Neutronics analysis of full MOX BWR core simulation experiments – FUBILA: Part 2

Abstract

A part of the experimental program FUBILA was dedicated to the study of core physics characteristics of full MOX BWR cores by testing five experimental cores: a core inserted with a B₄C control blade, a core loaded with UO₂ fuel rods, the core loaded with Gd₂O₃-UO₂ fuel rods, a core loaded with 10 × 10 MOX assemblies and a core loaded with time-elapsed MOX fuel. The present article describes analysis results of the experimental data with deterministic analysis codes and a continuous energy Monte Carlo code coupled with major nuclear data libraries. The calculated critical k _eff's with the Monte Carlo calculations range from 0.999 to 1.007. Those of the transport calculations with sixteen energy groups are close to those of the Monte Carlo calculations while those of diffusion calculations with the same sixteen energy groups are systematically smaller by −0.3 to −0.5% Δk than those of the Monte Carlo calculations. The RMSs of differences between the calculated and measured core radial fission rates are 2 to 3%, 1 to 2%, and 1 to 2% for the diffusion, transport and Monte Carlo calculations, respectively. For the analysis of the Gd₂O₃-UO₂ fuel rod loaded core, the C/Es of the radial fission rates were improved by adopting a detailed lattice calculation model and a newly measured thermal and resonance cross-sections of Gadolinium.

Keywords:

• full MOX BWR core
• core physics experiment
• FUBILA
• B₄C control blade
• UO₂ rod
• Gd₂O₃-UO₂ rod
• 10 × 10 MOX assembly
• core analysis
• Monte Carlo calculation
• deterministic calculation
• nuclear data library

1. Introduction

The Nuclear Safety Commission of Japan (NSC) suggested in the report, “Full core loading of mixed oxide fuels in advanced boiling water reactors,” [1] that the current analysis methods for safety design and evaluation of light water reactors should be advanced based on the systematic studies: (i) validation of burnup calculations by using fuel postirradiation examination data, (ii) that of nuclear design calculations by using critical experiments, and (iii) that of nuclear design calculations of irradiated fuel. In order to extend database for the above validation study (ii), the Japan Nuclear Energy Safety Organization (JNES) has implemented a core physics experimental program FUBILA [2–12]. 1 It aimed at obtaining the core physics characteristics of full MOX BWR cores, which consisted of high Pu-content BWR MOX assemblies for achieving high burnup of fuel. Part of the program was the measurements of various types of reactivity effects, which were performed using a neutron source multiplication method in sub-critical experimental cores. Yamamoto et al. [11] have reported the experimental method and results, and the neutronics analysis results. The other part of the experiments was conducted at the critical state of the experimental cores, where critical masses, core fission rate distributions and other core characteristics were obtained. The first series of the critical experiments was on in-channel void fractions in MOX assemblies at power operating conditions, and the neutronics analysis of the experimental data was reported in Yamamoto et al. [9]. The second series of the experiments was on a core inserted with a B₄C control blade, a core loaded with UO₂ fuel rods, a core loaded with Gd₂O₃-UO₂ fuel rods, a core loaded with 10 × 10 MOX assemblies and a core loaded with time-elapsed MOX fuel, which are hereafter referred to as “control blade core,” “UO₂ rod core,” “Gd₂O₃-UO₂ rod core,” “10 × 10 MOX assembly core” and “time-elapsed reference core,” respectively. The neutronics analysis of the second series of the experimental cores was performed using typical diffusion and transport calculation codes in order to evaluate the calculation accuracies. The analysis was also conducted using a continuous energy Monte Carlo calculation code with major nuclear data libraries to obtain the reference calculation results to be compared with those of the deterministic calculation codes and also evaluate accuracies of the nuclear data libraries.

The first part of the present article describes the core configurations and measurements, and the second part the neutronics analysis methods and results. The third part shows the conclusions.

2. Core configurations and measurements

2.1. Core configurations

The experimental core was contained in a double-wall cylindrical tank made of AG5 (Aluminum alloy) with 1.15 m inner diameter and 1.08 m height, which was filled with light water, as shown in Figure 1 . The inner tank was held in an outer cylindrical tank, and water always overflowed out from the top of the inner tank through the gap between the two tanks and was led to a temperature-regulation system during the experiments. The test fuel rods were supported by four grids, two of which were set at the level higher than the top of the effective part (80 cm) of the fuel rods, and two of which at the level lower than the bottom of the effective part. Four safety clusters, which were each inside of the four cylindrical tubes on the top grid in Figure 1, were arranged to be inserted into the driver region. They were completely withdrawn from the core during the experiments. The number of the MOX rods was adjusted so that the core reached an almost critical condition. A pilot rod, which was installed in the driver region (not seen in Figure 1) and had reactivity worth less than a half dollar, was used for finally approaching criticality by adjusting the inserted length. The axial configuration of the core structures and the fuel rods are shown in Refs. [4] and [10].

Figure 1. Picture of a core tank loaded with MOX rods and structures for the control blade core [4]. Note: the number of MOX rods in the driver region is not the same as that of the critical core.

The center part of the experimental cores was a test region consisting of four mockup BWR MOX assemblies, which were composed of BWR 9 × 9 type test MOX rods. Their total Pu contents 2 in the fuel pellets were 3.0, 5.0, 8.5, and 11.5 wt% in a depleted UO₂ (0.25 wt% ²³⁵U) matrix. Typical Pu and Am atomic compositions were ²³⁸Pu/²³⁹Pu/²⁴⁰Pu/²⁴¹Pu/²⁴²Pu/²⁴¹Am: 2/56/25/9/7/2% (as of February 2005) [9]. The test region realized mockup assembly configurations depending on the aims of the experimental cores. The driver region surrounding the test region was composed of full or partial 9 × 9 mockup MOX assemblies consisting of PWR type test MOX rods, which had been installed and used in the critical facility before the FUBILA program. Their total Pu content 3 in the pellets was 7.0 wt% in a depleted UO₂ (0.24 wt% ²³⁵U) matrix. Plutonium and Am atomic compositions were ²³⁸Pu/²³⁹Pu/²⁴⁰Pu/²⁴¹Pu/²⁴²Pu/²⁴¹Am: 1/58/25/5/5/6% (as of February 2005) [9].

