neutron transport benchmark problem proposal for fast critical assembly without homogenizations

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Annals of Nuclear Energy 34 (2007) 443–448

annals of

NUCLEAR ENERGY

Neutron transport benchmark problem proposal for fastcritical assembly without homogenizations

Go Chiba a,*, Kazuyuki Numata b

a Japan Atomic Energy Agency, 4002 Narita-cho, O-arai-machi, Ibaraki 311-1311, Japanb NESI, 4002 Narita-cho, O-arai-machi, Ibaraki 311-1311, Japan

Received 26 December 2006; received in revised form 23 February 2007; accepted 27 February 2007Available online 20 April 2007

Abstract

In the present paper, we propose a neutron transport benchmark problem for fast critical assembly without homogenizations. Withthis problem, we can validate applicability of neutron transport codes when employed in highly heterogeneous fast critical assembly anal-yses. In addition, this benchmark problem can be used to validate homogenization procedures for slab lattices.

Detailed configurations of the cores and the lattices and cross-section data are provided in this paper. Reference solutions obtainedwith a Monte Carlo code are also provided.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

The nuclear fuel cycle with fast breeder reactors is one ofthe most promising candidates for future energy resources.There have been many studies worldwide aimed at thedevelopment of this resource. Efficient numerical methodsfor neutron transport and accurate nuclear data have beendeveloped by many researchers and engineers in the field ofreactor physics to accurately predict the nuclear character-istics of fast reactors. And development has benefited fromthe efficient utilization of experimental data obtained fromfast critical assemblies.

Recently, it was pointed out that significant errors wereobserved in the homogenization of fast critical assemblylattices (Chiba, 2006). The errors were observed only inthe fast critical assemblies that contain low density regions.This indicates that the homogenization deteriorates theprediction accuracy for the coolant voided reactivity. Thedata obtained from fast critical assemblies are importantsince they are used to validate the applicability of numeri-

0306-4549/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2007.02.018

* Corresponding author. Tel.: +81 29 267 4141; fax: +81 29 266 2904.E-mail address: [email protected] (G. Chiba).

cal methods for neutron transport and nuclear data intopower reactors. Hence, it is essential to reduce the errorsobserved in fast critical assembly analyses in order toimprove the prediction accuracy for nuclear characteristicsof power reactors.

One method to eliminate the errors caused by latticehomogenization is to analyze fast critical assemblies with-out homogenizations. Such calculations with deterministictransport codes have not been attempted due to limitedcomputer power. Computer power has advanced greatlyin this decade, however, and it is thought that reactor cal-culations without homogenizations are now possible.

Here, we propose a neutron transport benchmark prob-lem for fast critical assembly without homogenizations.This benchmark problem can be utilized to validate theapplicability of neutron transport codes into such highlyheterogeneous problems.

Homogenization for slab lattices has been one of themost important topics in the field of reactor physicsbecause of the difficulty of taking into account the anisot-ropy of neutron streaming in lattice-homogenized reactorcalculations. This benchmark problem is also useful to val-idate methods to homogenize lattices and to consider theanisotropic neutron streaming effect.

mailto:[email protected]

Table 1Fission spectrum

Group Fission spectrum

1 0.885292 0.113293 0.001424 0.00000

444 G. Chiba, K. Numata / Annals of Nuclear Energy 34 (2007) 443–448

2. Specification of benchmark problems

The reactor core of this benchmark problem is the sim-plified MZA core (Kaise and Osada, 2003). The MZA corewas constructed at the ZEBRA critical assembly in theUnited Kingdom. The unit fuel lattice of this benchmarkproblem is composed of six sodium plates, four uranium-dioxide plates, two plutonium plates and two carbon andsteel plates. These material plates are piled along to theaxial direction. In addition, a voided-fuel lattice is also pre-pared to simulate a sodium-voided situation. All thesodium nuclides are removed in the sodium plate in thisvoided-fuel lattice. Since the sodium plate includes othernuclides such as iron, the voided-sodium plate is not rigor-ously void. Fig. 1 shows specifications of the unit fuel lat-tice and the unit voided-fuel lattice. Blanket and reflectorlattices are treated homogeneously in this problem.

