Implementation of a direct perturbation method in MCNP and application to SCALE verification

Annals of Nuclear Energy 62 (2013) 291–297

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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

Implementation of a direct perturbation method in MCNP and application to SCALE verification A. Trottier ⇑, L. Blomeley, J.C. Chow, A. Colton, E. Masala, B. Shukhman, D. Watts, B. Wilkin Atomic Energy of Canada Limited, Chalk River Laboratories, Ontario, Canada

a r t i c l e

i n f o

Article history: Received 3 October 2012 Received in revised form 14 June 2013 Accepted 18 June 2013 Available online 14 July 2013 Keywords: Sensitivity and uncertainty Nuclear data Reactor physics Monte Carlo

a b s t r a c t Many nuclear data uncertainty propagation methods are implemented on the basis of first-order perturbation theory. These methods are complex and integral verification using direct perturbation of the nuclear cross section data is difficult due to the structure of nuclear data files. We present a new implementation of the direct perturbation method, which eliminates the need to modify these files. The method was implemented in DPERT, a patched version of MCNP5. Using the DPERT patch, we present an integral verification of TSURFER based on 77 heavy water moderated ZED-2 critical experiments. TSURFER is a module of the SCALE code suite and applies first-order perturbation theory to propagate nuclear data uncertainties. The experiments were modeled using the standard MCNP5 code to establish the a priori keff calculation biases. TSURFER was used to minimize these biases by adjusting the underlying nuclear data. The proposed cross section alterations were then applied to the experiment models, and the DPERT patch was used to verify TSURFER’s evaluation of the a posteriori keff biases. The study confirmed the TSURFER bias reduction prediction, but suggests TSURFER may underestimate the impact of the nuclear data corrections by 1.35 ± 0.05 mk on average. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction Critical experiment facilities are rapidly disappearing, yet demands on uncertainty assessments for nuclear systems are increasing (Aliberti et al., 2006). Nuclear data uncertainty is a key contributor to the total uncertainty quantification problem. Current approaches used for nuclear data uncertainty analysis have recently been reviewed by the NEA (de Saint-Jean et al., 2011). The SCALE package (Bowman, 2011; ORNL, 2009) offers several tools for nuclear data sensitivity analysis (Rearden et al., 2011). The TSUNAMI approach, implemented in SCALE6, is based on first-order perturbation theory. A set of critical experiments is analyzed using the TSUNAMI module, which models the critical systems using the KENO V.a transport code (Goluoglu et al., 2011). The forward and adjoint fluxes are evaluated and the sensitivity of keff to nuclear data perturbations is determined. Using a generalized least squares algorithm, TSURFER can produce a set of adjustments to the nuclear data that reduces the calculated keff biases for a given set of experiments. The algorithm is constrained by the nuclear data and experimental keff covariance matrices. TSURFER does not perform a transport calculation using altered cross sections. This framework is complex, and it is desirable to have independent verification methods. The simplest approach is to perform a ⇑ Corresponding author. E-mail address: [email protected] (A. Trottier).

parallel analysis using a different neutron transport code. To make an independent estimate of the a posteriori keff bias would then require that the proposed nuclear data adjustments be applied to the a priori models, and that a new neutron transport calculation be executed. The application of the adjustments would therefore require that the nuclear data files be modified. These modifications would need to be repeated each time a different set of nuclear data adjustments was to be studied. The difficulty that arises is that the formats of the nuclear data files are complex and significant effort would be required to confirm that the alterations had been implemented in conformance with the proposed alterations. It is therefore desirable to eliminate the need to manipulate the nuclear data files, and simply specify the desired alterations within the problem input. MCNP5 (X-5 Monte Carlo Team, 2003) was modified to implement such an independent verification capability. The DPERT1 patch provides an integral testing tool for TSURFER. On the basis of 77 heavy water moderated critical experiments conducted at the ZED-2 facility, we present code-to-code comparisons of the a priori keff biases evaluated using MCNP5 and KENO V.a, and a detailed verification of TSURFER bias reduction predictions using the DPERT patch.

1 Direct cross-section PERTurbation, developed at Atomic Energy of Canada Ltd. (AECL). ‘‘Patch’’ is the term used in the MCNP5 manual for a modification to the MCNP5 code.