Table 1 shows the geometrical and material specifications of the mockup assemblies [3,4], which were selected to simulate commercial 9 × 9 and 10 × 10 type MOX assemblies. Figure 2 schematically shows radial core configurations of the critical cores [3,4], for which critical mass was measured. Table 2 shows the number of MOX, UO₂, and Gd₂O₃-UO₂ rods, other core structures loaded in the core, and other characteristics of the cores. Figure 2 and Table 2 also include the information of a 9 × 9 reference core of the FUBILA program for comparison [3,4].

Figure 2. Schematic diagram of core radial configurations of critical cores [3,4]. (a) 9 × 9 reference core (0% void), (b) Control blade core (40% void), (c) UO₂ rod core (40% void), (d) Gd₂O₃-UO₂ rod core (40% void), (e) 10 × 10 assembly core (40% void), (f) Time-elapsed reference core (0% void). Note: “Pu(t) 3.0wt%” means a total-Pu content (include ²⁴¹Am) of MOX pellets is 3.0 wt%, “AG3” is Aluminum alloy.

Table 1. Geometrical and material specifications of the mockup assemblies in the FUBILA cores [3,4]. (Geometrical unit: mm).

	Test assembly						Driver assembly
Item	9 × 9 reference core (0% void)	Control blade core (40% void)	UO2 rod core (40% void)	Gd2O3-UO2 rod core (40% void)	10 × 10 MOX assembly core (40% void)	Time-elapsed reference core (0% void)
Assembly lattice	9 × 9	9 × 9	9 × 9	9 × 9	10 × 10	9 × 9	9 × 9
MOX fuel rod
Pellet outer diameter	9.4	9.4	9.4	9.4	9.4	9.4	8.2
Fuel rod outer diameter	11.0	11.0	11.0	11.0	11.0	11.0	9.5
Thickness of fuel cladding	0.7	0.7	0.7	0.7	0.7	0.7	0.6
Material of fuel cladding	Zry-2	Zry-2	Zry-2	Zry-2	Zry-2	Zry-2	Zry-4
Overcladding outer/inner diameter	12.8/11.3	12.8/11.3	12.8/11.3	12.8/11.3	12.3/11.3	12.8/11.3	12.6/9.8
Material of overcladding	AG3^a	AG3	AG3	AG3	AG3	AG3	AG3
UO2 fuel rod	–	–	–	–	–	–	–
Pellet outer diameter			8.2	8.2
Fuel rod outer diameter			9.5	9.5
Thickness of fuel cladding			0.6	0.6
Materila of fuel cladding	–	–	Zry-4	Zry-4	–	–	–
Overcladding outer/inner diameter			12.8/9.8	12.8/9.8
Materila of overcladding			AG3	AG3
Gd2O3-UO2 fuel rod	–	–	–	–	–	–	–
Pellet outer diameter	–			8.2	–	–	–
Fuel rod outer diameter				9.5
Thickness of fuel cladding				0.6
Materila of fuel cladding			6.7	Zry-4
Overcladding outer/inner diameter			14.9	12.8/9.8
Materila of overcladding				AG3
AG3 rod outer diameter	–	6.7	6.7	6.7	–	–	–
Fuel rod pitch	14.9	14.9	14.9	14.9	13.4	14.9	14.9
Water rod							–
Number	9	9	9	9	9	9
AG3 tube outer/innner diameter	12.8/8.8	12.8/10.2	12.8/10.2	12.8/10.2	12.3/9.6	12.8/8.8
Out-channel water gap
Width of water gap^b	20.9	20.9	20.9	20.9	21.0	20.9	20.9
AG3 rod	13.2	11.4	11.4	11.4	10.4	13.2	–
Assembly pitch	155.0	155.0	155.0	155.0	155.0	155.0	155.0
B4C control blade	–	Figure 2	–	–	–	–	–
Note: ^aAluminum alloy, ^bGap between the outer surfaces of fuel cells, “–”: not applied.

Table 2. Core configurations of the FUBILA cores for critical mass measurement [3,4].

Core	9 × 9 reference core (0% void)	Control blade core (40% void)	UO2 rod core (40% void)	Gd2O3-UO2 rod core (40% void)	10 × 10 MOX assembly core (40% void)	Time-elapsed reference core (0% void)
Measurement date	2005/2/14	2006/2/7	2006/3/24	2006/4/27	2006/1/4	2006/7/12
Number of fuel rods and structures
Test region
MOX 3 wt%	16(4 × 4)	16(4 × 4)	8	–	–	16(4 × 4)
MOX 5 wt%	32(4 × 8)	32(4 × 8)	28	24(4 × 6)	–	32(4 × 8)
MOX 8.5 wt%	112(4 × 28)	112(4 × 28)	92	72(4 × 18)	112(4 × 28)	112(4 × 28)
MOX 11.5 wt%	128(4 × 32)	128(4 × 32)	128	128(4 × 32)	256(4 × 64)	128(4 × 32)
UO2	–	–	32(2 × 16)	32(2 × 16)	–	–
Gd2O3-UO2	–	–	–	32(2 × 16)	–	–
AG3 tube for water rod	36(4 × 9)	36(4 × 9)	36(4 × 9)	36(4 × 9)	32(4 × 8)	36(4 × 9)
AG3 rod	–	400(4 × 100)	400(4 × 100)	400(4 × 100)	–	–
AG3 rod for out-channel	117	117	117	117	129	117
Driver region
MOX 7wt%	598	1474	910	1226	617	661
Safety rod guide tube	20	20	20	20	20	20
Pilot rod guide tube	1	1	1	1	1	1
Fission chamber guide tube	1	1	1	1	1	2
Core temperature (°C)^a	19.8 ± 0.1	20.0 ± 0.1	19.6 ± 0.1	19.9 ± 0.1	20.0 ± 0.1	20.0 ± 0.1
Doubling time (s)^a	39.5 ± 0.2	21.0 ± 0.2	30.4 ± 0.3	28.0 ± 0.3	39.7 ± 0.3	49.3 ± 0.3
Excess reactivity ($)^a	0.143 ± 0.001	0.216 ± 0.008	0.170 ± 0.003	0.180 ± 0.003	0.141 ± 0.003	0.122 ± 0.004
Excess reactivity (pcm Δ k/k)^b	49.7 ± 0.5	71.0 ± 2.5	59.2 ± 1.1	61.6 ± 1.1	49.5 ± 1.1	42.1 ± 1.4
β eff (pcm)^c	348.6	328.0	349.0	342.4	351.8	344.7
Note: ^aError in 1 σ ; ^bExcess reactivity calculated using  β  eff in the bottom low of the table (unit: ×10^−5 Δ k/k); ^cCalculated by three dimensional diffusion calculations (see text).

