Acta Polytechnica CTU Proceedings DOI:10.14311/APP.2020.28.0023 Acta Polytechnica CTU Proceedings 28(0):23–31, 2020 © Czech Technical University in Prague, 2020 available online at http://ojs.cvut.cz/ojs/index.php/app METHODS OF XS DATA PREPARATION FOR GEOMETRY WITH FUEL DUMMY Pavel Suk Czech Technical University in Prague, Faculty of Nuclear Science and Physical Engineering, Department of Nuclear Reactors, V Holešovičkách 2, 180 00 Prague 8, Czech Republic correspondence: sukpave2@fjfi.cvut.cz Abstract. 3D deterministic core calculation represents important category of the nuclear fuel cycle and safe Nuclear Power Plant operation. The appropriate solution was not published yet. Data preparation process for non-fuel elements of the core represents the challenge for scientists. This report briefly introduce the problem of the data preparation process and gives the information about new input format for macrocode PARCS (PMAXS). The best homogenization process approach is to prepare data in infinite lattice cell for fuel assemblies, which are placed next to the another fuel assembly. Data for fuel assembly located next to the non-fuel region are better with preparation in the real geometry with the real boundary conditions. Results of the neutron spectra study show that the PMAXS file format is well prepared for the 2 group calculation, but it is not well prepared for the multigroup calculations, however the XSEC file format still gave reasonable results. Keywords: Eigenvalue, flux, fuel dummy, homogenization, SCALE, Serpent, PARCS. 1. Data preparation process and reflector calculation Nuclear Power Plant (NPP) cores are in general too large to calculate them with exact approach of the transport equation in acceptable calculation time. Based on this fact, the data preparation process and full core calculation via deterministic macrocodes are still remaining comprehensive parts of the fuel loading patterns preparation. These two processes are, in general, connected together, but in the reality, the complexity of this problem is replaced by the eas- ier and faster approach. This approach is based on the separation of the data preparation process for each part of the core and 3D full core deterministic calculation. The research reactor cores are sometimes also cal- culated with the simplified methods. Especially the fuel loading pattern in the high power research reac- tors can be designed via simplified homogenization approach. The total neutron-termohydraulic solvers are also based on this methodology. Therefore the homogenization process is also important during the analysis of research reactors. The data preparation process for the fuel elements is historically based on the exact transport simulation of the fuel assembly with reflective boundary condi- tions [1]. The main complication of this approach is fact that the fuel assemblies are not in the infinite medium environment during the 3D calculation. The reactor core does not consist of the same fuel assem- blies, some flux tilt inside the fuel assemblies can be presented. Some advantages of this approach can be found in the rehomogenization process, which is de- scribed in papers [2] or [3]. The research reactors and NPPs are almost always operated in the critical state, but during the data preparation process, the infinite medium of the the fuel assemblies can be sub-critical or super-critical in case of their enrichment and ge- ometry. Some solution of these issues can be found in the B1 approximation, described bellow or in the article [1]. The comparison of B1, P1 and CASMO methods can be found in the paper [1]. The data preparation process for non-fuel elements represents more challenge problem. The optimal solu- tion of that problem has not been found yet, however some approaches are developed for data preparation. The main complication of the data preparation process for the non-fuel elements is based on the non-leakage medium and impossibility to calculate exact neutron spectrum during the data preparation process. The most codes are based on the diffusion solver, but the diffusion theory is not met near the boundaries [4]. Data for non-fuel regions, or reflector regions can be prepared by various approaches. For instance 1D approach or more dimension approach [5]. For bet- ter criticality and power distribution prediction, the macroscopic data are sometimes optimized. The main complication of this approach is in the high calculation time. Next complication of the fuel loading pattern optimization is in the uniqueness of the pattern. The optimization should to be made for each pattern sep- arately. The optimization method can be found, for instance, in the paper [6]. Numerous details of data preparation for nodal cal- culation codes can be found also in the paper [7]. The author explain complications of the B1 method, diffusion coefficient calculation, and spatial homoge- nization for data preparation process there. 