ESTIMATION OF ENGINEERING FACTORS FOR FUEL BURNUP OF VVER

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1 ESIMAION OF ENGINEERING FACORS FOR FUEL BURNUP OF VVER A. Gagarisi, S. sygaov, L. Shishov Russia Research Cetre Kurchatov Istitute Russia Federatio, Moscow ABSRAC Util recetly, computer code certificates idicated o fuel burup calculatio errors. I the same time, these errors should be tae ito accout while checig the discharged fuel burup limit, as well as while checig the local liear heat rate ad local liear heat rate ramp limits depedig o fuel burup. his paper iteds to coect the fuel burup calculatio error with the calculatio error related to fuel power distributio for the cotemporary VVER fuel cycle. his paper also attempts to aalyze the burup calculatio error i the framewor of radom fuctios theory, o the basis of power desity correlatio fuctio. his paper also cotais examples of estimated errors for burup calculatios performed for characteristic VVER fuel loads. Fuel burup is icreasig steadily with the developmet of uclear power idustry ad the improvemet of its fuel cycles. I this coectio, the issue of egieerig margi factor for fuel burup rate (below referred to as B burup is becomig icreasigly vital. Burup egieerig margi factor should be used for checig the limitatio of burup rates of fuel elemets, pellets or assemblies agaist the admissible burup rate, as well as for checig the limitatio of liear power desity ad liear power desity jump as a fuctio of burup, if such limitatios were specified. It is ow that margi factors are itroduced because calculated values of project-relevat parameters differ from those emergig i the process of operatio for three reasos (groups of factors: - error of the methodology used met ; - deviatio of fuel compositios ad sizes from desig values durig fabricatio mech ; - deviatio of reactor operatio mode parameters (maily itegral power from desig values durig operatio. I Sice the burup is a time-itegral parameter of fuel, ad the factors listed above are radom ad have almost o correlatio, the burup egieerig margi factor could be preseted as:

2 ( ( ( eg ( 1 met B I B mech B K B = B B B, ( where met B (, I B, mech( B are relative errors coected with the three abovemetioed groups of factors ifluecig the burup. I the process of desig, umeric values B B B of these errors are determied by their admissible excess probability, which is usually assumed as 5% (i.e. the probability that the error corridor will ot be exceeded reaches 95%. Assumig the distributio of radom deviatios is close to ormal oe, the give probability would be met by coectig these errors with the followig respective roof-mea-suare (RMS deviatios: ( B = σ ( B. Let us successively cosider the ways of assessig each error compoet. Sice ualificatio certificates of BIPR ad PERMAK codes util recetly icluded o burup calculatio errors, it would be atural to attempt to coect these errors with the calculatio error related to power desity. o mae the case more specific, let us cosider fuel elemet burup. he followig relatio is true: 1 = b( rod + rod +... rod, (1 where B rod is the level of fuel elemet burup for operatig lifetime; b is ormalizig factor; rod K K is the average fuel elemet power for operatig lifetime; K is the average relative power of fuel assembly for operatig lifetime; K is the average relative power of fuel elemet iside the assembly for operatig lifetime. For cases whe each lifetime has a defied legth of, accordig to (1, ( ca be determied from the followig relatio: the error ( ( = b ( K ( K + ( K ( K,

3 from relative error of of fuel elemet burup ( ca be derived i the followig form: ( K B rod K K K ( K K ( ( ( + ( ( =. ( Similarly, relative errors for a pellet ad a fuel assembly have the forms of: ( K B tab KV K K V Btab ( KV K ( ( ( + ( ( =, (3 FA, (4 FA K ( B ( K = B K V w here is the average relative power of a fuel assembly fractio durig lifetime. hus, havig determied ( K, ( K V ad ( K, ad havig followed the history of ay fuel elemet durig lifetimes (by fixig the values of K, K V ad K oe ca assess the relative burup rate error level. I the same time, sice this procedure is very buly, oe ca use a simpler oe, assessig just the average value for all lifetimes: ( K K = + ( ( K K. he ( could be writte dow i the followig form

4 ( K ( K + ( K K rod ( B K K = ( K K. aig ito accout that if > 1, ( K K ( K K 1, we fid that: 1 rod K K rod ( B ( ( 1 +. (5 B K K 1 For = 1, we assume that 1 1. Similarly, Btab K V K Btab KV K ( ( ( (6 FA ( B B FA ( K 1 =. (7 K 1 o calculate errors accordig (5, 6 ad 7, it would be ecessary to determie average relative ( K ( K errors, ad V for several lifecycles. K K ( K K V Recet comparisos of calculated data with measured values show that coservative approaches towards processig such comparisos o the basis of a σ-priciple allow the followig uiform values to be assumed for absolute errors:

5 ( K 0,06, ( K V 0,1 ad ( K 0,05. Durig operatio, a fuel assembly chages its positio i the core, so iducig its average value As cocers assembly parts, whose K to vary betwee 0,8 ad 1,0. K chages i a maer K V, whether burup limitatios will be met is determied by those fuel coservatively that for the purposes of fidig ad KV 1, i.e. K exceeded 1 durig at least several lifecycles. hus, we assume V ( K = 0,067, K ( K K ad ( KV KV respectively, K 0,9 ( KV =0,1. For K, as well as for KV, we ca assume KV that burup limitatio will be determied by the areas, where K K is close to oe. Hece, to assess the methodical error of burup rates for fuel elemets ad pellets with accout tae of ueve coolat temperature distributio over the fuel assembly sectio, we ca assume that K ( K = 0,05. Coseuetly, K + K ( KV ( K V = 0,130; K K ( K ( + K = 0,083; ( K K = 0,067. Based o these data, methodical burup errors could be calculated usig (5, 6 ad 7. It is iterestig to compare the results of approximate burup error assessmets calculated usig (5, 6 ad 7 with more precise results obtaied usig (, 3 ad 4. his compariso was performed for fuel burup i typical VVER-440 ad VVER-1000 fuel cycles. ables 1 ad show parameters of fuel elemets with highest burup rates.

6 able 1. Parameters of the fuel elemet with the highest burup obtaied i a 5-year fuel cycle of VVER-440 Parameter Lifecycle No Burup, MW day/g U 15,7 30,0 4, 49, 53,8 K start/ed of lifecycle K start/ed of lifecycle 1,35/ 1,36 1,0/ 1,04 1,3/ 1, 1,04/ 1,01 1,1/ 1,09 1,0/ 1,01 0,40/ 0,48 1,50/ 1,41 0,5/ 0,30 1,57/ 1,48 able. Parameters of the fuel elemet with the highest burup obtaied i a 4-year fuel cycle of VVER-1000 Parameter Lifecycle No Burup, MW day/g U 14,4 8,0 40,1 49,7 K start/ed of lifecycle K start/ed of lifecycle 0,93/ 1,01 1,3/ 1,0 1,9/ 1,15 1,01/ 0,99 1,04/ 1,00 0,98/ 0,99 0,83/ 0,83 1,01/ 1,01 For VVER-440, the use of ( gives the followig burup error assessmet: met( = 0,04, while the use of (5 gives: met( = 0,041. For VVER-1000, the use of ( gives the followig burup error assessmet:

7 met( =0.037, while the use of (5 gives: met( =0,048. he deviatio of itegral thermal power from its omial value i case this deviatio is characteristic for the lifecycle as a whole ad, i the same time, occurs radomly from oe lifecycle to aother should be tae ito accout similarly to the methodical error: I ( B B 1 I( N, 1 N where I ( N N = % - is the error of itegral thermal power measurig durig operatio. Fially, it is proposed that the mechaical burup error caused by deviatio occurrig i the process of fuel fabricatio should be accouted for usig the method of variace: B B ( p = B i p i mech i σ ( p, i where σ ( p i is the variace of deviatio of p i factor ifluecig the deviatio of burup; B( pi is the sesitivity factor of burup for lifecycles to the variatio of p pi before the start of the first lifecycle. Sesitivity factors are estimated by imitatig burup processes with omial factor values, ad with successive deviatios of each importat factor. his produces a statistical sum of

8 impacts caused by deviatios of importat factors both i the very object of studies ad i its eighbors. hat is, if a fuel elemet is explored, accout is tae of the impact o its burup give by possible deviatios of for istace erichmet of this particular elemet, as well as of all other fuel elemets i the core. I their tur, ifluece factors are determied for characteristic fuel elemets, ad the highest of these factors are selected. Calculatios show that the most importat factors ifluecig the burup are: fuel erichmet ad desity; exteral diameter of fuel elemet; pitch betwee fuel elemets i the assembly; ad the distace betwee fuel assemblies. I coclusio, here are the characteristic errors for a 5-year fuel cycle (=5 of VVER-440: met( 0,04, I( 0,01, mech( 0,03. It should be oted that further precisio of the above relatios is possible. his precisio is based o the fact that the assumed suppositio of a costat methodical error of power distributio, (K =cost for example, does t mea that ( K = K. o obtai ew estimatios, let us use the relatios from the radom fuctios theory. Let s assume that the burup of a core sectio (assembly, elemet or pellet durig a sigle lifecycle is: B( = ( t dt, ( 8 0 where (t is power distributio i this sectio ad is the duratio of lifecycle; moreover, is a fixed value, while (t is a radom fuctio. 1 Itroducig the average lifecycle power desity as = ( t dt, we get B ( =. 0 he the variace of burup i the selected sectio is defied as: σ ( B B σ ( =, (9 which correspods to ( 4 above. Let us ow cosider statistical parameters of time-depedet (t fuctio i the framewor of the radom fuctios theory [1]. For the purpose of solvig the tass of burup variace

