Applying the continuous-energy Monte Carlo calculation code, MVP3, to analysis of kinetic parameters measured for light-water moderated UO₂ and MOX cores of the TCA and EOLE critical facilities

Figures & data

Table 1. Summary of the major characteristics of the experimental cores for the measurements of the kinetic parameters [5–7].

Critical facility		TCA																					EOLE
Core ID		TCA1	TCA2	TCA3	TCA4	TCA5	TCA6	TCA7		TCA8	TCA9	TCA10	TCA11	TCA12	TCA13	TCA14	TCA15	TCA16	TCA17		TCA18		MISTRAL1	MISTRAL2
Core configuration		One-region core								Two-region core											One-region core
		Core region
M/F ratio		1.50	1.83	1.83	1.83	1.83	1.83	2.48		1.76	2.00	2.38	2.95	1.76	2.00	2.38	2.95	2.40	2.96		1.83		1.75	1.76
Fuel pitch (cm)		1.85	1.96	1.96	1.96	1.96	1.96	2.15		1.66	1.73	1.82	1.96	1.66	1.73	1.82	1.96	1.48	1.58		1.96		1.32	1.32
Enrichment of UO2 fuel (wt%)		2.60	2.60	2.60	2.60	2.60	2.60	2.60		2.59	2.59	2.59	2.59								2.60		3.7
Number of UO2 fuel rods		361	309	342	361	484	625	225		49	49	49	49								292		744
Pu content of MOX fuel (wt%Pu)														3.4	3.4	3.4	3.4	4.9	4.9					7.0/8.7
Number of MOX fuel rods														49	49	49	49	100	100					1572/88
Water gap between core and driver regions (cm)										0.06	1.79	1.45	0.00	0.06	1.79	1.45	0.00	0.03	0.42
		Driver region
M/F ratio										1.83	1.83	1.83	1.83	1.83	1.83	1.83	1.83	1.50	1.50
Fuel pitch (cm)										1.96	1.96	1.96	1.96	1.96	1.96	1.96	1.96	1.85	1.85
Enrichment of UO2 fuel (wt%)										2.60	2.60	2.60	2.60	2.60	2.60	2.60	2.60	2.60	2.60
Number of UO2 fuel rods										296	284	276	272	288	260	236	244	224	164
Core total fuel rods		361	309	342	361	484	625	225		345	333	325	321	337	309	285	293	324	264		292		744	1660
Fraction of MOX fuel rods														0.15	0.16	0.17	0.17	0.31	0.38					1.00
Water temperature (℃)		22.7	23.0	22.9	23.9	23.0	24.8	20^a		20.5	20.8	20.9	13.7	18.9	20.6	16.2	20.4	21.0	21.0		20^a		20.0	20.0
Boron content in water (ppm)				72.34	146.6	345	554																～ 300
Critical water height (cm)		99.20	81.28	85.78	96.85	84.99	85.79	148.18		69.28	68.80	68.32	67.84	68.72	68.15	69.28	69.26	83.66	87.30		92.24		80.00	80.00
Measured kinetic parameter		βeff/ℓ																			βeff

Table 2. Major specifications of the UO₂ and MOX fuel rods loaded in the TCA cores [5].

Fuel rod type	2.60 UO2	2.59 UO2	3.4 MOX	4.9 MOX
Pellet diameter (cm)	1.25	1.07	1.07	0.86
Pellet density (g/cm^3)	10.4	10.4	6.8	10.1
Enrichment (wt%)	2.60	2.59	0.71(Nat. U)	0.71(Nat. U)
Pu content (wt%)			3.4	4.9
Pu isotope ratio (Pu-238/239/240/241/242) (wt%)			0.01/90.64/8.46/0.84/0.05	0.05/90.24/8.48/1.15/0.09
Cladding inner dia. (cm)	1.27	1.08	1.08	0.87
Cladding outer dia. (cm)	1.42	1.22	1.22	1.00
Cladding material	Al	Al	Zry-2	Zry-2

Figure 1. Calculated results of β_eff and ℓ for the TCA and EOLE cores.

Table 3. Comparison of the calculated and measured β_eff/ℓ values in C/E-1% for the TCA cores.

