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Assessment of the traditional neutron-diffusion core-analysis method for the analysis of the Super Critical Water Reactor
Annals of Nuclear Energy 45 (2012) 1–7
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Assessment of the traditional neutron-diffusion core-analysis method for the analysis of the Super Critical Water Reactor W. Shen ⇑ Candu Energy Inc., 2285 Speakman Dr., Mississauga, ON, Canada L5B 1K1
a r t i c l e
i n f o
Article history: Received 12 October 2011 Received in revised form 3 February 2012 Accepted 10 February 2012 Available online 14 March 2012 Keywords: SCWR SERPENT Lattice homogenization Neutron diffusion
a b s t r a c t The key design quantities of the pressure-tube-based (PT-based) Super Critical Water Reactor (SCWR) core design are expected to be computed with the traditional core-analysis code which solves the two-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the lattice code. Two issues may affect the accuracy of these computed quantities for the SCWR core: one is the two-energy-group neutron-diffusion theory; the other is the generation of lattice-homogenized properties with the lattice code based on the single-lattice-cell model without considering the effects of the environment. It has been illustrated that the single-lattice-cell method is not sufficiently accurate for heterogeneous core configurations when adjacent channels experience significant spectrum interaction. To ensure the qualification of these computed quantities for the SCWR core, a 2-D SCWR benchmark problem was setup (with the reference solution provided by the continuous energy Monte-Carlo code SERPENT) to assess the traditional neutron-diffusion core-analysis method. The assessment shows that the traditional two-group neutron-diffusion theory with the single-lattice-cell-based lattice properties is not sufficient to capture either the spectral change or the environment effect for the SCWR core. The solution of the eight-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the multicell model is considered appropriate for the analysis of the PT-based SCWR core. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.
1. Introduction The Super Critical Water Reactor (SCWR) concept has been developed to take advantage of the improvement in thermal efficiency of a reactor through the use of supercritical water as a coolant. Although the concept is not new, interest has recently been renewed through the Generation IV (Gen IV) program. The Gen IV program is an international collaboration whose focus is the identification and development of next-generation nuclear energy systems which could be deployed to meet growing worldwide energy demands. The main Canadian contribution to the Gen IV program is the development of a pressure-tube-based (PT-based) SCWR. The PT-based SCWR (Boczar et al., 2010) shares features with the conventional CANDU such as heavy water moderator and pressurized fuel channels, rather than a single pressure vessel. However, in order to meet the material challenges associated with the higher pressures and corrosive environment associated with supercritical water, there are a number of key differences between the SCWR and the CANDU reactor. Features being considered for the SCWR design that differ from conventional CANDU designs include: supercritical water coolant, enriched fuel (Pu ⇑ Fax: +1 905 822 0567. E-mail address: [email protected]
and Th mixture fuel in this paper) in a 54-element bundle with no fuel in the central pin, a tight lattice pitch (25 cm lattice pitch), vertical fuel channels, and off-line batch refuelling. These core design features result in an under-moderated lattice, and the neutronic behavior is considerably different from that in the current CANDU reactors. The key elements of the SCWR core design are flux and power distributions, peak channel and bundle powers, and core reactivity coefficients for the nominal core conditions and for postulated accident conditions, such as the loss-of-coolant accident (LOCA) transient. These quantities are expected to be computed with the traditional core-analysis code such as RFSP (Jenkins et al., 2010) which solves the two-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the lattice code such as WIMS-AECL (Altiparmakov, 2008) based on the singlelattice-cell model. Two issues may affect the accuracy of the RFSP calculations of these quantities for the SCWR core: one is the two-energy-group neutron-diffusion theory; the other is the generation of latticehomogenized properties with the lattice code based on the single-lattice-cell model without considering the effects of the environment. It has been illustrated that the single-lattice-cell method is not sufficiently accurate for heterogeneous core configurations when adjacent channels experience significant spectrum interaction (Shen, 2006). The multicell method (Shen,
0306-4549/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2012.02.009
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
2006) developed and implemented in WIMS-AECL/RFSP allows the use of WIMS-AECL multicell-based fuel properties with the effects of the environment being considered. Note that the above two issues are not SCWR specific, however, their effect may be more significant for SCWR. To ensure the qualification of RFSP calculations of these quantities for the SCWR core, a 2-D SCWR benchmark problem was setup to assess the traditional two-group neutron-diffusion theory with single-lattice-based lattice properties for the analysis of the SCWR core. This paper documents preliminary assessment of the applicability of the traditional neutron-diffusion method for the analysis of the SCWR core. The assessment performed in this paper is so general that the assessment conclusions are expected to be applicable to any core-analysis code which solves the two-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the single-lattice-cell model. 2. Description of the SCWR benchmark problem To assess the applicability of using the 2-group single-latticecell model in the diffusion theory for the SCWR core analysis, a 2-D 22 22 SCWR benchmark problem was setup from scratch. Fig. 1 illustrates the geometry of the SCWR fuel lattice. The specifications for the fuel channel are given in Table 1. The SCWR fuel lattice modeled has the following features:
Super-critical water coolant. 54-element fuel of ThO2 with 14 wt.% PuO2. Porous zirconia insulator. Stainless-steel (SS310) cladding. 25 cm square lattice pitch.
The 2-D 22 22 SCWR benchmark problem consists of an 18 18 array of SCWR fuel lattices, with a lattice pitch of 25 cm, surrounded by a two-lattice-pitch heavy-water reflector boundary. The vacuum boundary condition is used at the four external surfaces in the XY plane and the reflective condition is used in the axial direction. Fig. 2 illustrates the geometry of the 22 22 core which consists of three batches of SCWR fuel with the fuel burnup of 20181.53 MWd/T, 40527.81 MWd/T, and 57817.85 MWd/T, respectively. The burnup values of the three fuel batches represent the mid-plane of the SCWR core at the end-of-cycle core
Fig. 1. Illustration of the SCWR fuel lattice.
Table 1 Specification for the 54-element SCWR fuel bundle. Parameter
Value
Lattice pitch (square) Coolant Moderator Fuel elements per bundle Fuel stack length
25 cm Super-critical water 99.91 at.% D2O 54 500 cm
Fuel element Elements in rings 1, 2, 3 Pitch circle radius of rings 1, 2, 3 Radius Outer radius of pin cladding Composition Liner tube Inner radius Thickness Composition Insulator Inner radius Thickness Composition
12, 18, 24 2.8755 cm, 4.3305 cm, 5.8000 cm 0.62 cm 0.68 cm 14 wt.% PuO2 in ThO2 6.8 cm 0.1 cm 50 vol.% of SS310 and 50 vol.% of coolant 6.9 cm 1.33 cm 30 vol.% of zirconia and 70 vol.% of coolant
Pressure tube Inner radius Thickness Composition
8.23 cm 1.4 cm CANDU zirc-4
Center element Composition Radius
Zirconia 1.94 cm
condition without soluble poison. The loading pattern of the three batches of fuel shown in Fig. 3 was selected to have a flat power shape. 3. Description of computational codes and models 3.1. SERPENT SERPENT (Leppanen, 2007) is a three-dimensional (3-D) continuous-energy Monte-Carlo reactor-physics burnup-calculation code, specifically designed for lattice physics applications. The code uses built-in calculation routines for generating homogenized multigroup constants for deterministic reactor simulator calculations. The standard output includes effective and infinite multiplication factors, homogenized cross sections, scattering matrices, diffusion coefficients, assembly discontinuity factors, and kinetic parameters. SERPENT can be used for various reactor-physics calculations at pin, lattice, and core levels. The continuous-energy Monte-Carlo method allows the modelling of any critical reactor type, including both thermal and fast neutron systems. All SERPENT calculations were executed with SERPENT 1.1.14 with the parallel mode in this study. The ENDF/B-VII nuclear data library with continuous energy contained in the SERPENT package was used directly in the assessment. Note that the choice of nuclear data library does not impact the assessment as long as the library used in the SERPENT full-core calculation and the cross-section preparation is consistent. There are two objectives of using SERPENT in the assessment: one is to provide reference power distributions for the 22 22 SCWR benchmark problem; the other is to provide lattice-homogenized cross sections for diffusion calculations. To obtain statistically converged flux/power distributions and lattice-homogenized cross sections for the benchmark problem with SERPENT, extensive assessment with various histories was performed with SERPENT for a CANDU benchmark problem with
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
Fig. 2. Illustration of the 2-D 22 22 SCWR benchmark problem.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
4 4 1 1 1 3 1 2 1 1 1 1 1 1 2 1 3 1 1 1 4 4
4 4 1 3 2 1 3 1 2 2 3 3 2 2 1 3 1 2 3 1 4 4
4 4 1 2 3 2 1 3 1 3 2 2 3 1 3 1 2 3 2 1 4 4
4 4 3 1 2 3 2 3 2 1 3 3 1 2 3 2 3 2 1 3 4 4
4 4 1 3 1 2 3 1 3 3 1 1 3 3 1 3 2 1 3 1 4 4
4 4 2 1 3 3 1 3 2 3 3 3 3 2 3 1 3 3 1 2 4 4
4 4 1 2 1 2 3 2 1 2 2 2 2 1 2 3 2 1 2 1 4 4
4 4 1 2 3 1 3 3 2 3 2 2 3 2 3 3 1 3 2 1 4 4
4 4 1 3 2 3 1 3 2 2 2 2 2 2 3 1 3 2 3 1 4 4
4 4 1 3 2 3 1 3 2 2 2 2 2 2 3 1 3 2 3 1 4 4
4 4 1 2 3 1 3 3 2 3 2 2 3 2 3 3 1 3 2 1 4 4
4 4 1 2 1 2 3 2 1 2 2 2 2 1 2 3 2 1 2 1 4 4
4 4 2 1 3 3 1 3 2 3 3 3 3 2 3 1 3 3 1 2 4 4
4 4 1 3 1 2 3 1 3 3 1 1 3 3 1 3 2 1 3 1 4 4
4 4 3 1 2 3 2 3 2 1 3 3 1 2 3 2 3 2 1 3 4 4
4 4 1 2 3 2 1 3 1 3 2 2 3 1 3 1 2 3 2 1 4 4
4 4 1 3 2 1 3 1 2 2 3 3 2 2 1 3 1 2 3 1 4 4
4 4 1 1 1 3 1 2 1 1 1 1 1 1 2 1 3 1 1 1 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
1 2 3 4
-----
Once-Burnt Fuel Twice Burnt Fuel Thrice Burnt Fuel Reflector
Fig. 3. Fuel loading pattern for the 2-D 22 22 SCWR benchmark problem.
