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Initial verification of AZNHEX hexagonal-z neutron diffusion code with MCNP6 for two different study cases
Progress in Nuclear Energy 106 (2018) 284–292
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Initial verification of AZNHEX hexagonal-z neutron diffusion code with MCNP6 for two different study cases
T
Juan Galicia-Aragóna, Juan-Luis Françoisa,∗, Guillermo E. Bastida-Ortiza, Cecilia Martín-del-Campoa, Julio A. Vallejo-Quinteroa, Edmundo del-Valle-Gallegosb a
Universidad Nacional Autónoma de México, Facultad de Ingeniería, Departamento de Sistemas Energéticos, Av. Universidad 3000, C.U., 04510 Ciudad de México, Mexico b Instituto Politécnico Nacional, Escuela Superior de Física y Matemáticas, Av. IPN s/n, San Pedro Zacatenco, 07738 Ciudad de México, Mexico
A R T I C LE I N FO
A B S T R A C T
Keywords: Neutron diffusion code Reactor analysis Hexagonal-z geometry Finite element method
The core of the AZTLAN Mexican reactor analysis platform consists of a thermo-hydraulic code (AZTHECA), a neutron transport code (AZTRAN) and two neutron diffusion codes, namely AZKIND and AZNHEX. These codes are currently in the testing phase by simulating a variety of fuel assemblies and nuclear reactor cores to compare and verify their results with those obtained by codes globally used in the nuclear community such as SERPENT, CASMO/SIMULATE, MCNP and others. The main objective of this task is to improve future versions of the AZTLAN platform codes to obtain reliable results for the user. To verify the current version of AZNHEX neutron diffusion code, two cases were considered. The simulation of a WWER-440 fuel assembly was the first case analyzed, and the second case was the core of a sodium-cooled fast reactor in steady state, with control rods fully withdrawn. The comparison and verification of the results (neutron multiplication factor, axial and radial power distributions and radial neutron flux distribution) for both cases were done employing MCNP6 Monte Carlo code. Certain deviations were found between AZNHEX and MCNP6, since AZNHEX is a diffusion code and it is being compared against a Monte Carlo code. However, it is shown that the results provided to the user are reliable since they exhibit a good degree of fidelity.
1. Introduction Nuclear community has the goal to develop computer platforms for analysis and design of nuclear reactors to represent the different physical phenomena that occur in a nuclear reactor. Since 2005, the European Reference Simulation Platform for Nuclear Reactors, called NURESIM, has been developed (Chauliac et al., 2011). This platform provides an accurate representation of the physical phenomena by promoting and incorporating the latest advances in core physics, twophase thermal-hydraulics and fuel modeling. It includes multi-scale and multi-physics features, especially for coupling core physics and thermal-hydraulics models for reactor safety analysis. Quantitative deterministic and statistical sensitivity and uncertainty analyses tools are developed and provided through the platform. NURESIM includes generic pre-processing, post-processing and supervisory functions through the open-source SALOME software, which make the codes more user-friendly. In order to accelerate nuclear energy modeling and simulation (M&
S), the U.S. Department of Energy (DOE) established the Consortium for Advanced Simulation of Light Water Reactors (CASL) as its first DOE Energy Innovation Hub in July 2010 (CASL, 2017). This project was established to provide leading edge modeling and simulation capability to improve the performance of currently operating light water reactors. CASL is developing the Virtual Environment for Reactor Applications, VERA. CASL's VERA software simulates nuclear reactor physical phenomena using coupled multi-physics models. VERA's current physics capabilities include neutron transport, thermal-hydraulics, fuel performance, and coolant chemistry. The AZTLAN platform project (Gómez-Torres et al., 2015) emerged as an initiative to rank Mexico at a competitive level in software development for the analysis of nuclear reactors, and therefore, achieve autonomy in this area. The AZTLAN platform is composed of a thermohydraulic code (AZTHECA), a neutron transport code (AZTRAN) and two neutron diffusion codes (AZKIND, AZNHEX). AZNHEX (Esquivel and López, 2016) is a hexagonal-z time-dependent 3D diffusion code whose brief description will be given in the next
∗
Corresponding author. E-mail addresses: [email protected] (J. Galicia-Aragón), jlfl@fi-b.unam.mx (J.-L. François), [email protected] (G.E. Bastida-Ortiz), [email protected] (C. Martín-del-Campo), [email protected] (J.A. Vallejo-Quintero), [email protected] (E. del-Valle-Gallegos). https://doi.org/10.1016/j.pnucene.2018.03.017 Received 15 October 2017; Received in revised form 19 March 2018; Accepted 22 March 2018 0149-1970/ © 2018 Elsevier Ltd. All rights reserved.
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section. The starting version of AZNHEX arose as a product of a master's thesis carried out in 2015 (Esquivel, 2015), although it is an extension to 3D cores of the original method developed in 1997 (Hennart et al., 1997) for 2D cores. Since 2015, AZNHEX has been improved thanks to the feedback provided to the developers from the AZTLAN User's group. With the current version of this code, it is possible at the present to simulate and analyze light water reactors, such as the Russian WWER and a variety of fast reactors. Nonetheless, to guarantee its fidelity it is necessary to perform several tests with the current version of AZNHEX. This is important for the suggestions that the User's group may give to the developers. The verification of AZNHEX will help identify deficiencies in the methods and/or the implementation, thereby improving the quality of the code. The structure of the article is as follows. A brief description of AZNHEX is given in Section 2. The test cases with results are presented in Section 3. Finally, conclusions and future directions are given in Section 4. 2. AZNHEX description 2.1. Methodology AZNHEX code relies on several works headed by Professor J.P. Hennart related with nodal methods (Hennart, 1986; Hennart et al., 1988; Hennart and Del Valle, 1993; Hennart et al., 1996; Hennart et al., 1997; Hennart et al., 2003). It is based on a composite nodal method for prismatic hexagons, applied to solve the neutron diffusion equations for cores built by the union, side by side, of hexagonal prisms; each prismatic hexagon represents a sub-assembly. A standard finite element method is used starting by the strong form to get the corresponding weak form. A detailed description of the nodal method used in AZNHEX can be found in Del Valle-Gallegos et al. (2018). AZNHEX solves numerically the time dependent neutron diffusion equations in Hexagonal-Z geometry. The equations are given by a set of G + I partial differential equations (Duderstadt and Hamilton, 1976). The first equations correspond to the neutron density rate balance for each one of the G energy groups:
1 ∂ϕg − ∇ ·Dg ∇ϕg + vg ∂t
G
∑Rg ϕg = ∑
Fig. 1. Hexagonal-z array for AZNHEX.
