[95/95] APPROACH FOR DESIGN LIMITS ANALYSIS IN VVER. Shishkov L., Tsyganov S. Russian Research Centre Kurchatov Institute Russian Federation, Moscow

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1 [95/95] APPROACH FOR DESIGN LIMITS ANALYSIS IN VVER Shishkov L., Tsyganov S. Russian Research Centre Kurchatov Institute Russian Federation, Moscow ABSTRACT The aer discusses a well-known condition [95%/95%], which is iortant for onitoring soe liits of core araeters in the course of designing the reactors (such as PWR or VVER). The condition ensures the ostulate there is at least a 95 % robability at a 95 % confidence level that soe araeter does not exceed the liit. Such conditions are stated, for instance, in US standards and IAEA nors as recoendations for DNBR and fuel teerature. A question ay arise: why can such aroach for the liits be only alied to these araeters, while not norally alied to any other araeters? What is the way to ensure the liits in design ractice? Using the general stateents of atheatical statistics the authors interret the [95/95] aroach as alied to VVER design liits. Aroach 95/95 is stated in soe western reactor regulatory docuents, and aied at avoiding the dearture to nucleate boiling and fuel eltdown. For instance, NUREG says: At least a 95% robability at a 95% confidence level, that the eak centreline teerature of the fuel rods will not exceed the UO elting teerature. In the theral-hydraulic analysis to calculate the DNBR, uncertainties in the values of rocess araeters should be treated with at least a 95% robability at a 95% confidence level. Long ago the authors were concerned by the roble: why was figure 95% chosen? And why were only the DNBR and fuel teerature were chosen? Most likely, the condition (hereinafter, referred to as condition 95/95) aeared due to alication of interval estiates in atheatical statistics. Such estiates allow evaluation of the araeters that characterise a rando quantity y roceeding fro evaluation of liited, including sall, sale of the values out of nuber of this agnitude values: y 1, y,... y. Thus, if the deviation distribution is close to noral, average y and disersion σ ( y) ay be evaluated using the average value and disersion value obtained for the sales of : y and σ ( y ) in nuber. In this case [1, ], the accuracy of estiates deends on the diensions of

2 sale and rated reliability (confidence level), which could be easily denoted as (1 - ). The estiates can be exressed as σ( y) σ( y) y t(1 / ) y y + t(1 / ), (1) ( 1) σ ( σ ( y) y ), () χ where: t(1 / ) and χ are the quantiles of Student and Pirson distributions stated in statistics reference books. At the sae tie, should the distribution of rando quantity significantly differ fro the noral law, the atheatical statistics ay offer alternative estiates, for exale, the estiate based on the following Chebyshev inequation σ( y) σ( y) y y y +. (3) Further along, the exales of interval estiates of robability are given with the assution that the deviations are distributed in accordance with the noral law. Let us assue that effectiveness of reactor eergency rotection was easured. There were three easureents conducted, and the average effectiveness was deterined as ρ3 = 10β ef. The disersion was deterined on the basis of three easureents and akes σ 3 ( ρ) = 0.5β ef. If assue (1 - )=0.95 as reliability index of obtained data, quantile will be t1 / = t0.975 = 4.3, and, in accordance with (1), we obtain the following estiate 0,5β 0,5β 10β 4.3 ρ 10β + 4.3, which is true for ρ - easured effectiveness of 3 3 eergency rotection with a 95% confidence, that is, 8.76β ρ 11.4β. With (1 - )=0.99 the estiate will be 7.14β ρ 1.86β. As another exale of interval estiate of robability, let s discuss the case of siulated reactor oeration. Let it be the axiu teerature of fuel in design basis accident to be deterined. In this case it is required to account a ethodological error, and any robable technological or oeration deviations. Fuel teerature shall be deterined with reliability of (1 - )=0.95 and error within 30 С.