SERPENT-calculated core k-effective as the function of number of histories for the CANDU benchmark problem. Fig. 5 shows the differences in channel powers between SERPENT (with 800 million histories) and MCNP5 (with 2300 million histories) for the CANDU benchmark problem. Overall the assessment shows that SERPENT’s prediction of core k-effective and channel powers is very close to those predicted with MCNP5, with a difference in the core k-effective of about 0.3 mk and a RMS difference in the channel power of about 0.2% . 3.2. Miner
Fig. 4. Convergence in SERPENT-calculated core k-effective as a function of number of histories for the CANDU benchmark problem.
the same nuclear data library as that of MCNP5 (X-5 Monte Carlo Team, 2006). Fig. 4 shows the convergence of the
Because of the limitation of the 2-group approximation used in RFSP diffusion calculations, MINER (Phelps et al., 2009) was used to model the 22 22 SCWR benchmark problem. MINER is an in-house-developed 3-D multi-group neutron-diffusion code which contains two multi-group diffusion solvers based on the finitedifference method (FDM) and modern nodal method (Shen et al., 1995). Different numbers of energy groups and different mesh sizes can be used in the diffusion calculation. Assembly Discontinuity
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
A B C D E F G H J K L M N O P Q R S
1 0.2 0.3 0.2 0.2 0.3 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.0 -0.1 -0.1 0.1 -0.1
2 0.3 0.2 0.1 0.3 0.2 0.5 0.4 0.5 0.4 0.2 0.4 0.2 0.3 0.0 0.0 -0.1 0.0 -0.2
3 0.3 0.0 0.0 0.1 0.1 0.4 0.3 0.4 0.5 0.4 0.4 0.2 0.2 0.0 -0.1 -0.2 -0.1 0.0
4 0.2 -0.1 0.0 0.1 0.1 0.2 0.3 0.3 0.5 0.5 0.3 0.1 0.1 0.0 0.0 -0.2 -0.1 -0.1
5 0.1 0.2 0.1 0.0 0.0 0.2 0.3 0.3 0.4 0.4 0.4 0.2 0.0 -0.1 -0.1 -0.1 0.0 -0.1
6 0.1 0.0 0.0 0.0 0.0 0.1 0.3 0.2 0.3 0.3 0.2 0.2 0.0 -0.1 0.0 0.0 0.0 -0.2
7 0.0 0.0 0.0 0.0 -0.1 0.1 0.2 0.2 0.1 0.1 0.2 0.0 -0.1 -0.1 -0.1 -0.2 -0.1 -0.1
8 -0.1 0.0 -0.1 -0.1 -0.1 0.0 0.1 0.0 0.0 0.0 0.0 0.1 0.0 -0.2 -0.2 -0.2 -0.2 -0.3
9 -0.1 -0.2 -0.1 -0.2 -0.1 0.0 0.1 0.0 0.0 -0.1 0.0 0.1 0.0 -0.1 -0.1 -0.2 -0.4 -0.2
10 -0.2 -0.1 -0.1 -0.1 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0 -0.2 -0.2 -0.3 -0.3
11 -0.2 0.0 0.0 0.0 -0.1 0.0 0.1 -0.1 0.0 0.0 0.0 0.0 -0.1 -0.1 0.0 -0.1 -0.2 -0.2
12 0.0 -0.2 -0.1 0.0 0.0 0.0 0.0 0.0 -0.1 0.0 0.0 0.1 0.0 -0.2 0.0 -0.3 -0.3 -0.1
13 0.0 0.0 -0.1 0.0 -0.1 -0.1 0.0 -0.1 -0.1 -0.1 0.0 0.0 -0.2 -0.2 -0.2 -0.3 -0.3 -0.4
14 0.1 0.0 -0.1 -0.2 -0.2 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 -0.2 -0.3 -0.2 -0.4
15 0.0 -0.1 -0.2 -0.1 -0.1 -0.1 -0.1 0.0 0.0 0.0 0.0 0.1 -0.1 -0.2 -0.2 -0.3 -0.5 -0.2
16 0.0 -0.2 0.0 -0.1 -0.1 -0.1 -0.1 0.0 -0.1 0.0 0.0 0.0 -0.1 -0.2 -0.2 -0.4 -0.5 -0.1
17 -0.2 -0.1 -0.1 0.0 0.0 0.0 -0.2 0.0 -0.1 -0.1 0.1 -0.1 -0.2 -0.3 -0.1 -0.3 -0.3 -0.4
18 -0.1 -0.1 0.0 -0.3 -0.1 -0.1 -0.1 -0.2 -0.1 0.1 -0.1 -0.1 -0.2 -0.2 -0.4 -0.3 -0.2 0.0
Fig. 5. Differences (%) in channel powers between SERPENT (800 million histories) and MCNP (2300 million histories) calculations for the CANDU benchmark problem.
Table 2 Differences in k-effective between MINER and SERPENT for the 2-D 22 22 SCWR benchmark problem. Diffusion (FDM, 2 2 meshes/lattice) vs. reference
kEffective
Delta–rho (mk)
SERPENT 2000 M With 2-group single-cell-based With 4-group single-cell-based With 6-group single-cell-based With 8-group single-cell-based
1.007720 1.008274 1.007279 1.007096 1.006493
0.00 0.55 0.43 0.61 1.21
1.007720 1.005917 1.004882 1.004639 1.004391
0.00 1.78 2.80 3.04 3.29
SERPENT 2000 M With 2-group multicell-based With 4-group multicell-based With 6-group multicell-based With 8-group multicell-based
Fig. 6. The reference channel powers calculated with SERPENT for the 2-D 22 22 SCWR benchmark problem.