The 9 × 9 reference core was a basic mockup of a full BWR core with an in-channel void fraction of 0%, which was selected as a reference void fraction for studying effect on core characteristics by a change in in-channel void fractions at power operation [9]. In the control blade core, a mockup cruciform control blade with B₄C absorber rods in four wings was inserted in the center of the test region, and the absorber rods covered the full axial length of the effective fuel part of the test MOX rods. The cross-section of the control blade is schematically shown in Figure 3 . Figure 1 shows a picture of the control blade core and the structures of the core tank. In the UO₂ rod core, 32 MOX rods in two MOX assemblies in the test region were replaced by UO₂ rods. In the Gd₂O₃-UO₂ rod core, 32 MOX rods in two assemblies in the test region were replaced by the UO₂ rods and 32 MOX rods in the other assemblies in the test region by the Gd₂O₃-UO₂ rods. The pellet ²³⁵U enrichment of the UO₂ rods was 3.71 wt% and that of Gd₂O₃-UO₂ rods 4.95 wt%. The concentration of Gd₂O₃ in the pellets of Gd₂O₃-UO₂ rods was 2.55 wt%. The outer diameter of those fuel rods was 9.5 mm, which was smaller than that of the MOX rods in the test region and the same as that of MOX rods in the driver region. In the 10 × 10 assembly core, the test region was loaded with four mockup 10 × 10 MOX assemblies. The fuel rod diameters of typical 10 × 10 assemblies are generally smaller than those of 9 × 9 assemblies; however, in this experiment mockup 10 × 10 assemblies were composed of the BWR 9 × 9 type test MOX rods, and the local distribution of moderator-to-fuel ratios in the mockup 10 × 10 assemblies were adjusted by properly selecting the specifications of the overcladdings for the MOX rods and the other structures. The mockup 10 × 10 assemblies were composed of MOX rods of 8.5 and 11.5 wt% in total Pu content. The rod pitch was adjusted to that of a typical BWR 10 × 10 assembly. The selection of the high Pu content MOX rods was to obtain an assembly average total Pu content of 10.6 wt% which was higher than 9.1 wt% of the 9 × 9 MOX assemblies in the program. The average Pu contents of the mockup 9 × 9 and 10 × 10 assemblies approximately correspond to those of fuel assemblies which are necessary for the fuel assemblies to achieve the burnups 55 and 65 GWd/t, respectively. In the time-elapsed reference core, the test region is the same as the 9 × 9 reference core, and the number of MOX rods in the driver region was increased to compensate the reactivity loss due to decay of ²⁴¹Pu to ²⁴¹Am in the MOX fuel for 17 months.

Figure 3. Schematic drawing of cross-section of cruciform control blade with B₄C absorber rods [7].

The lattice configuration of the test assemblies except those for the time-elapsed reference core simulated the hot operating condition with the in-channel void fraction of 40% by adjusting a local moderator-to-fuel ratio distribution with properly selecting the specifications of AG3 (Aluminum alloy) overcladdings for the test MOX rods, AG3 rods at the corners of the MOX cells, AG3 tubes for the water rods and AG3 rods in the out-channel gaps. Figure 4 shows a configuration of the fuel assembly with the in-channel void fraction of 40% in the test region. The AG3 rods are explicitly shown in Figure 2; however, the overcladdings and AG3 tubes are not shown in Figure 2. The specifications of the assemblies in the driver region were mockup 9 × 9 MOX assemblies in a cold state of a BWR core, which was selected to make the cores critical by compensating various reactivities in the test region.

Figure 4. Configuration of a mockup fuel assembly with an in-channel void fraction of 40%.

2.2. Measurements

The critical mass was measured for each core. After the number of MOX rods in the driver region was adjusted to make the core critical, the pilot rod was withdrawn to make the core slightly supercritical. The doubling time was measured to determine the excess reactivity. Table 2 shows the measured doubling time and the excess reactivity in the units of dollar.

The fission rate distributions in the core radial and axial directions were measured by integral gamma scanning. Germanium gamma ray detectors were used to count gamma rays emitted from a specific axial part of the fuel rod after a short period irradiation. Integral counts of gamma ray pulses larger than 550 keV were used to obtain the relative fission rates. A middle part of the fuel rod was measured for the measurement of the core radial fission rate distributions. For the axial fission rate distribution, the integral gamma scanning measurement was applied with about 1-cm step in the axial direction of the fuel rod. The fission rate distribution measurement was applied to the control blade, Gd₂O₃-UO₂ rod and 10 × 10 cores and not applied to the UO₂ rod and time-elapsed reference cores as shown in Table 3 . In the core radial fission rate measurement of the Gd₂O₃-UO₂ rod core, gamma ray spectrometry on a specific gamma ray (1.596 keV) from the decay of ¹⁴⁰La was applied to represent MOX, UO₂, and Gd₂O₃-UO₂ rods. It was necessary to obtain the relative fission rate distribution in assemblies mix-loaded with MOX, UO₂, and Gd₂O₃-UO₂ fuel rods. The method was also applied to the 10 × 10 core to obtain the whole core radial fission rate distribution including the test and driver region by correlating the relative fission rates independently measured for the test and driver regions by integral gamma scanning. The Appendix describes the radial locations of the measured fuel rods and the measurement results of the core radial fission rate distributions.