0306-4549/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.anucene.2013.06.020

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A. Trottier et al. / Annals of Nuclear Energy 62 (2013) 291–297

Table 1 Summary fuel cluster data for the ZED-2 critical experiments. All fuels are sintered powders, with the exception of Type J fuel which is formed of natural Uranium metal. LEU fuels list enrichments in units of wt.% 235U/U. MOX fuels specify Pu content by wt.% Pu/(heavy element mass). NU fuels refer to natural Uranium fuel. Neutron absorbers include Dy, Gd and Hf. Fuel densities are averaged, but exclude any dedicated neutron absorbing elements. Type Fuel mass (kg)

Clad

Composition Enrich. (wt.%)

A B C D E F G H I J

Zirc-4 Zirc-4 Zirc-4 Zirc-4 Zirc-4 Zirc-4 Zirc-2 Zirc-2 Zirc-4 Al

LEU LEU LEU MOX LEU MOX NU NU LEU NU metal

21.0 18.7 20.9 21.1 20.2 16.1 22.1 15.0 21.2 45.5

Absorbers (g)

0.95 0 1.70 190 0.96 0 0.37, 0.30 10 0.96 70 1.5, 1.3 220 None 0 None 0 1.35 125 None 0

Density (g/cm3) 10.52 10.64 10.45 10.52 10.28 10.16 10.45 10.45 10.51 18.95

Table 2 Summary data on core parameters for ZED-2 experiments 1–20. Key: experiment index (EX), lattice pitch and symmetry (LP), moderator purity as wt. %D2O (MP), moderator temperature (MT), fuel temperature (FT). See Section 2 for details. EX

LP cm

Fuel load

MP %D2O

MT (°C)

FT (°C)

Coolant

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.5 24.5 24.5 24.5 24.0 24.0 24.0 24.0 24.0 24.0 24.0

200A 200A 200A 200A 104A 104A 104A 104A 104A 240A 240A 240A 240A 160A 155A 208A 208A 208A 208A 240A

99.82 99.82 99.82 99.83 99.54 99.53 99.52 99.52 99.52 99.61 99.61 99.76 99.58 98.55 98.55 99.11 99.15 99.12 99.14 99.48

24.0 23.9 24.0 23.6 21.9 22.7 23.0 23.3 23.3 23.1 23.3 20.7 21.0 24.3 24.2 25.5 24.5 23.3 25.5 24.0

– – – – – – – – – 24.3 301.0 18.0 300.0 – – – – – – –

H2O Air H2O/Air Air Air H2O/Air H2O H2O H2O CO2 CO2 H2O H2O H2O H2O H2O D2O Air H2O/Air H2O

sq. sq. sq. sq. sq. sq. sq. sq. sq. hex hex hex hex sq. sq. sq. sq. sq. sq. sq.

40B 20C 40B 20C 40B 20C 40B 20C 156B 156B 156B 156B 156B 35D 35D 35D 35D 40C 40C 16C 36E 16C 36E 16C 36E 16C 36E 20C

Section 2 describes the experiments. Section 3 presents the DPERT patch. The verification methodology is outlined in Section 4. Results are presented in Section 5 and discussed in Section 6. Section 7 summarizes the conclusions. Subsequent references to ‘‘MCNP’’ and ‘‘KENO’’ refer specifically to MCNP5 version 1.40 and KENO V.a respectively.

2. ZED-2 experiments The experiments were performed at the ZED-2 critical facility, located at Chalk River Laboratories (Canada). Atfield (2011) presents a detailed description of the ZED-2 reactor. All experiments were heavy water moderated. Table 1 summarizes the fuel cluster types used in the experiments, which spanned a wide range of core configurations (see Tables 2–5). These core configurations were typically large and complex, having hundreds of individual fuel bundles arranged within a moderator vessel over 3 m in size. Detailed specifications of some fuels are available in the open literature.2 2 For type G and I fuels, see Atfield (2011, 2012); type H, see Green and Bigham (1963).

Table 3 Summary data on core parameters for ZED-2 experiments 21–40. Explanatory notes in Table 2. EX

LP cm

Fuel load

MP %D2O

MT (°C)

FT (°C)

Coolant

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.5 24.0 24.5 24.0 24.0 24.0 24.0 24.0

200A 200A 200A 200A 200A 200A 200A 240A 240A 240A 240A 240A 240A 240A 240A 240A 240A 240A 240A 240A

99.45 99.44 99.43 99.45 99.46 99.44 99.44 98.53 99.87 98.52 99.88 99.38 99.24 99.41 99.25 98.84 98.84 98.84 98.83 98.89

25.1 23.1 22.2 25.0 24.8 24.1 21.3 23.9 22.2 23.4 22.8 26.0 24.5 23.9 24.4 24.8 24.8 24.8 24.5 23.6

– – – – – – – – – – – – – – – – – – – –

H2O D2O D2O H2O/Air Air H2O/D2O H2O/D2O H2O H2O Air Air H2O H2O Air Air H2O H2O H2O H2O Air

sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq.