23 http://dx.doi.org/10.14311/APP.2020.28.0023 http://ojs.cvut.cz/ojs/index.php/app Pavel Suk Acta Polytechnica CTU Proceedings 1.1. Macroscopic cross section preparation Macroscopic cross sections (XS) are, in general, quan- tities which provide information about the influence of that part of the core to the calculation. XS are defined in simple approach: Σi,g,r = Ni · σi,g,r, (1) where Ni is atomic density of the region i and σi,g,r is microscopic cross section of region i, energy group g, and reaction r. The more rigorous approach is based on the type of the calculation code. The lattice codes can be divided into the discrete energy codes and continuous energy codes. In these two cathegories, the macroscopic cross sections are generated: Σi,g,r = ∑Eg,max h=Eg,min Ni · σi,h,r · Φi,h∑Eg,max h=Eg,min Φi,h , (2) for lattice codes which calculated with the discrete energies. Φi,h is the neutron flux inside region i and energy group h. For the continuous energy calculation codes, the macroscopic cross sections are calculated via: Σi,g,r = ∫ Eg,max Eg,min Ni · σi,r(E) · Φi(E)dE∫ Eg,max Eg,min Φi(E)dE , (3) where σi,r(E) is the microscopic cross section of the region i, reaction r, and energy E and Φi(E) is the neutron flux inside region i, energy E. 1.2. B1 correction B1 correction, also called critical spectra correction, is a method used for treating the leakage of neutrons in the fuel assemblies places in the real core pattern. As was mentioned above, the 3D deterministic cal- culations are based on the two level approach. The macroscopic data for fuel assemblies are prepared for the fuel assemblies in the infinite lattice of the same assemblies, thus, the neutron leakage from the fuel assembly is zero. The B1 method prepares data based on the calculation of critical spectra in the fuel as- sembly and due to it treats the non zero neutron leakage. The main idea of the B1 approximation is to find appropriate buckling factor and thus appropriate neu- tron spectra for XS preparation. The B1 corrected homogenised XS are then prepared with the new neu- tron spectra. The buckling factor adds or removes additional neutron leakage in the fuel assemblies based on the equation: Σt,gΦg ± iBJg = χg ∑ h Σf,hΦh + ∑ h Σ0s,h→gΦh, (4) where Σt,g is the total macroscopic cross section in energy group g, Φg is the neutron flux in energy group g, B is the buckling factor, Jg is the neutron current in energy group g, χg is the neutron spectra in the energy group g, Σf,h is the fission macroscopic cross section in energy group h and Σ0s,h→g is the zero moment of scattering macroscopic cross section in energy group g. The critical spectra is to find iteratively by increas- ing buckling factor. The first step is based on the calculation with B2 = 0, the second step is based on the calculation with B2 = 10−6 and the next steps are calculated with extrapolated buckling factor from the previous calculation. The whole method is well described in the paper [1] and [7]. When the fuel assemblies in the infinite lattice have the eigenvalue less than 1, the neutron leakage is decreasing, when the eigenvalue is larger than 1, the leakage is increasing, in practical sense. 2. Calculation codes Calculation codes are essential for nuclear reactor op- eration. The codes can be divided into two main groups, macrocodes and microcodes (lattice codes). Macrocodes are used for the full core calculations with diffusion or simplified transport solution [8]. Mi- crocodes are used for exact full core transport calcula- tion or data preparation for macrocodes. Microcodes can be than divided into the deterministic codes [9] and stochastic codes [10]. PARCS PARCS calculation code is developed at Purdue Uni- versity [11] for 3D NPP core calculations. PARCS is Purdue Advanced Neutron Core Simulator code which offers various calculation methods based on the diffusion solver or simplified transport solution (SP3). Some