Table 3. Fission rate distribution measurements and measured buckling values.

Core/Measurement	Fission rate distribution measurement	Measurement of buckling
		Buckling (×10^−3 cm^−2)	Error (1σ) (×10^−3 cm^−2)
Control blade core	A	0.982	0.003
UO2 rod core	N/A	–	–
Gd2O3-UO2 rod core	A	1.007	0.002
10 × 10 assembly core	A	1.027	0.003
Time-elapsed reference core	N/A	–	–
Note: A: measurements were applied, N/A: measurements were not applied.

Axial fission rate distribution measurement was applied to two MOX rods in the test region and one MOX rod in the driver region for the control blade and Gd₂O₃-UO₂ rod cores and two MOX rods in the test region for the 10 × 10 core. The measured axial distributions were used to obtain the core axial buckling by fitting the measured data with a cosine function. Table 3 shows the average values of the three or two measurement values with the measurement errors.

3. Analysis of experimental cores

3.1. Analytical method

Neutronics analysis was applied to the experimental cores to obtain the calculated results of the effective neutron multiplication factors k _eff's and core radial fission rate distributions. One of the analysis systems was deterministic calculation methods, which are commonly used in LWR core design. The other was a continuous energy Monte Carlo calculation method for obtaining the reference calculation results to be compared with those of the deterministic calculation methods.

3.1.1. Deterministic calculations

The calculations were performed in two steps, cell calculations followed by core calculations. Two dimensional infinite square lattice calculations were applied to the component cells such as the fuel cells each typically consisting of a fuel pellet, a fuel cladding, an overcladding, water and parts of an AG3 rod located at four cell corners. A collision probability module of SRAC [13] was used in the lattice calculations with the 107-neutron-energy-group library based on the evaluated nuclear data library JENDL-3.3 [14], and the resonance cross-sections obtained by the ultra-fine-energy-group-calculation module PEACO [13]. A thermal cutoff energy was set to be 1.8554 eV, which was determined so that an infinite neutron multiplication factor obtained using SRAC coupled with PEACO agreed to that obtained using MVP [15] for a MOX fuel cell consisting of the MOX fuel rod of the total Pu content of 7.0 wt% and a moderator of water in the analysis study of the MISTRAL program [16].

A cell calculation for the out-channel gap, into which the control rod was inserted, was performed with a multi-cell model indicated in Ref. [13]. Figure 5 shows the specific model in the present study, where the control rod structures were exactly treated, and the fuel rod cells and the out-channel gap located in the opposite side of the out-channel gap with the control rod were homogenized in terms of the atomic densities. The row of MOX fuel rods in the model were set to the second row of MOX fuel rods from the outermost side of the MOX assembly facing the control blade. Sensitivity analysis was also performed with different rows of MOX fuel rods. The effect of the different rows of MOX fuel rods on the effective neutron multiplication factor k _eff of the core was 0.18 and 0.02% Δk, for a diffusion and transport calculations, respectively. The effect on the fission rate distributions will be discussed in the later section.

Figure 5. Multi-cell model for out-channel gap inserted with a cruciform control blade. Note: Ex. MOX 5.0wt%: homogenized cell of MOX rod of 5.0 (total Pu) wt%, moderator and structures.

In the cell calculation of a Gd₂O₃-UO₂ rod cell in the Gd₂O₃-UO₂ rod core, an infinite multi-cell model of 3 × 3 fuel rods was adopted, where the rod of the center position was the relevant Gd₂O₃-UO₂ rod and eight neighboring rods were the MOX rods of 7.0 (total Pu) wt%, which were loaded in the driver region. The aim of this model is for taking into account the spectrum interaction from the MOX rods on the Gd₂O₃-UO₂ rod in the process to generate the nuclear constants for the core calculation. As seen in Figure 2, the Gd₂O₃-UO₂ are adjoined by various cells such as MOX rods of 5.0, 8.5, and 11.5 (total Pu) wt%, Gd₂O₃-UO₂ rods, the water rods and water gaps so that the selection of the neighboring rods was a tentative one and more elaborate treatment will be discussed in the later section. After the cell calculations, the effective cross-sections were spatially averaged and also collapsed into 16 energy groups [16] for each cell.

In the core calculations, the diffusion calculation code CITATION [17] was used with the 16 neutron energy groups under two and/or three dimensional geometries. The transport calculation code DANTSYS [18] were also applied to the core calculations. In the transport calculations, the 16 neutron energy groups and the P₀–S₈ approximation were adopted.

In order to convert the unit of excess reactivity from dollars to Δk, an effective delayed neutron fraction was calculated for each core. This calculation adopted 21 neutron energy groups based on JENDL-3.3 and a three-dimensional geometrical model with CITATION. The 21 neutron energy groups were adopted to take into account the dominant neutron energy range of delayed neutrons from 40 keV to 2.0 MeV.

The minimum thickness of the water reflector in the core radial direction was from 15 cm for the control blade core to 27 cm for the 10 × 10 assembly core. The core radial geometrical model took into account the water reflector in the outside of the driver region up to the inner diameter of 115.4 cm of the core inner tank. Two kinds of treatments were applied to the axial modeling. The first one was a two dimensional geometrical model where the axial leakage of neutrons was calculated by applying the measured axial buckling values shown in Table 3. The second was the three dimensional geometrical model, which covers from the bottom core grid to the top core grid. The thickness of the water reflector, which was measured from the end of the effective part of the fuel rod to the surface of the bottom core grid and the bottom surface of the top core grid, was 13 cm and 15 cm in the bottom and top of the core, respectively. Both diffusion and transport calculations were basically applied to a one-fourth core model (except for the UO₂ and Gd₂O₃-UO₂ rod cores) since the core basically had a one-fourth symmetry, and small effects of a pilot rod and a fission chamber in the driver region can be neglected. In the radial model, two meshes were assigned to each cell including a fuel rod, and four meshes were assigned to each out-channel water gap, since a thermal neutron flux steeply changes in the out-channel water gap. The number of meshes in this mesh model was also applied to the out-channel water gap including the control blade. A mesh size in a water reflector region in the outside of the driver region was set to the same for the cell including a fuel rod. In the axial geometrical model, a mesh size in a fuel effective part was set to be 1 cm, and slightly larger mesh sizes were assigned to the reflector regions.

