Computaton of Ex-Core Detector Weghtng Functons for VVER-440 Usng MCNP5 Gabrel Farkas, Jozef Lpka, Ján Haščík, Vladmír Slugeň Slovak Unversty of Technology, Faculty of Electrcal Engneerng and Informaton Technology, Insttute of Nuclear and Physcal Engneerng Ilkovčova 3, SK-81 19 Bratslava, Slovaka gabrel.farkas@stuba.sk ABSTRACT The paper deals wth the problem of weghtng factor calculaton and the determnaton of spatal weghtng functons of ex-core detectors for VVER-440 usng the Monte Carlo method. The computatonal results were obtaned by the well-known code MCNP5 allowng hgh performance three-dmensonal modelng of complex geometry of the n-vessel and exvessel reactor parts. Despte the fact that adjont methods domnate n practce, forward mode of code runnng was chosen and appled to provde more accurate results contrary to the adjont one. The calculaton was performed for a boron lned proportonal counter CPNB44 nstalled at the 3 rd unt of NPP Jaslovské Bohunce. The base s the calculaton of ex-core detector reacton rate nduced by a neutron generated n a gven volume element of a fuel pn. All the geometrcal detals and arsng space heterogenetes were taken nto account wth the hghest accuracy n the complex reactor model. Havng obtaned computatonal results, the weghted least square method was used to ft axal weghtng functons. Wth respect to horzontal drecton, the polyhedral approxmaton of closed Jordan surfaces was used to fnd the proper shape of horzontal weghtng factor dstrbuton. Senstvty and parametrc analyss was performed to evaluate the nfluence of varous reactor operatonal parameters as well as the ex-core detector postonng on the weghtng functon values. 1 INTRODUCTION The contrbuton of fuel assembles to the ex-core detector response depends not only on the power, but also on the poston of the gven assembly n the core. The weght of the nner assembles s several orders of magntude lower than the outer ones. Consequently, the detector response for a gven reactor power s strongly nfluenced by the spatal power dstrbuton and ndrectly by the parameters determnng the dstrbuton, such as core loadng pattern, tme elapsed, poston of control assembly, coolant temperature, etc. Precse knowledge of the spatal weghtng functons can be very benefcal for the solvng of varous reactor physcal, operatonal, and safety problems. It can be useful for the proper nterpretaton of the startup test measurements, e. g. determnaton of whether the measured detector response durng the rod-drop experment supports the predcted reactvty value from the loadng pattern calculatons. Another use may be the calculaton of ex-core detector response n deep subcrtcal reactor states, evaluaton of nfluence of the core loadng pattern on the detector response, and the detector calbraton. A very mportant applcaton area s the reactor safety analyss. In ths case the elaborate weghtng functons can be helpful 317.1
317. n nvestgatng how effcently certan transents whch cause a sudden change n the power densty dstrbuton can be detected by the ex-core detectors. The work was based on the prevous study of Csom, Czfrus, and Fehér [1]. In ths paper, the developed calculatonal model and method s descrbed to determne the spatal weghtng functons of ex-core detectors for the VVER-440 reactor type, takng nto account dfferent operatonal parameters such as reactor power, burnup, borc acd concentraton, poston of the control assembly, etc. THE PROBLEM OF WEIGHTING FUNCTION The spatal weghtng functon gves a relatonshp between the spatal power dstrbuton n the core and the ex-core detector response. The weghtng functon can be defned n varous ways n dependence on the problem to be solved. In general, t s a poston-dependent contrbuton of a gven part of the core (e. g. fuel assembly, fuel pn or a porton of the fuel pn) to the ex-core detector response. In ths partcular case the weghng functon value represents the average number of reactons occurred n the ex-core detector per one source neutron created n one-twenteth of a fuel pn heght. It follows from the above defnton that real ex-core detector response D s obtaned by convoluton of two spatal functons - the power dstrbuton functon and the weghtng functon, and ntegraton of the response functon over the whole core regon n nterest. The correspondng ntegral equaton s the followng: D = C P p( r )w( r )dr, (1) V where w ( r ) s the spatal weghtng functon, p ( r ) - the relatve reactor power dstrbuton, P - the total reactor power, C - a constant, V - the relevant core volume. The response of the ex-core detector can be smplfed as the followng sum approxmaton: D C P W P, () where W s the spatal weghtng factor of th volume element of the core (.e. the twenteth of fuel pn heght), P - the correspondng relatve power of ths volume. 