of the solvers are based on the finite difference methods, others are based on the nodal methods. [8] In the study presented below, the finite difference method based on the diffusion solver (FDM, NEMMG) or simplified transport solver (SP3) are analysed. All cases were calculated with PARCS v3.3.1 code version. SCALE SCALE is a comprehensive calculation package which contain deterministic codes (NEWT, TRITON) and a stochastic code (KENO) [9]. The deterministic codes NEWT and TRITON are commonly used for the data preparation. Both these codes are based on the solu- tion of the transport equation in multigroup approach. XS can be prepared with actual flux spectrum or with B1 approximation. TRITON code is able to carried out burnup calculations in contrast with NEWT. All cases were calculated with SCALE 6.2.3 code ver- sion [12]. 3. Test case The model consisted of the simplified fuel assemblies and simplified fuel dummy filled with water was devel- oped as a test case. This model is not usual, however 24 vol. 28/2020 XS data it includes region for that the data preparation is com- plicated. The visualisation of the test case is shown in figure 1. This geometry can be found in the research reactors, where the water can be replaced by the re- search equipment like irradiation channels, detectors or other devices. Figure 1. Visualisation of the test case model Test case consists of the 3 elements types. The whole structure is placed in the infinite lattice of the same structures. Fuel assembly 1 (FA1) seems to be like a fuel assembly in the infinite lattice of the same assemblies, because each boundary is connected to the fuel assembly. Fuel assembly 2 (FA2) has three sides connected with the fuel assembly and one side is con- nected to the water fuel dummy. Water fuel dummy (DUM) is simulated with the same temperature and density conditions as the moderator inside the fuel assemblies (ρ = 0.71667 g/cm3, T = 578 K). Fuel assemblies consist of 17×17 fuel pins. Descrip- tion of the material and geometry of the test case is given in table 1. Two different enrichments of the fuel pins were simulated: 0.7% - sub-critical system and 4.3% super-critical system. Test case parameters Fuel diameter 0.82 cm Cladding diameter 0.95 cm Pin pitch 1.26 cm Fuel density 10.219 g/cm3 Moderator density 0.71667 g/cm3 Fuel temperature 1100 K Cladding temperature 600 K Moderator temperature 578 K Table 1. Description of the material and geometry of the test case 4. Calculation Many different approaches were realised during the analysis of behaviour in the test case. In the first stage, the influence of B1 approximation and noncritical flux were analysed. The next part of the study was focused on the downscatter treatment. Last part of the study deals with the multigroup approach and spectra comparison. Some of the diffusion codes are designed to use only downscattering, therefore the scattering cross sections are modified via: Σ̂g,h = Σg,h − Φh Φg Σh,g, (5) where the Σg,h, respective Σh,g is scattering cross section from group g to group h, respective from h to g. Φh, respective Φg is the neutron flux of group h, respective g. The compliance of the simplified solution calculated via PARCS was rated by the eigenvalue compliance, see equation (6) and the neutron flux values inside homogenised areas, see equation (7). The reference so- lution was obtained by the whole geometry simulation in the SCALE 6.2.3 calculation code. ∆keff = (keff,SCALE − keff,PARCS) · 105 (6) ∆Φg = √√√√ 1 N N∑ i=0 ( Φig,SCALE − Φ i g,PARCS Φig,SCALE 100 )2 (7) 4.1. Two group approach The two group XS were prepared with the SCALE TRITON calculation code in this section. The data for macrocode PARCS can be prepared by 2 ways, with PMAXS files and XSEC files. The data from SCALE TRITON calculation code was prepared by the GenPMAXS [13] program, which prepares PMAXS files for each region. The obtained results with PMAXS files showed good agreement, therefore the XSEC file format was not used during two group analyses. Many calculation modes and homogenization ap- proaches were carried during this analysis: • CASE1 - Data for each different part of model was prepared by simulation in the real geometry with the real boundary conditions with B1 correction. • CASE1_COR - CASE1 with downscatter correc- tion via equation (5). • CASE2 - Data for the fuel assemblies was prepared by simulation with the reflective boundary condi- tion, data