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Sensitivity and Uncertainty Analysis of the VENUS-F Critical Core
Available online at www.sciencedirect.com
ScienceDirect Energy Procedia 71 (2015) 33 – 41
The 4th International Symposium on Innovative Nuclear Energy Systems, INES-4
Sensitivity and uncertainty analysis of the VENUS-F critical core Hiroki Iwamotoa,b,∗, Alexey Stankovskiyb , Wim Uyttenhoveb a Japan
Atomic Energy Agency (JAEA), 2-2, Shirakatashirane, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan b Belgian Nuclear Research Centre (SCK•CEN), Boeretang 200, BE-2400 Mol, Belgium
Abstract A sensitivity and uncertainty analysis was performed for a critical configuration at the VENUS-F facility using three major nuclear data libraries (JENDL-4.0, ENDF/B-VII.1, and JEFF-3.1.2) and the JENDL-4.0 covariance data. Differences in keff between the nuclear data libraries are shown to be smaller than those between the geometric models/calculation methodologies. Both the diffusion and transport methodologies in the criticality calculation provide similar results for the differences in keff between JENDL-4.0 and ENDF/B-VII.1 (∼450 pcm) and between JENDL-4.0 and JEFF- 3.1.2 (∼270 pcm). Major contributors to these differences are identified along with sensitivity coefficients. Among nuclear data parameters, the keff of the VENUS-F core is sensitive to the fission cross section, fission neutron multiplicity, and capture cross section of uranium as well as the elastic scattering parameters of lead and 56 Fe. It was found that these parameters have a considerable impact on the differences in keff . However, their positive and negative elements cancel each other out, thereby resulting in a relatively small difference. Based on the JENDL-4.0 covariance data, the total uncertainty induced by the nuclear data is evaluated to exhibit a 0.833% δk/k (1σ). © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license c 2014 Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection andpeer-review peer-review under responsibility of Tokyo the Tokyo Institute of Technology. Selection and under responsibility of the Institute of Technology
Keywords: VENUS-F, JENDL-4.0, ENDF/B-VII.1, JEFF-3.1.2, sensitivity coefficient, uncertainty induced by nuclear data
1. Introduction The Belgian Nuclear Research Centre (SCK•CEN) has conducted research and development activities on the Multi-purpose hYbrid Research Reactor for High-tech Applications (MYRRHA) facility, [1] and has continued the design studies on the technical feasibility of the transmutation in the framework of the Integrated Project EUROpean research program for the TRANSmutation of high level nuclear waste in an accelerator driven system project (IP-EUROTRANS). Under the framework of the GUINEVERE project (2006–2011), as a part of IP-EUROTRANS project, the VENUS reactor at the SCK•CEN Mol site has been revised to a zero-power fast lead-based reactor (VENUS-F) coupled with a deuteron accelerator (GENEPI3C [2]). For the neutronics design of such nuclear systems, it is important to determine the safety margin ∗ Corresponding
author. Tel.: +81-29-282-5329 ; fax: +81-29-282-5671 Email address: [email protected] (Hiroki Iwamoto)
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Tokyo Institute of Technology doi:10.1016/j.egypro.2014.11.852
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
in the criticality calculations. Among various possible sources of uncertainty in the criticality (keff ), the uncertainty of the nuclear data is considered to have a significant impact on the total uncertainty value. To quantify these nuclear data uncertainties, much effort has been devoted to evaluating the covariance data of the nuclear data parameters. Thanks to the recent evaluation of covariance data for lead isotopes in JENDL-4.0 [3], the uncertainties induced by nuclear data can be comprehensively estimated for lead-based fast reactors. In this analysis, we focus on a critical configuration of the VENUS-F core as a representative reactor of lead- or lead-bismuth-based fast reactors such as MYRRHA and future accelerator-driven systems, and estimate its effective multiplication factor (keff ) using existing nuclear data libraries: JENDL-4.0, ENDF/B-VII.1 [4], and JEFF-3.1.2 [5]. The variation in keff for typical deterministic methodologies (the diffusion and transport theories) and geometric modeling (three-dimensional XYZ and two-dimensional RZ models) is also investigated. Furthermore, we will investigate the cause for the differences in keff between the nuclear data libraries using sensitivity coefficients computed with a sensitivity analysis code SAGEP [6]. Finally, the uncertainty induced by the nuclear data is evaluated using the JENDL-4.0 covariance data. 2. The VENUS-F critical core and geometric modeling Because a detailed description of the configuration is given in Refs [7, 8], only a brief explanation is provided here. The reactor core contains 30 wt% enriched uranium and cylindrical-structured lead reflector, and consists of a 12×12 grid filled with fuel and lead assemblies, control rods, and safety rods. The concept of the fuel assemblies is based on the experience in the MASURCA facility at CEA Cadarache. In order to allow a safe shutdown in all situations, six safety rods consisting of boron carbide with a fuel/lead follower are positioned inside the reactor core. The control rods also consist of boron carbide. The analysis performed with homogenized cells for two geometric models: three-dimensional XYZ and simplified two-dimensional RZ models. A horizontal cross sectional view of the XYZ model of the VENUS core is shown in Fig. 1, and the simplified RZ model of the VENUS-F core is shown in Fig. 2. 