Cross-section data are given in the four-group structure.The fission spectrum given in Table 1 is used commonly forthe plutonium plate, the uranium-dioxide plate and theblanket region. The scattering matrices are given only forthe P0 order. The anisotropic scattering is taken intoaccount by the transport approximation. Cross-sectionscorrected by the transport approximation are shown inTables 2–8.

In addition to the cross-sections of each material plate,homogenized cross-sections for the unit fuel lattice andvoided fuel lattice are given in Tables 9 and 10. With thesehomogenized cross-sections, we can validate the applicabil-ity of the neutron transport codes when employed in the‘lattice-homogenized’ problems, and it may be possible toextract problems specific to the lattice-heterogeneousproblems.

In this benchmark problem, four core configurations arecreated. Case 1 is a reference configuration while others are

Fuel unit cell

VoidedSodiumPu

VoidedSodium

C+SS

VoidedSodium

Voided fuel unit cell0

0.581.28

1.82

2.14

2.543.17

Sodium

UO2

Pu

Sodium

C+SS

Sodium

0

0.581.28

1.82

2.142.543.17

3.75

C+SS

Pu

Sodium

Sodium

Sodium

4.334.96

5.36

5.686.22

6.92

7.5 (cm)

C+SS

Pu

VoidedSodium

VoidedSodium

VoidedSodium

3.75

4.334.96

5.36

5.686.22

6.92

7.5 (cm)

UO2

UO2

UO2

UO2

UO2

UO2

UO2

Fig. 1. Unit lattice specification.

sodium-voided configurations. In case 2, fuel latticeslocated at core center are voided. In cases 3 and 4, fuel lat-tices located near boundaries to the blanket region arevoided. These core configurations are shown in Figs. 2–5.From the difference between keff of the reference case andthat of the sodium-voided case, a sodium-voided reactivitycan be defined. The spectrum component, which is positivereactivity caused by a change in the neutron spectrum, isdominant in the sodium-voided reactivity of case 2 whilethe leakage component, which is negative reactivity causedby an increase of neutron leakage, is dominant in the reac-tivity of case 3 or case 4. In the reactivity of case 3, the leak-age effect perpendicular to material plates is dominant,while the leakage effect parallel to plates is dominant inthe reactivity of case 4. The component-wise sodium-voided reactivities will be approximately quantified withthe diffusion-based perturbation theory in Section 4.

To validate the neutron transport codes only for two-dimensional problems, two-dimensional core configura-tions are also created. The lattice configurations and thecross-sections are same as those of the three-dimensionalproblem. The two-dimensional core configurations areshown in Figs. 6–9.

3. Reference solutions obtained with the Monte-Carlo code

To obtain reference solutions of this benchmark prob-lem, calculations with the multi-group Monte-Carlo codeGMVP (Nagaya et al., 2005) are performed. The totalnumber of histories in these calculations is about5,000,000,000 for each case. The obtained keffs are shownin Table 11. Their statistical uncertainties, shown in Table12, are outputs of the GMVP code. In addition, referencevalues of sodium voided reactivities are also calculated.The obtained sodium voided reactivities are shown inTable 12 along with their statistical uncertainties.