for DUM was prepared by simulation in the real geometry with the real boundary conditions with B1 correction. • CASE2_COR - CASE2 with downscatter correc- tion via equation (5). • CASE3 - Data for FA1 was prepared by simulation with the reflective boundary condition, other data was prepared by simulation in the real geometry with the real boundary conditions with B1 correc- tion. 25 Pavel Suk Acta Polytechnica CTU Proceedings • CASE3_COR - CASE3 with downscatter correc- tion via equation (5). • CASE4 - CASE1 without B1 correction. • CASE5 - CASE2 without B1 correction. • CASE6 - CASE3 without B1 correction. • CASE7 - Data for fuel assemblies was prepared by the simulation with the reflective boundary con- dition with B1, data for DUM was prepared by the simulation in the real geometry with the real boundary conditions without B1 correction. • CASE8 - Data for FA1 was prepared by the simu- lation with the reflective boundary condition with B1, data for other regions were prepared by the sim- ulation in the real geometry with the real boundary conditions without B1 correction. 4.1.1. 0.7 % enrichment Fuel enriched only with 0.7% was simulated to in- vestigate agreement for the sub-critical core regions. The results of eigenvalue calculation and relative flux differences are shown in table 2, visualised in figures 2 and 3. Modes ∆keff (pcm) ∆Φ1 (%) ∆Φ2 (%) CASE1 51.8 4.52 1.85 CASE1_COR 71.7 4.56 1.92 CASE2 91.4 3.47 1.40 CASE2_COR -46.0 8.60 1.92 CASE3 8.6 4.23 1.67 CASE3_COR 26.0 4.26 1.73 CASE4_COR 189.5 7.92 2.61 CASE5_COR 183.7 6.15 2.00 CASE6_COR 127.6 7.34 2.30 CASE7_COR 66.3 4.11 1.15 CASE8_COR 59.1 6.15 2.00 Table 2. Two group results of the the test case with fuel enrichment 0.7% (reference keff = 0.82853) -50 -25 0 25 50 75 100 125 150 175 200 CASE1 CASE1_COR CASE2 CASE2_COR CASE3 CASE3_COR CASE4_COR CASE5_COR CASE6_COR CASE7_COR CASE8_COR k e ff d if fe re n ce ( p cm ) keff relative difference in each case for 0.7 % enrichment fuel type Figure 2. Relative eigenvalue differences in the cal- culation modes with 0.7% fuel enrichment test case In case of eigenvalue, the CASE1, CASE2_COR, CASE3 and CASE3_COR were in the best agreement with the lattice code. On the opposite, the cases with- out B1 correction (CASE4, CASE5 and CASE6) were 0 1 2 3 4 5 6 7 8 9 CASE1 CASE1_COR CASE2 CASE2_COR CASE3 CASE3_COR CASE4_COR CASE5_COR CASE6_COR CASE7_COR CASE8_COR P d if fe re n ce ( % ) Relative power difference in each case for 0.7 % enrichment fuel type g1 g2 Figure 3. Relative power differences in the calcula- tion modes with 0.7% fuel enrichment test case 1 4 -2.19 0.49 4 5 -1.88 0.14 7 6 -2.19 0.49 2 3 -1.88 0.14 5 10 -11.30 -4.89 8 7 -1.88 0.14 3 2 -2.19 0.49 6 1 -1.84 0.15 9 8 -2.19 0.49 1 2 3 1 2 3 Legend N Type Group 1 Group 2 Figure 4. 2D flux differences for each part of the test case in CASE3 and enrichment 0.7% in the worse agreement. The neutron flux compari- son showed that the thermal neutron group deviates around 2% in all cases, but the fast neutron group flux difference reached up to 9% in CASE2_COR, CASE4_COR and CASE6_COR. The best results in case of eigenvalue and Φg are in CASE3. 2D flux differences of CASE3 are shown in figure 4. Figure 4 shows the best agreement in the thermal group (only 0.49%) for fuel assemblies and only 2% deviation in the fast energy group. The neutron fluxes in the fuel dummy region inside the test case were determined with worse agreement. Up to 5% for thermal energy group and up to 11.5% for the fast energy group. 4.1.2. 4.3 % enrichment The nuclear fuel in the NPP cores is in general en- riched up to 5%. The fuel pellets with enrichment 4.3% are usually in the active length of the fuel pins and, moreover, they make the simulated system super- critical. The results of the eigenvalue and the relative flux differences are shown in the table 3, visualised in figures 5 and 6. The best agreement in case of eigenvalue were reached in the cases without B1 correction (CASE4_COR, CASE5_COR, and CASE6_COR). On the other hand, the fast neutron flux differences 26 vol. 28/2020 XS data Modes ∆keff (pcm) ∆Φ1 (%) ∆Φ2 (%) CASE1 688.6 5.17 4.89 CASE1_COR 690.6 5.14 4.97 CASE2 767.6 5.43 4.74 CASE2_COR 730.6 5.49 4.87 CASE3 625.6 5.16 4.81 CASE3_COR 626.6 5.14 4.89 CASE4_COR 173.6 7.82 4.61 CASE5_COR 121.6 9.61 4.52 CASE6_COR 63.6 8.23 4.41 CASE7_COR 792.6 5.15 5.84 CASE8_COR 418.6 5.93 6.36 Table 3. Two group results of the test case