3. Calculation methods The deterministic calculation of keff , which includes the sensitivity and uncertainty analysis was performed with the MARBLE code system developed in JAEA [9]. In the MARBLE system, neutronics calculation modules such as CITATION-FBR [10], PARTISN [11], SRAROM-UF [12] and SAGEP were encapsulated in a user-friendly manner. Macroscopic effective cross sections were calculated using the cell calculation module, SLAROM-UF, in the MARBLE system containing a homogeneous cell model in a 70energy-group structure with energies ranging from 0.1 eV to 20 MeV. For the nuclear data library, the fast reactor group constant sets of JENDL-4.0 (UFLIB.J40 [13]), ENDF/B-VII.1 (UFLIB.E71), and JEFF-3.1.2 (UFLIB-F312) were used. The keff values were calculated with different types of deterministic models: the diffusion and transport theories by using CITATION-FBR and PARTISN, respectively. The sensitivity coefficients of keff were computed using SAGEP. 4. Results and discussion 4.1. Criticality The results of the keff value are listed in Table 1. Note that the differences in keff between the nuclear data libraries are smaller than those between the geometric models/calculation methodologies. With respect to the nuclear data libraries, ENDF/B-VII.1 yields the highest keff , and the difference from JENDL-4.0 is approximately +450 pcm for any of the models/methodologies. Conversely, the keff calculated with JEFF3.1.2 is 270 pcm higher than that calculated using JENDL-4.0.
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
s s s s s s , , , D D , , , s s s s s s
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Fig. 1. Schematic of the horizontal cross section of the VENUS-F core (XYZ model).
4.2. Cause of the difference in criticality To investigate the cause of the differences in keff between the nuclear data libraries, the sensitivity coefficients of keff were calculated using SAGEP. The sensitivity coefficient of keff to perturbations in a single energy group g of a particular nuclear data parameter σ is defined as δkeff keff (1) S keff = δσg σg The SAGEP code computes the sensitivity coefficients on the basis of the diffusion theory for the following parameters: capture, σcap ; fission, σfis ; elastic scattering, σel ; inelastic scattering, σinl ; (n,2n) reaction, σn2n ; average cosine of scattered neutrons, μ; ¯ fission neutron multiplicity, ν; and fission neutron spectrum, χ. The sensitivity coefficients of keff calculated using JENDL-4.0 for major isotopes are shown in Fig. 3. The fission cross section and ν value of 235 U and 238 U show very high sensitivity coefficients in the energy range from 10 keV to 10 MeV and above 1 MeV, respectively, which correspond to large positive effects on the reactivity. The elastic scattering cross section of the lead isotopes and that of the stainless steel constituents exhibit positive sensitivities (from 10 keV to 10 MeV). In contrast, the ν values of these isotopes exhibit negative sensitivities. These sensitive parameters can cause significant differences in keff for changes in the cross sections. Contributions to the difference in keff between the different nuclear data libraries were analyzed using the sensitivity method. From Eq. (1), the relative change by replacing a particular nuclear data parameter of
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
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Fig. 2. Simplified RZ model of the VENUS-F core (Unit is cm.).
JENDL-4.0 to that of ENDF/B-VII.1 can be written as − σJ40 σE71 δkeff g g = S gkeff ,J40 · . J40 keff σJ40→E71 σ g g For the scattering matrix such as elastic scattering, the relative change is expressed as J40 σE71 δkeff i, j − σi, j = S i,keffj ,J40 · . keff σJ40→E71 σJ40 i, j i j
(2)
(3)
where i and j indicate the energy group of the incoming and outgoing neutrons, respectively. Table 2 lists the major contributions to the differences in keff of ENDF/B-VII.1 and JEFF-3.1.2 from that of JENDL-4.0. Because the difference in keff between the nuclear data libraries contains positive and negative elements, some contributions cancel each other out, thereby resulting in a reduction of the total value. Figure 4 shows the sensitivity coefficients of keff with respect to the fission cross section of 235 U, relative differences in keff from JENDL-4.0, and relative differences in the fission cross section of 235 U from JENDL-4.0 and 1σ standard deviation of JENDL-4.0. Although the standard deviations are relatively small for energies above 10 keV, a considerable difference is observed in this energy region. This difference from 10 keV to 1 MeV affects the difference in keff . Figure 5 shows the sensitivity coefficients with respect to the ν value of 235 U, relative differences in keff from JENDL-4.0, and relative differences in the ν value of 235 U from JENDL-4.0 and 1σ standard deviation of JENDL-4.0. Even a slight difference in the ν value below 1% at the energies of several hundred keV affects the difference in keff between JENDL-4.0 and ENDF/B-VII.1.
Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
Table 1. Summary of keff .
keff
RZ model Δρ (pcm) ∗
Diffusion JENDL-4.0 1.00020 ENDF/B-VII.1 1.00457 JEFF-3.1.2 1.00301 Transport† JENDL-4.0 1.01071 ENDF/B-VII.1 1.01548 JEFF-3.1.2 1.01360 ∗ Difference from JENDL-4.0. † P0 -S4 approximation.