G
Σsg ′→ g ϕg ′ + χg (1 − β )
∑ ν ∑fg ′ ϕg ′
g ′= 1
g ′≠ g I
+
∑ χgi λi Ci, i=1
g = 1, .., G (1a)
Then, for the I groups of delayed neutron precursor concentrations we have:
∂Ci = βi ∂t
G
∑ g ′= 1
νΣfg ′ ϕg ′ − λi Ci, i = 1, ..., I
(1b)
Where:
• Φg: scalar neutron flux corresponding to the energy-group g [n/cm s] • Dg: diffusion coefficient corresponding to the energy-group g [cm] • G: total number of energy-groups • Σ : removal cross section corresponding to the energy-group g [cm ] • Σ : scattering cross section from energy-group g' to g [cm ] • ν: average number of neutrons produced per fission • Σ : fission cross section corresponding to the energy-group g [cm ] χ • : probability that prompt neutrons arising from fissions appear in energy-group g • λ : decay constant of the delayed neutron precursor i • β: delayed neutron fraction 2
Rg
AZNHEX solves Eq. (1) for hexagonal-z domains, as it shown in Fig. 1 (3D) and Fig. 2 (2D). In order to accomplish this, the method first applies a Gordon-Hall transfinite interpolation (Gordon and Hall, 1973a, 1973b) to each one of the four quadrants shown in Fig. 3a, to transform it into a cube as it is shown in Fig. 3b. Once this is done, the classic Galerkin's Finite Element Method is applied to approximate the scalar neutron flux, and the neutron precursor concentrations, by a polynomial interpolation over a reference cell [-1,+1]x [-1,+1]x [-1, +1]. This interpolation is known as the Raviart-Thomas-Nédelec interpolation of index zero: RTN-0 (Hennart et al., 1988). For the time variable, AZNHEX has implemented themethod. The resulting algebraic system is solved on each time step by using the BiCGSTAB method (Van
−1
sg´ → g
fg
Fig. 2. A cross section view of the hexagonal array for AZNHEX.
−1
−1
g
i
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Fig. 3. Gordon-Hall transformation for a hexagonal prism. a) Hexagonal prism is divided in four quadrants; b) One quadrant is transformed in a cube.
homogenized multi-group cross sections and the diffusion coefficients needed by AZNHEX to solve the nodal diffusion equations in G energy groups. For the test cases presented in this article SERPENT v.2.1.28 (Leppänen et al., 2015) was used. One of the main advantage of SERPENT over other Monte Carlo codes is its proved capability for producing homogenized group constants in a user-defined energy multi-group structure. The methodology used in Serpent for spatial homogenization is described in Leppänen et al. (2016). Several studies have been published showing the effectiveness of SERPENT as a tool to generate nuclear data libraries for nodal diffusion codes (Rachamin et al., 2013; Fridman and Shwageraus, 2013; Nikitin et al., 2015; Smith, 2017; Jo et al., 2018). It is out of the scope of this work to do an investigation on the best SERPENT's option for cross section homogenization, therefore, we used the infinite spectrum homogenization method. For the WWER fuel lattice test case, the cross sections set (XS data) used by AZNHEX was produced from a heterogeneous model of the fuel lattice in SERPENT (like that showed in Fig. 5). XS data were generated for the homogenized unit fuel cells and the central water thimble, in two energy groups, using a cut-off at 0.625 eV and the JEFF-3.1 library. Fig. 4 shows the procedure to transfer the XS data from SERPENT to AZNHEX. Similarly, for the sodium-cooled fast reactor core case, a detailed heterogeneous full core model was set up in SERPENT to produce assembly-wise XS data for AZNHEX in seven energy groups, according to Palmiotti (2011) energy structure. In this case, the JEFF-3.2 library was used. Once this calculation was done, the homogenized XS data for each assembly type were transferred to AZNHEX.
der Vorst, H. A., 1992) once the initial condition is obtained and a given transient is defined. The θ-method, also known as weighted method, was introduced by Henry and Vota (1965) and then some other authors applied this method to discretize the time dependence of the neutron diffusion or transport kinetic equations (Hansen et al., 1967; Yasinsky et al., 1968; Stacey, 1969) and there have been several review papers that mention the θ-method, for instance Chae (1979) and Sutton and Aviles (1996). The method as such covers the so called fully explicit (θ = 0), implicit (θ = 1), and semi-implicit (θ = 0.5) methods. Regarding the eigenvalue problem it is solved by using the standard power iterations method when there is no up-scattering, and with the Wielandt accelerated power iterations when there is up-scattering. Other acceleration techniques like the Coarse Mesh Finite Difference method (CMFD) will be included in a new version of AZNHEX. It is worthwhile to mention that at this stage of the code verification that only the steady state capability is tested and presented in this article. Further work will be devoted to test the transient mode of AZNHEX.
2.2. Cross section generation Given that AZNHEX is a nodal code, no heterogeneous pin by pin calculations are possible; this is a calculation of a lattice of homogeneous unit cells composed of fuel pin, clad and water, i.e. a pincellhomogenized assembly calculation. Due to this, homogeneous unit cells nuclear properties (see Fig. 4) must be provided to it. In our methodology, the Monte Carlo SERPENT code is used to generate spatially-
Fig. 4. XS data generation using SERPENT and its transfer to AZNHEX.
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Fig. 5. WWER-440 fuel assembly, MCNP6 heterogeneous model.
Fig. 6. WWER-440 fuel assembly, AZNHEX homogeneous model.
Table 1 Fuel characteristics of the WWER-440 fuel assembly. Number of fuel rods
Composition (wt %)
Temperature (K)
Pitch (cm)
126
3.175 (U-235) 84.980 (U-238) 11.850 (O-16)
900
1.23
Table 3 Values of the kinf obtained for the WWER-440 reactor assembly.a
k-inf
AZNHEX
MCNP6
Relative difference (pcm)
1.31786
1.31478 ± 0.00007
−234
a For the simulation with MCNP6, 500,000 neutron histories with the first 50 cycles discarded and 200 active cycles were used.
3. Test cases SERPENT-2 for AZNHEX are shown in Table 2. Fig. 6 shows the AZNHEX homogeneous model of the WWER-440 fuel lattice. The specular reflection boundary condition is applied in each of the outer borders of the lattice. To make a more accurate comparison, the assembly was simulated without channel and without water in the boarders, since they cannot be modeled in AZNHEX.