4. Estimation of component-wise sodium-voided reactivity

In order to quantify which component is dominant inthe sodium-voided reactivities of this benchmark problem,component-wise reactivities are calculated with the diffu-sion-based exact perturbation theory in the lattice-homog-enized problems. A spectrum component is defined as

qs ¼Z X

g

Xg0

/0g0 /þg0 � /þg

� �dRg0!g d~r ð1Þ

Table 2Cross-sections of sodium plate

Group Ra mRf Rtr Rs,i! 1 Rs,i! 2 Rs,i! 3 Rs,i! 4

1 1.33000e�4a 0.0 7.42500e�2 6.58626e�2 8.22400e�3 3.04000e�5 0.02 1.12000e�4 0.0 1.07400e�1 0.0 1.05428e�1 1.86000e�3 0.03 4.37000e�4 0.0 1.97300e�1 0.0 0.0 1.96477e�1 3.86000e�44 1.10600e�3 0.0 1.84100e�1 0.0 0.0 0.0 1.82994e�1

a Read as 1.33000 · 10�4.

Table 3Cross-sections of uranium-dioxide plate


1 4.49300e�3 8.08800e�3 1.83200e�1 1.53539e�1 2.49900e�2 1.78000e�4 0.02 3.24700e�3 3.86000e�4 3.34100e�1 0.0 3.26596e�1 4.25700e�3 0.03 1.15700e�2 9.95900e�4 4.56300e�1 0.0 0.0 4.43740e�1 9.90000e�44 2.79300e�2 4.17200e�3 5.51100e�1 0.0 0.0 0.0 5.23170e�1

Table 4Cross-sections of plutonium plate



Table 5Cross-sections of carbon and steel plate


1 6.49000e�4 0.0 1.88700e�1 1.65248e�1 2.26500e�2 1.53000e�4 0.02 6.30000e�4 0.0 3.26400e�1 0.0 3.18630e�1 7.14000e�3 0.03 2.41800e�3 0.0 5.62400e�1 0.0 0.0 5.58057e�1 1.92500e�34 8.30700e�3 0.0 7.72000e�1 0.0 0.0 0.0 7.63693e�1

Table 6Cross-sections of blanket region



Table 7Cross-sections of reflector region



G. Chiba, K. Numata / Annals of Nuclear Energy 34 (2007) 443–448 445

and a leakage component is defined as

ql ¼Z X

g

dDg

o/þgox�o/0goxþ

o/þgoy�o/0goyþ

o/þgoz�o/0goz

!d~r;

ð2Þ

where / 0 is the forward neutron flux in the sodium-voidedsituation, /+ is the adjoint neutron flux in the reference sit-uation and the index g corresponds to the energy group. InEq. (2), diffusion coefficient D is defined as D = 1/(3Rtr). Asshown in Eq. (2), the leakage component can be split into

Table 10Cross-sections of homogenized unit voided-fuel lattice



Fuel

Blanket

Reflector

X

Y

Reflective

Reflective Vacuum

Vacuum

5.4cm

5.4cm

X

Z

0

45

80

115 (cm)

Reflective

Vacuum

Z=0

Y=0

Reflector

Blanket

Fuel

Fig. 2. Core specification (3D, case 1).

Blanket

Reflector

X

Y

Reflective

Reflective Vacuum

Vacuum

X

Z

0

45

80

115 (cm)

Reflective

Vacuum

Z=0

Y=0

Reflector

Blanket

Fuel

Fuel

Voided fuel

22.5

Voided fuel


Table 8Cross-sections of sodium-voided plate



Table 9Cross-sections of homogenized unit fuel lattice




Blanket

Reflector

X

Y

Reflective

Reflective Vacuum

Vacuum

X

Z

0

45

80

115 (cm)

Reflective

Vacuum

Z=0

Y=0

Reflector

Blanket

Fuel

Fuel

Voided fuel

30

Voided fuel


Blanket

Reflector

X

Y

Reflective

Reflective Vacuum

Vacuum

X

Z

0

45

80

115 (cm)

Reflective

Vacuum

Z=0

Y=0

Reflector

Blanket

Fuel

Fuel

22.5

Voided fuel

Voided fuel


Fuel

Blanket

Reflector

0

45

82.5

120 (cm)

Reflective

Vacuum

82.545 120 (cm)