with fuel enrichment 4.3% (reference keff = 1.32588) 0 100 200 300 400 500 600 700 800 CASE1 CASE1_COR CASE2 CASE2_COR CASE3 CASE3_COR CASE4_COR CASE5_COR CASE6_COR CASE7_COR CASE8_COR k e ff d if fe re n ce ( p cm ) keff relative difference in each case for 4.3 % enrichment fuel type Figure 5. Relative eigenvalue differences in the cal- culation modes with 4.3% enrichment fuel test case 0 1 2 3 4 5 6 7 8 9 10 11 CASE1 CASE1_COR CASE2 CASE2_COR CASE3 CASE3_COR CASE4_COR CASE5_COR CASE6_COR CASE7_COR CASE8_COR P d if fe re n ce ( % ) Relative power difference in each case for 4.3 % enrichment fuel type g1 g2 Figure 6. Relative power differences in the calcula- tion modes with 4.3% enrichment fuel test case were the highest in that three cases. The 2D flux differences of CASE6_COR is shown in figure 7. CASE6_COR model showed the best result in case of eigenvalue, with difference only 63 pcm. Figure 7 shows relatively good agreement in thermal neutron flux in fuel assemblies and fast neutron flux in DUM, only up to 3%. Fast neutron flux in fuel assemblies and thermal neutron flux in the DUM were determined with worse agreement, up to 11%. 1 4 8.63 -2.28 4 5 8.76 -3.21 7 6 8.63 -2.28 2 3 8.76 -3.21 5 10 -2.08 -10.62 8 7 8.76 -3.21 3 2 8.63 -2.28 6 1 8.78 -3.23 9 8 8.63 -2.28 1 2 3 1 2 3 Legend N Type Group 1 Group 2 Figure 7. 2D flux differences for each part of the test case in CASE6_COR and enrichment 4.3% 4.1.3. Discussion The above results showed that the data preparation based on the actual boundary condition cannot bring good results, because the PARCS calculation code does not know the actual neutron leakage from the system and due to it there is lack of the informations to properly simulate the system. The best agreement in the lower enriched case was obtained with the B1 correction. This situation cor- responds with the physical solution. If the solved problem is sub-critical, the B1 approximation simu- lates less neutron leakage to reach criticality during the data preparation process. Because the each part of the test case is "locally" sub-critical, it means that the neutron spectra is changed in each part with the same direction and due to that the results are better with B1 approximation. The situation in the higher enriched case is different. The whole test case is super-critical, but the water fuel dummy is "locally" sub-critical. The B1 correction increases neutron leakage from the whole system, but the water fuel dummy is "locally" sub-critical. The B1 correction changes the neutron spectra in both regions with the same direction and due to that the neutron spectra in the water fuel dummy is very different during the XS preparation. The better results were obtained without B1 correction and it also corresponds with the above theory. 4.2. Multigroup approach To better understand the behaviour of the water-fuel neutron transport in the macrocodes, there was pres- sure to analyse and compare multigroup neutron flux in the PARCS and SCALE code systems. The fuel assembly with the mirror boundary conditions was analysed in the multi group approach. The neutron flux is constant in the assembly with the mirror boundary condition and due to it only eigen- value was analysed. As it was told before, PARCS calculation code offers to use two types of libraries structures (PMAXS and XSEC). The eigenvalues were calculated with many different ways: • XSEC - XSEC without downscatter correction, • XSEC COR - XSEC with downscatter correction, 27 Pavel Suk Acta Polytechnica CTU Proceedings -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 2 4 8 16 23 40 56 R el at iv e d if fe re n ce ( p cm ) Group number keff relative difference between SCALE and PARCS XSEC CORR B1 XSEC B1 XSEC CORR without B1 XSEC without B1 Figure 8. keff relative deviation with XSEC input files • PMAXS - PMAXS without downscatter correction, • PMAXS COR - PMAXS with downscatter correc- tion. Results of the simulations are in table 4. The PMAXS input format brings better results in two group approach, but the large deviations from the reference solution was found in many energy group analysis. The reference solution of the test case geom- etry was calculated with SCALE TRITON sequence keff = 0.86041. It seems that the PMAXS input for- mat is well prepared for two group calculations, which are most common in the full core calculations, but it is absolutely inappropriate for multi groups calculation. The