3 THE CALCULATION METHOD Consderng the complcated geometry of the space between the core volume elements and the ex-core detector, relable neutron transport calculaton can only be performed usng a technque enablng the treatment of complex three-dmensonal geometry. Therefore the MCNP5 Monte Carlo code was chosen []. Calculaton of the spatal weghtng functons of ex-core detectors usng the Monte Carlo method s not novel. The applcaton of the MCNP code for determnaton of weghtng functons s reported n the lterature [1] and [3]. For
317.3 several reasons, the forward method was appled contrary to the tme-savng adjont method. For example, the use of forward mode of run makes t possble to elmnate errors due to the homogenzaton of the assembly as well as the use of group-wse nuclear data. Neutron transport calculaton was performed by fx-source method. Ths method enables to compute sngle weghtng factor values from separate fuel pn elements of the source regon. 4 THE CALCULATION MODEL Regardng the objectve of the work to determne the weghtng functons wth hgh accuracy and relablty, the geometrc and materal part of the reactor model had to be created n the fnest possble detals. Precse 3D whole-core model of the VVER-440 was developed wth MCNP5. The model has a wde range of applcatons n varous areas of reactorphyscal, operatonal and safety calculatons and analyses. Horzontal and vertcal secton of the VVER-440 model s shown n Fg. and 5. The next step was the adjustment of created unversal model to the effcent neutron transport calculaton from defned core source regon to the selected ex-core detector. Spatal source regon boundary depends on the requred accuracy of the calculaton. However, reducton n uncertanty below a gven threshold can be unreasonable because of hgh computaton tme demands. The geometrc bounds of the source regon set n the model cover all of those fuel pn elements whose weghts exceed 0.5 % of the calculated maxmum weghtng factor value. The total number of fuel assembles to be taken nto account s 54, see Fg. 1. The calculaton of weghtng functons presented n ths paper was performed for a boron-lned proportonal counter CPNB44 nstalled at the 3 rd unt of NPP Jaslovské Bohunce. Ths source-range onzaton chamber s used for the startup reactvty measurements. Nuclear data used n the MCNP calculatons were manly based on the ENDF/B-VI lbrary. In order to mnmze the varance as well as CPU tme, a combnaton of the followng varance reducton technques was appled: cut-off energes, source varable basng, spatal mportance treatment n the space between the core and the ex-core detector.
317.4 Fgure 1: Placement of the VVER-440 ex-core detectors Fgure : Horzontal and vertcal secton of the VVER-440 model n MCNP5 Fgure 3: Indexng of fuel assembles n the core Fgure 4: Numberng of fuel pns n the fuel assembly
317.5 5 THE RESULTS Fgure 5: Plots from 3D whole-core VVER-440 model n MCNP5 The paper presents selected calculatonal results of the axal and horzontal weghtng factor dstrbuton n the source regon of the core. Approxmaton of analytcal functon was performed n the axal drecton so far. Weghtng factor dstrbuton n the horzontal drecton s shown for two selected core layers. 5.1 Axal weghtng factor dstrbuton Axal weghtng factor dstrbuton was calculated n the source fuel pns of the core. In the calculaton, 193 source pns were consdered all together. Those pns are dstrbuted n a quas-regular grd whch covers 54 fuel assembles of the core, see Fg. 1. Axal functons of four selected fuel assembly source pns are presented n ths paper. The objectve of the axal approxmaton was to fnd an analytcal functon whch properly descrbes weghtng factor dstrbuton n vertcal drecton that s along the source fuel pns. Dfferent approxmaton functons were nvestgated and the followng polynomal functon was proposed as the most approprate one: w ( z) = a + a z + a z +... + a a z M 0 1 M, (3)
where w a (z) s 317.6 axal weghtng functon of ex-core detector related to a gven fuel pn, z - heght coordnate of a pn element (fractonal dstance between the central plane of symmetry of the core and the centre of pn element, the value s postve above the plane of symmetry), a 0,..., a M are approxmaton constants. Degree of the polynomal functon depends on the poston of gven source pns n the core. It changes from a degree of 8 for closer perpheral pns up to a degree of 4 for dstant ones. Indexng of fuel assembles s shown n Fg. 3 and fuel pns n Fg. 4. Another applcaton of the axal approxmaton s the smoothng of calculated MCNP weghng factor values for a horzontal approxmaton. The am s to mprove the qualty of approxmaton n horzontal drecton (n a gven layer of the core). In terms of ths procedure the horzontal approxmaton s appled not on the calculated weghtng factor values, but on the axally smoothed values. Weghted least square method (WLSM) was chosen for the vertcal fttng [4], [5]. Fttng program based on the WLSM uses Sngle Value Decomposton method (SVD) to determne optmal approxmaton constants values arsng from Eq. 1. The program performs weghted fttng, whch means that ponts havng smaller varance wll appear n the approxmaton wth hgher statstcal weght. Ths mnmzaton can be expressed by the followng formula: N W χ = = 1 (4) M k = 0 σ a k Z k ( z ) mn, where χ s the ch-square, W - the calculated weghtng factor value of data pont z, σ - the standard devaton of the -th data pont, ak are approxmaton parameters, Z z ) - arbtrary fxed functons of z, called the bass functons. k ( The shapes of approxmaton curves correspond to the theoretcally expected weghtng factor dstrbuton n vertcal drecton. As t follows from Fg. 6, the axal weghtng factor dstrbuton n the closest pn (-11,4,070) to the ex-core detector as well as n the selected close pn (-10,4,065) s descrbed by a polynomal functon of the 8 th order, whch s a typcal bell curve. Wth respect to the dstant pn (-4,-7,17), axal dstrbuton s expressed by a polynomal functon of the 4 th order wth the shape of arc curve. Dstrbuton n the medum-dstant pn (-8,6,065), gven by a polynomal functon of the 6 th order, shows transent character between the above two curve shapes. Extenson of the curves that s ncrease n the FWHM (Full Wdth at Half Maxmum) wth ncreasng dstance of a gven fuel pn from the ex-core detector s manly caused by geometrcal reasons. In the case of dstant pns, the dstance rato of mddle and upper/lower pn elements from the ex-core detector s smaller than for closer pns. Axal curves are symmetrcal n respect of the mddle plane of symmetry of the core because of symmetrcal ex-core detector postonng. Axal weghtng factor dstrbuton of selected two fuel assembly source pns are shown n Fg. 7 and 8.
317.7 Relatve Weghtng Factor 1, 1,0 0,8 0,6 0,4 Pn (-11,4,070) - MCNP Pn (-11,4,070) - Ft Pn (-10,4,065) - MCNP Pn (-10,4,065) - Ft Pn (-4,-7,17) - MCNP Pn (-4,-7,17) - Ft Pn (-8,6,065) - MCNP Pn (-8,6,065) - Ft 0, 0,0 0 0,1 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Fractonal Core Heght Fgure 6: Relatve axal weghtng factors of four selected fuel assembly pns Weghtng Factor 3,0E-05,5E-05,0E-05 1,5E-05 1,0E-05 Pn 070 - MCNP Pn 070 - Ft Pn 065 - MCNP Pn 065 - Ft Pn 058 - MCNP Pn 058 - Ft Pn 17 - MCNP Pn 17 - Ft Pn 11 - MCNP Pn 11 - Ft Pn 007 - MCNP Pn 007 - Ft Pn 001 - MCNP Pn 001 - Ft 5,0E-06 0,0E+00 0 0,1 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Fractonal Core Heght Fgure 7: Axal weghtng factors of the fuel assembly (-11,4) source pns