XYZ model keff Δρ (pcm) ∗
— +435 +280
0.99791 1.00213 1.00064
— +422 +274
— +465 +282
1.00322 1.00774 1.00594
— +447 +269
Table 2. Major contributiors to the difference in keff from JENDL-4.0.
ENDF/B-VII.1 parameter value (pcm) 235 U, σfis +502 208 Pb, μ¯ −237 238 U, σinl +191 206 Pb, σel −148 235 U, σcap +143 52 Cr, σcap −110 204 Pb, σcap +101
JEFF-3.1.2 parameter value (pcm) 235 U, σfis +513 235 U, σfis −221 56 Fe, μ¯ −185 206 Pb, σel −148 238 U, σinl −143 235 U, σcap +132 204 Pb, σcap +101
Fig. 3. Sensitivity coefficients of keff calculated with JENDL-4.0.
37
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Fig. 4. Sensitivity coefficients with respect to the fission cross section of 235 U (top), relative difference in keff from that of JENDL-4.0 (center), and differences in the fission cross section of 235 U from that of JENDL-4.0 (bottom).
Fig. 5. Sensitivity coefficients with respect to the ν value of 235 U (top), relative difference in keff from JENDL-4.0 (center), and differences in the ν value of 235 U from that of JENDL-4.0 (bottom).
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4.3. Uncertainty induced by nuclear data The uncertainty induced by nuclear data was calculated with the sensitivity coefficient and covariance data of JENDL-4.0. The relative uncertainty in keff is given by
δk k
2 = GMGT
(4)
where G is the sensitivity coefficient vector, M is the cross-section covariance matrix, and T denotes the transpose of the matrix. In this analysis, ERRORJ [15] was used to process the JENDL-4.0 covariance data to a 70-energy-group matrix structure. The list of nuclides used in this analysis are as follows: 235 U, 238 U, 10 B, 11 B, 52 Cr, 53 Cr, 55 Mn, 56 Fe, 58 Ni, 60 Ni, 204 Pb, 206 Pb, 207 Pb, and 208 Pb. As given in Table 3, the total uncertainty in keff was 0.833% δk/k (1σ). This uncertainty was largely dominated by 235 U, 238 U, 208 Pb, and 56 Fe. As for the nuclear data parameters, the parameters related to the elastic scattering (σel and μ) ¯ of lead were the most dominant for the total uncertainty. Figure 6 shows the energy breakdown of the uncertainty in keff and the JENDL-4.0 covariance data (1σ standard deviations) with respect to the three representative parameters (208 Pb elastic scattering, 235 U fission, and 238 U capture). Although relatively large standard deviations can be observed above 1 MeV or below 1 keV, the uncertainty in keff is mainly caused by the covariance data in the energy range from 1 keV to 1 MeV.
235
U U 10 B 11 B 52 Cr 53 Cr 55 Mn 56 Fe 58 Ni 60 Ni 204 Pb 206 Pb 207 Pb 208 Pb Total 238
σcap 0.182 0.223 0.003 0.000 0.003 0.004 0.020 0.065 0.025 0.004 0.017 0.018 0.019 0.011 0.229
Table 3. Profile of the uncertainty in keff (1σ, unit: %δk/k).
σfis 0.269 0.031 — — — — — — — — — — — — 0.212
ν 0.205 0.055 — — — — — — — — — — — — 0.156
σel 0.031 0.050 0.000 0.009 0.030 0.017 0.018 0.226 0.008 0.003 0.011 0.067 0.072 0.388 0.466
σinl 0.089 0.188 N/A N/A 0.003 0.002 0.005 0.028 0.002 0.001 0.004 0.045 0.051 0.024 0.223
μ¯ 0.023 0.029 N/A N/A 0.018 0.010 N/A 0.065 0.007 0.003 0.008 0.134 0.135 0.375 0.428
σn2n 0.000 0.001 N/A N/A 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001
χ 0.155 0.015 — — — — — — — — — — — — —
Total 0.426 0.305 0.003 0.009 0.036 0.020 0.027 0.243 0.028 0.006 0.023 0.165 0.162 0.541 0.833
5. Summary and conclusion A sensitivity and uncertainty analysis was performed for a critical configuration at the VENUS-F facility with three major nuclear data libraries (JENDL-4.0, ENDF/B-VII.1, and JEFF-3.1.2) and the JENDL-4.0 covariance data. As the result of the criticality calculation, differences in keff between the nuclear data libraries were smaller than those between the geometric models/calculation methodologies. Both the diffusion and transport methodologies in the criticality calculation showed similar differences in keff between the JENDL-4.0 and ENDF/B-VII.1 (∼450 pcm) and between JENDL-4.0 and JEFF-3.1.2 (∼270 pcm). The major contributors to these differences were identified with the use of the sensitivity coefficients. Among the nuclear data parameters, the criticality of the VENUS-F core is sensitive to the fission cross section, fission neutron multiplicity, and capture cross section of uranium as well as the elastic scattering cross section and average cosine of the scattered neutrons of lead and 56 Fe. The analysis with the sensitivity coefficients
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
Fig. 6. Energy breakdown of uncertainty in keff and JENDL-4.0 covariance data with respect to representative parameters (208 Pb elastic scattering, 235 U fission, and 238 U capture).