XSEC input format seems to be more appro- priate for multigroup analysis. There are only small deviations between diffusion solution and appropriate transport solution. The highest deviation with the B1 correction is in case of XSEC for two group cal- culation. This deviation is caused by the fact, that PARCS was used in the diffusion mode and the in- puts scatter matrix is limited into the downscatter from fast to thermal group. Due to that fact, the two group solution needs to be corrected by downscatter- ing via the equation (5). When the downscatter was corrected, the results were similar to those obtained with PMAXS COR. This situation can be seen in both simulations (with B1 and without B1 approach). Deviations between the XSEC and XSEC COR for more group approach are smaller, because for more group calculations, there are total scatter matrix input data. Due to it there is no necessary to use correction via equation (5). Based on this fact, the XSEC without downscatter correction input format is used for other more group calculations. The relative difference of keff between SCALE and PARCS for XSEC input format is in figure 8. The convergence problem with the calculation were found in case of higher group number. The calculation did not converge automatically, but after it reached maximum number of iteration, it finished and gave a result without any warning message. The influence of B1 correction decreases with higher number of energy groups. This behaviour begins dur- ing the data preparation process by microcode SCALE TRITON, because the group structure for the data preparation process was 56 group ENDF/B-VII.0 li- brary and microcode SCALE prepares critical spectra in 56 groups. When user wants to calculate XS with B1 leakage correction in 56 groups, the collapsing to the energy groups is independent on the neutron spectra. 4.3. Neutron flux spectra comparison The neutron spectra was created to better understand the behaviour in the fuel regions near to the water fuel dummy region. The spectra was calculated with 56 energy groups. There was also the same convergence complications as were described in the previous sec- tions. Because of the results of two group calculation, there was analysed only three cases without B1 leak- age correction (CASE4, CASE5 and CASE6). The data for scattering were not corrected for downscat- tering, because the results from section 4.2 showed that the downscattering correction is not necessary. The neutron flux spectra comparison gives the in- formation about the spectra and slowing down in the SCALE TRITON and PARCS codes. The spectra were analysed separately in fuel regions and in the DUM region. 4.3.1. 0.7 % Enrichment The test case with 0.7% fuel enrichment represents the sub-critical problem. The neutron spectra in the fuel assembly FA1, respective in the DUM region are in figures 9, respective 10. The neutron flux spectra 28 vol. 28/2020 XS data Groups With B1 Without B1 PMAX COR PMAX XSEC COR XSEC PMAX COR PMAX XSEC COR XSEC 2 -38 -47 -38 -389 25 25 25 -323 3 -2296 -2307 -93 -106 -1950 -1950 229 228 4 -2168 -2174 40 33 -1952 -1952 231 230 8 -2192 -2195 -7 -10 -1954 -1953 206 207 16 -2110 -2112 77 74 -1955 -1955 205 205 23 -31510 -31512 112 117 -31479 -31479 141 147 40 -31495 -31496 63 68 -31479 -31479 76 83 56 -31985 -31987 -21 -20 -31987 -31987 -21 -20 Table 4. keff relative difference between SCALE and PARCS with multigroup approach for 0.7 % enriched fuel assembly with mirror boundary condition 1x10-12 1x10-10 1x10-8 1x10-6 0.0001 0.01 0.01 0.1 1 10 100 1000 10000 100000 1x10 6 1x10 7 Fl u x (c m -2 s- 1 ) Energy (eV) Neutron spectra in FA1 SCALE CASE4 CASE5 CASE6 Figure 9. Neutron flux spectra in the fuel assembly FA1 with 0.7% fuel enrichment 1x10-12 1x10-10 1x10-8 1x10-6 0.0001 0.01 0.01 0.1 1 10 100 1000 10000 100000 1x10 6 1x10 7 Fl u x (c m -2 s- 1 ) Energy (eV) Neutron spectra in DUM SCALE CASE4 CASE5 CASE6 Figure 10. Neutron flux spectra in the DUM with 0.7% fuel enrichment in FA1 and FA2 are similar and the differences are not visible in the spectra comparison. The eigenvalues and their differences from the reference calculation are in table 5. Main result from figures 9 and 10 is that the fuel spectrum in the fuel assemblies is well calculated with the PARCS calculation code. Differences between the PARCS and SCALE calcu- lation were analysed via equation: ∆Φg = ΦSCALEg − ΦPARCSg ΦSCALEg (8) The neutron