317.8 Weghtng Factor 6,0E-07 5,0E-07 4,0E-07 3,0E-07,0E-07 Pn 17 - MCNP Pn 17 - Ft Pn 17 - MCNP Pn 17 - Ft Pn 065 - MCNP Pn 065 - Ft Pn 058 - MCNP Pn 058 - Ft Pn 001 - MCNP Pn 001 - Ft 1,0E-07 0,0E+00 0 0,1 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Fractonal Core Heght Fgure 8: Axal weghtng factors of the fuel assembly (-4,-7) source pns 5. Horzontal weghtng factor dstrbuton The paper presents computatonal results obtaned for two selected layers of the twenteth of fuel pn heght. The frst layer (No. 11) s stuated mmedately above the central plane of symmetry of the core. The second one (No. 0) s the upper perpheral layer of the core. As t was mentoned earler, the twenteths of fuel pns were selected n such a way that they cover the model n a quas regular grd. At average, 3 or 4 twenteths of fuel pns (elements) were calculated for each fuel assembly, wth the excepton of perpheral core regon beng closest to the ex-core detector where 5-7 elements per assembly were selected. All together 193 fuel pn elements were calculated. The horzontal weghtng factor dstrbuton for two selected layers of the core s shown n Fg. 9 and 10. From the obtaned dstrbutons results: the major porton of the ex-core detector response s determned by a few perpheral fuel assembles located n the closest vcnty of the ex-core detector, the weghtng factor values vary sgnfcantly wthn a gven fuel assembly, the dfference between the mnmal and maxmal weghtng factor value n the gven fuel assembly s wthn one order of magntude, n the closest fuel assembly regon to the ex-core detector, n the drecton from the maxmum weght poston to the core centre, approxmately eght-fold decrease of weghtng factor values s observed, based on the weghtng factor dstrbuton n the gven layers t can be expected that horzontal weghtng functon wll warp n the upper and lower part of the source regon,.e. ther values wll not be manly determned by the dstance from the ex-core detector, weghtng contrbuton of the closest 0 perpheral fuel assembles to the ex-core detector sgnal represents 9 % from the sum of the core weghtng factor values,
317.9 t can be expected that the horzontal weghtng functon wll have a character of exponental polynomal functon of the 3 rd - 8 th order of magntude, generally expressed by Eq. 5 whle Eq. 6 can be desgned as a specfc form. w ( x, y) h exp K L = = 0 j= 0 a j x y j (5) w ( h (6) x, y ) = exp( a0 + a1x + a y + a3x + a4 xy + a5 y 5 5 6 6 + a x + a y + a x + a y ), 10 11 1 13 + a 6 x 3 + where w h ( x, y) s the horzontal weghtng functon of the ex-core detector related to a gven core layer, x, y are the coordnates of a pn element (fractonal dstance between the central plane of symmetry of the core and the centre of pn element, the value s postve above the plane of symmetry), a j, resp. a0,..., a13 - approxmaton constants. Fgure 9: Weghtng factor dstrbuton n the central layer of the core No. 11 Fgure 10: Weghtng factor dstrbuton n the upper perpheral layer of the core No. 0 6 CONCLUSION Present practce stll ponts out the need for the determnaton of spatal weghtng functons of ex-core detectors. The exact knowledge of these functons can be very useful for
317.10 the solvng of varous reactor physcal, operatonal and safety problems especally for nterpretaton of reloads start-up measurements. The objectve of the work was to present the methodology for determnaton of weghtng factors and functons at the twenteth of fuel pns level usng Monte Carlo computatonal approach. Ths can be acheved even wthn a statstcal uncertanty of 1 % at acceptable CPU tme. The calculatonal results show that the major porton of the VVER-440 ex-core detector response (more than 9 %) s attrbuted to less than 0 fuel assembles beng located closest to the gven detector REFERENCES [1] Gy. Csom, Sz. Czfrus, S. Fehér, T. Berk, Calculaton of Spatal Weght Functons for VVER-440 Ex-Core Neutron Detectors, Proc. Int. Conf. The 11 th Symposum of AER, Csopak, Hungary, September 4 8, 001, pp. 711 715. [] X-5 Monte Carlo Team, MCNP A general N-Partcle Transport Code, Verson 5 Volume I: Overvew and Theory, LA-UR-03-1987, Los Alamos Natonal Laboratory, 003. [3] E. Kalonen, R. Kyrk-Rajamäk, F. Wasastjerna, Smulaton of Rod Drop Experments n the Intal Cores of Lovsa and Mochovce, Proc. Int. Conf. The 9 th Symposum of AER, Demänovská Dolna, Slovaka, October 4 8, 1999, pp. 367. [4] W. H. Press, S. A. Teukolsky, W. T. Vetterlng, B. P. Flannery, Numercal Recpes n FORTRAN: The Art of Scentfc Computng, nd ed., Cambrdge Unversty Press, Cambrdge, U. K., 199. [5] W. T. Vetterlng, S. A. Teukolsky, W. H. Press, B. P. Flannery, Numercal Recpes Example Book (FORTRAN), nd ed., Cambrdge Unversty Press, Cambrdge, U. K., 199. [6] J. G. Ahn, et al., Generaton of Spatal Weghtng Functons for Ex-core Detectors by Adjont Transport Calculaton, Nuclear Technology, 103, 1993.