A proposal on accuracy estimation method for the sampling-based uncertainty analysis with Monte Carlo simulation technique

Figures & data

Figure 1. A group-wise calculation strategy for calculating $\mathrm{cov}[S_T,\overline{S_{MC}}]$.

Figure 2. Radial views of the benchmark problems.

Table 1. Details of the benchmark problems used in this study.

Problem name	Classification	Variable	Value
GODIVA	Geometry	Radius of uranium sphere	8.741 cm
	Density	U-sphere density	18.74 g/cc
	Material composition	U-235 (mass fraction)	94.73%
		U-238 (mass fraction)	5.27%
	Geometry	Radial arrangement	Infinite
		Radius of fuel cylinder	0.41275 cm
		Radius of inner cladding	0.42164 cm
		Radius of outer cladding	0.48514
		Height	Infinite
		x-max. of water moderator	0.6425 cm
		x-min. of water moderator	−0.6425 cm
		y-max. of water moderator	0.6425 cm
		y-min. of water moderator	−0.6425 cm
PWR Pin-cell	Density	Fuel	10.176 g/cc
		Gap	0.001 g/cc
		Cladding	6.55 g/cc
		Water	0.7 g/cc
	Material composition	U-234 in fuel (mass fraction)	0.0237%
		U-235 in fuel (mass fraction)	14.9618%
		U-238 in fuel (mass fraction)	66.1645%
		O-16 in fuel (mass fraction)	11.8500%
		O-16 in gap (mass fraction)	100%
		Natural Zr in cladding (mass   fraction)	100%
		H-1 in water (mass fraction)	11.19%
		O-16 in water (mass fraction)	88.81%
		B-10 in water (mass fraction)	0.02%
		B-11 in water (mass fraction)	0.08%
	Geometry	Radius of fuel sphere	0.02125 cm
		Radius of buffer	0.03125 cm
		Radius of inner PyC	0.03475 cm
		Radius of SiC	0.03825 cm
		Radius of outer PyC	0.04225 cm
		x-max. of graphite moderator	0.08 cm
		x-min. of graphite moderator	−0.08 cm
		y-max. of graphite moderator	0.08 cm
		y-min. of graphite moderator	−0.08 cm
		z-max. of graphite moderator	0.08 cm
		z-min. of graphite moderator	−0.08 cm
TRISO Pin-cell	Density	Density of fuel sphere	10.5 g/cc
		Density of buffer	1 g/cc
		Density of inner PyC	1.9 g/cc
		Density of SiC	3.2 g/cc
		Density of outer PyC	1.9 g/cc
		Density of graphite moderator	1.9 g/cc
	Material composition	U-235 in fuel sphere (atomic   fraction)	5.22%
		U-238 in fuel sphere (atomic   fraction)	28.07%
		O-16 in fuel sphere (atomic   fraction)	50.07%
		C-Nat in fuel sphere (atomic   fraction)	16.64%
		C-Nat in buffer (atomic fraction)	100%
		C-Nat in PyC (atomic fraction)	100%
		Si-28 in SiC (atomic fraction)	46.08%
		Si-29 in SiC (atomic fraction)	2.34%
		Si-30 in SiC (atomic fraction)	1.55%
		C-Nat in SiC (atomic fraction)	50.03%
		C-Nat in graphite moderator   (atomic fraction)	100%

Table 2. Monte Carlo simulation conditions of the benchmark problems for S/U analysis.

	Problem		Active	Particle
No.	name	J	cycles	history
1-1	GODIVA #1	1000	10,000	350
1-2	GODIVA #2	1000	1000	350
1-3/1-4	GODIVA #3	300	100	350
2-1	PWR Pin-cell #1	1000	5000	300
2-2	PWR Pin-cell #2	1000	500	300
2-3/2-4	PWR Pin-cell #3	300	100	300
3-1	TRISO Pin-cell #1	1000	2000	100
3-2	TRISO Pin-cell #2	1000	390	100
3-3/3-4	TRISO Pin-cell #3	300	190	100

Figure 3. Uncertainty results of responses (k_eff) with 95% confidence intervals as the number of cross section sets is increased.

Figure 4. Uncertainties of k_eff responses with 95% confidence intervals for each problem case.

Table 3. Results of the standard deviation and standard error with the proposed method.

	Standard	Standard	Average Monte
Problem	deviation,	error,	Carlo stochastic
number	SR	S[SR]	uncertainty, $\overline{S_{\mathrm{MC}}}$
1-1	0.015961	0.000357	0.000324
1-2	0.015947	0.000359	0.001024
1-3	0.015590	0.000725	0.003228
1-4	0.017197	0.000726	0.003259
2-1	0.022488	0.000504	0.000519
2-2	0.022254	0.000504	0.001643
2-3	0.020162	0.000869	0.003712
2-4	0.023951	0.001009	0.003678
3-1	0.017032	0.000386	0.001730
3-2	0.017273	0.000423	0.003920
3-3	0.016344	0.000650	0.005668
3-4	0.018383	0.000818	0.005645