showed that these parameters have a considerable impact on the differences in keff . However their positive and negative elements cancel each other out, thereby resulting in a relatively small difference. Based on the JENDL-4.0 covariance data, the total uncertainty in keff induced by nuclear data was evaluated as 0.833% δk/k (1σ). Among many contributors to the uncertainty of the criticality, we identified the following major parameters: • Fission cross section of 235 U (from 1 keV to 1 MeV) • Capture cross section of 238 U (from 1 keV to 1 MeV) • Elastic scattering cross section of 208 Pb (from 40 keV to 1 MeV). To reduce the uncertainty in keff of lead fast reactors with uranium fuel, a concentrated effort should be devoted to reducing the nuclear data uncertainties in this energy region. Acknowledgments One of the authors (H. Iwamoto) would like to thank the support of SCK•CEN and JAEA. He is also grateful to Dr. K. Tsujimoto of JAEA for his encouragement of this study. References [1] Technology and Components of Accelerator-driven Systems, OECD-NEA Workshop Proceedings, ISBN 978-92-64-11727-3, Karlsruhe, Germany; 2010. [2] Baylac M, Abderrahim HA, Baeten P, Billebaud A, Bergmans G, Boge P, Bondoux D, Bouvier J, Cabanel T, Carcagno Y, Dargaud G, De Conto JM, Gaudiot G, Gautier JM, Gomez Martinez Y, Granget G, Heitz G, Heusch M, Laune B, Marchand D, Mellier D, Merrer Y, Micoud R, Perbet E, Planet M, Poussot P, Reynet D, Ruescas C, Tourres D, Vermeersch F, Vittiglio G, Vermeersch F, and Vittigli G, “The GENEPI-3C Accelerator for the GUINEVERE Project,” Proceedings of the International Topical Meeting on Nuclear Research Applications and Utilization of Accelerators, IAEA, Vienna, Austria; 2010. [3] Shibata K, Iwamoto O, Nakagawa T, Iwamoto N, Ichihara A, Kunieda S, Chiba S, Furutaka K, Otuka N, Ohsawa T, Murata T, Matsunobu H, Zukeran A, Kamada S, and Kurata J, “JENDL-4.0: A New Library for Innovative Nuclear Science and Engineering,” Journal of Nuclear Science and Technology 2011;48[1]:1–30.
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[4] Chadwick MB, Herman M, Oblozinsky P, Dumn ME, Danon Y, Kahler AC, Smith DL, Pritychenko B, Arbanas G, Arcilla R, Brewer R, Brown DA, Capote R, Carlson AD, Cho YS, Derrien H, Guber K, Hale GM, Hoblit S, Holloway S, Johnson TD, Kawano T, Kiedrowski BC, Kim H, Kunieda S, Larson NM, Leal L, Mattoon CM, McKnight RD, Mughabghab SF, Nobre GPA, Palmiotti G, Palumbo A, Pigni MT, Pronyaev VG, Sayer RO, Sonozogni AA, Summers NC, Talou P, Thompson IJ, Trkov A, Vogt RL, Van der Marck SC, Wallner A, White MC, Wiarda D, and Young PG, “ENDF/B-VII.1 Nuclear Data for Science and Technology: Cross Sections, Covariances, Fission Product Yields and Decay Data,” Nuclear Data Sheets 2011;112,12:2887– 2996. [5] www.oecd-nea.org/dbforms/data/eva/evatapes/jeff_31/JEFF312 [6] Hara A, Takeda T, and Kikuchi Y, SAGEP: Two-Dimensional Sensitivity Analysis Code Based on Generalized Perturbation Theory, JAERI-M, 84-027; 1984 [in Japanese]. [7] Uyttenhove W, Baeten P, Van den Eynde G, Kochetkov A, Lathouwers D, and Carta M, “The neutronic design of a critical lead reflected zero-power reference core for on-line subcriticality measurements in Accelerator-Driven Systems,” Annals of Nuclear Energy 2011;38:1519-1526. [8] Bianchini G, Carta M, Pisacane F, Frisoni M, Peluso V, “Set-Up of a Deterministic Model for the Analysis of the Guinevere Experience,” PHYSOR 2010, Pittsburgh, Pennsylvania, USA, May 9–14, 2010, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2010). [9] Yokoyama K, Hirai Y, Tatsumi M, Hyoudou H, Chiba G, Hazama T, Nagaya Y, and Ishikawa M, “MARBLE: A Next Generation Neutronics Analysis Code System for Fast Reactors,” Proceedings of International Conference on the Physics of Reactors (PHYSOR2008) (CD-ROM), September 14–19, Switzerland; 2008. [10] Fowler TB and Vondy DR, Nuclear Reactor Core Analysis Code: CITATION, USAEC Report ORNL-TM-2496, Oak Ridge National Laboratory; 1969. [11] Alcouffe RE, Baker RS, Dahl JA, Turner SA, and Ward R, PARTISN: A Time-Dependent, Parallel Neutral Particle Transport Code System, LA-UR-05-3925; 2005. [12] Hazama T, Chiba G, Sato W, and Numata K, SLAROM-UF: Ultra Fine Group Cell Calculation Code for Fast Reactor – Version 20090113 – (translated Document), JAEA-Review, 2009-003; 2009. [13] Sugino K, Jin T, Hazama T, and Numata K, Preparation of Fast Reactor Group Constant Sets UFLIB.J40 and JFS-3-J4.0 Based on the JENDL-4.0 Data, JAEA-Data/Code, 2011-017; 2012 [in Japanese]. [14] Chiba G, ERRORJ: A Code to Process Neutron-nuclide Reaction Cross Section Covariance, Version 2.3, JAEA-Data/Code, 2007-007; 2007.
