A note on application of superhomogénéisation factors to integro-differential neutron transport equations

Figures & data

Figure 1. Calculation configurations.

Table 1. List of cross-sections.

Material	Cross-section (1/cm)
	Absorption	Total
4.95 wt.% UO2	0.38	0.75
2.00 wt.% UO2	0.18	0.55
RCC	7.6	7.6
MOX (8.3 wt.% Puf)	2.0	2.3
H2O	0.019	1.3

Table 2. Maximum and mean values of errors in cell-wise absorption reaction rate (without SPH factors, unit:%).

Problem	P1		P3		P5		S16
	Maximum	Mean	Maximum	Mean	Maximum	Mean	Maximum	Mean
1	12.5	4.8	6.4	2.7	5.9	2.5	5.6	2.4
2	8.5	2.2	5.2	1.4	4.8	1.3	4.7	1.3
3	20.5	4.6	13.7	3.2	12.8	3.0	12.4	3.0
4	20.5	4.7	13.7	3.2	12.8	3.1	12.4	3.0
5	6.8	1.9	3.9	1.1	3.8	1.1	3.7	1.0
6	2.5	0.4	1.8	0.3	1.7	0.3	1.7	0.3
7	9.8	1.3	7.5	1.0	7.2	1.0	7.0	1.0

Table 3. Maximum and mean values of errors in cell-wise absorption reaction rate (with SPH factors, direct correction, unit:%).

Problem	P1		P3		P5		S16
	Maximum	Mean	Maximum	Mean	Maximum	Mean	Maximum	Mean
1	1.9	0.32	1.0	0.22	1.1	0.22	1.0	0.22
2	2.1	0.37	1.5	0.23	1.4	0.22	1.4	0.22
3	7.8	1.10	5.6	0.80	5.3	0.78	5.2	0.76
4	4.7	0.62	3.1	0.42	2.9	0.40	2.8	0.39
5	2.0	0.34	1.4	0.21	1.3	0.21	1.3	0.20
6	2.5	0.42	1.8	0.28	1.7	0.27	1.7	0.26
7	9.8	1.33	7.5	1.03	7.2	1.00	7.0	0.98

Table 4. Maximum and mean values of errors in cell-wise absorption reaction rate (with SPH factors, simultaneous correction, unit:%).

Problem	P1		P3		P5		S16
	Maximum	Mean	Maximum	Mean	Maximum	Mean	Maximum	Mean
1	1.6	0.30	1.1	0.24	1.1	0.25	1.1	0.25
2	2.0	0.33	1.4	0.21	1.4	0.20	1.3	0.20
3	4.7	0.56	3.0	0.37	3.1	0.36	2.8	0.35
4	2.8	0.32	1.5	0.19	1.6	0.18	1.4	0.17
5	1.8	0.29	1.3	0.19	1.3	0.18	1.2	0.18

Table 5. Maximum and mean values of errors in cell-wise absorption reaction rate (with SPH factors, simultaneous correction, P₅, unit:%).

Problem	Present		Yamamoto		No correction
	Maximum	Mean	Maximum	Mean	Maximum	Mean
1	1.1	0.25	1.1	0.23	1.1	0.24
2	1.4	0.20	1.3	0.20	1.3	0.20
3	2.9	0.36	2.5	0.33	2.8	0.33
4	1.5	0.18	1.2	0.19	1.4	0.18
5	1.2	0.18	1.2	0.18	1.2	0.18

Table 6. Errors in cell-wise absorption reaction rate (with SPH factors, simultaneous correction, P₅, unit:%).

Cell ID	Problem
	1	2	3	4	5	6	7
1	−0.00	0.01	0.03	0.01	0.01	0.01	−0.00
2	−0.00	0.01	0.02	0.01	0.01	0.00	−0.01
3	0.05	0.02	0.06	0.03	−0.00	0.00	−0.04
4	0.15	−0.01	0.02	0.04	−0.02	−0.01	−0.12
5	0.33	−0.05	−0.03	0.05	0.01	−0.06	−0.30
6	0.05	−0.00	0.15	0.21	0.01	−0.17	−0.77
7	0.53	−0.35	−0.48	0.03	−0.30	−0.50	−2.05
8	1.13	−1.35	−2.94	−1.50	−1.24	−1.70	−7.15
9	−1.12	1.18	0.40	0.11	1.07	1.48	3.30
10	−0.40	0.14	1.08	0.56	0.10	0.29	1.73
11	−0.08	−0.10	0.34	0.18	−0.10	0.04	0.37
12	−0.08	−0.01	0.12	0.06	−0.06	−0.01	0.06
13	−0.03	−0.02	0.02	0.01	−0.01	−0.01	0.01
14	0.00	−0.02	0.00	−0.00	−0.01	−0.01	0.00
15	−0.00	−0.01	0.00	0.00	−0.01	−0.01	0.00
16	0.00	−0.01	0.01	0.01	−0.00	−0.00	0.00

Table 7. Maximum and mean values of errors in cell-wise absorption reaction rate (with SPH factors, improved methods, simultaneous correction, P₅, unit:%).

Problem	Improved SPH		Selengut normalization		Flux-volume normalization
	Maximum	Mean	Maximum	Mean	Maximum	Mean
1	0.6	0.18	0.5	0.13	1.1	0.25
2	0.7	0.15	0.3	0.11	1.3	0.20
3	1.7	0.49	1.5	0.27	2.8	0.35
4	1.7	0.42	1.1	0.18	1.4	0.17
5	0.6	0.14	0.2	0.11	1.2	0.18
6	0.3	0.09	0.3	0.09	1.7	0.26
7	1.4	0.24	1.4	0.24	7.0	0.98

Table 8. Errors in cell-wise absorption reaction rate (with SPH factors, Selengut normalization, S₁₆, unit:%).

Cell ID	Problem
	1	2	3	4	5	6	7
1	−0.00	0.02	0.05	0.01	0.02	0.02	0.04
2	0.00	0.03	0.06	0.02	0.03	0.02	0.06
3	0.04	0.06	0.14	0.05	0.04	0.03	0.09
4	0.13	0.06	0.15	0.07	0.05	0.05	0.15
5	0.30	0.07	0.22	0.13	0.14	0.08	0.25
6	−0.11	0.24	0.66	0.39	0.24	0.11	0.40
7	0.17	0.13	0.58	0.46	0.16	0.09	0.58
8	0.52	−0.21	0.05	0.07	−0.15	−0.28	0.14
9	−0.53	0.24	−1.49	−1.11	0.19	0.39	−1.39
10	−0.13	−0.21	0.66	0.33	−0.23	−0.17	0.54
11	0.06	−0.27	0.23	0.12	−0.25	−0.15	0.12
12	−0.00	−0.09	0.08	0.04	−0.14	−0.09	0.02
13	0.01	−0.05	0.01	0.01	−0.06	−0.05	0.00
14	0.02	−0.04	0.00	0.00	−0.04	−0.02	−0.00
15	0.01	−0.02	0.00	0.00	−0.02	−0.01	0.00
16	0.01	−0.01	0.00	0.00	−0.02	−0.01	0.00