CROSS SECTION WEIGHTING SPECTRUM FOR FAST REACTOR ANALYSIS

2009 International Nuclear Atlantic Conference - INAC 2009 Rio de Janeiro,RJ, Brazil, September27 to October 2, 2009 ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN ISBN: 978-85-99141-03-8 CROSS SECTION WEIGHTING SPECTRUM FOR FAST REACTOR ANALYSIS Jamil A. do Nascimento 1, Shizuca Ono 1 and Lamartine N. F. Guimarães 1 1 Divisão de Energia Nuclear - Instituto de Estudos Avançados (IEAv) Comando-Geral de Tecnologia Aeroespacial (CTA) Rodovia dos Tamoios, km 5,5 12228-001 São José dos Campos, SP jamil@ieav.cta.br shizuca@ieav.cta.br guimarae@ieav.cta.br ABSTRACT Preparation of a nuclear data library is the first task that a reactor analyst needs to perform a neutronic analysis of a reactor type. Today, in fast reactor area, the scheme used to generate this library includes the processing of an evaluated nuclear data file to obtain cross sections, in thousands of groups. Sequentially, the nuclear data are processed by a cell code to obtain neutron flux that is used to condense the large amount of energy groups to a practical calculation number of groups that can be used in reactor analysis. In the first step of the scheme it is necessary a weighting spectrum to generate the nuclear data. Here, it is proposed to use the flux estimated by Monte Carlo code using cell or the exact geometries and actual composition of the problem to obtain the main portion of the weighting spectrum instead of a code built-in function. As an example, it is presented the differences between selected pins spectrums obtained with MCNP5 calculation of a hexagonal fast reactor fuel assembly. Also, it is showed a comparison between these spectra and the one obtained in the representative unitcell model of this fuel assembly. The comparisons support that the proposed procedure, problem dependent, may be more accurate and a good choice to generate weighting spectrum in ultra-fine energy structure for fast reactor analysis. The proposed method will be used in space reactor neutronic analysis. 1. INTRODUCTION The core of a space reactor presents a complicated geometry and a diameter and height of tenths of centimeters. Neutronics analysis of this reactor type needs the preparation of suitable cross sections data library. The first step of a scheme to obtain this library is the processing of an evaluated nuclear data file with code like NJOY [1] or ETOE2 [2] to generate cross sections in thousands of energy groups. For example, each group may have a lethargy width of about 1/120, from 15-20 MeV to 0.215 ev. Next, for each core region, the generated data is processed by a cell code, like MC 2 2 [3], utilizing a simplified cell fuel geometry and the multigroup approach, to obtain a neutron flux to collapse the large amount of energy groups, i. e. 2082 in the ETOE2/MC 2 2 data conversion sequence, to a practical calculation number of groups (< 100), that will be used in the whole core analysis. In the first step of the scheme the processing code needs a weighting spectrum to generate cross sections in a fine or ultra-fine energy structure as mentioned below. Usually the