Factors (ADFs) can be used with both the finite-difference method and nodal method. Like other finite-difference codes, latticehomogenized cross sections and associated ADFs need to be provided for each lattice cell. The accuracy of the MINER solvers has been demonstrated (Phelps et al., 2009) by using it to solve a set of standard benchmark problems with both the FDM and the nodal method. The benchmarking results show that MINER is as accurate as RFSP and NESTLE, when the same calculation options are used. In order to perform full-core calculations with the diffusion code MINER, lattice-homogenized cross sections for each fuel channel and reflector channel need to be generated in advance. The singlelattice-cell-based lattice-homogenized cross sections for each fuel batch of the 22 22 SCWR benchmark problem were generated from the SERPENT single-lattice-cell transport calculations. The multicell-based lattice-homogenized cross sections for each fuel lattice location of the 22 22 SCWR benchmark problem were generated from the SERPENT full-core transport calculations. 4. Assessment results To assess the adequacy of using the 2-group single-lattice-cellbased lattice properties in the diffusion code, MINER calculations were performed with 2, 4, 6 and 8 groups, based on the single-lattice-cell-based lattice properties and multicell-based lattice
cross cross cross cross
cross cross cross cross
sections sections sections sections
sections sections sections sections
Table 3 Differences (%) in channel powers between MINER and SERPENT for the 2-D 22 22 SCWR benchmark problem. Diffusion (FDM, 2 2 meshes/lattice) vs. reference
Emax (center)
Emax (edge)
RMS
With 2-group sections With 4-group sections With 6-group sections With 8-group sections
single-cell-based cross
15.99
24.27
9.65
single-cell-based cross
11.86
18.15
7.21
single-cell-based cross
11.52
17.66
7.03
single-cell-based cross
9.62
14.21
5.75
7.57 3.38 2.50 1.64
7.80 3.98 2.69 1.45
3.91 1.79 1.33 0.90
With 2-group multicell-based cross sections With 4-group multicell-based cross sections With 6-group multicell-based cross sections With 8-group multicell-based cross sections
properties respectively. Considering the heterogeneous cluster geometry of the SCWR fuel bundle design, the option of 2 2 meshes per lattice with the FDM is considered most appropriate (Shen and Phelps, 2009) and was used in all MINER calculations. The MINER-calculated core k-effectives and channel powers with various options for the 22 22 SCWR benchmark problem were compared with the reference results. The reference results were obtained from the SERPENT full-core calculations with 4000,000 particles per cycle, 100 inactive cycles and 500 active cycles for a total of 2 billion active histories. The SERPENT uncertainty on the core k-effective value is 0.01 mk. The SERPENT uncertainty on the channel power is about 0.1% on average. Fig. 6
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
with 2-Group Single-Cell-Based Cross Sections A B C D E F G H J K L M N O P Q R S
1 24.3 19.6 16.1 9.0 12.1 10.1 12.9 13.4 13.4 13.4 13.4 12.9 10.1 12.1 9.0 16.1 19.6 24.3
2 3 19.6 16.0 8.2 6.8 6.9 0.9 7.3 1.4 2.2 2.4 5.6 -2.4 3.8 1.3 2.8 -3.5 0.5 -2.7 0.5 -2.7 2.8 -3.5 3.8 1.3 5.6 -2.4 2.2 2.4 7.3 1.4 6.9 0.9 8.2 6.8 19.6 16.0
4 8.7 7.0 1.2 -3.5 -3.2 -6.1 -4.5 -3.0 -7.5 -7.5 -3.0 -4.5 -6.1 -3.2 -3.5 1.2 7.0 8.7
5 11.8 1.8 2.2 -3.3 -6.7 -4.5 -9.5 -10.0 -6.1 -6.1 -10.0 -9.5 -4.5 -6.7 -3.3 2.2 1.8 11.8
6 9.5 5.2 -2.9 -6.4 -4.7 -10.0 -10.1 -13.2 -13.3 -13.3 -13.2 -10.1 -10.0 -4.7 -6.4 -2.9 5.2 9.5
7 12.2 3.2 0.8 -4.8 -9.7 -10.2 -8.9 -12.7 -13.3 -13.3 -12.7 -8.9 -10.2 -9.7 -4.8 0.8 3.2 12.2
8 12.8 2.3 -4.1 -3.5 -10.4 -13.5 -12.9 -16.0 -14.8 -14.8 -16.0 -12.9 -13.5 -10.4 -3.5 -4.1 2.3 12.8
9 12.9 -0.1 -3.2 -8.0 -6.6 -13.7 -13.6 -14.9 -15.3 -15.3 -14.9 -13.6 -13.7 -6.6 -8.0 -3.2 -0.1 12.9
10 12.9 -0.1 -3.2 -8.0 -6.6 -13.7 -13.6 -14.9 -15.3 -15.3 -14.9 -13.6 -13.7 -6.6 -8.0 -3.2 -0.1 12.9
11 12.8 2.3 -4.1 -3.5 -10.4 -13.5 -12.9 -16.0 -14.8 -14.8 -16.0 -12.9 -13.5 -10.4 -3.5 -4.1 2.3 12.8
12 12.2 3.2 0.8 -4.8 -9.7 -10.2 -8.9 -12.7 -13.3 -13.3 -12.7 -8.9 -10.2 -9.7 -4.8 0.8 3.2 12.2
13 9.5 5.2 -2.9 -6.4 -4.7 -10.0 -10.1 -13.2 -13.3 -13.3 -13.2 -10.1 -10.0 -4.7 -6.4 -2.9 5.2 9.5
14 11.8 1.8 2.2 -3.3 -6.7 -4.5 -9.5 -10.0 -6.1 -6.1 -10.0 -9.5 -4.5 -6.7 -3.3 2.2 1.8 11.8
15 8.7 7.0 1.2 -3.5 -3.2 -6.1 -4.5 -3.0 -7.5 -7.5 -3.0 -4.5 -6.1 -3.2 -3.5 1.2 7.0 8.7
16 17 16.0 19.6 6.8 8.2 0.9 6.9 1.4 7.3 2.4 2.2 -2.4 5.6 1.3 3.8 -3.5 2.8 -2.7 0.5 -2.7 0.5 -3.5 2.8 1.3 3.8 -2.4 5.6 2.4 2.2 1.4 7.3 0.9 6.9 6.8 8.2 16.0 19.6
18 24.3 19.6 16.1 9.0 12.1 10.1 12.9 13.4 13.4 13.4 13.4 12.9 10.1 12.1 9.0 16.1 19.6 24.3
Fig. 7. Differences (%) in channel powers between MINER (with the 2-group single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
with 8-Group Single-Cell-Based Cross Sections A B C D E F G H J K L M N O P Q R S
1 14.2 11.6 9.8 6.0 7.4 6.2 7.4 7.6 7.7 7.7 7.6 7.4 6.2 7.4 6.0 9.8 11.6 14.2
2 11.6 5.3 4.6 4.5 1.5 3.2 2.2 1.8 0.5 0.5 1.8 2.2 3.2 1.5 4.5 4.6 5.3 11.6
3 9.7 4.5 1.0 1.1 1.5 -1.4 0.7 -2.0 -1.2 -1.2 -2.0 0.7 -1.4 1.5 1.1 1.0 4.5 9.7
4 5.8 4.2 0.9 -1.9 -1.7 -3.6 -2.7 -1.8 -4.4 -4.4 -1.8 -2.7 -3.6 -1.7 -1.9 0.9 4.2 5.8
5 7.1 1.1 1.2 -1.9 -4.0 -2.7 -5.6 -5.9 -3.7 -3.7 -5.9 -5.6 -2.7 -4.0 -1.9 1.2 1.1 7.1
6 5.7 2.9 -1.9 -3.9 -2.9 -6.1 -5.9 -7.8 -7.9 -7.9 -7.8 -5.9 -6.1 -2.9 -3.9 -1.9 2.9 5.7
7 6.8 1.7 0.2 -3.0 -5.8 -6.0 -5.4 -7.6 -7.8 -7.8 -7.6 -5.4 -6.0 -5.8 -3.0 0.2 1.6 6.8
8 7.1 1.3 -2.5 -2.3 -6.3 -8.1 -7.7 -9.6 -8.8 -8.8 -9.6 -7.7 -8.1 -6.3 -2.3 -2.5 1.3 7.1
9 7.2 0.0 -1.7 -4.9 -4.2 -8.3 -8.0 -8.9 -9.1 -9.1 -8.9 -8.0 -8.3 -4.2 -4.9 -1.7 0.0 7.2
10 7.2 0.0 -1.7 -4.9 -4.2 -8.3 -8.0 -8.9 -9.1 -9.1 -8.9 -8.0 -8.3 -4.2 -4.9 -1.7 0.0 7.2
11 7.1 1.3 -2.5 -2.3 -6.3 -8.1 -7.7 -9.6 -8.8 -8.8 -9.6 -7.7 -8.1 -6.3 -2.3 -2.5 1.3 7.1
12 6.8 1.7 0.2 -3.0 -5.8 -6.0 -5.4 -7.6 -7.8 -7.8 -7.6 -5.4 -6.0 -5.8 -3.0 0.2 1.6 6.8
13 5.7 2.9 -1.8 -3.9 -2.9 -6.1 -5.9 -7.8 -7.9 -7.9 -7.8 -5.9 -6.1 -2.9 -3.9 -1.9 2.9 5.7
14 7.1 1.1 1.2 -1.9 -4.0 -2.7 -5.6 -5.9 -3.7 -3.7 -5.9 -5.6 -2.7 -4.0 -1.9 1.2 1.1 7.1
15 5.8 4.3 0.9 -1.9 -1.7 -3.6 -2.7 -1.8 -4.4 -4.4 -1.8 -2.7 -3.6 -1.7 -1.9 0.9 4.2 5.8
16 9.7 4.5 1.0 1.1 1.5 -1.4 0.7 -2.0 -1.2 -1.2 -2.0 0.7 -1.4 1.5 1.1 1.0 4.5 9.7
17 11.6 5.3 4.6 4.6 1.5 3.3 2.2 1.8 0.6 0.6 1.8 2.2 3.2 1.5 4.5 4.6 5.3 11.6
18 14.2 11.6 9.8 6.0 7.5 6.2 7.4 7.7 7.7 7.7 7.7 7.4 6.2 7.4 6.0 9.8 11.6 14.2
Fig. 8. Differences (%) in channel powers between MINER (with the 8-group single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
with 8-Group Multicell-Based Cross Sections A B C D E F G H J K L M N O P Q R S
1 1.2 1.1 1.0 -1.1 0.9 0.1 1.2 1.4 1.4 1.4 1.4 1.2 0.1 0.9 -1.1 1.0 1.1 1.2
2 1.1 -0.6 0.3 1.3 -0.6 1.2 0.5 0.4 -0.5 -0.5 0.4 0.5 1.2 -0.6 1.2 0.3 -0.7 1.1
3 0.9 0.2 -1.0 0.0 1.0 -0.8 1.2 -0.7 0.3 0.3 -0.7 1.2 -0.8 1.0 0.0 -1.0 0.1 0.9
4 -1.3 1.0 -0.2 -1.2 -0.2 -1.1 0.0 1.1 -0.8 -0.8 1.1 0.0 -1.1 -0.2 -1.2 -0.2 1.0 -1.3
5 0.6 -1.0 0.7 -0.3 -1.0 0.9 -1.0 -0.9 1.1 1.1 -0.9 -1.0 0.9 -1.0 -0.3 0.7 -1.0 0.6
6 -0.4 0.8 -1.3 -1.3 0.6 -1.2 -0.2 -1.3 -1.2 -1.2 -1.3 -0.2 -1.2 0.6 -1.4 -1.3 0.8 -0.4
7 0.7 -0.1 0.7 -0.3 -1.2 -0.3 0.9 -0.3 -0.1 -0.1 -0.3 0.9 -0.3 -1.2 -0.3 0.7 -0.1 0.7
8 0.8 -0.1 -1.2 0.6 -1.3 -1.6 -0.5 -1.5 -0.4 -0.4 -1.5 -0.5 -1.6 -1.3 0.6 -1.2 -0.1 0.8
9 1.0 -1.0 -0.2 -1.3 0.5 -1.6 -0.4 -0.6 -0.4 -0.4 -0.6 -0.4 -1.6 0.5 -1.3 -0.2 -1.0 1.0
10 1.0 -1.0 -0.2 -1.3 0.5 -1.6 -0.4 -0.6 -0.4 -0.4 -0.6 -0.4 -1.6 0.5 -1.3 -0.2 -1.0 1.0
11 0.8 -0.1 -1.2 0.6 -1.3 -1.6 -0.5 -1.5 -0.4 -0.4 -1.5 -0.5 -1.6 -1.3 0.6 -1.2 -0.1 0.8
12 0.7 -0.1 0.7 -0.3 -1.2 -0.3 0.9 -0.3 -0.1 -0.1 -0.3 0.9 -0.3 -1.2 -0.3 0.7 -0.1 0.7
13 -0.4 0.8 -1.3 -1.3 0.6 -1.1 -0.2 -1.3 -1.2 -1.2 -1.3 -0.2 -1.2 0.6 -1.3 -1.3 0.8 -0.4
14 0.6 -1.0 0.7 -0.3 -1.0 0.9 -1.0 -0.9 1.1 1.1 -0.9 -1.0 0.9 -1.0 -0.3 0.7 -1.0 0.6
15 -1.3 1.0 -0.2 -1.2 -0.2 -1.1 0.0 1.1 -0.8 -0.8 1.1 0.0 -1.1 -0.2 -1.2 -0.2 1.0 -1.3
16 0.9 0.2 -1.0 0.0 1.0 -0.8 1.2 -0.7 0.3 0.3 -0.7 1.2 -0.8 1.0 0.0 -1.0 0.1 0.9
17 1.1 -0.6 0.3 1.3 -0.6 1.2 0.5 0.4 -0.5 -0.5 0.4 0.5 1.2 -0.6 1.3 0.3 -0.6 1.1
18 1.2 1.1 1.0 -1.1 0.9 0.1 1.2 1.4 1.4 1.4 1.4 1.2 0.1 0.9 -1.1 1.0 1.1 1.2
Fig. 9. Differences (%) in channel powers between MINER (with the 8-group multicell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