9 assessmet, (t fuctio ca be cosidered as statioary. From the statistical viewpoit, that meas the (t distributio stays costat for all t ad complies with the suppositio of costat methodical error of power distributio, such as (K =cost. he correlatio fuctio is a importat parameter describig radom processes. his fuctio is defied as: {[ X ( t X ( t ] [ X ( t X ( ]} K( t1, t = M 1 1 t, ad determies the rate of liear depedece betwee the values the radom fuctio ca tae i two differet momets of time. he defiitio (9 implies that K ( t1, t1 = D( t1 - variace for the momet of t1. For statioary radom fuctios, the correlatio fuctio has the followig importat feature: K( t = τ τ = t t, (10 1, t1 K( t1 t = K(, 1 i.e., the correlatio fuctio s value depeds oly o the iterval betwee two momets of time. For the eergy distributio case from (10 we obtai K(0 = D = σ (. (11 Accordig to the theorem prove i [1] for the case of a fuctio beig a itegral of aother radom fuctio with ow K ( t, t 1 1, the variace of power averaged by lifecycle ca be estimated as: t 1 σ ( = K( t s dsdt. (1 00 Idetifyig the exact form of the correlatio fuctio for (t reuires some special statistical aalysis. I the absece of the latter, however, let us try to idetify this form o the basis of geeral cosideratios. I all real physical processes the correlatio fuctio is decreasig, because the farther two momets are from each other, the weaer their coectio is. Figure 1 shows examples of various correlatio fuctios. he fuctio i Figure 1a correspods to the case of a absolutely radom fuctio (such as white oise, which has o correlatio betwee its differet sectios. he fuctio i Figure 1b correspods to the case of wea correlatio, whe the coectio betwee momets decreases promptly with growig time itervals. he liear fuctio i Figure 1c correspods to the case of strog correlatio.

10 Figure 1. Forms of correlatio fuctios I the case of (t, o the basis of geeral coservative cosideratio, the form of correlatio fuctios for (t characteristic for the case of strog correlatio betwee differet momets of time was cosidered. his case is described by the followig liear fuctio: K ( τ = σ ( (1 βτ. (13 It is obvious that the selectio of β value determies the result of variace calculatios accordig to (1. Let us suppose, for example, that the mutual ifluece of lifecycle momets is limited to the period betwee refueligs. he the factor β = 1 ca be determied ad, from (1: t 3 1 σ ( β σ ( = ( (1 t s dsdt + = = 00 σ σ β β (, 6 3 i.e. the burup error measured by the ed of lifecycle is 3 times less that the power desity error. A more coservative case is whe i β = 1 is uderstood as a complete period of fuel stay i the core 5 years, for istace. A reverse approach to fidig the correlatio fuctio could also be proposed. Let us assume a specific type of correlatio fuctio for example, a liear fuctio of (13 type, which is coservative. he, o the basis of some additioal data allowig simultaeous assessmet of σ( ad σ(b, oe could determie their ratio coected with the duratio of fuel stay i the core. Ad afterwards, o the basis of this coectio, it would be possible to restore the value of β from (13, which could be used for subseuet ew estimatios. Usig the burup error reassessmet procedure described above would allow a well-grouded reductio of coservatism i determiig the margi factor for burup.

11 REFERENCES 1. A.D. Vetzel, A Course of Radom Process heory. M.: Naua, Fizmatlit, 1996 (i Russia.. Lui G., Noviov A., Pavlov V., Pavlovichev A., Advaced Fuel Cycle of WWER Reactor. Proceedig of the Fourth Iteratioal Coferece, 9 Sept. 3 Oct. 003, Albea, Bulgaria 3. Гагаринский А.А., Брик А.Н., Лизоркин М.П, и др. Топливные циклы для АЭС с реакторами ВВЭР-440. Состояние и перспективы. Словацко-российскочешский семинар, 6-7 сентября, 006 г, Смоленице, Словакия

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