4 Effective Delayed Neutron Fraction
89
Table 4.4 Comparison between β eff by MCNP6.1, β eff,eigen by the k-ratio method in Eq. (4.8),
β RR
eigen by the k-ratio method with reaction rate calculations (eigenvalue calculation) in Eq. (4.12),
β RR
source by the k-ratio method with reaction rate calculations (fixed-source calculation) in Eq. (4.20),
and β (eigenvalue and fixed-source calculations) in Eq. (4.1) (Ref. [2])
Case
Effective delayed neutron fraction (and delayed neutron fraction) [pcm]
β eff
(MCNP6.1)
β eff,eigen
(MCNPX-2.5.0)
β RR
eigen
(MCNPX-2.5.0)
β RR
source
(MCNPX-2.5.0)
I-1
817 ± 11
797 ± 9
791 ± 13
(649 ± 22)
809 ± 43
(637 ± 32)
I-2
794 ± 11
801 ± 9
795 ± 13
(649 ± 22)
873 ± 45
(636 ± 33)
I-3
814 ± 11
811 ± 9
807 ± 13
(649 ± 22)
880 ± 46
(636 ± 33)
I-4
806 ± 11
810 ± 9
809 ± 13
(649 ± 22)
826 ± 55
(637 ± 36)
I-5
822 ± 11
825 ± 9
838 ± 13
(649 ± 22)
819 ± 57
(637 ± 37)
I-6
808 ± 11
824 ± 9
806 ± 13
(649 ± 22)
865 ± 81
(636 ± 44)
I-7
829 ± 11
801 ± 9
783 ± 13
(649 ± 22)
792 ± 86
(635 ± 45)
II-1
808 ± 11
795 ± 9
803 ± 9
(652 ± 24)
910 ± 11
(645 ± 14)
II-2
798 ± 11
802 ± 9
802 ± 9
(653 ± 24)
881 ± 7
(635 ± 9)
results in Table 4.4, the proposed methodology by the k-ratio method was confirmed
valid on the basis of the reaction rates by the eigenvalue calculations.
4.1.3.2 Reaction Rates by Fixed-Source Calculations
From the definition of β
RR
eigen shown in Eq. (4.12), the methodology using eigenvalue
calculations is extended into a methodology using the fixed-source calculations. In
the fixed-source problem, the transport equation is expressed by introducing source
term s to acquire neutron flux formed in the subcritical core φ sub as follows:
L φ sub = F φ sub + s,
(4.13)
where L and F indicate the loss and fission operators, respectively. To distinguish
source and fission neutrons, φ sub was assumed to be expressed in terms of φ source and
φ core as follows:
89
Table 4.4 Comparison between β eff by MCNP6.1, β eff,eigen by the k-ratio method in Eq. (4.8),
β RR
eigen by the k-ratio method with reaction rate calculations (eigenvalue calculation) in Eq. (4.12),
β RR
source by the k-ratio method with reaction rate calculations (fixed-source calculation) in Eq. (4.20),
and β (eigenvalue and fixed-source calculations) in Eq. (4.1) (Ref. [2])
Case
Effective delayed neutron fraction (and delayed neutron fraction) [pcm]
β eff
(MCNP6.1)
β eff,eigen
(MCNPX-2.5.0)
β RR
eigen
(MCNPX-2.5.0)
β RR
source
(MCNPX-2.5.0)
I-1
817 ± 11
797 ± 9
791 ± 13
(649 ± 22)
809 ± 43
(637 ± 32)
I-2
794 ± 11
801 ± 9
795 ± 13
(649 ± 22)
873 ± 45
(636 ± 33)
I-3
814 ± 11
811 ± 9
807 ± 13
(649 ± 22)
880 ± 46
(636 ± 33)
I-4
806 ± 11
810 ± 9
809 ± 13
(649 ± 22)
826 ± 55
(637 ± 36)
I-5
822 ± 11
825 ± 9
838 ± 13
(649 ± 22)
819 ± 57
(637 ± 37)
I-6
808 ± 11
824 ± 9
806 ± 13
(649 ± 22)
865 ± 81
(636 ± 44)
I-7
829 ± 11
801 ± 9
783 ± 13
(649 ± 22)
792 ± 86
(635 ± 45)
II-1
808 ± 11
795 ± 9
803 ± 9
(652 ± 24)
910 ± 11
(645 ± 14)
II-2
798 ± 11
802 ± 9
802 ± 9
(653 ± 24)
881 ± 7
(635 ± 9)
results in Table 4.4, the proposed methodology by the k-ratio method was confirmed
valid on the basis of the reaction rates by the eigenvalue calculations.
4.1.3.2 Reaction Rates by Fixed-Source Calculations
From the definition of β
RR
eigen shown in Eq. (4.12), the methodology using eigenvalue
calculations is extended into a methodology using the fixed-source calculations. In
the fixed-source problem, the transport equation is expressed by introducing source
term s to acquire neutron flux formed in the subcritical core φ sub as follows:
L φ sub = F φ sub + s,
(4.13)
where L and F indicate the loss and fission operators, respectively. To distinguish
source and fission neutrons, φ sub was assumed to be expressed in terms of φ source and
φ core as follows: