ML19282D171

Axial Power Distributions from Fourier Fitting of Fixed Incore Detector Powers, Presented at ANS Winter Meeting in San Francisco,Ca on 751116-21

ML19282D171
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Site:	Palisades
Issue date:	11/21/1975
From:	Marks G, Terney W, Williamson E CONSUMERS ENERGY CO. (FORMERLY CONSUMERS POWER CO.)
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ML18043A689	List: ML18043A688 ML19282D171
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NUDOCS 7905230104
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a A INi'l: ACT l reliable melhud is neealedfor synthesi:ingfl ux drier ior readings into spatially dependent it. rial paarer sleipes trilh a limited nunder etifired on-core neutron detectors m an arial string. In this paper. the Fourirr expansion tech-vi<prefor obtaining arial p<arer diviributions is e.rantined.

\ tride raricly of representalire arial sleipes are sindied seith four, fire and six d+!ccior sydems. The results sh<ne all the systen;s perf >rm n ell. The use of Jiie detectors instead roffour increaxes the accuracy. trhile the use of sit detectors yire< lillie furtho r improrrment over Ibe fire-deterior syslern. \\~ilh Jire deleefors and fire Fourier modes. the standard deviation in the error in predicting the arial peak lo arcrage poorer ratio is aboni U.N'.k .

e

AXIAL POWER DISTRIBUTIONS FROM FOURIER FITTING OF FIXED IN-CORE DETECTC R POWERS INTilODl?CTION In this stmly, Ix>th the overlapping and continuous General schemes were investigated. Also, various arrangements With a limital number of Ilsed in-core neu t ron of four, live and six eletector strings were examined to detectors, a .cliable mellual is needal to synthnze the deterniine the liest arrangement of each. The study detector remlings into spatially depeinient inm er dis. was carrint out by testing the various systems on a tributions. Cornhustion Engineering's in-core detector large numlier of typical l'Wil axial ixmer shapes analysis system (INCA) is sm h a mellnul.* D .\ n generated from one- and three-dimensional dilfusion integral part of the melluul is the procedure used to theory calculations at different times in life for dilferent synthesize axial p<mer distributions from the reading conditions.

of a few detectors in an axial string. In this system, a metful based on expanding the axiali xmcr distribution llESUllI'S in terms of a few axially dependent fourier modes is Initial studies were done on a sample of 17 skeweil u sed .( 1 + p<m er shapes from a set of 170 typical and highly sLened besinning-of-life (U()l.' mid !!c-of-life OlOI.).

Formuhili"" mul ciul-of-life (13)la first cy cle one-dimen sional The l a-ie po red m e is to a-ome lhat the axial si.s pe Ty ph al emn.p'e, of he 4ain or gi$un m power distribution in an aswmbly may be reprewnted I ig.1. Tbc use of four and live equally spaced de-as the sum of the first N Fourier nuules:

n .- - - - , , . - - .

l'(z) ={

,, i a sin n r lle (1) n-l -

I nhere ~ i/

I l' is the p<mer per unit length,  !

z is the axial elevation in percent of the core heicht g (11), uron! 2{

5 J

j l

a,, are the unknow n combining coeflicients, and  ;

u -

.1 11 = (2) o, ,, 4--A A A--g-4 A A g f PACT! A G CW 4MT Note that 6 is the extrapolation distance, u hich usually Fig. b Representative shapes from one-dimensional onelyses is determined empirically.

The N combining n>cilicients are obtainni by match- tectors with lengths equal to 12r; of the core height ing the imer read by each of the N dc tectors to the was investigated, as well as using subsets of three integral of Eq (1) mer the axial extent of each of the detector readings for the fitting. The pseudo detector N detectors readings were obtained by integrating thr given shains

• n,g '

over the detector lengths. Then the fitting was done def de Ta sin n r N and compared to the given shapet The boundary o (3) i,..o ,,,n he . .

conditions (o or U) were chosen to yichi a mean error of

-_1, . . N, near zero in fitting axial peak-to-average poner ratios.

Table I gives the results for these 17 one-dimensional where d is the iwmcr read by the i"' detector and z/""

3 axial shapes with the best four and five detector ar-and z/'"""'" are the axial elex ations of the top and rangements. With four detectors, hecating the centers luittom of the i"' detectors, re3peelively- of the segments at 20, 10, 60, and 30% of the core This can be done for all the detectors in a string, or height Inl to the minimum uncertainty in the litted for sulisels. l'or instance, with four detectors, four axial peak, in the five detector system, the center <

imx!cs anild 1.c nxwl t i inatch all f<oir detectairs sienil- were locatal at 3 0, .10, ,30, 70. aiul 90% of t he nire-taneously. Alternalisely, the top three detectors coohl heicht. Two points are immediately apparent: One is he mali hed with three nauh s, and the bot tom three that using the madonnn possible number of modes is detecters uit h thrn nu.drs. Th" actual p.m cr dislii- hetter than using groups of subseis. For four delectors.

bution uonhl then be made of Iop and lottom segments using four nmdes is slighily het ter than using ino sets of from t!- tu o !L tiu ee nnedes snmimiy ior ine detecons ami liw nu,<ies.

1

.'s TABLEI ERROR ANALYSIS OF THE AXtAL PEAK TO AVERAGE POWER RATIO FOR 1.D AXtAL SHAPES Case Mean Error Standard Deviation Maximum Error 4 detectors centered at 20,40. 60,80% of core height 2 sets of 3 modes. 0.1% 3.5% 9.3%

1 set of 4 modes. 0.5% 3.1% 8.4%

5 detectors centered at 10,30,50,70,90% of core height 2 sets of 3 modes. O% 1.2% 2.7%

1 set of 5 modes. 0.2% 0.7% 1.5%

reat. fit

% error = X100 real l'urther, it is clear that a live detector sptem is an span 41 detectors.This subset of reenlar skewed, hichly-improvenn nt over ihe four detector system. The rrason peaked rmbled mul unrmhled distributions uas taken for this is twofold: from a gioup of blh shapes :tenerated durina fir t and (1) With th e detectors. the peaks near tl.e end of the hiter ey la t hrecalimen-ional ca!i ula tions. 4 me of um core are seen hot t"r: u h . a- w it h four d tceti.: y pu al sh, ors are show n in Fr - i. F H ii i r. ~ t w h .. .

th ~ areh.udl3 -cen. lend h of F of t h - o , ' , A Tb to -

(2) % it h th e detectors. In e modes can he o wd . the sarne as before nith t he m delcetois 1, ens. i enten d which gises a better chance of hasing a com- at 10. 26 12. 51 71. mal 9ng of the core height, ponent with a peak in th" richi lo"ation. h"i" 'I" houndars cornlitions w ere s"I"eted to cive a

,I,lu. s is d. lustrated in f_ir. 2 mul l. I more 2 3 hows mean crior in the asial prak-to-as erage power ratio of ainut /cro.

the worst curse that occurred durinc a transent with four deteriors, mni Fig. 3, the same worst curve nitb 14 h,ve n_etxtors. , , , , r ,

q l

In siew of this success. a representatise smnple of ze -

4 25 axial power sliapes from three-dirnensional calcula-tions were analped with four, five, mal sis e<pmily '

r

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08 1

09 1:

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Fig. 4: Ty," col shapes frorn three-dimensional analyses c4 4 m,, , , , , N The results of the analpis are shown in Table 11.

0

, 4 g,-p,-pyg p.

rasaw,a cm mcer

%-d A gain the improvement in goine frorn fou r to fin dein lors is apparent, as well as the liinited estra caiti Fig. 2: Transient shape, four detectors in goin;: to six detectors. Tin largest error occurs in 'a hos which has scry low power. since it is abnost f ully 7' ' '

rochhx!, but which has distorted power distributions in 20 o the hottom 100 of the core below the rmL This is illustrated in Fig. 5 for the various ca<es. Such a box

" 16-g _

would not he a limiting case.

12- o o oxie nt These results indicate that a live detector syste m is E

o heller than a four detector systern, and that a six 8'-

detertwr splem does not give significant further gains.