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ScienceDirect Energy Procedia 71 (2015) 33 – 41
The 4th International Symposium on Innovative Nuclear Energy Systems, INES-4
Sensitivity and uncertainty analysis of the VENUS-F critical core Hiroki Iwamotoa,b,∗, Alexey Stankovskiyb , Wim Uyttenhoveb a Japan
Atomic Energy Agency (JAEA), 2-2, Shirakatashirane, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan b Belgian Nuclear Research Centre (SCK•CEN), Boeretang 200, BE-2400 Mol, Belgium
Abstract A sensitivity and uncertainty analysis was performed for a critical configuration at the VENUS-F facility using three major nuclear data libraries (JENDL-4.0, ENDF/B-VII.1, and JEFF-3.1.2) and the JENDL-4.0 covariance data. Differences in keff between the nuclear data libraries are shown to be smaller than those between the geometric models/calculation methodologies. Both the diffusion and transport methodologies in the criticality calculation provide similar results for the differences in keff between JENDL-4.0 and ENDF/B-VII.1 (∼450 pcm) and between JENDL-4.0 and JEFF- 3.1.2 (∼270 pcm). Major contributors to these differences are identified along with sensitivity coefficients. Among nuclear data parameters, the keff of the VENUS-F core is sensitive to the fission cross section, fission neutron multiplicity, and capture cross section of uranium as well as the elastic scattering parameters of lead and 56 Fe. It was found that these parameters have a considerable impact on the differences in keff . However, their positive and negative elements cancel each other out, thereby resulting in a relatively small difference. Based on the JENDL-4.0 covariance data, the total uncertainty induced by the nuclear data is evaluated to exhibit a 0.833% δk/k (1σ). © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license c 2014 Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection andpeer-review peer-review under responsibility of Tokyo the Tokyo Institute of Technology. Selection and under responsibility of the Institute of Technology
Keywords: VENUS-F, JENDL-4.0, ENDF/B-VII.1, JEFF-3.1.2, sensitivity coefficient, uncertainty induced by nuclear data
1. Introduction The Belgian Nuclear Research Centre (SCK•CEN) has conducted research and development activities on the Multi-purpose hYbrid Research Reactor for High-tech Applications (MYRRHA) facility, [1] and has continued the design studies on the technical feasibility of the transmutation in the framework of the Integrated Project EUROpean research program for the TRANSmutation of high level nuclear waste in an accelerator driven system project (IP-EUROTRANS). Under the framework of the GUINEVERE project (2006–2011), as a part of IP-EUROTRANS project, the VENUS reactor at the SCK•CEN Mol site has been revised to a zero-power fast lead-based reactor (VENUS-F) coupled with a deuteron accelerator (GENEPI3C [2]). For the neutronics design of such nuclear systems, it is important to determine the safety margin ∗ Corresponding
author. Tel.: +81-29-282-5329 ; fax: +81-29-282-5671 Email address: [email protected] (Hiroki Iwamoto)
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Tokyo Institute of Technology doi:10.1016/j.egypro.2014.11.852
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
in the criticality calculations. Among various possible sources of uncertainty in the criticality (keff ), the uncertainty of the nuclear data is considered to have a significant impact on the total uncertainty value. To quantify these nuclear data uncertainties, much effort has been devoted to evaluating the covariance data of the nuclear data parameters. Thanks to the recent evaluation of covariance data for lead isotopes in JENDL-4.0 [3], the uncertainties induced by nuclear data can be comprehensively estimated for lead-based fast reactors. In this analysis, we focus on a critical configuration of the VENUS-F core as a representative reactor of lead- or lead-bismuth-based fast reactors such as MYRRHA and future accelerator-driven systems, and estimate its effective multiplication factor (keff ) using existing nuclear data libraries: JENDL-4.0, ENDF/B-VII.1 [4], and JEFF-3.1.2 [5]. The variation in keff for typical deterministic methodologies (the diffusion and transport theories) and geometric modeling (three-dimensional XYZ and two-dimensional RZ models) is also investigated. Furthermore, we will investigate the cause for the differences in keff between the nuclear data libraries using sensitivity coefficients computed with a sensitivity analysis code SAGEP [6]. Finally, the uncertainty induced by the nuclear data is evaluated using the JENDL-4.0 covariance data. 2. The VENUS-F critical core and geometric modeling Because a detailed description of the configuration is given in Refs [7, 8], only a brief explanation is provided here. The reactor core contains 30 wt% enriched uranium and cylindrical-structured lead reflector, and consists of a 12×12 grid filled with fuel and lead assemblies, control rods, and safety rods. The concept of the fuel assemblies is based on the experience in the MASURCA facility at CEA Cadarache. In order to allow a safe shutdown in all situations, six safety rods consisting of boron carbide with a fuel/lead follower are positioned inside the reactor core. The control rods also consist of boron carbide. The analysis performed with homogenized cells for two geometric models: three-dimensional XYZ and simplified two-dimensional RZ models. A horizontal cross sectional view of the XYZ model of the VENUS core is shown in Fig. 1, and the simplified RZ model of the VENUS-F core is shown in Fig. 2. 