illustrates the reference channel-power shape calculated with SERPENT. The relative difference in the core reactivity (in mk)
and the relative difference in the channel powers reported in this paper are defined as follows:
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
Difference (%) in Channel Power
Difference (%) in Channel Power
1 15
with 2-Group Single-Cell-Based Cross Sections with 4-Group Single-Cell-Based Cross Sections with 6-Group Single-Cell-Based Cross Sections with 8-Group Single-Cell-Based Cross Sections
10 5 0 -5 -10 -15 -20
1
2
8 6
Difference (%) in Channel Power
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
30 with 2-Group Single-Cell-Based Cross Sections with 4-Group Single-Cell-Based Cross Sections with 6-Group Single-Cell-Based Cross Sections with 8-Group Single-Cell-Based Cross Sections
25 20 15 10 5 0
Channel Number in Row A Fig. 11. Differences (%) in channel powers for row A between MINER (with the single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
Difference (%) in Channel Power
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
6 with 2-Group Multicell-Based Cross Sections with 4-Group Multicell-Based Cross Sections with 6-Group Multicell-Based Cross Sections with 8-Group Multicell-Based Cross Sections
4 2 0 -2 -4 -6 -8
Channel Number in Row J Fig. 12. Differences (%) in channel powers for row J between MINER (with the multicell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
SERPENT MINER 1000 Dq ¼ 1=k 1=k (
P MINER j
ð1Þ
)
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
with 2-Group Multicell-Based Cross Sections with 4-Group Multicell-Based Cross Sections with 6-Group Multicell-Based Cross Sections with 8-Group Multicell-Based Cross Sections
4 2 0 -2
Channel Number in Row J Fig. 10. Differences (%) in channel powers for row J between MINER (with the single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
3
10
Channel Number in Row A
Fig. 13. Differences (%) in channel powers for row A between MINER (with the multicell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
effectives and channel powers between MINER and SERPENT when different options are used in MINER. Table 2 shows that the impact of different options on the core k-effective is small (about 2 mk). It is interesting to note that, with the increasing complexity of the MINER calculation, the magnitude of the discrepancy in core keffective slightly increases. It is thought that the better agreement in the core k-effective with the simplest solution might be caused by cancelation of error. It should be noted that few milli-k discrepancy in k-effective can be easily accounted for in the core-analysis code with an allocated bias/uncertainty. Table 3 shows that both the multi-group model and the multicell model improve the accuracy in MINER-predicted channel powers, with the improvement made by the multicell model being more significant. For the cases with single-lattice-cell-based cross sections, the RMS differences in the channel power are 9.65%, 7.21%, 7.03% and 5.75% when 2, 4, 6 and 8 energy groups are used in MINER, respectively. For the cases with multicell-based cross sections, the RMS differences in the channel power are 3.91%, 1.79%, 1.33% and 0.90% when 2, 4, 6 and 8 energy groups are used in MINER, respectively. When the 8-group multicell model is used to replace the 2-group single-lattice-cell model, the RMS difference in the channel power reduces significantly from 9.65% to 0.90%. Figs. 7–9 show the differences in channel powers for the whole core between MINER and SERPENT when three different options are used in MINER. For the cases with the single-lattice-cell model, Figs. 10 and 11 show the differences in channel powers across the core for two rows of channels (one row at the inner region and another row at the edge region) between MINER and SERPENT. For the cases with the multicell model, Figs. 12 and 13 show the differences in channel powers across the core for the same two rows of channels between MINER and SERPENT. The results shown in these figures confirm the conclusion that both the multi-group model and the multicell model improve the accuracy in MINER-predicted channel powers, with the improvement made by the multicell model being more significant. 5. Conclusion
ð2Þ
Based on the assessment work presented above, the following conclusions can be made:
where kMINER and kSERPENT are MINER- and SERPENT-calculated core k-effectives. PMINER and P SERPENT are MINER- and SERPENT-calculated j j channel power at channel location j. Both the maximum differences at the edge and inner regions, and the RMS (Root Mean Square) differences in channel powers between MINER and SERPENT are reported for the benchmark problem. Tables 2 and 3 summarize the differences in core k-
A 2-D SCWR benchmark problem is setup (with the reference solution provided by the continuous energy Monte-Carlo code SERPENT) to assess the accuracy of the traditional neutron-diffusion core-analysis method. The Monte Carlo code SERPENT and the neutron-diffusion code MINER are appropriate tools for the assessment of reactor-physics methods for use in the SCWR core analysis.
ej ¼
1 SERPENT
Pj
100%
W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
The two-group neutron-diffusion theory is not sufficient to capture the spectral change for the analysis of the SCWR core, regardless of the use of single-lattice-cell- or multi-cell-based lattice properties. Multi-group neutron-diffusion theory is needed for the analysis of the SCWR core. The single-lattice-cell model is not sufficient to capture the environment effect for the generation of the lattice cross sections for the SCWR core. The multicell method, developed and implemented in RFSP, is essential for the analysis of the SCWR core. The coarse-mesh (2 2 meshes/lattice) finite-difference method solution of the eight-group neutron-diffusion equation with the multicell model is considered appropriate for the analysis of the SCWR core. In future work, it is recommended to extend the current 2-D benchmark problem to a 3-D benchmark problem in which absorber rods are included because absorbers usually introduce more heterogeneity into the calculation which may be more challenging for analysis. Acknowledgment The author is grateful to B. Hyland and J. Pencer for an enjoyable collaboration on the SCWR R&D. The funding of this research work from AECL Chalk River Nuclear Lab is acknowledged.
7
References Altiparmakov, D., 2008. New capabilities of the lattice code WIMS-AECL. In: Proceedings of the International Conference on the Physics of Reactors, PHYSOR-2008, Interlaken, Switzerland, September 14–19. Boczar, P.G., Shen, W., Pencer, J., Hyland, B., Chan, P.S.W., Dworshak, R.G., 2010. Reactor physics studies for a pressure-tube SCWR. In: Proceedings of the 2nd Canada–China Joint Workshop on Supercritical Water-Cooled Reactors (CCSC2010), Toronto, Ontario, Canada, April 25–28. Jenkins, D.A., Rouben, B., Shen, W., 2010. History of RFSP for CANDU fuel management and safety analysis. In: Proceedings of the 31st Annual Conference of the Canadian Nuclear Society, Montreal, Canada, May 24–27. Leppanen, J., 2007. Development of a new Monte Carlo Reactor Physics Code. D. Sc. Thesis, Helsinki University of Technology, VTT Publication 640, Finland, 2007. Phelps, B., Shen W., Varin, E., 2009. MINER: three-dimensional multi-group finitedifference and nodal method for CANDU applications. In: Proceedings of the 30th Annual Conference of Canadian Nuclear Society, Calgary, Canada, May 31– June 3. Shen, W., 2006. Development of a multicell methodology to account for heterogeneous core effects in the core-analysis diffusion code. In: Proceedings of the International Conference on the Advances in Nuclear Analysis and Simulation, PHYSOR-2006, Vancouver, September 10–14. Shen, W., Phelps, B., 2009. Assessment of the coarse-mesh finite-difference method with the multicell methodology in RFSP for ACR-1000. In: Proceedings of the 30th Annual Conference of the Canadian Nuclear Society, Calgary, Canada, May 31–June 3. Shen, W., Xie, Z., Yin, B., 1995. The Green’s function nodal expansion method for light water reactor diffusion calculations. Nucl. Sci. Eng. 121, 130. X-5 Monte Carlo Team, 2006. MCNP- A General Monte Carlo N-Particle Transport Code, Version 5, vol I: Overview and Theory, LA-UR-03-1987.