Since AZNHEX is intended to be used as the AZTLAN's lattice code and core simulator in hexagonal geometry, the current version was tested with two cases. The first one is a fuel lattice of a WWER-440 like fuel assembly. This case was considered because it represents a “classic” lattice calculation (2-D thermal reactor fuel lattice using two neutron energy groups). This WWER-440 fuel lattice was taken from the input exercises of SERPENT code. The second case is a sodium-cooled fast reactor core (López-Solis and François-Lacouture, 2015). This example represents a more complex case compared with the fuel lattice case. This is a relative small core with three different 235U enrichment zones (one blanket), and steel radial and axial reflectors. To perform the simulations of both cases, and asses the results of the current version of AZNHEX, the Monte Carlo code MCNP6 (MCNP6, 2014) was used. This code is one of the most recognized and employed in the nuclear industry; due to this, it was selected to calculate the reference parameters of this work.
3.1.1. Criticality calculations Table 3 shows the results of the infinite neutron multiplication factor (kinf) and the relative difference with respect to the value calculated with MCNP6. As it can be seen, there is a very good approximation of the kinf considering the assumptions considered in both models. 3.1.2. Radial power distribution Fig. 7 shows the results of the normalized radial power distribution calculated with AZNHEX (values in blue) and the relative errors in percent with respect to MCNP6 (values in red). Relative errors shown in Fig. 7 indicate the accuracy of AZNHEX of the power values of each fuel rod. The highest differences with respect to MCNP6 are in the central zone, around the water hole where a high neutron flux gradient is expected (see Fig. 8). However, relative differences do not exceed 1.9%.
3.1. Fuel assembly of a WWER-440 reactor Fig. 5 shows the fuel assembly simulated with MCNP6, which is our reference case in this study. The characteristics of the 126 fuels rods are shown in Table 1. The water and the fuel cladding have a temperature of 600 K. The libraries used in this case in MCNP6 were ENDF/B-VI.2 for some isotopes and JEFF-3.2 for others. The XS data generated with
3.1.3. Radial neutron flux distribution Fig. 8 shows the normalized one group neutron flux values obtained
Table 2 XS data generated with SERPENT-2 for the WWER-440 fuel assembly. MATERIAL
NEUTRON ENERGY GROUP
DIFF. COEFF.
REMOVAL XS
ν*FISSION XS
κ*FISSION XS
FISSION SPECTRUM (χ)
SCATT. MATRIX
Fuel cell
1 2 1 2
1.450 0.411 1.750 0.344
0.0252 0.0991 0.0249 0.0100
0.00801 0.1710 0.0000 0.0000
0.6410 14.200 0.0000 0.0000
1.0000 0.0000 0.0000 0.0000
0.4870 0.00270 0.5830 0.00187
Water
287
0.0144 1.0900 0.0243 2.5600
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Table 4 Characteristics of the sodium-cooled fast reactor. Materials
Enrichment (at%)
Temperature (K)
No. of assemblies
Fuel (zone 1-orange) Fuel (zone 2-red) Fuel (blanket-yellow) Steel Sodium
9.5
1200
18
14.25
1200
36
0.35
1200
66
– –
600
97
600
–
with AZNHEX (values in blue) and the relative errors in percent with respect to MCNP6 (values in red). As it can be seen in Fig. 8, the results of AZNHEX for the radial neutron flux distribution are very accurate, with errors below 0.123%. This shows that if the user desires to simulate the assembly of a thermal reactor, the results obtained will be highly reliable. 3.2. Sodium-cooled fast reactor core Table 4 presents the characteristics of a sodium-cooled fast reactor and Fig. 9 shows a view of the core (López-Solis and FrançoisLacouture, 2015). As it can be seen, this reactor has three different 235U enrichment zones (one blanket), steel radial and axial reflectors, and seven steel assemblies were considered in the control rods locations. The library used in this case in MCNP6 was JEFF-3.2. The XS data generated with SERPENT-2 are shown in Table 5.
Fig. 7. Normalized radial power distribution for the WWER-440 fuel assembly.
3.2.1. Criticality calculations Table 6 shows the results of the effective neutron multiplication factor (keff) and the relative difference with respect to the value calculated with MCNP6. As in the previous case, the keff value calculated with AZNHEX presents an acceptable approximation having a difference with respect to MCNP6 of 358 pcm. A quarter of the core was also simulated, and the neutron multiplication factor was compared with that obtained considering full core. The motivation for running a quarter-core geometry is to simplify the geometry by taking advantage of symmetry. This will reduce runtime and simplify the layout. Table 7 presents the comparison of the neutron multiplication factor. As shown in the previous table, the specular reflection condition works correctly if the user wants to simulate a simplified case due to the complexity of a symmetric reactor core. If the core is complex (i.e., asymmetric), then full-core representation is necessary. 3.2.2. Radial power distribution Fig. 10 shows the results obtained with MCNP6 (values on the top), AZNHEX (values on the middle) and the relative errors with respect to MCNP6 (values on the bottom). Due to the symmetry of this reactor,
Fig. 8. Normalized radial neutron flux distribution for the WWER-440 fuel assembly.
Fig. 9. Sodium-cooled fast reactor core. 288
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Table 5 XS data generated with SERPENT-2 for the sodium-cooled fast reactor.
only a quarter of the core is shown. The colors help to identify each zone. It should be noted that three assemblies (in gray) do not have data due to these are steel assemblies. The previous figure indicates that the values obtained with AZNHEX are very similar to those obtained with MCNP, especially the values located in zone 2 (see Table 4 for zone description and location). As it can be seen, these assemblies have the higher magnitude in both codes, since it corresponds to the higher enrichment zone. On the other hand, the blanket region (in yellow) shows the greater relative errors between AZNHEX and MCNP6. This is certainly due to the fact that the blanket is surrounded by the steel reflector and the neutron flux gradient is important in this zone, and therefore, not well reproduced by the diffusion theory code.
Table 6 Values of the keff obtained for the sodium-cooled fast reactor.a
k-inf
AZNHEX
MCNP6
Relative difference (pcm)
1.04107
1.03736 ± 0.00028
−358
a For the simulation with MCNP6, 100,000 neutron histories with the first 50 cycles discarded and 200 active cycles were used.