Reflective

Vacuum


Fuel

Blanket

Reflector

0

45

82.5

120 (cm)

Reflective

Vacuum

30

Voided fuel82.545 120 (cm)

Reflective Vacuum

30


Fuel

Blanket

Reflector

0

45

82.5

120 (cm)

Reflective

Vacuum

30

Voided fuel

82.545 120 (cm)

Reflective Vacuum


Fuel

Blanket

Reflector

0

45

82.5

120 (cm)

Reflective

Vacuum

Voided fuel82.545 120 (cm)

Reflective Vacuum

30


G. Chiba, K. Numata / Annals of Nuclear Energy 34 (2007) 443–448 447

Table 11Reference keffs obtained with GMVP

Dimension Lattice Case 1 Case 2 Case 3 Case 4

2 Hetero. 1.18616 (0.002%a) 1.19485 (0.002%) 1.18088 (0.002%) 1.17986 (0.002%)Homo. 1.18649 (0.002%) 1.19489 (0.002%) 1.17990 (0.002%) 1.17988 (0.002%)

3 Hetero. 1.02867 (0.003%) 1.03063 (0.003%) 1.02539 (0.003%) 1.01988 (0.003%)Homo. 1.02777 (0.003%) 1.02962 (0.003%) 1.02371 (0.003%) 1.01884 (0.003%)

a Statistical uncertainty (1r).

Table 12Reference sodium voided reactivities obtained with GMVP

Dimension Lattice Case 2 Case 3 Case 4

2 Hetero. 0.00613 (0.6%a) �0.00377 (1.0%) �0.00450 (0.8%)Homo. 0.00592 (0.6%) �0.00471 (0.8%) �0.00472 (0.8%)

3 Hetero. 0.00185 (2.0%) �0.00311 (1.2%) �0.00838 (0.4%)Homo. 0.00175 (2.1%) �0.00386 (0.9%) �0.00853 (0.4%)

a Statistical uncertainty (1r).

Table 13Component-wise sodium voided reactivities calculated with diffusiontheory (unit: 10�2Dk/kk 0)

Dimension Component Case 2 Case 3 Case 4

2 Spectrum +1.14 +0.32 +0.32Leakage (x) �0.29 �0.19 �0.66Leakage (y) �0.29 �0.66 �0.19

3 Spectrum +0.49 +0.18 +0.61Leakage (xy) �0.23 �0.12 �1.45Leakage (z) �0.09 �0.49 �0.10


each directional component. The obtained component-wisereactivities are shown in Table 13.

5. Conclusions

In the present paper, we proposed a neutron transportbenchmark problem for fast critical assembly withouthomogenizations. With this problem, we can validateapplicability of neutron transport codes when employedin highly heterogeneous fast critical assemblies. In addi-

tion, this benchmark problem can be used to validatehomogenization procedures for slab lattices.

Acknowledgements

The authors wish to express their deep gratitudes to Dr.A. Yamamoto and Dr. T. Endo who attempted to utilizethis benchmark problem with the deterministic transportcodes based on the method of characteristics and the dis-crete ordinates method.

References

Chiba, G., 2006. Overestimation in parallel component of neutron leakageobserved in sodium void reactivity worth calculation for fast criticalassemblies. J. Nucl. Sci. Technol. 43 (8), 946–949.

Kaise, Y., Osada, H., 2003. Investigation of MOZART experimental dataand analysis of MOZART experiment using JFS-3-J3.2R groupconstant. JNC TJ9400 2003-009, Japan Nuclear Cycle DevelopmentInstitute.

Nagaya, Y. et al., 2005. MVP/GMVP-II: General purpose Monte Carlocodes for neutron and photon transport calculations based oncontinuous energy and multigroup methods. JAERI 1348, JapanAtomic Energy Research Institute.

neutron transport benchmark problem proposal for fast critical assembly without homogenizations

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