Contents lists available at SciVerse ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Assessment of the traditional neutron-diffusion core-analysis method for the analysis of the Super Critical Water Reactor W. Shen ⇑ Candu Energy Inc., 2285 Speakman Dr., Mississauga, ON, Canada L5B 1K1
a r t i c l e
i n f o
Article history: Received 12 October 2011 Received in revised form 3 February 2012 Accepted 10 February 2012 Available online 14 March 2012 Keywords: SCWR SERPENT Lattice homogenization Neutron diffusion
a b s t r a c t The key design quantities of the pressure-tube-based (PT-based) Super Critical Water Reactor (SCWR) core design are expected to be computed with the traditional core-analysis code which solves the two-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the lattice code. Two issues may affect the accuracy of these computed quantities for the SCWR core: one is the two-energy-group neutron-diffusion theory; the other is the generation of lattice-homogenized properties with the lattice code based on the single-lattice-cell model without considering the effects of the environment. It has been illustrated that the single-lattice-cell method is not sufficiently accurate for heterogeneous core configurations when adjacent channels experience significant spectrum interaction. To ensure the qualification of these computed quantities for the SCWR core, a 2-D SCWR benchmark problem was setup (with the reference solution provided by the continuous energy Monte-Carlo code SERPENT) to assess the traditional neutron-diffusion core-analysis method. The assessment shows that the traditional two-group neutron-diffusion theory with the single-lattice-cell-based lattice properties is not sufficient to capture either the spectral change or the environment effect for the SCWR core. The solution of the eight-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the multicell model is considered appropriate for the analysis of the PT-based SCWR core. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.
1. Introduction The Super Critical Water Reactor (SCWR) concept has been developed to take advantage of the improvement in thermal efficiency of a reactor through the use of supercritical water as a coolant. Although the concept is not new, interest has recently been renewed through the Generation IV (Gen IV) program. The Gen IV program is an international collaboration whose focus is the identification and development of next-generation nuclear energy systems which could be deployed to meet growing worldwide energy demands. The main Canadian contribution to the Gen IV program is the development of a pressure-tube-based (PT-based) SCWR. The PT-based SCWR (Boczar et al., 2010) shares features with the conventional CANDU such as heavy water moderator and pressurized fuel channels, rather than a single pressure vessel. However, in order to meet the material challenges associated with the higher pressures and corrosive environment associated with supercritical water, there are a number of key differences between the SCWR and the CANDU reactor. Features being considered for the SCWR design that differ from conventional CANDU designs include: supercritical water coolant, enriched fuel (Pu ⇑ Fax: +1 905 822 0567. E-mail address: [email protected]
and Th mixture fuel in this paper) in a 54-element bundle with no fuel in the central pin, a tight lattice pitch (25 cm lattice pitch), vertical fuel channels, and off-line batch refuelling. These core design features result in an under-moderated lattice, and the neutronic behavior is considerably different from that in the current CANDU reactors. The key elements of the SCWR core design are flux and power distributions, peak channel and bundle powers, and core reactivity coefficients for the nominal core conditions and for postulated accident conditions, such as the loss-of-coolant accident (LOCA) transient. These quantities are expected to be computed with the traditional core-analysis code such as RFSP (Jenkins et al., 2010) which solves the two-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the lattice code such as WIMS-AECL (Altiparmakov, 2008) based on the singlelattice-cell model. Two issues may affect the accuracy of the RFSP calculations of these quantities for the SCWR core: one is the two-energy-group neutron-diffusion theory; the other is the generation of latticehomogenized properties with the lattice code based on the single-lattice-cell model without considering the effects of the environment. It has been illustrated that the single-lattice-cell method is not sufficiently accurate for heterogeneous core configurations when adjacent channels experience significant spectrum interaction (Shen, 2006). The multicell method (Shen,
0306-4549/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2012.02.009
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
2006) developed and implemented in WIMS-AECL/RFSP allows the use of WIMS-AECL multicell-based fuel properties with the effects of the environment being considered. Note that the above two issues are not SCWR specific, however, their effect may be more significant for SCWR. To ensure the qualification of RFSP calculations of these quantities for the SCWR core, a 2-D SCWR benchmark problem was setup to assess the traditional two-group neutron-diffusion theory with single-lattice-based lattice properties for the analysis of the SCWR core. This paper documents preliminary assessment of the applicability of the traditional neutron-diffusion method for the analysis of the SCWR core. The assessment performed in this paper is so general that the assessment conclusions are expected to be applicable to any core-analysis code which solves the two-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the single-lattice-cell model. 2. Description of the SCWR benchmark problem To assess the applicability of using the 2-group single-latticecell model in the diffusion theory for the SCWR core analysis, a 2-D 22 22 SCWR benchmark problem was setup from scratch. Fig. 1 illustrates the geometry of the SCWR fuel lattice. The specifications for the fuel channel are given in Table 1. The SCWR fuel lattice modeled has the following features:
Super-critical water coolant. 54-element fuel of ThO2 with 14 wt.% PuO2. Porous zirconia insulator. Stainless-steel (SS310) cladding. 25 cm square lattice pitch.
The 2-D 22 22 SCWR benchmark problem consists of an 18 18 array of SCWR fuel lattices, with a lattice pitch of 25 cm, surrounded by a two-lattice-pitch heavy-water reflector boundary. The vacuum boundary condition is used at the four external surfaces in the XY plane and the reflective condition is used in the axial direction. Fig. 2 illustrates the geometry of the 22 22 core which consists of three batches of SCWR fuel with the fuel burnup of 20181.53 MWd/T, 40527.81 MWd/T, and 57817.85 MWd/T, respectively. The burnup values of the three fuel batches represent the mid-plane of the SCWR core at the end-of-cycle core
Fig. 1. Illustration of the SCWR fuel lattice.
Table 1 Specification for the 54-element SCWR fuel bundle. Parameter
Value
Lattice pitch (square) Coolant Moderator Fuel elements per bundle Fuel stack length
25 cm Super-critical water 99.91 at.% D2O 54 500 cm
Fuel element Elements in rings 1, 2, 3 Pitch circle radius of rings 1, 2, 3 Radius Outer radius of pin cladding Composition Liner tube Inner radius Thickness Composition Insulator Inner radius Thickness Composition
12, 18, 24 2.8755 cm, 4.3305 cm, 5.8000 cm 0.62 cm 0.68 cm 14 wt.% PuO2 in ThO2 6.8 cm 0.1 cm 50 vol.% of SS310 and 50 vol.% of coolant 6.9 cm 1.33 cm 30 vol.% of zirconia and 70 vol.% of coolant
Pressure tube Inner radius Thickness Composition
8.23 cm 1.4 cm CANDU zirc-4
Center element Composition Radius
Zirconia 1.94 cm
condition without soluble poison. The loading pattern of the three batches of fuel shown in Fig. 3 was selected to have a flat power shape. 3. Description of computational codes and models 3.1. SERPENT SERPENT (Leppanen, 2007) is a three-dimensional (3-D) continuous-energy Monte-Carlo reactor-physics burnup-calculation code, specifically designed for lattice physics applications. The code uses built-in calculation routines for generating homogenized multigroup constants for deterministic reactor simulator calculations. The standard output includes effective and infinite multiplication factors, homogenized cross sections, scattering matrices, diffusion coefficients, assembly discontinuity factors, and kinetic parameters. SERPENT can be used for various reactor-physics calculations at pin, lattice, and core levels. The continuous-energy Monte-Carlo method allows the modelling of any critical reactor type, including both thermal and fast neutron systems. All SERPENT calculations were executed with SERPENT 1.1.14 with the parallel mode in this study. The ENDF/B-VII nuclear data library with continuous energy contained in the SERPENT package was used directly in the assessment. Note that the choice of nuclear data library does not impact the assessment as long as the library used in the SERPENT full-core calculation and the cross-section preparation is consistent. There are two objectives of using SERPENT in the assessment: one is to provide reference power distributions for the 22 22 SCWR benchmark problem; the other is to provide lattice-homogenized cross sections for diffusion calculations. To obtain statistically converged flux/power distributions and lattice-homogenized cross sections for the benchmark problem with SERPENT, extensive assessment with various histories was performed with SERPENT for a CANDU benchmark problem with
3
W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
Fig. 2. Illustration of the 2-D 22 22 SCWR benchmark problem.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
4 4 1 1 1 3 1 2 1 1 1 1 1 1 2 1 3 1 1 1 4 4
4 4 1 3 2 1 3 1 2 2 3 3 2 2 1 3 1 2 3 1 4 4
4 4 1 2 3 2 1 3 1 3 2 2 3 1 3 1 2 3 2 1 4 4
4 4 3 1 2 3 2 3 2 1 3 3 1 2 3 2 3 2 1 3 4 4
4 4 1 3 1 2 3 1 3 3 1 1 3 3 1 3 2 1 3 1 4 4
4 4 2 1 3 3 1 3 2 3 3 3 3 2 3 1 3 3 1 2 4 4
4 4 1 2 1 2 3 2 1 2 2 2 2 1 2 3 2 1 2 1 4 4
4 4 1 2 3 1 3 3 2 3 2 2 3 2 3 3 1 3 2 1 4 4
4 4 1 3 2 3 1 3 2 2 2 2 2 2 3 1 3 2 3 1 4 4
4 4 1 3 2 3 1 3 2 2 2 2 2 2 3 1 3 2 3 1 4 4
4 4 1 2 3 1 3 3 2 3 2 2 3 2 3 3 1 3 2 1 4 4
4 4 1 2 1 2 3 2 1 2 2 2 2 1 2 3 2 1 2 1 4 4
4 4 2 1 3 3 1 3 2 3 3 3 3 2 3 1 3 3 1 2 4 4
4 4 1 3 1 2 3 1 3 3 1 1 3 3 1 3 2 1 3 1 4 4
4 4 3 1 2 3 2 3 2 1 3 3 1 2 3 2 3 2 1 3 4 4
4 4 1 2 3 2 1 3 1 3 2 2 3 1 3 1 2 3 2 1 4 4
4 4 1 3 2 1 3 1 2 2 3 3 2 2 1 3 1 2 3 1 4 4
4 4 1 1 1 3 1 2 1 1 1 1 1 1 2 1 3 1 1 1 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
1 2 3 4
-----
Once-Burnt Fuel Twice Burnt Fuel Thrice Burnt Fuel Reflector
Fig. 3. Fuel loading pattern for the 2-D 22 22 SCWR benchmark problem.