60B 60B 60B 60B 60B 60B 60B 20C 20C 20C 20C 20C 20C 20C 20C 20C 20C 20C 20C 20C

Fuel lattices were formed using square or hexagonal symmetry. With one exception (Type J fuel), fuel located at a lattice site is loaded into the core within assemblies containing five fuel clusters stacked vertically. The clusters are contained within Aluminum- or Zirconium-based test channels comprising concentric calandria and pressure tubes. Zirconium-based test channels are only present in dedicated fuel channel heating experiments. Type J fuel rods are formed of a single long element; their sheaths are in direct contact with the moderator. Type J fuel is used as a driver fuel in some experiments. Moderator temperature and purity are uniform within the core, but fuel temperature and coolant conditions are not always so, as some experiments are of the ‘‘substitution’’ type. In such experiments, the most abundant fuel clusters form a ‘‘reference’’ region which surrounds the ‘‘test’’ region in the inner part of the core. Fuel clusters in the reference region were always at the moderator temperature and, except for Type J, cooled with light water. Type J clusters are cooled by the heavy water moderator. Blank fuel temperature entries within the tables indicate that the fuel temperature was the same as the moderator temperature. The ‘‘coolant’’ column in the tables refer to the coolant in the test region. Multiple coolant entries indicate a checkerboard pattern of cooled fuel assemblies.

3. DPERT Patch DPERT is based on MCNP5 version 1.40. It accepts a set of nuclide-reaction pair adjustments specified on a user defined multigroup energy grid. The patch applies these adjustments to the continuous energy cross sections on a point-by-point basis, and executes a normal MCNP transport run using the altered data. The resulting neutron spectrum is representative of the altered cross sections, as all adjustments are applied simultaneously. In contrast, TSURFER assumes the neutron spectrum is invariant to cross section perturbations, which are applied one at a time. These differences in approaches offer complementary strenghts and weaknesses: cross section adjustments are modeled with greater fidelity in DPERT, but TSURFER provides a tool which can estimate the optimal cross section adjustments required to achieve a stated target. Using DPERT to implement a bias minimization algorithm would lead to a multi-dimensional forward Monte Carlo sampling approach.

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A. Trottier et al. / Annals of Nuclear Energy 62 (2013) 291–297 Table 4 Summary data on core parameters for ZED-2 experiments 41–60. Explanatory notes in Table 2. Fuel load

MP %D2O

MT (°C)

FT (°C)

Coolant

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.5 24.5 24.5 24.5

240A 20C 240A 20C 240A 20C 240A 20C 240A 20C 240A 20C 240A 20C 240A 20C 240A 20C 200A 20B 40F 200A 30B 30F 200A 20B 40F 200A 30B 30F 200A 30B 30F 200A 20B 40F 200A 30B 30F 240G 150H 35E 240G 150H 35E 240G 150H 35E 240G 150H 35E

98.80 99.04 99.03 99.03 99.03 99.06 99.06 99.06 99.05 99.31 99.29 99.32 99.30 99.29 99.34 99.30 99.73 99.73 99.73 99.73

23.9 20.2 34.7 40.6 50.0 21.5 34.4 40.1 49.3 22.0 20.0 22.0 20.1 20.1 22.2 20.3 24.9 24.9 24.9 24.9

– – – – – 25.1 25.6 31.2 31.0 23.2 18.8 22.1 19.4 20.0 22.6 18.7 25.0 25.0 50.0 50.0

Air H2O H2O H2O H2O Air Air Air Air H2O H2O H2O/Air H2O/Air H2O/Air Air Air CO2 H2O CO2 H2O

sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. hex hex hex hex

Table 5 Summary data on core parameters for ZED-2 experiments 61–77. Explanatory notes in Table 2. EX

LP cm

Fuel load

MP %D2O

MT (°C)

FT (°C)

Coolant

61

24.5 hex

99.73

24.9

250.0

CO2

62

24.5 hex

240G 150H 35E 240G 150H 35E 240G 150H 35E 240G 150H 35E 84J 35I

99.73

24.9

250.0

H2O

63

24.5 hex

64

24.5 hex

65

21.59 hex 21.59 hex 21.59 hex 24.5 hex 24.5 hex 24.5 hex 24.5 hex 24.5 hex 24.5 hex 24.5 hex 24.5 hex 24.5 hex 24.5 hex