3. Calculation methods The deterministic calculation of keff , which includes the sensitivity and uncertainty analysis was performed with the MARBLE code system developed in JAEA [9]. In the MARBLE system, neutronics calculation modules such as CITATION-FBR [10], PARTISN [11], SRAROM-UF [12] and SAGEP were encapsulated in a user-friendly manner. Macroscopic effective cross sections were calculated using the cell calculation module, SLAROM-UF, in the MARBLE system containing a homogeneous cell model in a 70energy-group structure with energies ranging from 0.1 eV to 20 MeV. For the nuclear data library, the fast reactor group constant sets of JENDL-4.0 (UFLIB.J40 [13]), ENDF/B-VII.1 (UFLIB.E71), and JEFF-3.1.2 (UFLIB-F312) were used. The keff values were calculated with different types of deterministic models: the diffusion and transport theories by using CITATION-FBR and PARTISN, respectively. The sensitivity coefficients of keff were computed using SAGEP. 4. Results and discussion 4.1. Criticality The results of the keff value are listed in Table 1. Note that the differences in keff between the nuclear data libraries are smaller than those between the geometric models/calculation methodologies. With respect to the nuclear data libraries, ENDF/B-VII.1 yields the highest keff , and the difference from JENDL-4.0 is approximately +450 pcm for any of the models/methodologies. Conversely, the keff calculated with JEFF3.1.2 is 270 pcm higher than that calculated using JENDL-4.0.
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
s s s s s s , , , D D , , , s s s s s s
s s s s , , D D D D D D D D , , s s s s
s s s , D D D D D D D D D D D D , s s s
s s , , s s
s , D > > > > > > > > > > > D , s
s , D > > > & & & & & & > > > D , s
, D D > > & & ^ & & ^ & & > > D D ,
, D D > & & & & & & & & & & > D D ,
, D D > & ^ & & & & & & & & > D D ,
D D D > & & & & & & & & & & > D D D
D D D > & & & & & & & & & & > D D D
, D D > & ^ & & & & & & & & > D D ,
, D D > & & & & & & & & & & > D D ,
, D D > > & & ^ & & ^ & & > > D D ,
s , D > > > & & & & & & > > > D , s
s , D > > > > > > > > > > > D , s
s s , , s s
s s s , D D D D D D D D D D D D , s s s
s s s s , , D D D D D D D D , , s s s s
s s s s s s , , , D D , , , s s s s s s
ϴ͘ϬϳĐŵ
>͗>ĞĂĚƌŽĚ D͗>ĞĂĚ &͗&ƵĞůƌŽĚ s͗ŝƌ ^͗^ĂĨĞƚLJƌŽĚ ,͗>ĞĂĚнsŽŝĚ ͗ŽŶƚƌŽůƌŽĚ ͗^^н>ĞĂĚ;ƐŝĚĞͿ
Fig. 1. Schematic of the horizontal cross section of the VENUS-F core (XYZ model).
4.2. Cause of the difference in criticality To investigate the cause of the differences in keff between the nuclear data libraries, the sensitivity coefficients of keff were calculated using SAGEP. The sensitivity coefficient of keff to perturbations in a single energy group g of a particular nuclear data parameter σ is defined as δkeff keff (1) S keff = δσg σg The SAGEP code computes the sensitivity coefficients on the basis of the diffusion theory for the following parameters: capture, σcap ; fission, σfis ; elastic scattering, σel ; inelastic scattering, σinl ; (n,2n) reaction, σn2n ; average cosine of scattered neutrons, μ; ¯ fission neutron multiplicity, ν; and fission neutron spectrum, χ. The sensitivity coefficients of keff calculated using JENDL-4.0 for major isotopes are shown in Fig. 3. The fission cross section and ν value of 235 U and 238 U show very high sensitivity coefficients in the energy range from 10 keV to 10 MeV and above 1 MeV, respectively, which correspond to large positive effects on the reactivity. The elastic scattering cross section of the lead isotopes and that of the stainless steel constituents exhibit positive sensitivities (from 10 keV to 10 MeV). In contrast, the ν values of these isotopes exhibit negative sensitivities. These sensitive parameters can cause significant differences in keff for changes in the cross sections. Contributions to the difference in keff between the different nuclear data libraries were analyzed using the sensitivity method. From Eq. (1), the relative change by replacing a particular nuclear data parameter of
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
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Fig. 2. Simplified RZ model of the VENUS-F core (Unit is cm.).