,,, q, Tin se results are borne out w hen the entire set of W

"mg Ul6 one- and t hree-dinn nsional shapes n ere considered.

o , rppgg q, 4 7 gQ With the de cetois, n sin..le s alue of 11 was owd to mcmwis trar o r obtain the result s civen in Table 111. For the fou r fig. 3: Tromient shape, five detectors detector splem. t he best s alues of 11 w ei" used for 2

TABLE 11 ERROR ANALYSISt OF THE AXIAL PEAK TO AVERAGE POWER RATIO FOR 3.D AXIAL SHAPES Case Mean Error, % Standard Deviation, % Maximum Error, %

4 detectors. . - 0.2 2.9 + 12.5 5 detectnrs. .. .. . . .. - 0.1 1.4 - 6.0 6 detectors. .. . - 0.1 1.2 -3.7 real-fit t % error = X 100 real TABLElli ERRORt ANALYSIS OF THE AXIAL PEAK TO AVERAGE POWER RATES FOR ALL SHAPES Case Mean Error, % Standard Deviation, % Maximum Error, %

4 detectors. 0.1 1.4 +12 5 detectors. + 0.1 0.8 -6 reaf fit v 100 t % error -

real no si/eahle local deprevia ns ilne to In. onel gride, etc.

If such grid clTects are present. the result s w ould

- deteriorate sognenhat. Starplard devi:itions of Ilie error N, in the tw ak-to-average g>oner ratio coilld increase hv

!y aE - j some 0.5 to IS'r .

g- /s

/'

~

. _ %Cs' CONCLUSIONS

_ _ i u cn.s

._3n, The concept of synthesizing asial pow er distributions l frorn a liinited nurnber of detector readings wilh Fourier a u u_.x ___w 1 2 w -- - l C \ P""55"" "" *d"5 iS " V I"hI" """C"P I

• A IIV" d"I""I"'

' i

,Mr;w$'w c jd r system leads to espected standard deviations in the Fig. S: Cornparison of four and five detector synthesis accu 7a,. og glu. peg go 99 74, pone 7 79g;n o( ag,,ut for a bottom peaked destribution 0.0%.... In addition, a unitpar fitt.ing pararneter in the each set of curves, i.e., a dill.erent value for each t.irne form of an extrapolation distance can 1,e determined w hich .is valid f.or all t.unes m life. ,I'he h,ve detector m life. W.ith f.ive detectors, t here is lit tle variat. ion of thc houndary eondition nith life. The cspei ted standard m trin. thus, represents an a6ance om the four e n or system, wMe a sh detector systeni does not deviation in the error in fitting the asial peak to II"E I"I"T sgnd.u ant gains.

average pow er is about 0.0'l.

These results were all obtained with the smooth gumer distributions ty pical of C-E reactors, which have * ",',di," l,0l"f,'.iJg,;.,l,"i"'L,',;,..'l"$1,"l7,.d*" ~" "" *"* *"*

3

ItEFEllENCFS

1. I1. l.. II :u.i:3.s, T. G. Oiu:n.11. D. Oiu:n. ".1 3 felhod of Analy:ing In-core I)ctector I)ala on l'ouer lle-actors," Trans .\ NS,12, !t20 (1969).

2. T. G. Oiu:n, P.11. G uix, "Use of in-core InsIrn-mentation in Combustion Engineering I'orcer fle-nelors.' Trans .\ NS. 19. 218 (19 1); for complete paper: d 'omlin-t ion Encincerin: Pnlilication TIS-1271.

1. T. (;. t hu.n. \\ . I: Tum v. G. II. \l e s 'I W l 31rthwi of .t wily:io,1 In-c.,re lieterior />ahr ni l'on er lleuctors." CEN PI).115. Comlnistion Engineering.

Inc. (1973).

1. W. II. Ti:nsi.v. T. G. ( >ni:n. E. .\. Wu u otsox, J n.,

"Three 1)inavnsional Co!culatica:al l'erification of C-E's in-core Instrumentation Sy>lem stilh 1.ifetime."

Trans .\NS. 21, (1975): for complete paper: Com-liustion Engineering Pul>lication TIS-1719.

5. \l. \1. I.r:visi:, f ). J. Ihotoso, "/!cuelor I'mver I)istribution from Analysis of in-core 1)elecir,r llead-ings." Nucl. Sci. Eng., 17, 115 (1972).

6. .\. Joswas, " Fourier E.rpansion as an Aid in the solution of the I)i.I.Tusion Equation for Three 1)imen-sional l'ou er I)istributions in 11 aler lleuclors."

.\nnals of Nuclear Energy, 2. 17 (1975).

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