SERPENT-calculated core k-effective as the function of number of histories for the CANDU benchmark problem. Fig. 5 shows the differences in channel powers between SERPENT (with 800 million histories) and MCNP5 (with 2300 million histories) for the CANDU benchmark problem. Overall the assessment shows that SERPENT’s prediction of core k-effective and channel powers is very close to those predicted with MCNP5, with a difference in the core k-effective of about 0.3 mk and a RMS difference in the channel power of about 0.2% . 3.2. Miner
Fig. 4. Convergence in SERPENT-calculated core k-effective as a function of number of histories for the CANDU benchmark problem.
the same nuclear data library as that of MCNP5 (X-5 Monte Carlo Team, 2006). Fig. 4 shows the convergence of the
Because of the limitation of the 2-group approximation used in RFSP diffusion calculations, MINER (Phelps et al., 2009) was used to model the 22 22 SCWR benchmark problem. MINER is an in-house-developed 3-D multi-group neutron-diffusion code which contains two multi-group diffusion solvers based on the finitedifference method (FDM) and modern nodal method (Shen et al., 1995). Different numbers of energy groups and different mesh sizes can be used in the diffusion calculation. Assembly Discontinuity
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
A B C D E F G H J K L M N O P Q R S
1 0.2 0.3 0.2 0.2 0.3 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.0 -0.1 -0.1 0.1 -0.1
2 0.3 0.2 0.1 0.3 0.2 0.5 0.4 0.5 0.4 0.2 0.4 0.2 0.3 0.0 0.0 -0.1 0.0 -0.2
3 0.3 0.0 0.0 0.1 0.1 0.4 0.3 0.4 0.5 0.4 0.4 0.2 0.2 0.0 -0.1 -0.2 -0.1 0.0
4 0.2 -0.1 0.0 0.1 0.1 0.2 0.3 0.3 0.5 0.5 0.3 0.1 0.1 0.0 0.0 -0.2 -0.1 -0.1
5 0.1 0.2 0.1 0.0 0.0 0.2 0.3 0.3 0.4 0.4 0.4 0.2 0.0 -0.1 -0.1 -0.1 0.0 -0.1
6 0.1 0.0 0.0 0.0 0.0 0.1 0.3 0.2 0.3 0.3 0.2 0.2 0.0 -0.1 0.0 0.0 0.0 -0.2
7 0.0 0.0 0.0 0.0 -0.1 0.1 0.2 0.2 0.1 0.1 0.2 0.0 -0.1 -0.1 -0.1 -0.2 -0.1 -0.1
8 -0.1 0.0 -0.1 -0.1 -0.1 0.0 0.1 0.0 0.0 0.0 0.0 0.1 0.0 -0.2 -0.2 -0.2 -0.2 -0.3
9 -0.1 -0.2 -0.1 -0.2 -0.1 0.0 0.1 0.0 0.0 -0.1 0.0 0.1 0.0 -0.1 -0.1 -0.2 -0.4 -0.2
10 -0.2 -0.1 -0.1 -0.1 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0 -0.2 -0.2 -0.3 -0.3
11 -0.2 0.0 0.0 0.0 -0.1 0.0 0.1 -0.1 0.0 0.0 0.0 0.0 -0.1 -0.1 0.0 -0.1 -0.2 -0.2
12 0.0 -0.2 -0.1 0.0 0.0 0.0 0.0 0.0 -0.1 0.0 0.0 0.1 0.0 -0.2 0.0 -0.3 -0.3 -0.1
13 0.0 0.0 -0.1 0.0 -0.1 -0.1 0.0 -0.1 -0.1 -0.1 0.0 0.0 -0.2 -0.2 -0.2 -0.3 -0.3 -0.4
14 0.1 0.0 -0.1 -0.2 -0.2 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 -0.2 -0.3 -0.2 -0.4
15 0.0 -0.1 -0.2 -0.1 -0.1 -0.1 -0.1 0.0 0.0 0.0 0.0 0.1 -0.1 -0.2 -0.2 -0.3 -0.5 -0.2
16 0.0 -0.2 0.0 -0.1 -0.1 -0.1 -0.1 0.0 -0.1 0.0 0.0 0.0 -0.1 -0.2 -0.2 -0.4 -0.5 -0.1
17 -0.2 -0.1 -0.1 0.0 0.0 0.0 -0.2 0.0 -0.1 -0.1 0.1 -0.1 -0.2 -0.3 -0.1 -0.3 -0.3 -0.4
18 -0.1 -0.1 0.0 -0.3 -0.1 -0.1 -0.1 -0.2 -0.1 0.1 -0.1 -0.1 -0.2 -0.2 -0.4 -0.3 -0.2 0.0
Fig. 5. Differences (%) in channel powers between SERPENT (800 million histories) and MCNP (2300 million histories) calculations for the CANDU benchmark problem.
Table 2 Differences in k-effective between MINER and SERPENT for the 2-D 22 22 SCWR benchmark problem. Diffusion (FDM, 2 2 meshes/lattice) vs. reference
kEffective
Delta–rho (mk)
SERPENT 2000 M With 2-group single-cell-based With 4-group single-cell-based With 6-group single-cell-based With 8-group single-cell-based
1.007720 1.008274 1.007279 1.007096 1.006493
0.00 0.55 0.43 0.61 1.21
1.007720 1.005917 1.004882 1.004639 1.004391
0.00 1.78 2.80 3.04 3.29
SERPENT 2000 M With 2-group multicell-based With 4-group multicell-based With 6-group multicell-based With 8-group multicell-based
Fig. 6. The reference channel powers calculated with SERPENT for the 2-D 22 22 SCWR benchmark problem.