Core	TCA1	TCA2	TCA3	TCA4	TCA5	TCA6	TCA7	TCA8	TCA9	TCA10	TCA11	TCA12	TCA13	TCA14	TCA15	TCA16	TCA17
This study (MVP3+ JENDL-4.0)	1.38	1.72	1.37	1.53	1.51	4.04	−1.08	0.98	0.35	0.71	0.49	1.46	−0.93	0.76	1.16	1.54	0.30
(Uncertainty (%) in C/E − 1 for this study ^a)	(0.68)	(1.70)	(2.68)	(0.47)	(0.80)	(0.72)	(1.95)	(0.36)	(1.00)	(0.33)	(1.18)	(1.56)	(0.66)	(0.58)	(0.95)	(0.83)	(0.95)
Tsuruta & Kitamoto [5]	−0.30	0.00	−0.70	−0.50	0.00	3.40	−3.90									−2.10	−3.90

^aUncertainty (1σ) in C/E-1 was composed of those in the calculated and measured results.

Figure 2. Comparison of the calculated and measured β_eff/ℓ values in C/E-1% for the TCA cores. This figure shows the results of this study (MVP3 + JENDL-4.0) and those of Tsuruta and Kitamoto [5]. The error bars for this study show the range of the uncertainties (1σ) in C/E-1 composed of those in the calculated and measured results.

Table 4. Comparison of the calculated and measured β_eff values in C/E-1%.

Core	TCA18	MISTRAL1	MISTRAL2
This study (MVP3+ JENDL-4.0)	1.21	−1.45	−4.61
(Uncertainty (%) in C/E-1 for this study ^a)	(2.25)	(1.51)	(1.59)
Nakajima [6]	2.9
Nagaya [2]	1.2
Litaize et al. [7]		2.4	0.1
Hibi et al. [12]		−0.8	−2.5
NUPEC [17]		−2	−3

^aUncertainty (1σ) in C/E-1 was composed of those in the calculated and measured results.

Table 5. Calculated results of β_eff, ℓ and β_eff /ℓ with the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries and the ratios to those obtained with the JENDL-4.0-based neutron library.

Core	Library	βeff	(βeff/JENDL-4.0–1)%	ℓ (s)	(ℓ/JENDL-4.0–1)%	βeff/ℓ (1/s)	(βeff/ℓ/JENDL-4.0–1)%
TCA1	JENDL-4.0	7.749E-03		3.494E-05		2.218E+02
	JENDL-3.3	7.788E-03	0.50	3.500E-05	0.18	2.225E+02	0.32
	ENDF/B-VII.1	7.702E-03	−0.61	3.497E-05	0.09	2.201E+02	−0.75
	JEFF-3.2	7.968E-03	2.82	3.487E-05	−0.20	2.285E+02	3.02
TCA6	JENDL-4.0	7.524E-03		2.839E-05		2.650E+02
	JENDL-3.3	7.539E-03	0.21	2.841E-05	0.07	2.653E+02	0.14
	ENDF/B-VII.1	7.484E-03	−0.54	2.836E-05	−0.12	2.639E+02	−0.41
	JEFF-3.2	7.736E-03	2.82	2.837E-05	−0.08	2.727E+02	2.90
TCA17	JENDL-4.0	5.378E-03		3.041E-05		1.769E+02
	JENDL-3.3	5.387E-03	0.17	3.049E-05	0.28	1.767E+02	−0.11
	ENDF/B-VII.1	5.437E-03	1.09	3.060E-05	0.62	1.777E+02	0.46
	JEFF-3.2	5.573E-03	3.61	3.042E-05	0.05	1.832E+02	3.56
TCA18	JENDL-4.0	7.803E-03		3.787E-05		2.060E+02
	JENDL-3.3	7.757E-03	−0.59	3.797E-05	0.25	2.043E+02	−0.84
	ENDF/B-VII.1	7.721E-03	−1.05	3.792E-05	0.12	2.036E+02	−1.17
	JEFF-3.2	8.003E-03	2.56	3.784E-05	−0.08	2.115E+02	2.65
MISTRAL1	JENDL-4.0	7.768E-03		2.727E-05		2.849E+02
	JENDL-3.3	7.785E-03	0.22	2.734E-05	0.27	2.847E+02	−0.07
	ENDF/B-VII.1	7.681E-03	−1.12	2.730E-05	0.10	2.814E+02	−1.24
	JEFF-3.2	7.934E-03	2.14	2.717E-05	−0.37	2.920E+02	2.50
MISTRAL2	JENDL-4.0	3.596E-03		1.387E-05		2.593E+02
	JENDL-3.3	3.595E-03	−0.04	1.389E-05	0.13	2.588E+02	−0.18
	ENDF/B-VII.1	3.612E-03	0.43	1.397E-05	0.71	2.585E+02	−0.28
	JEFF-3.2	3.722E-03	3.51	1.382E-05	−0.34	2.693E+02	3.86