Table 7 Comparison of results using specular reflection.
k-value
Full core
Quarter core
1.04107
1.04107
3.2.3. Axial power distribution AZNHEX provides to the user the axial power distribution in each assembly. To obtain the axial power distribution using MCNP, the FMESH card is used together with a tally multiplier (Van Veen, 2011), which is placed in each assembly. The axial power distribution obtained with both codes for specific assemblies, depending on its location, is shown in the following figures. The calculated data are also presented in the following tables. Fig. 11 shows the location of the assembly where the axial power distribution was calculated along eight axial nodes of height 10 cm. The relative power obtained for the assembly is presented in Table 8 and the axial power distribution for the assembly is shown in Fig. 12. The shape of the axial power distribution for the assembly is the expected with the highest value located in the middle of the assembly height. As shown in Table 8 and Fig. 12, both AZNHEX and MCNP produced similar values. As in the previous case, Fig. 13 shows the location of the assembly where the axial power distribution was calculated. The data obtained for this assembly are presented in Table 9 and the axial power distribution is shown in Fig. 14. This assembly is located in zone 2 (see Fig. 13). As shown in Fig. 14, the values obtained with AZNHEX in the middle zone are a bit lower than those calculated with MCNP6; this can be seen also in Table 9. However, the behavior obtained in both codes
Fig. 10. Normalized radial power distribution for the Sodium-cooled fast reactor. 289
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Fig. 13. Location of the assembly of interest.
Fig. 11. Location of the assembly of interest.
Table 8 Relative power for the assembly of interest.
Table 9 Relative power for the assembly of interest.
MCNP6
AZNHEX
Relative error (%)
MCNP6
AZNHEX
Relative error (%)
0.785 0.967 1.096 1.160 1.159 1.091 0.962 0.780
0.810 0.969 1.082 1.140 1.140 1.082 0.969 0.810
−3.148 −0.201 1.326 1.749 1.665 0.833 −0.717 −3.802
0.783 0.964 1.097 1.163 1.162 1.093 0.961 0.777
0.809 0.969 1.082 1.140 1.140 1.082 0.969 0.809
−3.269 −0.475 1.345 1.920 1.888 1.009 −0.787 −4.160
Fig. 12. Axial power distribution for the assembly of interest.
Fig. 14. Axial power distribution for the assembly interest.
are the same as in the previous case with the highest value located also in the middle part. Fig. 15 shows the location of the assembly (blanket zone) where the axial power distribution was calculated. The values obtained are presented in Table 10 and the axial power distribution is shown in Fig. 16. The behavior in this case is also the expected and. The agreement between AZNHEX and MCNP6 is good, except for the lower and the upper nodes, where the neutron flux gradient is significant. An extra analysis was performed by obtaining the average axial
power distribution in the reactor core. Table 11 shows the values calculated with both MCNP6 and AZNHEX. Fig. 17 shows the axial power distribution obtained with these codes. The previous figure indicates that the power distribution obtained from AZNHEX produces a good approximation relative to the one obtained with MCNP6. Again, the largest differences are obtained in the boundaries (lower and upper) between the core and the steel reflector, where the neutron flux gradient is significant.
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Fig. 17. Average axial power distribution in the core.
4. Conclusions and future work Certain deviations were found between AZNHEX and MCNP6, since AZNHEX is a diffusion theory code and it is being compared against a Monte Carlo code. However, the results provided to the user are reliable since they present a good degree of fidelity. Improvements are expected in future versions of AZNHEX. The homogenized multi-group cross sections obtained with SERPENT were very appropriate to perform calculations with AZNHEX, as the results presented in this work show. This proof that SERPENT is an adequate choice for the generation of cross sections. The specular reflection condition implemented in AZNHEX works as expected since satisfactory results were obtained in all the analyzed cases where it was used. The solution of the multigroup diffusion equation for seven energy groups appears to provide adequate results. AZNHEX code is in continuous improvement and anisotropy will be one of the first points to be considered. To face up this challenge the neutronics team-work is now dealing with a module allowing to solve the SPL equations, particularly the SP3 ones. Then, a module to solve the PN equations will be developed to make AZNHEX more versatile. The implementation of discontinuity factors and the Coarse Mesh Finite Difference acceleration technique are also considered as part of near future developments. Finally, the possibility to extend the method described in Del Valle and Mund (2004) to time-dependent hexagonal-z geometry is under analysis.
Fig. 15. Location of the assembly of interest.
Table 10 Relative power for the assembly of interest. MCNP6
AZNHEX
Relative error (%)
0.810 0.968 1.080 1.142 1.142 1.082 0.966 0.809
0.826 0.968 1.074 1.132 1.132 1.074 0.968 0.826
−1.903 0.026 0.567 0.893 0.861 0.753 −0.197 −2.133
Acknowledgments The authors acknowledge the financial support from the National Strategic Project No. 212602 (AZTLAN Platform) as part of the Sectorial Fund for Energetic Sustainability CONACYT - SENER. Prof. Edmundo del Valle-Gallegos acknowledges to ININ (National Nuclear Research Center) the Sabbatical Leave he spent there where part of this work was developed. Fig. 16. Axial power distribution for the assembly interest.
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AZNHEX
Relative errors (%)
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0.813 0.969 1.080 1.138 1.138 1.080 0.969 0.813
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Politécnico Nacional, ESFM, Physics Department. Esquivel, J., López, R.C., 2016. AZNHEX: AZtlan Neutronic HEXagonal Code. (User`s manual). Fridman, E., Shwageraus, E., 2013. Modeling of SFR cores with Serpent–DYN3D codes sequence. Ann. Nucl. Energy 53, 354–363. Gómez-Torres, A.-M., Puente Espel, F., Del Valle Gallegos, E., François, J.L., Martin-delCampo, C., Espinosa-Paredes, G., 2015. AZTLAN: mexican platform for analysis and design of nuclear reactors. In: Proceedings of the International Congress on Advances in Nuclear Power Plants ICAPP. May 03-06, 2015, Nice, France. Gordon, W.J., Hall, C.A., 1973a. Transfinite element methods: blending-function interpolation over arbitrary curved element domains. Number. Math. 21, 109–129. Gordon, W.J., Hall, C.A., 1973b. Construction of curvilinear coordinate systems and applications to mesh generation. Int. J. Numer. Meth. Eng. 7, 461–477. Hansen, K.F., et al., 1967. GAKIN; a One Dimensional Multigroup Kinetics Code. GA7543. Hennart, J.P., 1986. A general family of nodal schemes. SIAM J. Sci. Stat. Comput. 3, 264. Hennart, J.P., Jaffré, J., Roberts, J.E., 1988. A constructive method for deriving finite elements of nodal type. Numer. Math. 53 (6), 701–738. Hennart, J.P., Del Valle, E., 1993. On the relationship between nodal schemes and mixedhybrid finite elements. Numer. Meth. Part. Differ. Equ. 9, 411. Hennart, J.P., Malambu, E.M., Mund, E.H., Del Valle, E., 1996. Efficient higher order nodal finite element formulations for neutron multigroup diffusion equations. Nucl. Sci. Eng. 124, 97. Hennart, J.P., Mund, E.H., Del Valle, E., 1997. A composite nodal finite element for hexagons. Nucl. Sci. Eng. 127, 139–153. Hennart, J.P., Mund, E.H., Del Valle, E., 2003. Third order nodal finite element methods with transverse and reduced integration for elliptic problems. Appl. Numer. Math. 46 (2), 209. Henry, A.F., Vota, A.V., 1965. WIGL2-A Program for the Solution of the One-dimensional, Two-group, Space-time Diffusion Equations Accounting for Temperature, Xenon, and Control Feedback. WAPD-TM-532. .