spectra differences in figures 11, 12 -14 -12 -10 -8 -6 -4 -2 0 2 4 0.01 0.1 1 10 100 1000 10000 100000 1x10 6 1x10 7 fl u x d if fe re n ce ( % ) Energy (eV) Neutron spectra diferences between SCALE and PARCS in FA1 PARCS CASE4 PARCS CASE5 PARCS CASE6 Figure 11. Relative difference in spectra between SCALE and PARCS calculation for FA1 for 0.7% fuel enrichment -2 0 2 4 6 8 10 12 0.01 0.1 1 10 100 1000 10000 100000 1x10 6 1x10 7 fl u x d if fe re n ce ( % ) Energy (eV) Neutron spectra diferences between SCALE and PARCS in FA2 PARCS CASE4 PARCS CASE5 PARCS CASE6 Figure 12. Relative difference in spectra between SCALE and PARCS calculation for FA2 for 0.7% fuel enrichment and 13 show that the spectrum calculated in the FA1 has the large deviations in the slowing down region and particularly in the high energy neutron region. The FA2 has great deviations only in the high energy neutron region. In the fuel assemblies, the relative deviations are only up to 12% in the absolute val- ues, but the deviations in the DUM region are up to 225%. That means that the data prepared for the DUM region are not prepared well and the behaviour of neutrons near the DUM region is not described correctly. 29 Pavel Suk Acta Polytechnica CTU Proceedings -250 -200 -150 -100 -50 0 50 100 0.01 0.1 1 10 100 1000 10000 100000 1x10 6 1x10 7 fl u x d if fe re n ce ( % ) Energy (eV) Neutron spectra diferences between SCALE and PARCS in REF PARCS CASE4 PARCS CASE5 PARCS CASE6 Figure 13. Relative difference in spectra between SCALE and PARCS calculation for DUM for 0.7% fuel enrichment 4.3.2. 4.3 % Enrichment Fuel pins with 4.3% enrichment represent the super- critical problem, which, based on the information obtained during the two group study, can brings more deviated results. The eigenvalue differences are in table 5. The neutron spectra were similar to the neutron spectra in the lower enrichment case with little bit higher discrepancies. Mode ∆keff 0.7% (pcm) ∆keff 4.3% (pcm) CASE4 120 923 CASE5 24 783 CASE6 61 837 Table 5. keff relative difference between SCALE and PARCS with 56 groups calculation 4.4. SP3 multigroup solution The PARCS calculation code is able to calculate simpli- fied transport approach (SP3) instead of the diffusion approach. The NEMMG solver with the nspn 3 was used to calculate the test geometry. The only appropriate approach of CASE4, CASE5 and CASE6 were calculated. The results of the eigen- value for both enrichment cases are in table 6. Mode ∆keff 0.7% (pcm) ∆keff 4.3% (pcm) CASE4 -52 732 CASE5 -148 590 CASE6 -111 646 Table 6. keff relative difference between SCALE and PARCS with 56 groups calculation and SP3 The neutron spectra for SP3 calculations are very similar to the neutron spectra calculated with diffusion approach. The main difference is in the higher ener- gies in the FA2 fuel assembly with 0.7% enrichment. The neutron spectra are more accurate in the higher energies in comparison with the diffusion approach for higher enriched fuel. The eigenvalue deviations increased in comparison to the diffusion approach in lower enrichment case, but deviation decreased with higher enrichment case. 5. Conclusion Various different approaches were analysed in this pa- per. Two different enrichments were studied. Results of the two group calculation showed that in case of lower enrichment, the data prepared by the simulation of the FA1 fuel assembly in the infinite lattice and FA2 and DUM in the real geometry with B1 approximation (CASE3) well reproduce the eigenvalue. The relative flux differences were also calculated with the smallest difference in the CASE3. On the other hand, the cases, which were calculated without B1 approximations gave more-times worse results in case of lower enrichment. The results were opposite in the case of higher enrich- ment than in the lower enrichment calculation. The best agreement were obtained during the calculation without B1 approximation and the same data prepara- tion scheme (CASE6). The eigenvalue was bellow 100 pcm from reference calculation, but the fast group neu- tron fluxes were calculated with higher difference (up to 10%) in each calculation without B1 approximation. The upper results are consistent with B1 correction. In the case of lower enrichment, the situation corre- sponds with the physical solution. The whole geome- try is sub-critical and B1 approximation decreases the neutron leakage to reach criticality during the data