processing codes provide a several options of representative weighting spectrum of the main reactors types and the user needs to choice the one that is suitable for his case. Other possibility is to use programs like ACES [4] that compound a particular spectrum by coupling a group of theoretical functions commonly utilized in the nuclear area. Today, modern computer with multi-core processors, or computer cluster like the BEOWULF type, and large memory capacity had been enable the application of Monte Carlo (MC) technique to reactors calculations that are prohibitive until recent years. Particularly, for neutron spectrum calculation, the possibilities of the utilization of continuous cross section and exact geometry and materials compositions are very attractive. Therefore, considering fast reactor analysis, here is proposed to use the flux estimated by the code MCNP5 [5] in association with theoretical functions to compound a weighting spectrum for cross section generation. The paper was structured as follow. In Section 2 is presented the methodology and approaches used to obtain the neutron weighting spectrum. In Section 3 is shown the application of the methodology to obtain spectrums in selected pins of a fast reactor fuel assembly (FA). Also, the values of cross sections in one and seventy groups for 235 U calculated with these spectrums are presented. The final comments are made in Section 4. 2. METHODOLOGY The MCNP code can estimates the flux in any volume and bin energy structure specified by the user. To perform this estimative the code utilizes the tally F4 that is an estimative of the equation flux: φ V = 1 V de dt dv r Ψ(r, Ωˆ,E, t) dω, (1) p where: V, E, t, r e Ω are: volume, energy, time, position and solid angle respectively. If the user specifies an energy bin structure the code output provides the estimated flux in each bin. That is, the energy integration in Eq. 1 is realized considering the energy limits for each bin. In a reactor core, neutrons from fission emerge with average energy in the range of MeV. These neutrons loose energy when strike the atoms of core materials and slowing down to lower energy. The analog MC method follow the physic processes involved during the neutron slowing down until the neutron leaving the volume of interest or it is absorbed in a nuclear reaction. Considering a fast reactor, this method requires many histories to estimate the neutron flux with an acceptable statistical error in the bins that are located at the lower energy range of the spectrum. In the same manner the sample of high energy neutron source either needs many histories to enable acceptable flux statistical errors in the bins located at high energy range of the spectrum. Fortunately, the extremities of the spectrum are less important in the multiplication factor calculation of a fast reactor. These characteristics were demonstrated in reference [6] where were calculated three hard spectrum critical assemblies. The reference documented that 99.8% of the neutrons are distributed in the range 10 kev to 10 MeV. This means that for fast reactor k eff calculation the most important energies range points are within the specified interval and the fluxes errors in the tails of the spectrum are not a concern.

Another issue is the choice of the bin energy structure. Considering k ef evaluation, a base rule that may be used to define this structure is that the bin energy width must be narrowest than the neutron energy change in elastic collision with core materials. All neutron energy change is evaluated and the spectrum can be properly established in this case. To illustrate as narrow the bin width must be, is presented Table 1 that show the neutron lethargy increment in elastic collision with the atom of selected materials that may be present in core of fast reactors. As can be seen, neutron collision with fuel material and liquid metal coolant atoms slightly increases the lethargy. Therefore, the bin energy structure needs thousands of bins to cover the energy range of interest, from ev to MeV, in k eff evaluation. Table 1. Neutron lethargy increment (ζ ) in selected fast reactor material Material ζ Material ζ Uranium 0.00838 Sodium 0.08451 Lead 0.00963 Oxygen 0.12000 Iron 0.03529 Carbon 0.15789 In summary, the evaluation of the flux distribution (spectrum) require thousands of bins and the bins located in the spectrum tails may have negligible contribution [6] to k ef evaluation of fast reactors. Here, it is proposed the following scheme to compose the weighting spectrum for multigroup cross section generation: (i) the spectrum kernel is composed by the estimated fluxes of a MCNP calculation with statistical errors less than the recommended value of 0.1% [5]. The lower and upper energy breakpoints of the kernel are defined in this way; (ii) in the intermediate energy range, from 1.85 ev to the lower energy breakpoint of the kernel, where the fluxes statistical errors are > 0.1%, is assumed that the tendency of the estimated fluxes points is correct and it is adjusted, using the least-squares fit method, a n< 7 grade polynomial function; (iii) in the thermal energy range, from 1.0E-4 ev to 1.85 ev, it is utilized a Maxwellian function with fixed (0.025 ev) or adjusted thermal temperature; and (iv) in the fast energy range, from the upper energy breakpoint of the kernel to 18 MeV is utilized a fission Maxwellian function with fission temperature of 1.4E+6 ev. The spectrum values in the energy range where the Maxwellians functions are used are obtained by integration of these functions considering the bin energy boundaries. All coupling among theoretical functions and the kernel to compound the spectrum in the energy range of interest is made utilizing multiplier constants. These constants are obtained by imposing that the fluxes values at the joint bins are equals. To finalize, the flux distribution is normalized, Σ φ = 1.