3 In order to solve the roble it is ossible to use the ethods of statistical testing. Earlier, the ethods were discussed in [3-5]. Let s select all the araeters of calculation odel (let it be the araeter of code ackage), whose values are known with a certain error, and assign a certain distribution for these araeters. Let s deterine the distribution araeters for all the inut data, which were also deterined with soe error. Let s siulate the accident any ties ( ties) using the selected odel, every tie randoly siulating the odel araeters and inut data in accordance with their distribution. Each calculation deterines a axiu fuel teerature, then, the average fuel teerature T for this teerature selection, and standard deviation σ ( T ). Using (1), the argins of ossible teerature values shall be deterined σ( T) σ( T) T t(1 / ) T T + t(1 / ). (4) Let it be 30 calculations erfored (=30), which deterined T =500 С, σ ( T ) =80 С, σ t1 / =.04 and ( T ) t (1 / ) =8 С. Thus, the axiu teerature of fuel ranges between T with a 95% confidence. Let s refer to condition 95/95, which is stated by soe regulatory docuents. These conditions can be forulated anew as: the calculated value of the araeter shall be obtained using the ethods that ensure a 95 % confidence level confired by testing, as well as the statistical characteristics of robable deviations. Moreover, the calculated value of the araeter with allowance for deviation statistics shall ensure a 95 % robability and without exceeding a design liit. Let s discuss in ore details this interretation of the conditions. The calculation outcoe is a certain calculated estiate of araeter of quantity x. Actually, in the course of reactor x calc oeration, another genuine value of araeter x is ised. The difference ay be caused by the ethods inaccuracies, rocess deviations and oeration deviations of reactor araeters fro noinal values. If the statistical values of deviations are well known (for exale, average value and disersion value for noral distribution), a 100 % confidence level could be ensured for the satisfaction of established liits by eans of iosing sufficiently guaranteed argins. For exale, if we want the genuine value of the araeter to be in 95 % of istic conditions lower than the ultiately tolerable value, we ust ake a shift to the left fro it by 1.645σ, etc. The deviation statistics is not known exactly in every istic case. This can be esecially related to colicated functionals, such as dearture fro nucleate boiling. This is the case, lenty of factors, such as heat flux, teerature, coolant flow and coolant ressure, the channel configuration, etc. affect the result.

4 For the urose of required true estiates, the investigated case shall be referred to a set (a sale) of other siilar cases, which were studied earlier, i.e. which subjected to direct or indirect easureents for the araeter we are interested in. The case ay be understood as a su of all factors that affect the araeter we are interested in. With reactor alication, these araeters include the coosition of core section, fuel and coolant teerature, ressure and flow. After the saling is ade, the araeter value shall be deterined for each eleent of the sale. Then, a atheatical odel shall be lotted to deterine average value of the araeter as a function fro effective factors (in case of DNBR the odel is called correlation). Also, a disersion of the araeter deviations fro such average value shall be deterined. It is obvious, that the larger is the sale, the ore accurate will be the calculation odel, and the ore recise will be the calculated disersion of deviations. As it was above entioned, on the basis of a liited nuber of coarisons, atheatical statistics is caable of evaluating with a rated reliability the interval within which the genuine average values and disersion values lie. The quantitative robability is reflected by forulas (1-3). If the fuel teerature is discussed anew, it is required to carry out a certain nuber of easureents for close to istic conditions, lot a calculation odel, which will siulate both testing conditions, and istic reactor conditions. The coare the easured and calculated data using the odel (deterine a disersion of deviations), and using forulas (1) and () access the average fuel teerature and disersion. The genuine fuel teerature ay differ fro average fuel teerature by the value of rando deviation, so, in order to ensure a 95% confidence in the absence of eltdown, it is required to fulfil the following condition for a noral distribution of deviations t T T T (1 / ) calc + ( ) ( ) elt σ + σ T χ <. (5) A second coonent characterises a robable axiu error of the average value, a third coonent characterises 95 % robability. Siilar considerations ay be cited for dearture fro nucleate boiling. Let us reind that qcr (, T, v DNBR = ), where qcr is a critical heat flux on the surface of fuel in, q (, T, v ) which deends on ressure, coolant teerature T and coolant velocity v; q is a heat flux that asses to the coolant fro the surface of fuel in. To ensure a correct DNBR assessent, easured data q and q cr are iortant to be easured in the conditions close to the istic reactor conditions. Using the easured data it is necessary to lot a odel («correlation»), which could siulate both the istic reactor conditions, and testing facility conditions for the urose of calculating a heat flux. This is the way for deriving forula qcr = qcr (, T, v ), for exale. At that, the error obtained as a result of this calculation is ainly affected by the deviation of arguents, T and v fro genuine values, but not a selected functional deendence uon the arguents (however, the contribution of this coonent can not be rejected, either).