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Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene
Initial verification of AZNHEX hexagonal-z neutron diffusion code with MCNP6 for two different study cases
T
Juan Galicia-Aragóna, Juan-Luis Françoisa,∗, Guillermo E. Bastida-Ortiza, Cecilia Martín-del-Campoa, Julio A. Vallejo-Quinteroa, Edmundo del-Valle-Gallegosb a
Universidad Nacional Autónoma de México, Facultad de Ingeniería, Departamento de Sistemas Energéticos, Av. Universidad 3000, C.U., 04510 Ciudad de México, Mexico b Instituto Politécnico Nacional, Escuela Superior de Física y Matemáticas, Av. IPN s/n, San Pedro Zacatenco, 07738 Ciudad de México, Mexico
A R T I C LE I N FO
A B S T R A C T
Keywords: Neutron diffusion code Reactor analysis Hexagonal-z geometry Finite element method
The core of the AZTLAN Mexican reactor analysis platform consists of a thermo-hydraulic code (AZTHECA), a neutron transport code (AZTRAN) and two neutron diffusion codes, namely AZKIND and AZNHEX. These codes are currently in the testing phase by simulating a variety of fuel assemblies and nuclear reactor cores to compare and verify their results with those obtained by codes globally used in the nuclear community such as SERPENT, CASMO/SIMULATE, MCNP and others. The main objective of this task is to improve future versions of the AZTLAN platform codes to obtain reliable results for the user. To verify the current version of AZNHEX neutron diffusion code, two cases were considered. The simulation of a WWER-440 fuel assembly was the first case analyzed, and the second case was the core of a sodium-cooled fast reactor in steady state, with control rods fully withdrawn. The comparison and verification of the results (neutron multiplication factor, axial and radial power distributions and radial neutron flux distribution) for both cases were done employing MCNP6 Monte Carlo code. Certain deviations were found between AZNHEX and MCNP6, since AZNHEX is a diffusion code and it is being compared against a Monte Carlo code. However, it is shown that the results provided to the user are reliable since they exhibit a good degree of fidelity.
1. Introduction Nuclear community has the goal to develop computer platforms for analysis and design of nuclear reactors to represent the different physical phenomena that occur in a nuclear reactor. Since 2005, the European Reference Simulation Platform for Nuclear Reactors, called NURESIM, has been developed (Chauliac et al., 2011). This platform provides an accurate representation of the physical phenomena by promoting and incorporating the latest advances in core physics, twophase thermal-hydraulics and fuel modeling. It includes multi-scale and multi-physics features, especially for coupling core physics and thermal-hydraulics models for reactor safety analysis. Quantitative deterministic and statistical sensitivity and uncertainty analyses tools are developed and provided through the platform. NURESIM includes generic pre-processing, post-processing and supervisory functions through the open-source SALOME software, which make the codes more user-friendly. In order to accelerate nuclear energy modeling and simulation (M&
S), the U.S. Department of Energy (DOE) established the Consortium for Advanced Simulation of Light Water Reactors (CASL) as its first DOE Energy Innovation Hub in July 2010 (CASL, 2017). This project was established to provide leading edge modeling and simulation capability to improve the performance of currently operating light water reactors. CASL is developing the Virtual Environment for Reactor Applications, VERA. CASL's VERA software simulates nuclear reactor physical phenomena using coupled multi-physics models. VERA's current physics capabilities include neutron transport, thermal-hydraulics, fuel performance, and coolant chemistry. The AZTLAN platform project (Gómez-Torres et al., 2015) emerged as an initiative to rank Mexico at a competitive level in software development for the analysis of nuclear reactors, and therefore, achieve autonomy in this area. The AZTLAN platform is composed of a thermohydraulic code (AZTHECA), a neutron transport code (AZTRAN) and two neutron diffusion codes (AZKIND, AZNHEX). AZNHEX (Esquivel and López, 2016) is a hexagonal-z time-dependent 3D diffusion code whose brief description will be given in the next
∗
Corresponding author. E-mail addresses: [email protected] (J. Galicia-Aragón), jlfl@fi-b.unam.mx (J.-L. François), [email protected] (G.E. Bastida-Ortiz), [email protected] (C. Martín-del-Campo), [email protected] (J.A. Vallejo-Quintero), [email protected] (E. del-Valle-Gallegos). https://doi.org/10.1016/j.pnucene.2018.03.017 Received 15 October 2017; Received in revised form 19 March 2018; Accepted 22 March 2018 0149-1970/ © 2018 Elsevier Ltd. All rights reserved.