66 67 68 69 70 71 72 73 74 75 76 77

24.9

300.0

CO2

99.73

24.9

300.0

H2O

99.06

23.5

–

H2O

84J 35I

99.07

22.7

–

Air

84J 35I

99.06

23.4

–

H2O

99.24 99.24 99.24 99.24 99.24 99.24 99.24 99.24 99.24 99.24

20.3 20.3 20.3 20.3 20.3 21.0 21.0 21.0 21.0 21.0

17.2 50.6 100.6 200.1 300.0 19.2 51.5 101.0 200.7 301.3

H2O H2O H2O H2O H2O CO2 CO2 CO2 CO2 CO2

32C 32C 32C 32C 32C 32C 32C 32C 32C 32C

35F 35F 35F 35F 35F 35F 35F 35F 35F 35F

DPERT only increases or decreases cross sections for nuclidereaction pairs specified by the user in the patch input. The patch can be used to apply different temperature-dependent adjustments, provided appropriately broadened data is available in the ACE data files. For each nuclide-reaction pair, DPERT performs a common adjustment to the continuous energy cross section data for all points within range of each of the multigroup energy bins specified in the patch input. Cross section data can therefore become discontinuous at group boundaries. Resonance line-shapes within the group boundaries are altered and the interpolation error originally used to prepare the ACE3 data files is no longer preserved. See Fig. 1 for two examples. 3

ACE is an acronym for A Compact ENDF.

1 0.8 0.6 0.4 0.2 0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1.5

2

Energy axis 1.4 Original Adjusted, ±25%

1.2 1 0.8 0.6 0.4 0.2

99.73

208A 208A 208A 208A 208A 208A 208A 208A 208A 208A

Cross section data

LP cm

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Original Adjusted, +25%

1.2

Cross section data

EX

1.4

0 -2

-1.5

-1

-0.5

0

0.5

1

Energy axis Fig. 1. Examples of DPERT adjustment. Top: adjustment over full range of resonnance lineshape; the line width of the profile is impacted. Bottom: discontinuity at energy group boundary.

The total cross section is adjusted as each component reaction changes. The fission neutron energy distributions (v) are renormalized, with the outgoing energies adjusted the same for all incident energies. The modified nuclear data is not saved. ACE data points are neither added nor deleted. The patch is only implemented for continuous and discrete energy nuclear data – multigroup data is not supported. Thermal scattering data (Sab) can also be adjusted. Not all ACE file options are supported at present. The fission neutron energy distribution (v) can only be adjusted for cross section Þ can only be tables that use LAW = 4. Fission neutron production ðm adjusted for cross section tables using data option LNU = 2. Finally,  data; delayed m  data DPERT can only handle prompt and total m cannot be adjusted. In summary, DPERT provides a tool to evaluate the a posteriori keff bias on a case-by-case basis using direct continuous-energy transport calculations. This allows for an integral verification of TSURFER, which relies on first-order perturbation theory and forward/adjoint flux calculations derived from multigroup transport simulations.

4. Verification methodology The experiments were analyzed using both MCNP and TSUNAMI. The MCNP simulations were performed using continuous energy data, while TSUNAMI simulations were based on 238-group data. KENO V.a, which is invoked by TSUNAMI, was used for the neutron transport. Both MCNP and KENO simulations use ENDF/

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A. Trottier et al. / Annals of Nuclear Energy 62 (2013) 291–297

B-VI.8 nuclear data.4 The a priori keff calculation biases were evaluated as:

Dkeff ¼ keff  1

ð1Þ

This expression implicitly assumes that the experiment configuration, as described, was exactly critical. All subsequent references to keff biases should be understood to be calculation biases evaluated under this assumption. TSURFER is based on a generalized least squares algorithm. The response vector is formed of the keff biases and the parameter vector is defined by the nuclide-reaction cross section data over a multigroup energy grid. The nuclide-reaction parameters are adjusted by TSURFER within the limitations imposed by the experimental keff covariance and nuclear data covariance matrices. The adjustments are therefore sensitive to the experimental keff uncertainties and the correlations between experiments. For the purposes of the present work, a rigorous evaluation of the experimental keff covariance matrix, such as performed in Ivanova et al. (2003), is not required. Atfield (2011) presents an excellent experimental uncertainty analysis as applied to ZED-2 critical experiments. The moderator purity, moderator level, fuel isotopic compositions and fuel bundle geometry are the dominant parameters. Typical experimental uncertainties are below 2.5 mk,5 but experiments are strongly correlated. Experiments are grouped in sets of 2–6 according to their underlying study. An average correlation coefficient of 0.9 was applied to all experiment pairs within each group, and zero otherwise. A TSURFER calculation was then executed. The MCNP a priori keff biases were supplied as the biases to be reduced. Cross section sensitivity coefficients required by TSURFER were based on the a priori TSUNAMI calculations, condensed to a 44-group structure. The resulting TSURFER adjustments were copied into the original MCNP input files, with no other alterations to the input except those required to enable the DPERT patch. DPERT was then used to recalculate the a posteriori biases. Cross section adjustments supplied to DPERT are temperature independent: TSURFER does not report temperature-dependent cross section adjustments. The energy group boundaries used to adjust the nuclear data in DPERT follow the 44-energy group structure; discontinuities in the modified cross section data are therefore consistent with TSURFER calcualtions.