3.2.2. Monte Carlo calculations

The MVP code [15] was used for continuous energy Monte Carlo calculations, and the nuclear data libraries were prepared from the evaluated nuclear data libraries of JENDL-3.3 [14], JENDL-4.0 [19], ENDF/B-VI.8 [20], ENDF/B–VII [21], and JEFF-3.1 [22]. The three dimensional geometrical model included the core and water reflector inside the core tank in the core radial direction. The core axial model covers the same region as the deterministic calculations. The model explicitly simulated the fuel rods and other components including guide tubes for the pilot rod and the fission chamber. In order to obtain smaller calculation statistical errors in radial fission rate distributions, a whole effective fuel part was regarded as a tally region. In the radial direction, two or four fuel rods at symmetrical positions were combined as a single tally region. In this treatment, the small effect of the pilot rod and fission chamber, which asymmetrically located in the driver region, on the radial fission rate distribution in the test region was neglected.

Two thousand three hundred batch samplings with 20,000 neutron histories per batch were performed, and the initial 300 batches were skipped from the statistical process, which were selected to obtain the statistical errors less than 0.0002 Δk in the k _eff's.

3.2. Analysis results and discussion

3.2.1. Criticality

Effective neutron multiplication factor k _eff was calculated for each core at the critical mass measurement, which had small excess reactivity. In order to obtain critical k _eff, the excess reactivity was subtracted from it. The excess reactivity in the unit Δk/k was obtained from the measured excess reactivity in the unit dollar using the calculated effective delayed neutron fraction, which is shown in Table 2. Here, the excess reactivity was approximately regarded as that in the unit Δk since k _eff was very close to 1.0. Table 4 summarizes the critical k _eff's of the adopted calculation methods for the five cores. Figure 6 schematically shows those calculated using the deterministic and Monte Carlo calculation methods based on JENDL-3.3 against the number of MOX rods in the driver region; Figure 7 shows those calculated using MVP with the different nuclear data libraries. The statistical errors in the MVP calculations are far smaller than the sizes of symbols in the figure. Figures 6 and 7 also include the calculated k _eff's for other critical cores in FUBILA reported in Ref. [9] for comparison. As seen in Figures 6 and 7, the calculated critical k _eff's for the five cores are generally consistent with those in Ref. [9]. Discussions are following.

Figure 6. Critical k _eff's calculated with JENDL-3.3 for the FUBILA cores. Note: Critical k _eff's of bold–italic faced cores show the present results and others those in Ref. [9].

Figure 7. Critical k _eff's calculated by MVP. Note: Critical k _eff's of bold–italic faced cores show the present results and others those in Ref. [9].

Table 4. Critical k _eff of the FUBILA cores.

Core		Control blade core (40% void)	UO2 rod core (40% void)	Gd2O3-UO2 rod core (40% void)	10 × 10 MOX assembly core (40% void)	Time-elapsed reference core (0% void)
No. of MOX rods in driver region		1474	910	1226	618	661
Code	Library	Critical k eff
CITATION-2D^a	JENDL-3.3	1.0014	–	1.0000	0.9974	–
CITATION-3D^b		–	0.9974	–	0.9966	0.9980
CITATION-3D (21g)^c		0.9993	0.9980	0.9989	0.9969	0.9980
TWODANT		1.0031	–	1.0001	1.0001	–
THREEDANT		–	1.0010	1.0015	1.0021	1.0025
MVP^d	JENDL-3.3	1.0042	1.0024	1.0034	1.0010	1.0023
	JENDL-4.0	1.0035	1.0024	1.0029	1.0016	1.0022
	ENDF/B-IV.8	1.0025	1.0005	1.0012	0.9986	0.9996
	ENDF/B-VII	1.0071	1.0060	1.0061	1.0040	1.0051
	JEFF-3.1	1.0024	1.0020	1.0024	1.0012	1.0018
	JENDL-3.3 + ^157Gd^e	–	–	1.0038	–	–
Note: ^aTwo-dimensional calculations with 16 neutron energy groups; ^bThree-dimensional calculations with 16 neutron energy groups; ^cThree-dimensional calculations with 21 neutron energy groups; ^dStatistical error is ± 0.0002; ^eCalculated results with the JENDL-3.3 base nuclear data file which adopted the cross sections of ^157Gd of the new measurement data of Leinweber et al. [23].

3.2.1.1. Deterministic calculations

The critical k _eff's of the diffusion calculations (CITATION) with the JENDL-3.3-base library for the five cores are from 0.997 to 1.001, which are systematically lower than the Monte-Carlo calculations (MVP) by −0.3 to −0.5% Δk. This is attributed to the calculation model of the diffusion calculation model adopted in the present study. Those of the transport calculations (TWODANT and THREEDANT) are from 1.000 to 1.003, which are close to those of the Monte Carlo calculations. It indicates that the transport calculation model of the present study gives accurate k _eff's, which are close to those of the Monte Carlo calculations. These observations are consistent with those in Ref. [9]. This indicates that the features applied in the five cores, i.e. the B₄C control blade, the UO₂ and Gd₂O₃-UO₂ fuel rods in MOX assemblies, the 10 × 10 MOX assemblies, and the time-elapsed MOX core do not introduce excess calculation errors in the reactivity calculations of the present deterministic calculation methods.