Table 6. βvalues of ²³⁵U, ²³⁹Pu, and ²³⁸U for JENDL-4.0 [4], ENDF/B-VII.1 [9], JEFF-3.2 [10], and JEFF-3.3 [22].

Nuclide (neutron energy)	Nuclear data library	β	Ratio to JENDL-4.0
^235U (0.0253 eV)	JENDL-4.0	6.507 × 10^−3	-
	ENDF/B-VII.1	6.505 × 10^−3	1.000
	JEFF-3.2	6.650 × 10^−3	1.022
	JEFF-3.3	6.721 × 10^−3	1.033
^239Pu (0.0253 eV)	JENDL-4.0	2.161 × 10^−3	-
	ENDF/B-VII.1	2.240 × 10^−3	1.037
	JEFF-3.2	2.261 × 10^−3	1.046
	JEFF-3.3	2.268 × 10^−3	1.050
^238U (2.0 MeV)	JENDL-4.0	1.762 × 10^−2	-
	ENDF/B-VII.1	1.670 × 10^−2	0.948
	JEFF-3.2	1.858 × 10^−2	1.055
	JEFF-3.3	1.818 × 10^−2	1.030

Table 7. Comparison of the calculated and measured β_eff/ℓ and β_eff values in C/E − 1% for the MVP3 calculations with the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries.

	βeff/ℓ			βeff
Library/core	TCA1	TCA6	TCA17	TCA18	MISTRAL1	MISTRAL2
JENDL-4.0 (Uncertainty (%) in C/E-1 for this study ^a)	1.38 (0.68)	4.04 (0.72)	0.30 (0.95)	1.21 (2.25)	−1.45 (1.51)	−4.61 (1.59)
JENDL-3.3	1.70	4.18	0.19	0.61	−1.23	−4.65
ENDF/B-VII.1	0.61	3.61	0.76	0.14	−2.55	−4.20
JEFF-3.2	4.44	7.06	3.87	3.80	0.66	−1.26

^aUncertainty (1σ) in C/E-1 was composed of those in the calculated and measured results.

Figure A1. Ratios of the calculated β_eff obtained with the other methods in MVP3 [3] to the reference (the results with the differential-operator-sampling technique with the indirect-perturbed-source effect) for all the cores.

Figure A2. Ratios of the calculated ℓ obtained without the indirect-perturbed-source effect for the differential-operator-sampling technique to the reference (with the indirect-perturbed-source effect for the differential-operator-sampling technique) for all the cores.

Table A1. Calculated results of the kinetic parameters, β_eff and ℓ, with the methods [3] other than the differential-operator-sampling technique as the eigenvalue method taking into account the indirect-perturbed-source effect [1,2].