Table 6 keff biases and code-to-code differences for cases 1–20. Key: experiment index (EX), MCNP5 a priori bias (MC), KENO a priori bias (KE), TSURFER a posteriori bias (TS), DPERT a posteriori bias (DP). All entries in mk (1 mk = 100 pcm).

4 This evaluation yields a priori keff biases which are of greater magnitude than are obtained when using ENDF/B-VII.0 nuclear data, thus providing a more stringent verification test. 5 1 mk = 100 pcm.

MC

KE

TS

DP

MC-KE

DP-TS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

11.23 10.70 10.93 8.52 8.93 9.43 10.27 10.33 10.25 10.48 10.71 10.86 10.75 8.45 9.33 9.89 6.45 6.54 7.70 11.53

11.62 8.51 11.14 11.13 6.40 5.39 6.24 6.23 6.55 10.14 10.53 10.73 10.83 12.06 12.35 11.64 8.31 8.56 8.72 11.78

1.00 1.50 1.34 1.28 0.28 0.55 1.47 1.18 1.25 0.43 0.64 0.43 0.38 2.15 1.35 0.34 1.29 0.59 1.06 1.36

0.43 0.18 0.37 1.01 1.38 1.97 2.46 2.29 2.30 0.06 0.07 0.08 0.15 2.98 2.25 1.50 3.52 2.62 2.75 0.13

0.39 2.19 0.21 2.61 2.53 4.04 4.03 4.10 3.70 0.34 0.18 0.13 0.08 3.61 3.02 1.75 1.86 2.02 1.02 0.25

1.43 1.68 1.71 2.29 1.66 1.42 0.99 1.11 1.05 0.37 0.57 0.35 0.23 0.83 0.90 1.16 2.23 2.02 1.68 1.23

Table 7 keff biases and code-to-code differences for cases 21–40. All entries in mk. See Table 6 for explanatory notes.

5. Results Tables 6–9 present the raw simulation results, which have been plotted in Fig. 2. MCNP and KENO a priori keff biases are tabulated, as are the TSURFER and DPERT a posteriori biases. The MCNP-toKENO and DPERT-to-TSURFER code-to-code differences are also listed. Bias uncertainties for MCNP, KENO and DPERT do not exceed 0.2 mk and include only the Monte Carlo transport statistical uncertainties. The TSURFER bias uncertainties do not exceed 0.6 mk, and include nuclear data uncertainty contributions. The most significant nuclide-reaction adjustments proposed by TSURFER are summarized in Table 10, and the first two sample adjustments are plotted in Fig. 3. The list of TSURFER adjustments is significantly longer, but contributions decrease rapidly for items further down the list. Actual DPERT calculations were based on adjustments correcting up to 99.98% of the bias. Summary statistics and correlations of bias results are presented in Table 11. Histograms of these data are presented in Fig. 4. Fig. 5 presents correlation plots of particular interest.

EX

EX

MC

KE

TS

DP

MC-KE

DP-TS

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

11.79 11.34 11.41 11.06 10.99 11.52 11.71 11.66 11.46 8.40 7.75 10.75 10.88 8.78 7.74 11.71 11.88 11.75 12.08 7.86

9.25 9.22 9.53 9.96 9.58 9.96 10.12 12.24 7.78 6.98 11.34 11.91 11.70 11.70 6.63 11.83 13.26 13.40 13.42 7.97

0.98 1.53 1.23 0.91 1.42 1.19 1.22 1.12 1.48 0.85 0.71 0.07 0.49 0.79 0.38 1.28 1.46 1.29 1.55 0.48

0.06 0.04 0.11 0.12 0.03 0.07 0.03 0.18 0.51 0.68 1.13 0.82 0.60 0.54 1.55 0.24 0.26 0.23 0.38 0.39

2.54 2.12 1.88 1.10 1.41 1.56 1.59 0.58 3.68 1.42 3.59 1.16 0.82 2.92 1.11 0.12 1.38 1.65 1.34 0.11

