are examples of the above points taken from the appl i - cation of s t a t i s t i c a l methods t o specify technical operating l imi t s or evaluate production f u e l o

Quantifying Risk . . . .

.. . -. . The. design crl 'terion -which determines . the. fuel. pin ' power 1 imi.l;..,i.s, . . ., . . : . .. . .

"there should be a low probabili ty of fuel melting a t . 15% overpower . .

condition".(7) T h i s c r i te r ion i s not precise, andt 'hus i t could n o t be . .

,

shown that i t could be met. In f ac t the P-19 power-to-melt data( ' /) used in conjunction with the specified gap width (Figure 2 ) showed 'that i t could not be met.

However, recognize that there are very few pins, and t h u s very few pe l l e t s a t peak power. In addition, there i s not a great number of pel- l e t s a t the extreme of the gap specifica.tion. A s t a t i s t i c a l analysis was performed t o quantify the .potential of achieving both these events simul- taneously and re l a t e t o that to the tendency fo r fuel melting under these conditions, Figure 3.

. . . .

The application of p robab i l i s t i c analyses does not eliminate the low . . b u t f i n i t e probabili ty of fuel melting. However, i t does provide an opportunity t o . quantify the risk , and evaluate the 'consequence. The ' ' : rat ionale used to evaluate the conseqtlence (once t h e r isk i s quantified) was again placed o n . eng'ineering judgment and i t was determined tha t a s ingle pe l l e t a t incipi ent-fuel-melting could not s igni f icant ly a f f e c t ' .

the reactor safety or performance: . ,

. . . . Evaluatjon of Variable Interdependence

As FFTF vendor fuel was recieved, inspected and accepted i t became apparent t h a t some of the fabrication variables were not s t a t i s t i c a l l y . .

, independent:. Because of the fabrication process and specif icat ions i t was more convenient. f o r the manufacturer t o fabr ica te a pe l l e t t o some density; then, i f necessary to meet f i s s i l e specif icat ion, se lec t ive ly grind pe l le t s . ' This leads t o a pe l l e t density and fuel p e l l e t diameter interdependence, Figure 4. By performing simple 1i near regression t o find the dependence and. variation of the l o t average diameter and den- s i t y , a fur ther reduction i n the tendency for fuel melting was calculated fo r the "as-fabricated" fuel i n FTR, Figure 5.

Assistance i n Providing Specifications . .

Often a parameter which affects the fuel pin or fuel performance . .

cannot be measured d i rec t ly , and t h u s cannot be specified d i rec t ly . An example i s the- desire to keep fuel pin cladding f r ee from fabricat ion defects, one of which i s impressed par t ic les (cladding inclusions). The . ..

concern here is the depth of penetration' which cannot be observed nonde- s t ruct ively. However, the width-to-depth r a t i o can be obtained from a sample of observed par t ic les , Figure 6 .

. . . . . ..

. . . . D u r i n g f h e design phase, an allowance f o r cladding defect depth wil l . . . . be madeo This allowance, the -above width-to-depth relationship, and - some . .,

. ' indication of the size-frequency distri 'bution can be combined t o 'obtain a r e a l i s t i c inspection c r i t e r i a . As an example of the possible inspection c r i t e r i a t h a t can be used, Fi'gure 7 was obtained from typical FBR product i on f ue 1. . .

Hot Assembly Sodium Temperatures

a lo analysis of a typical LMFBR thermal-hydraulic design' A 'Onte F9Y i n comparison w i t h the current semi - s t a t i s t i ca l hot was performed

channel fac tor ( SSHCF) analysis. The more basic and comprehensive .nature of the Monte Carlo approach not only allows an estimate of the coolant temperature d is t r ibut ion , but a1 so an assessment of the SSHCF analysis. As shown i n Figure 8, the Monte Carlo analysis quant i f ies the design mar- gin known only qual i ta t ive ly in the SSHCF. For this example, the SSHCF temperature is shown to be a t the 3.50 confidence level.

CONCLUSION ' ' . . . .

The f i e l d of s t a t i s t i c s and probabil'ity, as applied t o physical ' . . . .

' - phenomena, is a natural extension of engineering i n .which'. r e a l i t y is modeled. Randomvariations and their,combined e f f e c t on system response ' ' . '

occur. T h i s paper summarized the a c t i v i t i e s a t th ree DOE sponsored . . '

1 aboratori es to es tab l i sh accepted methods to evaluate these .random '

variations, t h e i r impact on system performance and t o provide ass i s tance . i n establishing meaningful and quantified product specif icat ions and . .

operating 1 imi ts. Continued development and refinement of methods t o perform probabi l i s t ic analysis is required i n order f o r these techniques to become accepted design methods. . .