		Perturbation method								Next fission probability method
		Differential-operator-sampling				Correlated sampling
Core		With indirect-perturbed-source effect		Without indirect-perturbed-source effect		With indirect-perturbed-source effect		Without indirect-perturbed-source effect		Meulekamp's method
TCA1	βeff	7.749E-03	(0.30)	7.692E-03	(0.12)	7.748E-03	(0.31)	7.692E-03	(0.12)	7.76E-03	(0.21)
	Ratio to Reference^a	1.000		0.993		1.000		0.993		1.002
	ℓ (s^−1)	3.494E-05	(0.05)	3.944E-05	(0.02)
	Ratio to Reference^a	1.000		1.129
TCA6	βeff	7.524E-03	(0.29)	7.436E-03	(0.12)	7.531E-03	(0.30)	7.436E-03	(0.12)	7.51E-03	(0.21)
	Ratio to Reference^a	1.000		0.988		1.001		0.988		0.999
	ℓ (s^−1)	2.839E-05	(0.03)	2.985E-05	(0.01)
	Ratio to Reference^a	1.000		1.051
TCA17	βeff	5.378E-03	(0.36)	5.559E-03	(0.14)	5.380E-03	(0.36)	5.559E-03	(0.14)	5.63E-03	(0.25)
	Ratio to Reference^a	1.000		1.034		1.000		1.034		1.047
	ℓ (s^−1)	3.041E-05	(0.05)	3.634E-05	(0.02)
	Ratio to Reference^a	1.000		1.195
TCA18	βeff	7.803E-03	(0.30)	7.698E-03	(0.12)	7.810E-03	(0.31)	7.698E-03	(0.12)	7.75E-03	(0.21)
	Ratio to Reference^a	1.000		0.987		1.001		0.987		0.993
	ℓ (s^−1)	3.79E-05	(0.04)	4.20E-05	(0.02)
	Ratio to Reference^a	1.000		1.110
MISTRAL1	βeff	7.77E-03	(0.31)	7.67E-03	(0.13)	7.77E-03	(0.32)	7.67E-03	(0.13)	7.74E-03	(0.41)
	Ratio to Reference^a	1.000		0.988		1.000		0.988		0.997
	ℓ (s^−1)	2.726E-05	(0.04)	3.052E-05	(0.01)
	Ratio to Reference^a	1.000		1.119
MISTRAL2	βeff	3.596E-03	(0.37)	3.561E-03	(0.16)	3.599E-03	(0.38)	3.561E-03	(0.15)	3.567E-03	(0.52)
	Ratio to Reference^a	1.000		0.990		1.001		0.990		0.992
	ℓ (s^−1)	1.387E-05	(0.07)	1.866E-05	(0.03)
	Ratio to Reference^a	1.000		1.345

(): Statistical uncertainties in the calculations (1σ) (%)

^aThe result with the differential-operator-sampling with the indirect-perturbed souce effect for the perturbation method.

Table A2. Calculated results of β_eff with the Meulekamp’s method, the importance function method, and the constant weight method in the next fission probability methods for MISTRAL1.

	Perturbation method		Next fission probability method
	Differential-operator-sampling with indirect-perturbed-source effect		Meulekamp’s method		Importance function weight method (Nauchi’s method)		Contant weight method (Non-adjointweighted method)
βeff	7.748E-03	(1.01)	7.741E-03	(0.43)	7.659E-03	(0.43)	7.021E-03	(0.0028)
Ratio to Reference^a	1.000		0.999		1.000		0.906

(): Statistical uncertainties in the calculations (1σ) (%)’

^aThe result with the differential-operator-sampling with the indirect-perturbed souce effect for the perturbation method.

Figure A3. K_eff values for the critical cores obtained with the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1-, and JEFF-3.2-based neutron libraries.

Table A3. Effective neutron multiplication factors for the critical cores with respect to the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries. The statistical uncertainties were less than 0.005 to 0.006%Δk_eff.

Core	TCA1	TCA2	TCA3	TCA4	TCA5	TCA6	TCA7	TCA8	TCA9	TCA10	TCA11
JENDL4.0	1.00026	1.00076	1.00067	1.00009	1.00258	1.00316	1.00072	0.99994	1.00006	0.99975	0.99995
JENDLE-3.3	0.99748					1.00072
ENDF/B-VII.1	1.00107					1.00335
JEFF-3.2	1.00046					1.00295
Core	TCA12	TCA13	TCA14	TCA15	TCA16	TCA17	TCA18	MISTRAL1	MISTRAL2
JENDL4.0	1.00016	0.99994	1.00023	1.00023	0.99934	1.00018	1.00057	1.00191	1.00684
JENDLE-3.3						0.99837	0.99795	0.99998	1.00717
ENDF/B-VII.1						0.99996	1.00105	1.00193	1.00711
JEFF-3.2						0.99947	1.00043	1.00283	1.00194

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