1.04 1.49 1.33 1.04 1.45 1.26 1.19 0.94 0.97 1.53 1.84 0.88 1.10 1.33 1.93 1.04 1.19 1.06 1.18 0.09

6. Discussion 6.1. Nuclear data adjustments and a priori biases Uranium adjustments account for close to 60% of the global bias corrections. By isotope, the rankings are as follows: 235U – 44.2%, 2 H – 19.5%, Al – 17.3 %, 238U – 15.5 %. Oxygen and Zirconium together contribute less than 2.5% to the bias correction. Corrections to Uranium cross sections are expected since these drive the neutron chain reaction. The relative importance of the Al (n, c) correction is likely due to the use of Aluminum in the ZED-2 test channels. The Al (n, c) adjustment is mainly constant6 at thermal neutron energies. Elastic scattering from Deuterium (2H) was expected to be significant given the composition of the moderator. The MCNP and KENO a priori keff bias distributions are generally similar (panels ‘a’ and ‘b’ of Fig. 4). The mean values are equal to

6

Roughly a 7% reduction.

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A. Trottier et al. / Annals of Nuclear Energy 62 (2013) 291–297 Table 8 keff biases and code-to-code differences for cases 41–60. All entries in mk. See Table 6 for explanatory notes. MC

KE

TS

DP

MC-KE

DP-TS

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

8.10 11.52 10.79 10.75 11.07 7.70 7.75 7.50 7.34 10.95 11.19 10.45 11.01 10.85 10.20 10.52 6.16 6.54 6.27 6.26

10.60 12.25 12.60 12.08 11.80 8.48 7.45 6.91 7.47 10.35 10.30 10.36 10.43 10.11 10.18 10.43 8.22 9.12 8.20 9.36

0.72 1.16 0.46 0.44 0.76 0.39 0.41 0.18 0.01 0.79 0.75 0.82 1.16 1.05 1.01 1.19 2.22 2.18 2.14 2.46

1.24 0.11 0.46 0.58 0.50 1.44 1.51 1.70 1.85 0.78 0.35 0.77 0.23 0.48 0.71 0.26 3.80 3.61 3.74 3.75

2.50 0.73 1.81 1.33 0.73 0.78 0.30 0.59 0.13 0.60 0.89 0.09 0.58 0.74 0.02 0.09 2.06 2.58 1.93 3.10

1.96 1.27 0.92 1.02 1.26 1.84 1.92 1.87 1.87 1.57 1.10 1.59 1.39 1.53 1.71 1.45 1.58 1.43 1.60 1.29

MC TS DP

4

2

0

-2

Bias (mk)

EX

6

-4

-6

-8 Table 9 keff biases and code-to-code differences for cases 61–77. All entries in mk. See Table 6 for explanatory notes. EX

MC

KE

TS

DP

MC-KE

DP-TS

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77

6.14 6.16 6.18 6.39 2.63 2.58 2.95 8.30 8.07 8.15 8.28 8.62 9.04 9.08 8.86 8.77 8.69

8.20 9.36 8.20 9.34 4.54 4.16 4.38 8.39 8.39 8.72 9.10 9.37 9.03 9.14 9.06 8.93 8.99

2.23 2.52 2.23 2.28 3.94 3.36 3.47 1.03 1.27 1.23 1.13 0.77 0.02 0.03 0.18 0.31 0.43

3.87 3.69 3.71 3.53 5.45 5.58 4.96 2.33 2.58 2.61 2.47 2.18 1.15 1.60 1.55 1.76 2.02

2.06 3.20 2.02 2.95 1.91 1.58 1.43 0.09 0.32 0.57 0.82 0.75 0.01 0.06 0.20 0.16 0.30

1.64 1.17 1.48 1.25 1.51 2.22 1.49 1.30 1.31 1.38 1.34 1.41 1.17 1.63 1.37 1.45 1.59

within twice the standard error on the mean. However the correlation coefficient between these biases is only 0.64 (panel ‘a’, Fig. 5). On a case-by-case basis, the maximum absolute difference was found to be 4.1 mk. The MCNP and KENO biases do not correlate to any given core condition, such as moderator purity, temperature, fuel isotopics or poison concentration. The distribution of MCNP-to-KENO differences (Fig. 4, panel ‘c’) also fails to correlate to these variables. 6.2. Impact of nuclear data adjustments Three main observations can be made regarding the impact of the nuclear data adjustments. Firstly, TSURFER largely corrects the a priori keff biases. This is clearly demonstrated by the difference in TSURFER and MCNP bias results in Fig. 2, as well as in panel ‘b’ from Fig. 4. From Table 11, we find that the bias shift due to nuclear data adjustments was from 9.19(25) mk to 0.08(16) mk. Secondly, DPERT confirms the TSURFER prediction that biases would be reduced. DPERT bias results in Figs. 2 and 4 display shifts from the a priori MCNP biases which mirror those seen in the TSURFER results. From Table 11, the bias shift was from 9.19(25) mk to 1.43(17) mk. In addition, the DPERT and TSURFER