3.2.1.2. Trend with number of MOX rods in driver region

A general trend of k _eff's with the number of MOX rods in the driver region is seen in those of the five cores in both deterministic and Monte Carlo calculations. The correlation of the trend with the fuel compositions in the MOX rods was discussed in Ref. [9] on the uncertainty in the thermal and/or resonance cross-sections of ²⁴¹Am and ²⁴¹Pu. Beside this general trend, it is also observed in Figures 6 and 7 that there are small divergences in k _eff's from the general increasing trend for the 10 × 10, UO₂ rod, and Gd₂O₃-UO₂ rod cores. This would be related to the higher Pu contents in the 10 × 10 assemblies and the existence of UO₂ rod and Gd₂O₃-UO₂ rods in the UO₂ rod and Gd₂O₃-UO₂ cores. It is typically seen in the results of MVP based on JENDL-3.3 in Figure 7 that the increase in the k _eff's from the 9 × 9 reference to time-elapsed cores is larger than the general increase with the number of MOX rods in the driver region. This would be due to the fact that the increase in the k _eff's is caused by time-elapsed effect in the Pu isotopic composition of the MOX rods in the test region in 17 months in addition to the change in the number of MOX rods in the driver region.

3.2.1.3. Nuclear data library

The critical k _eff's of the Monte Carlo calculations range from 1.000 to 1.007 as seen in Figure 7. The values based on JENDL-3.3, JENDL-4.0, and JEFF-3.1 are closely plotted; on the other hand, those based on ENDF/B-VI.8 and ENDF/B-VII disperse from them. The k _eff's based on ENDF/B-VII show from 0.5 to 0.6% Δk higher than those based on ENDF/B-VI.8. It is noted that the k _eff's with JEFF-3.1 show the smallest trend against the change in the number of the MOX rods in the driver region.

Blaise et al. [6] have reported the critical k _eff's calculated by the continuous energy Monte Carlo calculation code TRIPOLI-4.5 for the 9 × 9 reference, 40% void, and 70% void cores of the FUBILA program. The critical k _eff's are 1.001 to 1.002 for JEFF-3.1 and 1.0045 to 1.007 for ENDF/B-VII. These results agree with those of the present calculations of MVP based on the same nuclear data libraries for all the FUBILA critical cores shown in Figure 7.

The critical k _eff's of JENDL-3.3 and JENDL-4.0 show close results, which was also reported in Ref. [24] for those of the 9 × 9 reference core of FUBILA. One of major difference in cross-sections relevant to the core analysis of the FUBILA program between JENDL-3.3 and JENDL-4.0 is those of ²⁴¹Am. The thermal capture cross-section and resonance integral cross-sections of ²⁴¹Am in JENDL-4.0 are larger by 7 and 9% than those in JENDL-3.3, respectively. This makes the k _eff's for JENDL-4.0 smaller than those for JENDL-3.3. The changes would be almost compensated by the other increasing effect for the k _eff's from the other differences between JENDL-3.3 and JENDL-4.0 such as decreases in the thermal capture cross-sections of ²³⁸U and the resonance integral cross-sections of ²³⁹Pu. It should, however, be confirmed in the further study.

3.2.2. Core radial fission rate distribution

3.2.2.1. Control blade core

Figures 8 and 9 show the relative deviations of the calculated radial fission rates from the measured results in the test region as defined (Calculated fission rate–measured fission rate)/(measured fission rate) × 100. The calculated results are those of the diffusion calculation (CITATION-2D) and the transport calculation (TWODANT) based on JENDL-3.3. Figure 10 schematically shows the comparison between the calculated results of CITATION-2D and the measurements in the diagonal direction of the test region. The measurement errors of 1.2% are also attached to the measurements. Table 5 also shows the root mean squares (RMSs) of the relative deviations, and the maximum (Max) and minimum (Min) deviations. Table 5 includes those of the results calculated by using MVP based on JENDL-3.3 [4]. The table also shows the measurement errors which were evaluated using the differences in the measured radial fission rates on the fuel rods at the geometrically symmetrical locations in the test region [4]. Here, the deviations were defined as those from the average fission rates of the two fuel rods at the geometrically symmetrical locations. The RMSs of the deviations were calculated and converted to the measurement error of the fission rate measurement. The analysis errors slightly depend on the types of the analysis methods. The general trends in the errors are represented by the parameters shown in Table 5. The RMS of the diffusion calculation is two times of the measurement error, that of the transport calculation 1.6 times, and that of the Monte Carlo calculation 1.4 times. In the deterministic calculations of the control blade core, the differences in the RMSs were less than 0.1% for adopting different rows of MOX fuel rods in the multi-cell model for the out-channel gap inserted with the control blade.

Figure 8. Deviation of calculated radial fission rates of CITATION-2D calculation from measurements for the test region of the control blade core. Note: Deviation (%): (calculated fission rate −measured fission rate)/(measured fission rate) × 100.

Figure 9. Deviation of calculated radial fission rates of TWODANT calculation from measurements for the test region of the control blade core. Note: Deviation (%): (calculated fission rate −measured fission rate)/(measured fission rate) × 100

Table 5. Root-mean-square (RMS), maximum (Max) and minimum (Mini) values of (calculated–measured)/measured in % for radial fission rates in the test region.

Core	Control blade core (Test region)			Gd2O3-UO2 rod core (Test region)			10 × 10 MOX assembly core
Measurement error (%)	1.2			1.3			1.3
Code^a	RMS	Max	Min	RMS	Max	Min	RMS	Max	Min
CITATION-2D	2.44	5.68	−6.45	3.00	7.24	−8.15	2.28^b	4.55^b	−3.72^b
TWODANT	2.01	4.39	−6.43	2.03	4.59	−4.89	1.48^b	2.89^b	−5.97^b
MVP^c	1.65	4.61	−3.15	2.20	3.47	−7.42	1.17	2.55	−3.24
Note: ^aThe calculations were performed using the nuclear data library JENDL-3.3; ^bThe calculated results of the 10 × 10 assembly core are those calculated using CITATION-3D and THREEDANT; ^cThe statistical errors in the calculated fission rates are about 0.5%.