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section. The starting version of AZNHEX arose as a product of a master's thesis carried out in 2015 (Esquivel, 2015), although it is an extension to 3D cores of the original method developed in 1997 (Hennart et al., 1997) for 2D cores. Since 2015, AZNHEX has been improved thanks to the feedback provided to the developers from the AZTLAN User's group. With the current version of this code, it is possible at the present to simulate and analyze light water reactors, such as the Russian WWER and a variety of fast reactors. Nonetheless, to guarantee its fidelity it is necessary to perform several tests with the current version of AZNHEX. This is important for the suggestions that the User's group may give to the developers. The verification of AZNHEX will help identify deficiencies in the methods and/or the implementation, thereby improving the quality of the code. The structure of the article is as follows. A brief description of AZNHEX is given in Section 2. The test cases with results are presented in Section 3. Finally, conclusions and future directions are given in Section 4. 2. AZNHEX description 2.1. Methodology AZNHEX code relies on several works headed by Professor J.P. Hennart related with nodal methods (Hennart, 1986; Hennart et al., 1988; Hennart and Del Valle, 1993; Hennart et al., 1996; Hennart et al., 1997; Hennart et al., 2003). It is based on a composite nodal method for prismatic hexagons, applied to solve the neutron diffusion equations for cores built by the union, side by side, of hexagonal prisms; each prismatic hexagon represents a sub-assembly. A standard finite element method is used starting by the strong form to get the corresponding weak form. A detailed description of the nodal method used in AZNHEX can be found in Del Valle-Gallegos et al. (2018). AZNHEX solves numerically the time dependent neutron diffusion equations in Hexagonal-Z geometry. The equations are given by a set of G + I partial differential equations (Duderstadt and Hamilton, 1976). The first equations correspond to the neutron density rate balance for each one of the G energy groups:
1 ∂ϕg − ∇ ·Dg ∇ϕg + vg ∂t
G
∑Rg ϕg = ∑
Fig. 1. Hexagonal-z array for AZNHEX.
G
Σsg ′→ g ϕg ′ + χg (1 − β )
∑ ν ∑fg ′ ϕg ′
g ′= 1
g ′≠ g I
+
∑ χgi λi Ci, i=1
g = 1, .., G (1a)
Then, for the I groups of delayed neutron precursor concentrations we have:
∂Ci = βi ∂t
G
∑ g ′= 1
νΣfg ′ ϕg ′ − λi Ci, i = 1, ..., I
(1b)
Where:
• Φg: scalar neutron flux corresponding to the energy-group g [n/cm s] • Dg: diffusion coefficient corresponding to the energy-group g [cm] • G: total number of energy-groups • Σ : removal cross section corresponding to the energy-group g [cm ] • Σ : scattering cross section from energy-group g' to g [cm ] • ν: average number of neutrons produced per fission • Σ : fission cross section corresponding to the energy-group g [cm ] χ • : probability that prompt neutrons arising from fissions appear in energy-group g • λ : decay constant of the delayed neutron precursor i • β: delayed neutron fraction 2
Rg
AZNHEX solves Eq. (1) for hexagonal-z domains, as it shown in Fig. 1 (3D) and Fig. 2 (2D). In order to accomplish this, the method first applies a Gordon-Hall transfinite interpolation (Gordon and Hall, 1973a, 1973b) to each one of the four quadrants shown in Fig. 3a, to transform it into a cube as it is shown in Fig. 3b. Once this is done, the classic Galerkin's Finite Element Method is applied to approximate the scalar neutron flux, and the neutron precursor concentrations, by a polynomial interpolation over a reference cell [-1,+1]x [-1,+1]x [-1, +1]. This interpolation is known as the Raviart-Thomas-Nédelec interpolation of index zero: RTN-0 (Hennart et al., 1988). For the time variable, AZNHEX has implemented themethod. The resulting algebraic system is solved on each time step by using the BiCGSTAB method (Van
−1
sg´ → g
fg
Fig. 2. A cross section view of the hexagonal array for AZNHEX.
−1
−1
g
i
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Fig. 3. Gordon-Hall transformation for a hexagonal prism. a) Hexagonal prism is divided in four quadrants; b) One quadrant is transformed in a cube.
homogenized multi-group cross sections and the diffusion coefficients needed by AZNHEX to solve the nodal diffusion equations in G energy groups. For the test cases presented in this article SERPENT v.2.1.28 (Leppänen et al., 2015) was used. One of the main advantage of SERPENT over other Monte Carlo codes is its proved capability for producing homogenized group constants in a user-defined energy multi-group structure. The methodology used in Serpent for spatial homogenization is described in Leppänen et al. (2016). Several studies have been published showing the effectiveness of SERPENT as a tool to generate nuclear data libraries for nodal diffusion codes (Rachamin et al., 2013; Fridman and Shwageraus, 2013; Nikitin et al., 2015; Smith, 2017; Jo et al., 2018). It is out of the scope of this work to do an investigation on the best SERPENT's option for cross section homogenization, therefore, we used the infinite spectrum homogenization method. For the WWER fuel lattice test case, the cross sections set (XS data) used by AZNHEX was produced from a heterogeneous model of the fuel lattice in SERPENT (like that showed in Fig. 5). XS data were generated for the homogenized unit fuel cells and the central water thimble, in two energy groups, using a cut-off at 0.625 eV and the JEFF-3.1 library. Fig. 4 shows the procedure to transfer the XS data from SERPENT to AZNHEX. Similarly, for the sodium-cooled fast reactor core case, a detailed heterogeneous full core model was set up in SERPENT to produce assembly-wise XS data for AZNHEX in seven energy groups, according to Palmiotti (2011) energy structure. In this case, the JEFF-3.2 library was used. Once this calculation was done, the homogenized XS data for each assembly type were transferred to AZNHEX.
der Vorst, H. A., 1992) once the initial condition is obtained and a given transient is defined. The θ-method, also known as weighted method, was introduced by Henry and Vota (1965) and then some other authors applied this method to discretize the time dependence of the neutron diffusion or transport kinetic equations (Hansen et al., 1967; Yasinsky et al., 1968; Stacey, 1969) and there have been several review papers that mention the θ-method, for instance Chae (1979) and Sutton and Aviles (1996). The method as such covers the so called fully explicit (θ = 0), implicit (θ = 1), and semi-implicit (θ = 0.5) methods. Regarding the eigenvalue problem it is solved by using the standard power iterations method when there is no up-scattering, and with the Wielandt accelerated power iterations when there is up-scattering. Other acceleration techniques like the Coarse Mesh Finite Difference method (CMFD) will be included in a new version of AZNHEX. It is worthwhile to mention that at this stage of the code verification that only the steady state capability is tested and presented in this article. Further work will be devoted to test the transient mode of AZNHEX.
2.2. Cross section generation Given that AZNHEX is a nodal code, no heterogeneous pin by pin calculations are possible; this is a calculation of a lattice of homogeneous unit cells composed of fuel pin, clad and water, i.e. a pincellhomogenized assembly calculation. Due to this, homogeneous unit cells nuclear properties (see Fig. 4) must be provided to it. In our methodology, the Monte Carlo SERPENT code is used to generate spatially-
Fig. 4. XS data generation using SERPENT and its transfer to AZNHEX.