-10

-12

-14 0

5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

Experiment Fig. 2. keff MCNP a priori biases (MC) and a posteriori bias predictions from TSURFER (TS) and DPERT (DP). Error bars omitted for clarity, see Section 5 for bias uncertainties.

Table 10 Adjusted nuclide-reaction pairs and their TSURFER-predicted contribution to the overall bias correction. This is a truncated listing. Zr adjustments are for all Zr isotopes within the 44-group ENDF/B-VI.8 SCALE nuclear data library. Fig. 3 presents sample Þ and Al (n, c). adjustments for 235U ðm Nuclide

Reaction

Bias correction (%)

Cumulative correction (%)

235

m

Al H

(n, c) (n, n) fission (n, c) (n, c) (n, 2n)

22.51 17.28 14.95 13.89 13.56 5.15 4.55 2.63 1.68 1.58 0.83 0.40

22.51 39.79 54.74 68.63 82.19 87.34 91.89 94.52 96.2 97.78 98.61 99.00

U

2

235

U U U

238 235 2

H

235

v

16

(n, (n, (n, (n,

U O 238 U Zr 238 U

n) n0 ) c) n)

bias distributions are highly correlated, as shown in panel ‘b’ of Fig. 5 and supported by the relative magnitudes of the standard deviations listed in Table 11. Indeed, the distribution of the DPERT-to-TSURFER code-to-code difference is very narrow (panel ‘d’, Fig. 4) when compared to the widths of the MCNP or KENO bias distributions or the MCNP-to-KENO code-to-code difference distribution (panel ‘c’ of same). Finally, TSURFER consistently underestimates the impact of the nuclear data adjustments which are proposed. Our assertion is supported by the fact that all DPERT keff biases are greater than

296

A. Trottier et al. / Annals of Nuclear Energy 62 (2013) 291–297

10

Frequency

20 MC DP 1.4, 1.5

(a)

15

-9.2, 2.2

10 5

1 0 -14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

6

8

Bias (mk)

Frequency

Adjustment (%)

20

0.1

KE TS 0.1, 1.4

(b)

15

-9.5, 2.1

10 5 0 -14

-12

-10

-8

-6

-4

-2

0

2

4

Bias (mk)

0.01

U-235 nubar |Al (n,γ)|

0.001 0.0001

0.01

1

100

10000

1e+06

1e+08

Energy (eV)

15

(c)

10 5

 and jAl (n, c)j. Al adjustments are negative. Um

MC

KE

TS

DP

MC-KE

DP-TS

9.19 0.25 2.22 12.08 2.58

9.49 0.24 2.11 13.42 4.16

0.08 0.16 1.38 1.55 3.94

1.43 0.17 1.50 0.51 5.58

0.30 0.21 1.83 4.10 3.61

1.35 0.05 0.43 0.09 2.29

Correlations MC 1.00 KE 0.64 TS 0.83 DP 0.90 MC-KE 0.47 DP-TS 0.45

1.00 0.56 0.61 0.37 0.34

1.00 0.96 0.37 0.13

1.00 0.39 0.40

1.00 0.15

Min Max

DP-TS

1.4, 0.4

20

0 -2

0

2

Difference (mk)

Statistic

30

(d)

10

0

235

Table 11 Summary statistics and correlation data for bias and code-to-code difference distributions. l, rl and r represent the mean, the standard error of the mean and the standard deviation. Column keys are as per Table 6. All entries in the top half are in mk (1 mk = 100 pcm). The correlation matrix is symmetric and dimensionless.

l rl r

40

0.3, 1.8

-4 Fig. 3.

MC-KE

Frequency

Frequency

20

4

-4

-2

0

2

4

Difference (mk)

Fig. 4. Biases and code-to-code differences. Numeric values are distribution means and standard deviations. Key: MCNP (MC), KENO (KE), TSURFER (TS), DPERT (DP).