3.2.2.2. Gd₂O₃-UO₂ core

Figures 11 and 12 present the relative deviations of the calculated radial fission rates from the measurements for the fuel rods in the test region. The calculated results are those for CITATION-2D and TWODANT based on JENDL-3.3. Table 5 also shows the root mean squares (RMSs) of the relative deviations, and the maximum (Max) and minimum (Min) deviations, where those of the results calculated using MVP based on JENDL-3.3 [4] are also included. Figures 11 and 12 indicate systematic deviations unique to the types of fuel rods. The average values of the deviations for the fuel rod types are shown in Table 6 , which include calculated results using MVP based on JENDL-3.3. The figures and Table 6 indicate that the diffusion calculation systematically underestimates the fission rates of UO₂ rods, and the transport and Monte Carlo calculations do those of Gd₂O₃-UO₂ and UO₂ rods. When the calculated results of the deterministic methods were compared with those of the Monte Carlo calculation as the reference calculations, the diffusion calculation underestimates the fission rates of UO₂ rods by 3% and overestimates those of Gd₂O₃-UO₂ rods by 4%; the transport calculation overestimates those of UO₂ rods by 2%. Further analysis was performed to know possible causes of the systematic trends.

Figure 10. Comparison between calculated fission rates of CITATION-2D and measurements in the diagonal direction of the test region of the control blade core. Note: The average values of fission rates are normalized to 1.0.

Figure 11. Deviation of calculated radial fission rates of CITATION-2D calculation from measurements for the test region of the Gd₂O₃-UO₂ core [12]. Note: Deviation (%): (calculated fission rate −measured fission rate)/(measured fission rate) × 100. Notation: Brown: UO₂ rod, dark blue: Gd₂O₃-UO₂ rod, light blue, white and yellow: MOX rods.

Figure 12. Deviation of calculated radial fission rates of TWODANT calculation from measurements for the test region of the Gd₂O₃-UO₂ core [12]. Deviation (%): (calculated fission rate −measured fission rate)/(measured fission rate) × 100. Notation: Brown: UO₂ rod, dark blue: Gd₂O₃-UO₂ rod, light blue, white and yellow: MOX rods.

Table 6. Average deviation of calculated fission rates from measurements as (calculated–measured)/measured in % for the Gd₂O₃-UO₂ rod core.

Type of fuel rod	UO2	Gd2O3-UO2	MOX
CITATION-2D (JENDL-3.3)	−5.29	−0.09	0.99
	(−3.37)^a	(0.17)^a	(0.67)^a
TWODANT (JENDL-3.3)	−2.32	−1.98	0.44
	(−1.25)^a	(−1.73)^a	(0.23)^a
MVP (JENDL-3.3)	−1.86	−4.09	0.58
MVP (ENDF/B-VII)	−1.90	−4.61	0.63
MVP (JEFF-3.1)	−2.26	−4.21	0.63
MVP (JENDL-3.3 + ^157Gd)^b	−2.11	−2.59	0.51
Note: ^aCalculated results by using the improved lattice calculation (see text); ^bCalculated results with the JENDL-3.3 base nuclear data file which adopted the cross sections of ^157Gd of the new measurement data of Leinweber et al. [23].

In order to improve the cell calculations of the cells of UO₂ and Gd₂O₃-UO₂ rods, infinite assembly models of the Gd₂O₃-UO₂ rod loaded assembly and the UO₂ rod loaded assembly shown in Figure 13 were adopted, which was aimed at taking into account the influence on the neutron spectrum calculation of the UO₂ and Gd₂O₃-UO₂ rod cells from adjoining cells with different characteristics. The cell calculations for the MOX rod cells remained the infinite one rod cell calculations. The deviations of the calculations with the improved lattice calculations from the measured fission rates are shown in Table 6. The significant improvement is seen in the deviations of the UO₂ rods. On the other hand, the change in the Gd₂O₃-UO₂ rods is small since the original lattice calculation model adopted the 3 × 3 multi-cell calculations in order to take into account the interaction effect from neighboring MOX rods, even though the selection of the MOX rods adjoining to the Gd₂O₃-UO₂ rod was a tentative one in the multi-cell mode. It is still observed that there exist the general deviations of the calculated fission rates of the UO₂ and Gd₂O₃-UO₂ rods in the diffusion and transport calculations from the Monte Carlo calculations even after improving the cell calculation. That would be due to modeling errors in the deterministic calculation methods.

Figure 13. Infinite assembly models in lattice calculations for Gd₂O₃-UO₂ rod cells and UO₂ rod cells in the Gd₂O₃-UO₂ Core. Note: Notation should be referred to Figure 2.

It was assumed that the relatively large underestimation of about 4% in the fission rates for the Gd₂O₃-UO₂ rods and about 2% for the UO₂ rods in the Monte Carlo calculations was caused by the uncertainties in the nuclear data libraries. This trend was almost common among the nuclear data libraries as shown in Table 6. New measurements of thermal and resonance cross-sections of Gadolinium have been reported by Leinweber et al. [23]. Their significant findings were that the thermal (2200 m/s) capture cross-section of ¹⁵⁷Gd was measured to be 11% smaller than that calculated from ENDF/B-VI [20]. Referring to the measured data, Jatuff et al. [25] have reported that the new cross-sections of gadolinium isotopes reduce the discrepancies in comparison between the measurement and neutronics analysis on the rod-by-rod fission rates and modified conversion ratios obtained in the critical experiments at the PROTEUS critical facility. In the process of evaluation work of JENDL-4.0, a temporary nuclear data library of MVP [26], was prepared, which replaced the cross-sections of ¹⁵⁷Gd by the new measurement data. The calculated results of MVP with the temporary nuclear data library are shown in Table 6 and Figure 14 . Figure 11 shows the deviations of the calculated fission rates from the measurements against the measured fission rates. They show that the degree of the underestimation of the fission rates by the MVP calculations for the Gd₂O₃-UO₂ rods is reduced from −4.09% to −2.59%, which makes the degree of the underestimation similar for both UO₂ and Gd₂O₃-UO₂ rods.

Figure 14. Deviation of calculated radial fission rates of MVP calculation with ¹⁵⁷Gd-modified JENDL-3.3 from measurements for the test region of the Gd₂O₃-UO₂ core.

Table 4 shows the calculated k _eff of MVP with the temporary nuclear data library of ¹⁵⁷Gd. The effect of the temporary nuclear data library on k _eff was +0.04 ± 0.02% Δk and not significant due to a small number of Gd₂O₃-UO₂ rods in the core. It should be noted that the finalized JENDL-4.0 adopted the resonance parameters based on the new measurements, and also background cross-sections aimed at making the capture cross-section at 2200 m/s of JENDL-4.0 coincide with that of JENDL-3.3 [19,24].