JENDL-4.0 to that of ENDF/B-VII.1 can be written as − σJ40 σE71 δkeff g g = S gkeff ,J40 · . J40 keff σJ40→E71 σ g g For the scattering matrix such as elastic scattering, the relative change is expressed as J40 σE71 δkeff i, j − σi, j = S i,keffj ,J40 · . keff σJ40→E71 σJ40 i, j i j
(2)
(3)
where i and j indicate the energy group of the incoming and outgoing neutrons, respectively. Table 2 lists the major contributions to the differences in keff of ENDF/B-VII.1 and JEFF-3.1.2 from that of JENDL-4.0. Because the difference in keff between the nuclear data libraries contains positive and negative elements, some contributions cancel each other out, thereby resulting in a reduction of the total value. Figure 4 shows the sensitivity coefficients of keff with respect to the fission cross section of 235 U, relative differences in keff from JENDL-4.0, and relative differences in the fission cross section of 235 U from JENDL-4.0 and 1σ standard deviation of JENDL-4.0. Although the standard deviations are relatively small for energies above 10 keV, a considerable difference is observed in this energy region. This difference from 10 keV to 1 MeV affects the difference in keff . Figure 5 shows the sensitivity coefficients with respect to the ν value of 235 U, relative differences in keff from JENDL-4.0, and relative differences in the ν value of 235 U from JENDL-4.0 and 1σ standard deviation of JENDL-4.0. Even a slight difference in the ν value below 1% at the energies of several hundred keV affects the difference in keff between JENDL-4.0 and ENDF/B-VII.1.
Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
Table 1. Summary of keff .
keff
RZ model Δρ (pcm) ∗
Diffusion JENDL-4.0 1.00020 ENDF/B-VII.1 1.00457 JEFF-3.1.2 1.00301 Transport† JENDL-4.0 1.01071 ENDF/B-VII.1 1.01548 JEFF-3.1.2 1.01360 ∗ Difference from JENDL-4.0. † P0 -S4 approximation.
XYZ model keff Δρ (pcm) ∗
— +435 +280
0.99791 1.00213 1.00064
— +422 +274
— +465 +282
1.00322 1.00774 1.00594
— +447 +269
Table 2. Major contributiors to the difference in keff from JENDL-4.0.
ENDF/B-VII.1 parameter value (pcm) 235 U, σfis +502 208 Pb, μ¯ −237 238 U, σinl +191 206 Pb, σel −148 235 U, σcap +143 52 Cr, σcap −110 204 Pb, σcap +101
JEFF-3.1.2 parameter value (pcm) 235 U, σfis +513 235 U, σfis −221 56 Fe, μ¯ −185 206 Pb, σel −148 238 U, σinl −143 235 U, σcap +132 204 Pb, σcap +101
Fig. 3. Sensitivity coefficients of keff calculated with JENDL-4.0.
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
Fig. 4. Sensitivity coefficients with respect to the fission cross section of 235 U (top), relative difference in keff from that of JENDL-4.0 (center), and differences in the fission cross section of 235 U from that of JENDL-4.0 (bottom).
Fig. 5. Sensitivity coefficients with respect to the ν value of 235 U (top), relative difference in keff from JENDL-4.0 (center), and differences in the ν value of 235 U from that of JENDL-4.0 (bottom).
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
4.3. Uncertainty induced by nuclear data The uncertainty induced by nuclear data was calculated with the sensitivity coefficient and covariance data of JENDL-4.0. The relative uncertainty in keff is given by
δk k
2 = GMGT
(4)
where G is the sensitivity coefficient vector, M is the cross-section covariance matrix, and T denotes the transpose of the matrix. In this analysis, ERRORJ [15] was used to process the JENDL-4.0 covariance data to a 70-energy-group matrix structure. The list of nuclides used in this analysis are as follows: 235 U, 238 U, 10 B, 11 B, 52 Cr, 53 Cr, 55 Mn, 56 Fe, 58 Ni, 60 Ni, 204 Pb, 206 Pb, 207 Pb, and 208 Pb. As given in Table 3, the total uncertainty in keff was 0.833% δk/k (1σ). This uncertainty was largely dominated by 235 U, 238 U, 208 Pb, and 56 Fe. As for the nuclear data parameters, the parameters related to the elastic scattering (σel and μ) ¯ of lead were the most dominant for the total uncertainty. Figure 6 shows the energy breakdown of the uncertainty in keff and the JENDL-4.0 covariance data (1σ standard deviations) with respect to the three representative parameters (208 Pb elastic scattering, 235 U fission, and 238 U capture). Although relatively large standard deviations can be observed above 1 MeV or below 1 keV, the uncertainty in keff is mainly caused by the covariance data in the energy range from 1 keV to 1 MeV.