3. APPLICATION TO FAST REACTOR FUEL ASSEMBLY Monte Carlo method needs a high computing time for reliable results. The analyst must establish a balance between precision needs and time requirement in the way that de calculations can be satisfactorily made. The energy range of interest and the lethargy increment in elastic collision for fast reactor analysis was shown below. As narrow is the bin width as more computation time is needed to obtain a statistical flux error lower than the recommended value. Considering these points was determined the following lethargies widths in the energy range of interest: 0.008 from 18 MeV to 10.5 MeV, 0.005 from 10.5 MeV to 10 kev, and 0.008 from 10 kev to 0.250 ev. With these conditions the energy range of interest is cover by 2786 bins. It was chousen a hexagonal FA, Fig. 1, of a lead reactor constituted by 127 pins of 1.0 cm diameter. The FA duct width is 0.356 cm, the inter-assembly gap width is 0.215 cm and the assembly pitch is 15.209 cm. The hexagonal unit cell models dimensions, shown in Table 2, was obtained from the actual geometry for comparison purpose. Cell model c1 is the actual fuel pin with coolant channel. Cell model c2 was obtained with volume conservation, all the FA material is heterogeneously distributed into the three regions of the cell as identified in Fig.1. The fuel material is the alloy U10Zr, enriched at 17.5 o /w, with smear density of 75 %, the structural material is the stainless steel HT9, the coolant is lead and T = 20 o C. The numbers inside the FA hexagonal geometry identify the pins where the spectrum was obtained. All the calculation was performed with the MCNP5 code. The number of cycles was 1,100, each one with 10 5 neutron histories. The 100 initial cycle calculations are not considered in the results. Nuclear cross sections are from ENDF/B-VI data file computed at the temperature of 293.15 K. Reflected boundary conditions were used in all direction. 7 5 3 2 6 4 8 1 U-10Zr 1 2 HT-9 3 Pb L Figure 1. Fast reactor fuel assembly and cell model

The actual FA and the cell geometries were codified in MCNP5 input format and processed to obtain the flux distribution. Fig. 2 presents the fluxes estimative with relative statistical error < 0.1 % obtained in pins #1 and #8 and the difference between then, taking the central pin #1 as the reference. Pin #8 is located at the external border of the FA and is strongly influenced by FA double heterogeneity resulting in the highest fluxes at the lower energy range of the spectrum than the any one obtained in the others pins. In other words, the pin #8 spectrum is the softly for this FA. This behavior is clearly viewed in the differences showed in Fig. 2. The deep in the spectrum at ~20 kev is due to the large capture resonances of iron around this energy. The MCNP5 results for k are shown in Table 3; all the values have standard deviation of 4 10-5. The k for the actual FA is the reference and it was obtained considering the exact geometry and composition. The comparison of the cells results shows that cell model c1 is a poor representation of the actual problem and that cell model c2 is a good model representation for k evaluation. The difference between the k of the actual FA and the cell model c2 is only 200 pcm. Table 2. Cell regions and dimensions (cm). Cell model c1 c2 Fuel (r1) 0.423 0.423 Cladding (r2) 0.500 0.546 Coolant (L) 1.308 1.451 Table 3. Fuel assembly and fuel cell model k results. Model k k (pcm) c1 1.39901-5324 c2 1.34377 200 Fuel Assembly 1.34577 ref. Figure 3 shows the FA central pin (#1) fluxes at the intermediate energy range with statistical errors higher than 0.1%. Also it shows the adjusted polynomial function of grade n = 5 and, for comparison purpose only, a straight line of the form C/E commonly used in this energy range. As can be seen, the polynomial function represents adequately the average behavior of the flux at this energy range. Figure 4 presents the compound spectrum for the FA pin #1 utilizing the proposed method. A measure of the difference between spectrums is the median energy that exactly divides in half the spectrum integral. This median energy was calculated for the selected FA pins