5 Thus, in order to calculate DNBR it is required to get true data for q and, while in order to get true data for q cr it is ost iortant to get true data for, T and v at the oint of in surface, which is interesting for us. Therefore, the basis for true DNBR calculation shall ake the easureents of active reactor factors, which enable the calculation of q and q cr, and, consequently, DNBR. q cr A set of such easureents carried out for the conditions identical to the case of our interest (siilar FA tyes, identical oeration conditions, siilar ower distribution) allow us to evaluate the interval, within which the genuine average DNBR value lies, as well as the statistics of robable deviations fro the average DNBR value (deviation disersion). Then, we shall assue that the condition of interest ay be characterised by the identical statistical data, and the condition araeters shall be selected in such a anner, that they could satisfy the following inequality q t cr (1 / ) 1 ( DNBR) ( DNBR) 1 q σ σ >, (6) χ here is the nuber of easured data used for assessent of deviation between calculated and easured DNBR, while all other sybols are siilar to (1 and ). Should the nuber of t easured data be large we have: (1 / ) 1 0, 1, (6) will shift to χ q cr q σ ( DNBR) > 1. (7) It is necessary to oint out that the Russian ractice of reactor design has been using a ore conservative assution, as coared to (7), such as qcr q [ 1 ασ ( qcr )] [ 1 + ασ ( q) ] > 1, (8) where, tyically, α=, or α=1.645 in soe odern designs. In order to ensure a true assessent of fuel teerature and DNBR a secific attention is given to transient conditions of reactor oeration with a develoed tie lag of detectors that show the ower density distribution and coolant araeters. The ethods of these araeters calculation shall take into consideration this tie lag and ust be tested in alicable exerients that shall be conducted in the ost istic conditions.

6 The selected 95 % used for calculating robability and confidence are, obviously, only reasoned by the eculiarities of eole s sychology: 90 % is too little; 99 % is not easy to achieve, while 95 % is quite sufficient. And one ore general reark to wind u. It goes without saying that the alication of robability interval estiates, even only in relation of soe safety iortant reactor araeters, is a good idea. However, it sees that the requireents shall be secified in ore detail and account coon secific features of reactor calculations and easureents. In this rocess, the defining eleent of safety assurances still reains the hysical intuition of design and testing ersonnel, when they select easured data to justify their calculation ethods. REFERENCES 1. Pustylnik Е.I. Methods of statistical analysis and evaluation of investigations. Publishing houses «Nauka», Moscow, Guran V.Е. Theory of robability and atheatical statistics. Publishing houses Vyschaya shkola, Moscow, Panka I., Kereszturi A. Uncertainty analysis for hot channel. Proceedings of sixteenth Syosiu of AER, Langenbuch S. Status reort on Uncertainty and Sensitivity Analysis. VALCO Worksho, Helsinki, Langenbuch S., Schidt K.D.. GRS Methodology for Uncertainty and Sensativity Analysis of Results fro Couter Model. VALCO Worksho, Garching, 003.