8 Sensitivity and Uncertainty of Criticality
223
ν = G tar M (G tar )
t
=
i
j
s i c i, j s j ≡
i
j
υ i, j (1 ≤ i, j ≤ p) ,
(8.15)
where G tar (1 × p) indicates the sensitivity vector of reactor physics parameters,
M (p × p) the covariance matrix of nuclear reaction parameters, s i the sensitivity
coefficient, c i, j the covariance, υ i, j the factor of uncertainty and p the number of
nuclear reactions including the nuclides. Thus, the contribution of uncertainty u i in
each nuclear reaction can be defined as follows:
u i ≡
i
υ i, j .
(8.16)
Generally, since sensitivity coefficient s i and covariance c i, j are dominant in
the energy group, the factor of uncertainty is finally expressed with the use of the
maximum number of energy group G as follows:
υ i, j =
g
g
s
i
g c
i, j
g, g s
j
g
1 ≤ g, g
≤ G
,
(8.17)
where g and g
indicate the energy groups.
8.2.3 Results and Discussion
8.2.3.1 Eigenvalue Calculations
In MCNP simulations, excess reactivity and control rod worth were obtained by
two eigenvalue calculations in critical and super-critical, and, critical and subcritical states, respectively: the difference between the inverse values of eigenvalue
calculations in the two states. Here, MCNP eigenvalue calculations were made with
2,000 active cycles of 50,000 histories, resulting in a standard deviation within 10
pcm. The numerical results of excess reactivity and control rod worth were obtained
by MCNP6.1 with JENDL-4.0 in EE1 and E3 cores, as shown in Table 8.3a, b,
respectively. Moreover, estimation of C/E (calculation/experiment) values revealed
an accuracy of around 5% error with the use of the experimental and numerical
results of excess reactivity and control rod worth, as shown in Eqs. (8.1) and (8.2),
respectively, excluding small values of excess reactivity and C2 control rod worth
in EE1 core. From the calculated results in Table 8.3a, b, the difference between
numerical analyses by MCNP6.1 with JENDL-4.0 and ENDF/B-VII.0 was found
within an error of 3% of the C/E value.
The ability of MCNP6.1 calculations was confirmed at a critical state in terms
of the eigenvalue bias by the MCNP approach, as shown in Table 8.4. In EE1 core,
eigenvalue bias by MCNP6.1 with JENDL-4.0 demonstrated a relatively small value
223
ν = G tar M (G tar )
t
=
i
j
s i c i, j s j ≡
i
j
υ i, j (1 ≤ i, j ≤ p) ,
(8.15)
where G tar (1 × p) indicates the sensitivity vector of reactor physics parameters,
M (p × p) the covariance matrix of nuclear reaction parameters, s i the sensitivity
coefficient, c i, j the covariance, υ i, j the factor of uncertainty and p the number of
nuclear reactions including the nuclides. Thus, the contribution of uncertainty u i in
each nuclear reaction can be defined as follows:
u i ≡
i
υ i, j .
(8.16)
Generally, since sensitivity coefficient s i and covariance c i, j are dominant in
the energy group, the factor of uncertainty is finally expressed with the use of the
maximum number of energy group G as follows:
υ i, j =
g
g
s
i
g c
i, j
g, g s
j
g
1 ≤ g, g
≤ G
,
(8.17)
where g and g
indicate the energy groups.
8.2.3 Results and Discussion
8.2.3.1 Eigenvalue Calculations
In MCNP simulations, excess reactivity and control rod worth were obtained by
two eigenvalue calculations in critical and super-critical, and, critical and subcritical states, respectively: the difference between the inverse values of eigenvalue
calculations in the two states. Here, MCNP eigenvalue calculations were made with
2,000 active cycles of 50,000 histories, resulting in a standard deviation within 10
pcm. The numerical results of excess reactivity and control rod worth were obtained
by MCNP6.1 with JENDL-4.0 in EE1 and E3 cores, as shown in Table 8.3a, b,
respectively. Moreover, estimation of C/E (calculation/experiment) values revealed
an accuracy of around 5% error with the use of the experimental and numerical
results of excess reactivity and control rod worth, as shown in Eqs. (8.1) and (8.2),
respectively, excluding small values of excess reactivity and C2 control rod worth
in EE1 core. From the calculated results in Table 8.3a, b, the difference between
numerical analyses by MCNP6.1 with JENDL-4.0 and ENDF/B-VII.0 was found
within an error of 3% of the C/E value.
The ability of MCNP6.1 calculations was confirmed at a critical state in terms
of the eigenvalue bias by the MCNP approach, as shown in Table 8.4. In EE1 core,
eigenvalue bias by MCNP6.1 with JENDL-4.0 demonstrated a relatively small value