spectrums identified in Fig.1 and for the fuel cell models. The median values and the kernel energy breakpoints of these spectrums are presented in Table 5. Table 5. Kernel breakpoint and median energies Kernel breakpoint energy Median energy Case (kev) Inferior Superior (ev) (MeV) Cell model c1 238.8 238.5 15.8 Cell model c2 217.2 166.4 15.8 (a) see Fig.1. Pin #1 a 219.7 539.3 10.5 Pin #4 218.6 539.3 10.5 Pin #7 215.8 522.3 10.5 Pin #8 214.7 539.3 10.5 10 1 10 0 10-1 spectrum/mev 10-2 10-3 10-4 10-5 pin 1 pin 8 difference (%) 10-6 40 20 0-20 -40-60 -80-100 10-4 10-3 10-2 10-1 10 0 10 1 energy (MeV) Figure 2. FA pins #1 and #8 spectrums and its differences The median energy values clearly show that the spectrum in the central pin (#1) is the hardest and that the spectrum softs slightly when the pin position change in direction to the FA

border. Two interesting results considering the spectrum of fuel cell model c2 are: (i) its median energy is between the median energies of the FA pins, and (ii) its kernel presents the greatest energy range among the kernels of the FA pins spectrums. The c2 spectrum characteristics (median and breakpoints kernel energies) emphasize that this model is a good approximation for the average pin of this FA from neutronic behavior point of view and it can be used for generating a representative average FA weighting spectrum. 1,0E-09 1,0E-10 1,0E-11 MCNP polinomial function c/e Flux 1,0E-12 1,0E-13 1,0E-14 1,0E-15 1,0E-16 1,3 2,4 4,6 8,8 16,6 31,5 59,8 113,3 214,9 407,6 Energy (ev) Figure 3. Fluxes at intermediate energy range for FA pin #1 and the adjusted polynomial function 1,0E-02 1,0E-03 1,0E-04 Normalized spectrum 1,0E-05 1,0E-06 1,0E-07 1,0E-08 1,0E-09 1,0E-10 1,0E-03 2,2E-02 4,7E-01 1,0E+01 2,2E+02 4,7E+03 4,2E+04 2,9E+05 2,0E+06 1,6E+07 Energy (ev) Figure 4. FA central pin compound spectrum

Also, the NJOY system with the ENDF/B-VII was processed to have a measure of difference between multigroup cross sections generated with different weighting spectrums. With the GROUPR module of this system it were calculated condensed infinity dilution cross sections at 293 K, in one and 70 energy groups, JFS-2 Japanese format [7], for the 235 U. The energy range considered was 0.215 ev to 10.5 MeV and were utilized the weighting spectrums of the pins #1, #8 and of the cell model c2. For comparison purpose it was generated the same cross sections with the weighting spectrum IWT=8 of NJOY system that can be used for fast reactor analysis. The one group cross sections are presented in Table 6. Fig 5 shows the 70 group fission and capture differences. In all cases the reference is the cross section condensed with the spectrum generated in FA pin #1. Table 6. 235 U one group cross section difference (%) Cross section σ f σ c σ e σ in υ Spectrum Pin #1 (barn) 1.23572 0.140522 5.05406 1.68812 2.60230 Pin #8-0.08 (a) -2.96-0.98 1.20 0.359 Cell model c2-0.036-1.43-0.47 0,58 0.176 NJOY (IWT =8) -73.96-374 -72.73 59.37 5.4098 (a) diff = ((ref.-case)/ref) 100 Difference (%) 10 5 0-5 -10-15 -20-25 10-1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Energy (ev) (a) cell model c2 NJOY-IWT=8 Difference (%) 20 15 10 5 0-5 -10-15 -20 10-1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Energy (ev) (b) cell model c2 NJOY-IWT=8 Figure 5. 235 U cross section seventy group difference in selected spectrum: (a) fission, (b) capture Table 6 shows that the cross sections differences are small, less than 3 % between the pin #1 (hardest spectrum) and the pin #8 (softest spectrum) of the FA. As expected the cell model c2