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Fig. 5. WWER-440 fuel assembly, MCNP6 heterogeneous model.
Fig. 6. WWER-440 fuel assembly, AZNHEX homogeneous model.
Table 1 Fuel characteristics of the WWER-440 fuel assembly. Number of fuel rods
Composition (wt %)
Temperature (K)
Pitch (cm)
126
3.175 (U-235) 84.980 (U-238) 11.850 (O-16)
900
1.23
Table 3 Values of the kinf obtained for the WWER-440 reactor assembly.a
k-inf
AZNHEX
MCNP6
Relative difference (pcm)
1.31786
1.31478 ± 0.00007
−234
a For the simulation with MCNP6, 500,000 neutron histories with the first 50 cycles discarded and 200 active cycles were used.
3. Test cases SERPENT-2 for AZNHEX are shown in Table 2. Fig. 6 shows the AZNHEX homogeneous model of the WWER-440 fuel lattice. The specular reflection boundary condition is applied in each of the outer borders of the lattice. To make a more accurate comparison, the assembly was simulated without channel and without water in the boarders, since they cannot be modeled in AZNHEX.
Since AZNHEX is intended to be used as the AZTLAN's lattice code and core simulator in hexagonal geometry, the current version was tested with two cases. The first one is a fuel lattice of a WWER-440 like fuel assembly. This case was considered because it represents a “classic” lattice calculation (2-D thermal reactor fuel lattice using two neutron energy groups). This WWER-440 fuel lattice was taken from the input exercises of SERPENT code. The second case is a sodium-cooled fast reactor core (López-Solis and François-Lacouture, 2015). This example represents a more complex case compared with the fuel lattice case. This is a relative small core with three different 235U enrichment zones (one blanket), and steel radial and axial reflectors. To perform the simulations of both cases, and asses the results of the current version of AZNHEX, the Monte Carlo code MCNP6 (MCNP6, 2014) was used. This code is one of the most recognized and employed in the nuclear industry; due to this, it was selected to calculate the reference parameters of this work.
3.1.1. Criticality calculations Table 3 shows the results of the infinite neutron multiplication factor (kinf) and the relative difference with respect to the value calculated with MCNP6. As it can be seen, there is a very good approximation of the kinf considering the assumptions considered in both models. 3.1.2. Radial power distribution Fig. 7 shows the results of the normalized radial power distribution calculated with AZNHEX (values in blue) and the relative errors in percent with respect to MCNP6 (values in red). Relative errors shown in Fig. 7 indicate the accuracy of AZNHEX of the power values of each fuel rod. The highest differences with respect to MCNP6 are in the central zone, around the water hole where a high neutron flux gradient is expected (see Fig. 8). However, relative differences do not exceed 1.9%.
3.1. Fuel assembly of a WWER-440 reactor Fig. 5 shows the fuel assembly simulated with MCNP6, which is our reference case in this study. The characteristics of the 126 fuels rods are shown in Table 1. The water and the fuel cladding have a temperature of 600 K. The libraries used in this case in MCNP6 were ENDF/B-VI.2 for some isotopes and JEFF-3.2 for others. The XS data generated with
3.1.3. Radial neutron flux distribution Fig. 8 shows the normalized one group neutron flux values obtained
Table 2 XS data generated with SERPENT-2 for the WWER-440 fuel assembly. MATERIAL
NEUTRON ENERGY GROUP
DIFF. COEFF.
REMOVAL XS
ν*FISSION XS
κ*FISSION XS
FISSION SPECTRUM (χ)
SCATT. MATRIX
Fuel cell
1 2 1 2
1.450 0.411 1.750 0.344
0.0252 0.0991 0.0249 0.0100
0.00801 0.1710 0.0000 0.0000
0.6410 14.200 0.0000 0.0000
1.0000 0.0000 0.0000 0.0000
0.4870 0.00270 0.5830 0.00187
Water
287
0.0144 1.0900 0.0243 2.5600
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Table 4 Characteristics of the sodium-cooled fast reactor. Materials
Enrichment (at%)
Temperature (K)
No. of assemblies
Fuel (zone 1-orange) Fuel (zone 2-red) Fuel (blanket-yellow) Steel Sodium
9.5
1200
18
14.25
1200
36
0.35
1200
66
– –
600
97
600
–
with AZNHEX (values in blue) and the relative errors in percent with respect to MCNP6 (values in red). As it can be seen in Fig. 8, the results of AZNHEX for the radial neutron flux distribution are very accurate, with errors below 0.123%. This shows that if the user desires to simulate the assembly of a thermal reactor, the results obtained will be highly reliable. 3.2. Sodium-cooled fast reactor core Table 4 presents the characteristics of a sodium-cooled fast reactor and Fig. 9 shows a view of the core (López-Solis and FrançoisLacouture, 2015). As it can be seen, this reactor has three different 235U enrichment zones (one blanket), steel radial and axial reflectors, and seven steel assemblies were considered in the control rods locations. The library used in this case in MCNP6 was JEFF-3.2. The XS data generated with SERPENT-2 are shown in Table 5.
Fig. 7. Normalized radial power distribution for the WWER-440 fuel assembly.
3.2.1. Criticality calculations Table 6 shows the results of the effective neutron multiplication factor (keff) and the relative difference with respect to the value calculated with MCNP6. As in the previous case, the keff value calculated with AZNHEX presents an acceptable approximation having a difference with respect to MCNP6 of 358 pcm. A quarter of the core was also simulated, and the neutron multiplication factor was compared with that obtained considering full core. The motivation for running a quarter-core geometry is to simplify the geometry by taking advantage of symmetry. This will reduce runtime and simplify the layout. Table 7 presents the comparison of the neutron multiplication factor. As shown in the previous table, the specular reflection condition works correctly if the user wants to simulate a simplified case due to the complexity of a symmetric reactor core. If the core is complex (i.e., asymmetric), then full-core representation is necessary. 3.2.2. Radial power distribution Fig. 10 shows the results obtained with MCNP6 (values on the top), AZNHEX (values on the middle) and the relative errors with respect to MCNP6 (values on the bottom). Due to the symmetry of this reactor,
Fig. 8. Normalized radial neutron flux distribution for the WWER-440 fuel assembly.