About remaining deviation in C/Es between the MOX and UO₂ fuel rods, Chauvin et al. [27] and Nagaya et al. [28] have also reported that their neutronics analysis results overestimate the measured fission rates of the MOX fuel rod region in the UO₂–MOX mixed cores of the EPICURE and VENUS-2 programs. The overestimation of the fission rates for the MOX rods, which means underestimation of the UO₂ and Gd₂O₃-UO₂ rods on the other side, would be related to uncertainties in nuclear data libraries.

3.2.2.3. 10 × 10 MOX assembly core

Figures 15 and 16 show the relative deviations of the calculated radial fission rates from the measurements. The calculated results are those of the diffusion calculation (CITATION-3D) and the transport calculation (THREEDANT) based on the nuclear data library JENDL-3.3. Figure 17 schematically shows the comparison between the calculated results of CITATION-3D and the measurements about the fission rates in the diagonal direction. Table 5 also shows the root mean squares (RMSs) of the relative deviations, and the maximum (Max) and minimum (Mini) deviations, where those of the results calculated by using MVP based on JENDL-3.3 [4]. are also included. The analysis errors slightly depend on the analysis methods, and the general trend is represented by the parameters shown in Table 5. The RMS of the diffusion calculation is 1.8 times of the measurement error, and those of the transport and Monte Carlo calculations are same order. The RMSs of the 10 × 10 MOX assembly core is comparable with those of the 40% void core (9 × 9 assembly) in Ref. [9], i.e. 1.81, 1.33, and 1.08% for the diffusion, transport and Monte Carlo calculation methods and 1.0% for the measurement error, which indicate that the higher Pu content assemblies with the 10 × 10 lattice do not introduce extra calculation errors in the fission rate calculation for the present deterministic and Monte Carlo calculation methods.

Figure 15. Deviation of calculated radial fission rates of CITATION-3D calculation from measurements for the 10 × 10 MOX assembly core.

Figure 16. Deviation of calculated radial fission rates of THREEDANT calculation from measurements for the 10 × 10 MOX assembly core.

Figure 17. Comparison between calculated fission rates of CITATION-3D and measurements in the diagonal direction of the 10 × 10 MOX assembly core. Note: The average values of fission rates are normalized to 1.0.

Figure A1. Measured relative radial fission rate distribution in the control blade core [4]. Note: The average value of measured 81 fuel rods in the test region is 1.0 and that of 7 fuel rods in the driver region 1.0.

Figure A2. Measured relative radial fission rate distribution in the test region of the Gd₂O₃-UO₂ rod core [4]. Note: The average value of measured 167 fuel rods in the test region is 1.0.

Figure A3. Measured radial fission rate distribution in the 10 × 10 MOX assembly core [4]. Note: The average value of measured 109 fuel rods in the test region is 1.0.

4. Conclusions

Major findings in the neutronics analysis of the control blade, UO₂ rod, Gd₂O₃-UO₂ rod, 10 × 10 MOX assembly and time-elapsed reference cores in the FUBILA program are the following. The k _eff's of the diffusion calculations in the present study are lower than those of the Monte-Carlo calculations by −0.3 to −0.5% Δk. On the other hand, those of the transport calculations are close to those of the Monte Carlo calculations, which means the model of the present transport calculations is accurate enough to calculate the k _eff's of the FUBILA critical cores. The Monte Carlo calculations based on the major nuclear date libraries show the k _eff's ranging from 0.999 to 1.007. The differences in the k _eff's between JENDL-3.3 and JENDL-4.0 are small; on the other hand, those of ENDF/B-VII are larger than those of ENDF/B-VI.8 by 0.5 to 0.6% Δk.

The analysis of the core radial fission rates indicates that the RMSs of differences between the calculated and measured core radial fission rates are 2 to 3%, 1 to 2%, and 1 to 2% for the diffusion, transport, and Monte Carlo calculations, respectively, which are one to three times larger than the measurement errors. They are generally smaller in the order of the Monte Carlo, transport and diffusion calculations under the present analysis models. The extended study on the Gd₂O₃-UO₂ core indicates that taking into account the adjoining cells in the cell calculations of the UO₂ fuel rods makes the radial fission rates of the UO₂ fuel rods in the deterministic calculations close to those in the Monte Carlo calculations. Applying the newly measured thermal and resonance cross-sections of ¹⁵⁷Gd in the Monte Carlo calculations also improves the agreement between the Monte Carlo calculations and the measurements for radial fission rates of the Gd₂O₃-UO₂ rods in the Gd₂O₃-UO₂ rod core.

Acknowledgments

The authors wish to thank the experimental team lead by Drs. P. Fougeras and P. Blaise for the implementation of the FUBILA experiments in the EOLE critical facility.

Notes

1. Full MOX core physics experiments of BWR Initiated for Lattice analysis method verification and improvement.

2. Including ²⁴¹Am.

3. Including ²⁴¹Am.

Appendix

The measured radial fission rate distributions are shown in Figure A1 for the control blade core [6], Figure A2 for the Gd ₂O₃-UO₂ core [6], and Figure A3 for the 10 × 10 MOX assembly core [6]. In order to estimate measurement errors in the radial fission rate distribution, the fission rates of fuel rods were compared with those of fuel rods at symmetrical positions in the left-bottom fuel assembly of the test region. It was found that the differences between them were generally larger than the counting statistical errors in the integral gamma scanning, which were from 0.2 to 0.3%. Using the information about the symmetry of fission rates, the study of measurement errors was performed to assess the overall measurement errors [6]. The estimated measurement errors of the radial fission rates were from 1.2 to 1.3% as shown in Table 5 in the text. The measurement errors of the Gd₂O₃-UO₂ and 10 × 10 MOX assembly cores include the measurement errors in the determination of the radial fission rate distributions using the spectrometry on the specific gamma ray (1.596 keV) from the decay of ¹⁴⁰La.