presents the small difference values. By the way, the difference values for the NJOY spectrum case are the greatest, approaching 374 % for the capture cross section (σ c ). The IWT=8 spectrum may be used as last alternative considering that the reactivity, or the cell spectrum, calculated by the codes may present a large uncertainty. The calculated seventy group differences, Fig 5, show that the higher values are in the energy range of resolved resonances (ev to kev). This energy range, where the MC results present high fluxes errors, theoretical functions is adjusted to obtain the composed spectrum. In fast reactor area this energy range do not affect seriously the calculation of k eff [6] but it can be important considering the evaluation of temperature dependent phenomena like Doppler effect. Fast reactor presents a small Doppler effect, therefore, an evaluation of the impact of the use of the proposed method needs a specific calculation. 4. FINAL REMARKS It was proposed a method to obtain a weighting spectrum for multigroup cross section generation for fast reactor analysis. The method utilizes Monte Carlo fluxes estimative, to obtain the kernel of the spectrum and theoretical functions, to compound a weighting spectrum. The aim is to use the method to prepare a cross section library for analysis of a space fast reactor. The work showed that a unit cell model of three regions: fuel, structure and coolant, where all the materials in the fuel assembly are heterogeneously distributed is a good model for an average spectrum evaluation. Therefore this model can be used to obtain weighting spectrum to generate cross section library for k eff analysis. The comparison between calculated microscopic cross section in one and seventy groups, condensed with theoretical code built-in function IWT=8 of NJOY code and the ones condensed with representatives pins spectrums of a fast fuel assembly show that the first originates higher differences. Therefore, the use of IWT=8 weighting spectrum will produce a higher uncertainty in reactor reactivity and this will impact the core project. Another issue is the cross section differences at lower spectrum energies, resolved resonance region. This range is important in the computation of temperature effects, like Doppler. Although in fast reactor calculation this effect is small it is important to know its magnitude. Consequently, a specific calculation may be necessary to clarify the impact of using the proposed method for this case. The next step of this work will be the evaluation, with deterministic transport theory codes, of the k eff sensibility to the multigroup cross sections generated with weighting spectrums obtained with the proposed method. REFERENCES 1. R. R. Mac Farlane, NJOY Version 99, up 259, Los Alamos National Laboratory (2005). 2. Computer Codes: ETOE-2, http://www.ne.anl.gov/codes/etoe-2 (2008) 3. H. Henryson II, B. J. Toppel, C. G. Stenberg, MC 2-2: A Code to Calculate Fast Neutron Spectra and Multigroup Cross Sections, Argonne National Laboratory, ANL-8144 (1976). 4. A. D. Caldeira, E. S. Chaloub, A Program for Generating Pointwise Weighting Functions, Annals of Nuclear Energy, Vol. 20, pp 605-609 (1993). 5. X5 Monte Carlo Team, MCNP A General Monte Carlo N-Particle Transport Code, Version 5, Los Alamos National Laboratory LA-UR-03-1987 (2003).

6. D. E. Cullen, R. N. Blomquist, P. N. Brown, C. J. Dean, M. E. Dunn, Y. K. Lee, R. MacFarlane, S. McKinley, E. F. Plechaty, J. C. Sublet, ENDF/B-VII.0 Data Testing for Three Fast Critical Assemblies, Lawrence Livermore National Laboratory, UCRL-TR- 233310 (2007). 7. H. Takano, A. Hasegawa, M. Nakagawa, Y. Ishiguro, S. Katsuragi, JAERI Fast Reactor Group Constants Set, Version II, Japan Atomic nergy Research Institute, JAERI 1255 (1978).