Fig. 9. Sodium-cooled fast reactor core. 288
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Table 5 XS data generated with SERPENT-2 for the sodium-cooled fast reactor.
only a quarter of the core is shown. The colors help to identify each zone. It should be noted that three assemblies (in gray) do not have data due to these are steel assemblies. The previous figure indicates that the values obtained with AZNHEX are very similar to those obtained with MCNP, especially the values located in zone 2 (see Table 4 for zone description and location). As it can be seen, these assemblies have the higher magnitude in both codes, since it corresponds to the higher enrichment zone. On the other hand, the blanket region (in yellow) shows the greater relative errors between AZNHEX and MCNP6. This is certainly due to the fact that the blanket is surrounded by the steel reflector and the neutron flux gradient is important in this zone, and therefore, not well reproduced by the diffusion theory code.
Table 6 Values of the keff obtained for the sodium-cooled fast reactor.a
k-inf
AZNHEX
MCNP6
Relative difference (pcm)
1.04107
1.03736 ± 0.00028
−358
a For the simulation with MCNP6, 100,000 neutron histories with the first 50 cycles discarded and 200 active cycles were used.
Table 7 Comparison of results using specular reflection.
k-value
Full core
Quarter core
1.04107
1.04107
3.2.3. Axial power distribution AZNHEX provides to the user the axial power distribution in each assembly. To obtain the axial power distribution using MCNP, the FMESH card is used together with a tally multiplier (Van Veen, 2011), which is placed in each assembly. The axial power distribution obtained with both codes for specific assemblies, depending on its location, is shown in the following figures. The calculated data are also presented in the following tables. Fig. 11 shows the location of the assembly where the axial power distribution was calculated along eight axial nodes of height 10 cm. The relative power obtained for the assembly is presented in Table 8 and the axial power distribution for the assembly is shown in Fig. 12. The shape of the axial power distribution for the assembly is the expected with the highest value located in the middle of the assembly height. As shown in Table 8 and Fig. 12, both AZNHEX and MCNP produced similar values. As in the previous case, Fig. 13 shows the location of the assembly where the axial power distribution was calculated. The data obtained for this assembly are presented in Table 9 and the axial power distribution is shown in Fig. 14. This assembly is located in zone 2 (see Fig. 13). As shown in Fig. 14, the values obtained with AZNHEX in the middle zone are a bit lower than those calculated with MCNP6; this can be seen also in Table 9. However, the behavior obtained in both codes
Fig. 10. Normalized radial power distribution for the Sodium-cooled fast reactor. 289
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Fig. 13. Location of the assembly of interest.
Fig. 11. Location of the assembly of interest.
Table 8 Relative power for the assembly of interest.
Table 9 Relative power for the assembly of interest.
MCNP6
AZNHEX
Relative error (%)
MCNP6
AZNHEX
Relative error (%)
0.785 0.967 1.096 1.160 1.159 1.091 0.962 0.780
0.810 0.969 1.082 1.140 1.140 1.082 0.969 0.810
−3.148 −0.201 1.326 1.749 1.665 0.833 −0.717 −3.802
0.783 0.964 1.097 1.163 1.162 1.093 0.961 0.777
0.809 0.969 1.082 1.140 1.140 1.082 0.969 0.809
−3.269 −0.475 1.345 1.920 1.888 1.009 −0.787 −4.160
Fig. 12. Axial power distribution for the assembly of interest.
Fig. 14. Axial power distribution for the assembly interest.
are the same as in the previous case with the highest value located also in the middle part. Fig. 15 shows the location of the assembly (blanket zone) where the axial power distribution was calculated. The values obtained are presented in Table 10 and the axial power distribution is shown in Fig. 16. The behavior in this case is also the expected and. The agreement between AZNHEX and MCNP6 is good, except for the lower and the upper nodes, where the neutron flux gradient is significant. An extra analysis was performed by obtaining the average axial
power distribution in the reactor core. Table 11 shows the values calculated with both MCNP6 and AZNHEX. Fig. 17 shows the axial power distribution obtained with these codes. The previous figure indicates that the power distribution obtained from AZNHEX produces a good approximation relative to the one obtained with MCNP6. Again, the largest differences are obtained in the boundaries (lower and upper) between the core and the steel reflector, where the neutron flux gradient is significant.
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Fig. 17. Average axial power distribution in the core.
4. Conclusions and future work Certain deviations were found between AZNHEX and MCNP6, since AZNHEX is a diffusion theory code and it is being compared against a Monte Carlo code. However, the results provided to the user are reliable since they present a good degree of fidelity. Improvements are expected in future versions of AZNHEX. The homogenized multi-group cross sections obtained with SERPENT were very appropriate to perform calculations with AZNHEX, as the results presented in this work show. This proof that SERPENT is an adequate choice for the generation of cross sections. The specular reflection condition implemented in AZNHEX works as expected since satisfactory results were obtained in all the analyzed cases where it was used. The solution of the multigroup diffusion equation for seven energy groups appears to provide adequate results. AZNHEX code is in continuous improvement and anisotropy will be one of the first points to be considered. To face up this challenge the neutronics team-work is now dealing with a module allowing to solve the SPL equations, particularly the SP3 ones. Then, a module to solve the PN equations will be developed to make AZNHEX more versatile. The implementation of discontinuity factors and the Coarse Mesh Finite Difference acceleration technique are also considered as part of near future developments. Finally, the possibility to extend the method described in Del Valle and Mund (2004) to time-dependent hexagonal-z geometry is under analysis.
Fig. 15. Location of the assembly of interest.
Table 10 Relative power for the assembly of interest. MCNP6
AZNHEX
Relative error (%)
0.810 0.968 1.080 1.142 1.142 1.082 0.966 0.809
0.826 0.968 1.074 1.132 1.132 1.074 0.968 0.826
−1.903 0.026 0.567 0.893 0.861 0.753 −0.197 −2.133
Acknowledgments The authors acknowledge the financial support from the National Strategic Project No. 212602 (AZTLAN Platform) as part of the Sectorial Fund for Energetic Sustainability CONACYT - SENER. Prof. Edmundo del Valle-Gallegos acknowledges to ININ (National Nuclear Research Center) the Sabbatical Leave he spent there where part of this work was developed. Fig. 16. Axial power distribution for the assembly interest.
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Table 11 Data obtained for the axial power distribution. MCNP6
AZNHEX
Relative errors (%)
0.787 0.966 1.093 1.158 1.158 1.091 0.963 0.785
0.813 0.969 1.080 1.138 1.138 1.080 0.969 0.813
−3.233 −0.326 1.133 1.675 1.676 0.964 −0.598 −3.482
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