235
U U 10 B 11 B 52 Cr 53 Cr 55 Mn 56 Fe 58 Ni 60 Ni 204 Pb 206 Pb 207 Pb 208 Pb Total 238
σcap 0.182 0.223 0.003 0.000 0.003 0.004 0.020 0.065 0.025 0.004 0.017 0.018 0.019 0.011 0.229
Table 3. Profile of the uncertainty in keff (1σ, unit: %δk/k).
σfis 0.269 0.031 — — — — — — — — — — — — 0.212
ν 0.205 0.055 — — — — — — — — — — — — 0.156
σel 0.031 0.050 0.000 0.009 0.030 0.017 0.018 0.226 0.008 0.003 0.011 0.067 0.072 0.388 0.466
σinl 0.089 0.188 N/A N/A 0.003 0.002 0.005 0.028 0.002 0.001 0.004 0.045 0.051 0.024 0.223
μ¯ 0.023 0.029 N/A N/A 0.018 0.010 N/A 0.065 0.007 0.003 0.008 0.134 0.135 0.375 0.428
σn2n 0.000 0.001 N/A N/A 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001
χ 0.155 0.015 — — — — — — — — — — — — —
Total 0.426 0.305 0.003 0.009 0.036 0.020 0.027 0.243 0.028 0.006 0.023 0.165 0.162 0.541 0.833
5. Summary and conclusion A sensitivity and uncertainty analysis was performed for a critical configuration at the VENUS-F facility with three major nuclear data libraries (JENDL-4.0, ENDF/B-VII.1, and JEFF-3.1.2) and the JENDL-4.0 covariance data. As the result of the criticality calculation, differences in keff between the nuclear data libraries were smaller than those between the geometric models/calculation methodologies. Both the diffusion and transport methodologies in the criticality calculation showed similar differences in keff between the JENDL-4.0 and ENDF/B-VII.1 (∼450 pcm) and between JENDL-4.0 and JEFF-3.1.2 (∼270 pcm). The major contributors to these differences were identified with the use of the sensitivity coefficients. Among the nuclear data parameters, the criticality of the VENUS-F core is sensitive to the fission cross section, fission neutron multiplicity, and capture cross section of uranium as well as the elastic scattering cross section and average cosine of the scattered neutrons of lead and 56 Fe. The analysis with the sensitivity coefficients
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Hiroki Iwamoto et al. / Energy Procedia 71 (2015) 33 – 41
Fig. 6. Energy breakdown of uncertainty in keff and JENDL-4.0 covariance data with respect to representative parameters (208 Pb elastic scattering, 235 U fission, and 238 U capture).
showed that these parameters have a considerable impact on the differences in keff . However their positive and negative elements cancel each other out, thereby resulting in a relatively small difference. Based on the JENDL-4.0 covariance data, the total uncertainty in keff induced by nuclear data was evaluated as 0.833% δk/k (1σ). Among many contributors to the uncertainty of the criticality, we identified the following major parameters: • Fission cross section of 235 U (from 1 keV to 1 MeV) • Capture cross section of 238 U (from 1 keV to 1 MeV) • Elastic scattering cross section of 208 Pb (from 40 keV to 1 MeV). To reduce the uncertainty in keff of lead fast reactors with uranium fuel, a concentrated effort should be devoted to reducing the nuclear data uncertainties in this energy region. Acknowledgments One of the authors (H. Iwamoto) would like to thank the support of SCK•CEN and JAEA. He is also grateful to Dr. K. Tsujimoto of JAEA for his encouragement of this study. References [1] Technology and Components of Accelerator-driven Systems, OECD-NEA Workshop Proceedings, ISBN 978-92-64-11727-3, Karlsruhe, Germany; 2010. [2] Baylac M, Abderrahim HA, Baeten P, Billebaud A, Bergmans G, Boge P, Bondoux D, Bouvier J, Cabanel T, Carcagno Y, Dargaud G, De Conto JM, Gaudiot G, Gautier JM, Gomez Martinez Y, Granget G, Heitz G, Heusch M, Laune B, Marchand D, Mellier D, Merrer Y, Micoud R, Perbet E, Planet M, Poussot P, Reynet D, Ruescas C, Tourres D, Vermeersch F, Vittiglio G, Vermeersch F, and Vittigli G, “The GENEPI-3C Accelerator for the GUINEVERE Project,” Proceedings of the International Topical Meeting on Nuclear Research Applications and Utilization of Accelerators, IAEA, Vienna, Austria; 2010. [3] Shibata K, Iwamoto O, Nakagawa T, Iwamoto N, Ichihara A, Kunieda S, Chiba S, Furutaka K, Otuka N, Ohsawa T, Murata T, Matsunobu H, Zukeran A, Kamada S, and Kurata J, “JENDL-4.0: A New Library for Innovative Nuclear Science and Engineering,” Journal of Nuclear Science and Technology 2011;48[1]:1–30.
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