preparation process. In case of higher enrichment, the whole geometry is super-critical, but there is part which is "locally" sub-critical - DUM. The B1 approximation increases neutron leakage from the system to reach the criticality, but this does not correspond with the physi- cal solution in the DUM part. The neutron spectra is changed there and due to that the B1 correction cannot brings better results in the case of higher enrichment. The main objective of the spectra study is that PMAXS files, which PARCS use newly, are not able to calculate core with multigroup approach. The eigen- value differences showed that there is any problem with data preparation for that calculation. The results of calculations with XSEC file format bring reasonably results of eigenvalue despite the convergence problems. Based on the results of multigroup study, the neutron spectra in multigroup approach were calculated for test cases without B1 approximation. Calculations without B1 approximation were chosen because the final geometry was not critical and there are still unsolved problems withB1 approximation. The paper [7] informs that the B1 calculation is not rigorous and due to the reaching critical spectra by the changing absorption in each energy group, the result are not representative for LWR calculations, where the criticality is reached by the absorption of only thermal neutrons. The neutron spectra in the fuel assemblies were de- termined with a good agreement with the reference 30 vol. 28/2020 XS data calculation, but the spectra in the DUM was deter- mined with larger discrepancies. It is well known, that the diffusion theory is not fulfilled in the water fuel dummy region and the discrepancies are mostly based on this fact. There are also methods, which can bring better results, for instance the influence of neutron scatter anisotropy in the DUM. The different approach of the DUM preparation data will be study in the future. 6. Acknowledgement This work was supported by the project no. CZ.02.1.01/0.0/0.0/16_013/0001790 supported by Op- erational Programme Research, Development and Ed- ucation co-financed from European Structural and In- vestment Funds and from state budget of the Czech Republic and it was supported by the Grant Agency of the Czech Technical University in Prague, grant no. SGS19/114/OHK4/2T/14. List of symbols Φi(E) Neutron flux of region i and energy E [cm−2s−1] Φi,g Neutron flux of region i and energy group g [cm−2s−1] σi,g Microscopic cross section of region i and group g [cm2] σi(E) Microscopic cross section of region i and energy E [cm2] Σi,g Macroscopic cross section of region i and group g [cm−1] Ni Atomic density of region i [cm−3] Eg, min Minimal energy of boundary related to the energy group g [eV] Eg, max Maximal energy of boundary related to the energy group g [eV] N Number of regions in the test case [–] Σt,g Total macroscopic cross section for energy group g [cm−1] B Buckling factor [cm−1] Jg Neutron current in group g [cm−2s−1] χg Neutron spectra in group g [–] Σf,h Fission macroscopic cross section [cm−1] Σ0s,h→g Scattering macroscopic cross section [cm −1] keff,X Eigenvalue calculated with mode X [–] ∆keff Difference of the eigenvalue [pcm] ∆Φg Relative difference of Φg for the whole test case [%] References [1] S. Choi, K. S. Smith, H. Kim, et al. 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Xu. GenPMAXS v6.2 Code for Generation the PARCS Cross Section Interface File PMAXS. RES/U.S. NRC, Rockville, Md, 2016. http: //nuram.engin.umich.edu/software/genpmaxs/. 31 https://doi.org/10.1080/00223131.2017.1299648 https://doi.org/10.1016/j.anucene.2006.05.008 https://doi.org/10.14311/APP.2018.19.0014 https://doi.org/10.1088/1742-6596/781/1/012029 https://doi.org/10.1016/j.nucengdes.2014.03.044 https://doi.org/10.1016/j.pnucene.2017.06.013 https://engineering.purdue.edu/PARCS http://nuram.engin.umich.edu/software/genpmaxs/ http://nuram.engin.umich.edu/software/genpmaxs/ Acta Polytechnica CTU Proceedings 28(0):1–9, 2020 1 Data preparation process and reflector calculation 1.1 Macroscopic cross section preparation 1.2 B1 correction 2 Calculation codes 3 Test case 4 Calculation 4.1 Two group approach 4.1.1 0.7 % enrichment 4.1.2 4.3 % enrichment 4.1.3 Discussion 4.2 Multigroup approach 4.3 Neutron flux spectra comparison 4.3.1 0.7 % Enrichment 4.3.2 4.3 % Enrichment 4.4 SP3 multigroup solution 5 Conclusion 6 Acknowledgement List of symbols References