1. TSURFER assumes neutron flux distribution invariance with respect to individual cross section adjustments. In DPERT, the neutron flux distribution reflects the impact of all cross section alterations applied simultaneously. 2. Differences between continuous-energy transport (DPERT) versus multigroup cross section (TSURFER), specifically the impact of the resonance treatment used to collapse multigroup cross sections for KENO transport. 3. Distortion of continuous-energy cross section resonance lineshapes due to uniform application of the adjustments in DPERT.

6.3. Adjustments are driven by the a priori biases 1.00

the corresponding TSURFER results, as seen in Fig. 2 and in panel ‘d’ of Fig. 4. Indeed, Table 11 indicates that, on average, DPERT keff bias results exceed the TSURFER predictions by 1.35 ± 0.05 mk. The DPERT-to-TSURFER code-to-code difference does not display any significant correlation to MCNP, KENO or MCNP-to-KENO distribution (panel ‘c’, Fig. 5), nor to any given core state parameter mentioned above. The origin of the systematic difference between DPERT and TSURFER results is not clear from the present data. Current hypotheses include:

A final observation to be drawn is that the nuclear data adjustments proposed by TSURFER are primarily driven by the a priori keff bias values, as opposed to the computer codes used to evaluate them. This conclusion is supported by examining the correlation diagrams presented in Fig. 5. Comparing panels ‘d’ and ‘e’, it is apparent that both DPERT and TSURFER a posteriori biases are correlated to the MCNP a priori biases.7 The TSURFER results themselves are more highly correlated to MCNP a priori biases than to the KENO a priori biases (Table 11, and panels ‘e’ and ‘f’ in Fig. 5). A similar result would be obtained if DPERT and KENO biases were compared. 7

These were used by TSURFER to derive the adjustments.

A. Trottier et al. / Annals of Nuclear Energy 62 (2013) 291–297

-4

6

(a) 0.64 -6

4

DP (mk)

KE (mk)

(b) 0.96

5

-8 -10

3 2 1 0

-12

-1 -14 -14 -12 -10 -8

-2 -6

-4

-2

-2

MC (mk)

2

4

TS (mk)

2.5

6

(c) 0.15

5

2

(d) 0.90

4

DP (mk)

DP-TS (mk)

0

1.5 1

3 2 1 0

0.5

297

The TSURFER module of the SCALE6 code suite was used to evaluate the nuclear data adjustments required to globally minimize the a priori biases. Uranium, Deuterium and Aluminum cross section adjustments dominate, as expected for the selected experiments. The TSURFER adjustments largely correct the a priori keff biases. The adjustments were provided to DPERT in order to test the bias reduction predictions using the DPERT patch. A detailed comparison of DPERT and TSURFER a posteriori keff biases was performed. The DPERT results confirm the TSURFER bias corrections. DPERT results are consistently higher than TSURFER results by 1.35 ± 0.05 mk, indicating that TSURFER underestimates the impact of its nuclear cross section data adjustments. The difference between DPERT and TSURFER results does not correlate to differences between a priori MCNP and KENO keff values. Comparisons of the TSURFER/MCNP and TSURFER/KENO correlation coefficients indicates that the nuclear data adjustments proposed by TSURFER are driven by the calculated a priori keff biases, as opposed to the algorithms employed by TSURFER.

-1 0 -6

-4

-2

0

2

-2 -14 -12 -10 -8

4

MC-KE (mk)

-2

6

(e) 0.83

5

(f) 0.56

Acknowledgments The authors thank Fred P. Adams (AECL) for his support in developing the DPERT concept. Erik G. Hagberg and Fook Choy Wong, both retired from AECL, developed several of the MCNP models used in this paper.

4

TS (mk)

4

TS (mk)

-4

MC (mk)

6 5

-6

3 2 1

3 1

0

0

-1

-1

-2 -14 -12 -10 -8

-6

MC (mk)

-4

-2

References

2

-2 -14 -12 -10 -8 -6 -4 -2

KE (mk)

Fig. 5. Correlations and correlation coefficients. Key: MCNP (MC), KENO (KE), TSURFER (TS), DPERT (DP). Error bars omitted for clarity, see Section 5 for bias uncertainties.

7. Conclusion An MCNP5 patch, DPERT, has been developed which allows users to modify nuclear cross section data and execute a transport calculation – without requiring that nuclear data files be modified. This provides a tool which may be used to verify predictions based on first-order perturbation theory. DPERT has been used to verify TSURFER keff bias reduction predictions. A set of 77 heavy water moderated critical experiments from ZED-2 have been analyzed using MCNP and TSUNAMI. The a priori keff biases have been compared. The distributions means agree within 2-r and the correlation coefficient between the MCNP and TSUNAMI a priori biases is 0.64.

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