Factors (ADFs) can be used with both the finite-difference method and nodal method. Like other finite-difference codes, latticehomogenized cross sections and associated ADFs need to be provided for each lattice cell. The accuracy of the MINER solvers has been demonstrated (Phelps et al., 2009) by using it to solve a set of standard benchmark problems with both the FDM and the nodal method. The benchmarking results show that MINER is as accurate as RFSP and NESTLE, when the same calculation options are used. In order to perform full-core calculations with the diffusion code MINER, lattice-homogenized cross sections for each fuel channel and reflector channel need to be generated in advance. The singlelattice-cell-based lattice-homogenized cross sections for each fuel batch of the 22 22 SCWR benchmark problem were generated from the SERPENT single-lattice-cell transport calculations. The multicell-based lattice-homogenized cross sections for each fuel lattice location of the 22 22 SCWR benchmark problem were generated from the SERPENT full-core transport calculations. 4. Assessment results To assess the adequacy of using the 2-group single-lattice-cellbased lattice properties in the diffusion code, MINER calculations were performed with 2, 4, 6 and 8 groups, based on the single-lattice-cell-based lattice properties and multicell-based lattice
cross cross cross cross
cross cross cross cross
sections sections sections sections
sections sections sections sections
Table 3 Differences (%) in channel powers between MINER and SERPENT for the 2-D 22 22 SCWR benchmark problem. Diffusion (FDM, 2 2 meshes/lattice) vs. reference
Emax (center)
Emax (edge)
RMS
With 2-group sections With 4-group sections With 6-group sections With 8-group sections
single-cell-based cross
15.99
24.27
9.65
single-cell-based cross
11.86
18.15
7.21
single-cell-based cross
11.52
17.66
7.03
single-cell-based cross
9.62
14.21
5.75
7.57 3.38 2.50 1.64
7.80 3.98 2.69 1.45
3.91 1.79 1.33 0.90
With 2-group multicell-based cross sections With 4-group multicell-based cross sections With 6-group multicell-based cross sections With 8-group multicell-based cross sections
properties respectively. Considering the heterogeneous cluster geometry of the SCWR fuel bundle design, the option of 2 2 meshes per lattice with the FDM is considered most appropriate (Shen and Phelps, 2009) and was used in all MINER calculations. The MINER-calculated core k-effectives and channel powers with various options for the 22 22 SCWR benchmark problem were compared with the reference results. The reference results were obtained from the SERPENT full-core calculations with 4000,000 particles per cycle, 100 inactive cycles and 500 active cycles for a total of 2 billion active histories. The SERPENT uncertainty on the core k-effective value is 0.01 mk. The SERPENT uncertainty on the channel power is about 0.1% on average. Fig. 6
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W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
with 2-Group Single-Cell-Based Cross Sections A B C D E F G H J K L M N O P Q R S
1 24.3 19.6 16.1 9.0 12.1 10.1 12.9 13.4 13.4 13.4 13.4 12.9 10.1 12.1 9.0 16.1 19.6 24.3
2 3 19.6 16.0 8.2 6.8 6.9 0.9 7.3 1.4 2.2 2.4 5.6 -2.4 3.8 1.3 2.8 -3.5 0.5 -2.7 0.5 -2.7 2.8 -3.5 3.8 1.3 5.6 -2.4 2.2 2.4 7.3 1.4 6.9 0.9 8.2 6.8 19.6 16.0
4 8.7 7.0 1.2 -3.5 -3.2 -6.1 -4.5 -3.0 -7.5 -7.5 -3.0 -4.5 -6.1 -3.2 -3.5 1.2 7.0 8.7
5 11.8 1.8 2.2 -3.3 -6.7 -4.5 -9.5 -10.0 -6.1 -6.1 -10.0 -9.5 -4.5 -6.7 -3.3 2.2 1.8 11.8
6 9.5 5.2 -2.9 -6.4 -4.7 -10.0 -10.1 -13.2 -13.3 -13.3 -13.2 -10.1 -10.0 -4.7 -6.4 -2.9 5.2 9.5
7 12.2 3.2 0.8 -4.8 -9.7 -10.2 -8.9 -12.7 -13.3 -13.3 -12.7 -8.9 -10.2 -9.7 -4.8 0.8 3.2 12.2
8 12.8 2.3 -4.1 -3.5 -10.4 -13.5 -12.9 -16.0 -14.8 -14.8 -16.0 -12.9 -13.5 -10.4 -3.5 -4.1 2.3 12.8
9 12.9 -0.1 -3.2 -8.0 -6.6 -13.7 -13.6 -14.9 -15.3 -15.3 -14.9 -13.6 -13.7 -6.6 -8.0 -3.2 -0.1 12.9
10 12.9 -0.1 -3.2 -8.0 -6.6 -13.7 -13.6 -14.9 -15.3 -15.3 -14.9 -13.6 -13.7 -6.6 -8.0 -3.2 -0.1 12.9
11 12.8 2.3 -4.1 -3.5 -10.4 -13.5 -12.9 -16.0 -14.8 -14.8 -16.0 -12.9 -13.5 -10.4 -3.5 -4.1 2.3 12.8
12 12.2 3.2 0.8 -4.8 -9.7 -10.2 -8.9 -12.7 -13.3 -13.3 -12.7 -8.9 -10.2 -9.7 -4.8 0.8 3.2 12.2
13 9.5 5.2 -2.9 -6.4 -4.7 -10.0 -10.1 -13.2 -13.3 -13.3 -13.2 -10.1 -10.0 -4.7 -6.4 -2.9 5.2 9.5
14 11.8 1.8 2.2 -3.3 -6.7 -4.5 -9.5 -10.0 -6.1 -6.1 -10.0 -9.5 -4.5 -6.7 -3.3 2.2 1.8 11.8
15 8.7 7.0 1.2 -3.5 -3.2 -6.1 -4.5 -3.0 -7.5 -7.5 -3.0 -4.5 -6.1 -3.2 -3.5 1.2 7.0 8.7
16 17 16.0 19.6 6.8 8.2 0.9 6.9 1.4 7.3 2.4 2.2 -2.4 5.6 1.3 3.8 -3.5 2.8 -2.7 0.5 -2.7 0.5 -3.5 2.8 1.3 3.8 -2.4 5.6 2.4 2.2 1.4 7.3 0.9 6.9 6.8 8.2 16.0 19.6
18 24.3 19.6 16.1 9.0 12.1 10.1 12.9 13.4 13.4 13.4 13.4 12.9 10.1 12.1 9.0 16.1 19.6 24.3
Fig. 7. Differences (%) in channel powers between MINER (with the 2-group single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
with 8-Group Single-Cell-Based Cross Sections A B C D E F G H J K L M N O P Q R S
1 14.2 11.6 9.8 6.0 7.4 6.2 7.4 7.6 7.7 7.7 7.6 7.4 6.2 7.4 6.0 9.8 11.6 14.2
2 11.6 5.3 4.6 4.5 1.5 3.2 2.2 1.8 0.5 0.5 1.8 2.2 3.2 1.5 4.5 4.6 5.3 11.6
3 9.7 4.5 1.0 1.1 1.5 -1.4 0.7 -2.0 -1.2 -1.2 -2.0 0.7 -1.4 1.5 1.1 1.0 4.5 9.7
4 5.8 4.2 0.9 -1.9 -1.7 -3.6 -2.7 -1.8 -4.4 -4.4 -1.8 -2.7 -3.6 -1.7 -1.9 0.9 4.2 5.8
5 7.1 1.1 1.2 -1.9 -4.0 -2.7 -5.6 -5.9 -3.7 -3.7 -5.9 -5.6 -2.7 -4.0 -1.9 1.2 1.1 7.1
6 5.7 2.9 -1.9 -3.9 -2.9 -6.1 -5.9 -7.8 -7.9 -7.9 -7.8 -5.9 -6.1 -2.9 -3.9 -1.9 2.9 5.7
7 6.8 1.7 0.2 -3.0 -5.8 -6.0 -5.4 -7.6 -7.8 -7.8 -7.6 -5.4 -6.0 -5.8 -3.0 0.2 1.6 6.8
8 7.1 1.3 -2.5 -2.3 -6.3 -8.1 -7.7 -9.6 -8.8 -8.8 -9.6 -7.7 -8.1 -6.3 -2.3 -2.5 1.3 7.1
9 7.2 0.0 -1.7 -4.9 -4.2 -8.3 -8.0 -8.9 -9.1 -9.1 -8.9 -8.0 -8.3 -4.2 -4.9 -1.7 0.0 7.2
10 7.2 0.0 -1.7 -4.9 -4.2 -8.3 -8.0 -8.9 -9.1 -9.1 -8.9 -8.0 -8.3 -4.2 -4.9 -1.7 0.0 7.2
11 7.1 1.3 -2.5 -2.3 -6.3 -8.1 -7.7 -9.6 -8.8 -8.8 -9.6 -7.7 -8.1 -6.3 -2.3 -2.5 1.3 7.1
12 6.8 1.7 0.2 -3.0 -5.8 -6.0 -5.4 -7.6 -7.8 -7.8 -7.6 -5.4 -6.0 -5.8 -3.0 0.2 1.6 6.8
13 5.7 2.9 -1.8 -3.9 -2.9 -6.1 -5.9 -7.8 -7.9 -7.9 -7.8 -5.9 -6.1 -2.9 -3.9 -1.9 2.9 5.7
14 7.1 1.1 1.2 -1.9 -4.0 -2.7 -5.6 -5.9 -3.7 -3.7 -5.9 -5.6 -2.7 -4.0 -1.9 1.2 1.1 7.1
15 5.8 4.3 0.9 -1.9 -1.7 -3.6 -2.7 -1.8 -4.4 -4.4 -1.8 -2.7 -3.6 -1.7 -1.9 0.9 4.2 5.8
16 9.7 4.5 1.0 1.1 1.5 -1.4 0.7 -2.0 -1.2 -1.2 -2.0 0.7 -1.4 1.5 1.1 1.0 4.5 9.7
17 11.6 5.3 4.6 4.6 1.5 3.3 2.2 1.8 0.6 0.6 1.8 2.2 3.2 1.5 4.5 4.6 5.3 11.6
18 14.2 11.6 9.8 6.0 7.5 6.2 7.4 7.7 7.7 7.7 7.7 7.4 6.2 7.4 6.0 9.8 11.6 14.2
Fig. 8. Differences (%) in channel powers between MINER (with the 8-group single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
with 8-Group Multicell-Based Cross Sections A B C D E F G H J K L M N O P Q R S
1 1.2 1.1 1.0 -1.1 0.9 0.1 1.2 1.4 1.4 1.4 1.4 1.2 0.1 0.9 -1.1 1.0 1.1 1.2
2 1.1 -0.6 0.3 1.3 -0.6 1.2 0.5 0.4 -0.5 -0.5 0.4 0.5 1.2 -0.6 1.2 0.3 -0.7 1.1
3 0.9 0.2 -1.0 0.0 1.0 -0.8 1.2 -0.7 0.3 0.3 -0.7 1.2 -0.8 1.0 0.0 -1.0 0.1 0.9
4 -1.3 1.0 -0.2 -1.2 -0.2 -1.1 0.0 1.1 -0.8 -0.8 1.1 0.0 -1.1 -0.2 -1.2 -0.2 1.0 -1.3
5 0.6 -1.0 0.7 -0.3 -1.0 0.9 -1.0 -0.9 1.1 1.1 -0.9 -1.0 0.9 -1.0 -0.3 0.7 -1.0 0.6
6 -0.4 0.8 -1.3 -1.3 0.6 -1.2 -0.2 -1.3 -1.2 -1.2 -1.3 -0.2 -1.2 0.6 -1.4 -1.3 0.8 -0.4
7 0.7 -0.1 0.7 -0.3 -1.2 -0.3 0.9 -0.3 -0.1 -0.1 -0.3 0.9 -0.3 -1.2 -0.3 0.7 -0.1 0.7
8 0.8 -0.1 -1.2 0.6 -1.3 -1.6 -0.5 -1.5 -0.4 -0.4 -1.5 -0.5 -1.6 -1.3 0.6 -1.2 -0.1 0.8
9 1.0 -1.0 -0.2 -1.3 0.5 -1.6 -0.4 -0.6 -0.4 -0.4 -0.6 -0.4 -1.6 0.5 -1.3 -0.2 -1.0 1.0
10 1.0 -1.0 -0.2 -1.3 0.5 -1.6 -0.4 -0.6 -0.4 -0.4 -0.6 -0.4 -1.6 0.5 -1.3 -0.2 -1.0 1.0
11 0.8 -0.1 -1.2 0.6 -1.3 -1.6 -0.5 -1.5 -0.4 -0.4 -1.5 -0.5 -1.6 -1.3 0.6 -1.2 -0.1 0.8
12 0.7 -0.1 0.7 -0.3 -1.2 -0.3 0.9 -0.3 -0.1 -0.1 -0.3 0.9 -0.3 -1.2 -0.3 0.7 -0.1 0.7
13 -0.4 0.8 -1.3 -1.3 0.6 -1.1 -0.2 -1.3 -1.2 -1.2 -1.3 -0.2 -1.2 0.6 -1.3 -1.3 0.8 -0.4
14 0.6 -1.0 0.7 -0.3 -1.0 0.9 -1.0 -0.9 1.1 1.1 -0.9 -1.0 0.9 -1.0 -0.3 0.7 -1.0 0.6
15 -1.3 1.0 -0.2 -1.2 -0.2 -1.1 0.0 1.1 -0.8 -0.8 1.1 0.0 -1.1 -0.2 -1.2 -0.2 1.0 -1.3
16 0.9 0.2 -1.0 0.0 1.0 -0.8 1.2 -0.7 0.3 0.3 -0.7 1.2 -0.8 1.0 0.0 -1.0 0.1 0.9
17 1.1 -0.6 0.3 1.3 -0.6 1.2 0.5 0.4 -0.5 -0.5 0.4 0.5 1.2 -0.6 1.3 0.3 -0.6 1.1
18 1.2 1.1 1.0 -1.1 0.9 0.1 1.2 1.4 1.4 1.4 1.4 1.2 0.1 0.9 -1.1 1.0 1.1 1.2
Fig. 9. Differences (%) in channel powers between MINER (with the 8-group multicell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
illustrates the reference channel-power shape calculated with SERPENT. The relative difference in the core reactivity (in mk)
and the relative difference in the channel powers reported in this paper are defined as follows:
6
W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
Difference (%) in Channel Power
Difference (%) in Channel Power
1 15
with 2-Group Single-Cell-Based Cross Sections with 4-Group Single-Cell-Based Cross Sections with 6-Group Single-Cell-Based Cross Sections with 8-Group Single-Cell-Based Cross Sections
10 5 0 -5 -10 -15 -20
1
2
8 6
Difference (%) in Channel Power
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
30 with 2-Group Single-Cell-Based Cross Sections with 4-Group Single-Cell-Based Cross Sections with 6-Group Single-Cell-Based Cross Sections with 8-Group Single-Cell-Based Cross Sections
25 20 15 10 5 0
Channel Number in Row A Fig. 11. Differences (%) in channel powers for row A between MINER (with the single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
Difference (%) in Channel Power
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
6 with 2-Group Multicell-Based Cross Sections with 4-Group Multicell-Based Cross Sections with 6-Group Multicell-Based Cross Sections with 8-Group Multicell-Based Cross Sections
4 2 0 -2 -4 -6 -8
Channel Number in Row J Fig. 12. Differences (%) in channel powers for row J between MINER (with the multicell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
SERPENT MINER 1000 Dq ¼ 1=k 1=k (
P MINER j
ð1Þ
)
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
with 2-Group Multicell-Based Cross Sections with 4-Group Multicell-Based Cross Sections with 6-Group Multicell-Based Cross Sections with 8-Group Multicell-Based Cross Sections
4 2 0 -2
Channel Number in Row J Fig. 10. Differences (%) in channel powers for row J between MINER (with the single-cell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
3
10
Channel Number in Row A
Fig. 13. Differences (%) in channel powers for row A between MINER (with the multicell model) and SERPENT for the 2-D 22 22 SCWR benchmark problem.
effectives and channel powers between MINER and SERPENT when different options are used in MINER. Table 2 shows that the impact of different options on the core k-effective is small (about 2 mk). It is interesting to note that, with the increasing complexity of the MINER calculation, the magnitude of the discrepancy in core keffective slightly increases. It is thought that the better agreement in the core k-effective with the simplest solution might be caused by cancelation of error. It should be noted that few milli-k discrepancy in k-effective can be easily accounted for in the core-analysis code with an allocated bias/uncertainty. Table 3 shows that both the multi-group model and the multicell model improve the accuracy in MINER-predicted channel powers, with the improvement made by the multicell model being more significant. For the cases with single-lattice-cell-based cross sections, the RMS differences in the channel power are 9.65%, 7.21%, 7.03% and 5.75% when 2, 4, 6 and 8 energy groups are used in MINER, respectively. For the cases with multicell-based cross sections, the RMS differences in the channel power are 3.91%, 1.79%, 1.33% and 0.90% when 2, 4, 6 and 8 energy groups are used in MINER, respectively. When the 8-group multicell model is used to replace the 2-group single-lattice-cell model, the RMS difference in the channel power reduces significantly from 9.65% to 0.90%. Figs. 7–9 show the differences in channel powers for the whole core between MINER and SERPENT when three different options are used in MINER. For the cases with the single-lattice-cell model, Figs. 10 and 11 show the differences in channel powers across the core for two rows of channels (one row at the inner region and another row at the edge region) between MINER and SERPENT. For the cases with the multicell model, Figs. 12 and 13 show the differences in channel powers across the core for the same two rows of channels between MINER and SERPENT. The results shown in these figures confirm the conclusion that both the multi-group model and the multicell model improve the accuracy in MINER-predicted channel powers, with the improvement made by the multicell model being more significant. 5. Conclusion
ð2Þ
Based on the assessment work presented above, the following conclusions can be made:
where kMINER and kSERPENT are MINER- and SERPENT-calculated core k-effectives. PMINER and P SERPENT are MINER- and SERPENT-calculated j j channel power at channel location j. Both the maximum differences at the edge and inner regions, and the RMS (Root Mean Square) differences in channel powers between MINER and SERPENT are reported for the benchmark problem. Tables 2 and 3 summarize the differences in core k-
A 2-D SCWR benchmark problem is setup (with the reference solution provided by the continuous energy Monte-Carlo code SERPENT) to assess the accuracy of the traditional neutron-diffusion core-analysis method. The Monte Carlo code SERPENT and the neutron-diffusion code MINER are appropriate tools for the assessment of reactor-physics methods for use in the SCWR core analysis.
ej ¼
1 SERPENT
Pj
100%
W. Shen / Annals of Nuclear Energy 45 (2012) 1–7
The two-group neutron-diffusion theory is not sufficient to capture the spectral change for the analysis of the SCWR core, regardless of the use of single-lattice-cell- or multi-cell-based lattice properties. Multi-group neutron-diffusion theory is needed for the analysis of the SCWR core. The single-lattice-cell model is not sufficient to capture the environment effect for the generation of the lattice cross sections for the SCWR core. The multicell method, developed and implemented in RFSP, is essential for the analysis of the SCWR core. The coarse-mesh (2 2 meshes/lattice) finite-difference method solution of the eight-group neutron-diffusion equation with the multicell model is considered appropriate for the analysis of the SCWR core. In future work, it is recommended to extend the current 2-D benchmark problem to a 3-D benchmark problem in which absorber rods are included because absorbers usually introduce more heterogeneity into the calculation which may be more challenging for analysis. Acknowledgment The author is grateful to B. Hyland and J. Pencer for an enjoyable collaboration on the SCWR R&D. The funding of this research work from AECL Chalk River Nuclear Lab is acknowledged.
7
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