REFERENCES

1) K. H. Chen and D. G. Hoover, "PACT 2 - Probabi l i s t ic ~ n a l y s i s of Coolant. and Cladding Temperatures," GEFR-00417, GE-ARSD, Sunnyvale, Cal if orni a, October 1978.

2) B. T. Feerick, e t al., "PEFT 2 - Probabi l is t ic Evaluation of Fuel Temperatures," GE-FBRD, . Sunnyvale, California, GEAP-14076, August

. , 1975.

3) F. Schmi t t r o t h , "Calculational Kethods fo r Power Capabili ty Studies, " HEIIL-TME 75-141, Hanf ord Engi neering ' Developmerit Laboratory, Richland, ' WA, February 1976. . .

4) W. S. Lovejoy, i t al., "PECS-111: ~ r o b a b i l i s t i c ~ v a l o a t i o n of Clad- ding Lifetime i n LMFBR Fuel Pins, GEFR-00256, GE-FBRD, Sunnyvale, California, October 1978. . ' . .

. . . . . . . . . . . . -. .. . . . .

. . . .

;* 5) B. D. OIReilly, "A Mathematical Probability Method for Predicting Fuel Pin Rel iab i l i ty ," Atomics International, Canoga Park, C A , TI-707-240-023, June 1975.

6 ) B. D. OIReilly and C. A - Wok," A Least Squares Algorithm f o r Com- bining Est imatesof Performance Model Parameters Extracted from More . .

Than One Set of Experimen.tal Data," Rockwell Energy Systems Group, Canoga Park, CA, N707TJ240033, December 1978, .

. . . . . .

7) "FFTF Fuel Safety ~ n a l ~ s i s Report", HEDL-TI-75001 ( ~ v a i l abi 1 i t y : U. S. DOE' Technical Information . . Center), September-October 1975.

8) R. B . Baker, "Integral Heat Rate-to-Incipient ' Melting i n U02-Pu02 Fast Reactor Fuel;" HEDL-ME 77-23, Hanford Engineering . .

Development Laboratory, Rich1 and, WA, May 1978, . .

9) K. H. Chen, "Comparison o f semi-stat is t ical Hot Channel Factor and Monte Carlo Analyses of Fuel' Assembly Cool ant Temperature Uncertain- t i .es , " GEFR-00354, GE-FBRD, Sunny(-NORTH-) i f orni a, June 1978.

MONTE. CARL0 SIMULATION

PHYSICAL

WHERE - I X =-(X, , X2, . . .; X,) . .

, . - - - - X = ( X I X2,, ..., X,) VECTORS DESCRIBIIQG THE ,

STATISTICAL DISTRIBUTION .. a = (a 1, a 2 , ..., a ,I I b~ THE INPUT VARIABLES

-F igu re 1. Schematic Represen t a t i on of Fe thods Used i n Probabi'l i s t i c Mode'l i rig'.'

-C-

AS-BUILT FUEL-TO-CUDDIRG GAP, MILS

. I ALLOWEC rua I -CLADDIN.: CAP-^ I . ,.

40 I I I I I 1 I

0.05 0.10 0.15 0.20 I

0.25

>

FIGURE " ...- 20; Coinpar$ son ,Power-to-Tiel "L Correlation to Fuel Spec i - ' . . - fict;:ons, ' .

. . REACTOR POWER, MW

- . .

FIGURE -,3~: ~xpec ted Number o f p e l l e t s a t . . .

. . Iiicipient Fuel Me1 t i n g for Core 1 Power D i s t r i b u t i o n s .

. . . . . . . . . . . . , . . . .

. . . . . . ,

. . . . : . .

, . . . . . . . . . . . . : : . ' . . .

. .

FIGURE'.?,-/ FFTF Core 1 and 2 h e l Pellet Dns i ty and ~iarneter ~ o k r e l a t i o n . . . "

. .

. . .

. . . . . .

A ... .r 1

. . . . 0.001 - I 1 1 1 1 1 450 . 670 en APO . .

. m.. . . . . . . . . . . . . PEAClO.2 P A E P , r?W . . . . . .

..... - .. . . - . . . . . . - ..... .......... FIGURE 5.1 ~ x ~ p r t e d Number of Pellets a t ~ n c i p i e n t ~ u e l Melting Using

.

Independent and Correlated Pe'll e t Density .and Gap Var.iab1 es .:

FIGURE ,.6 1

FIGURE 7.1 cumulative Probabi l i ty of Occurrence Versus Depth o f Penetra-tion.

. . . ,

. . COOLANT TEWIPERBTURS eel . , . . . ....

~igure8- . . '~oolant Probability ~istr ibution for t h e Hot channel by MO& carlo Analysis.