222
M. Yamanaka
where A and F indicate operators of transport and fission terms, respectively, and
φ the forward neutron flux. Multiplying Eq. (8.8) by adjoint neutron flux φ
* and
integrating over whole volume and energy, the following equation is obtained:
1
k eff
=
φ
∗ Aφ
φ ∗ Fφ
,
(8.9)
where brackets < > indicate integration over the whole volume and energy.
With the use of an operator B, Eq. (8.8) is expressed as follows:
A −
1
k eff
F
φ = Bφ = 0.
(8.10)
Here, assuming that parameter x, operator B, and neutron flux φ are changed into
x + δx, B + δB, and φ + δφ, respectively, in a critical state, the following equations
are obtained:
(B + δB) (φ + δφ) = 0 .
(8.11)
Neglecting second-order perturbation terms, Eq. (8.10) is expressed as follows:
B δφ + δB φ = 0 .
(8.12)
Introducing the generalized adjoint flux
* , the following equation is obtained
with the use of adjoint operator B
* and a certain adjoint source term q
* , defined as
reactivity in these analyses:
B
∗
Γ
∗
= q
∗
.
(8.13)
Considering the theoretical background [13–16] and using Eqs. (8.10) through
(8.13), the sensitivity coefficient in Eq. (8.7) is finally expressed by applying the
first-order perturbation approximation [17], as follows:
S =
φ
∗
−
∂B
∂ x
φ
φ ∗ F φ
k e f f .
(8.14)
Finally, applicability of sensitivity analyses to k eff was investigated for a thermal
spectrum core, such as the KUCA core, with the use of SAGEP that had been
originally developed for conducting the sensitivity analyses of fast reactors.
8.2.2.3 Uncertainty
In analyzing the cross-section uncertainty of nuclear data [18], the uncertainty of
reactor physics parameter ν is expressed as follows:
M. Yamanaka
where A and F indicate operators of transport and fission terms, respectively, and
φ the forward neutron flux. Multiplying Eq. (8.8) by adjoint neutron flux φ
* and
integrating over whole volume and energy, the following equation is obtained:
1
k eff
=
φ
∗ Aφ
φ ∗ Fφ
,
(8.9)
where brackets < > indicate integration over the whole volume and energy.
With the use of an operator B, Eq. (8.8) is expressed as follows:
A −
1
k eff
F
φ = Bφ = 0.
(8.10)
Here, assuming that parameter x, operator B, and neutron flux φ are changed into
x + δx, B + δB, and φ + δφ, respectively, in a critical state, the following equations
are obtained:
(B + δB) (φ + δφ) = 0 .
(8.11)
Neglecting second-order perturbation terms, Eq. (8.10) is expressed as follows:
B δφ + δB φ = 0 .
(8.12)
Introducing the generalized adjoint flux
* , the following equation is obtained
with the use of adjoint operator B
* and a certain adjoint source term q
* , defined as
reactivity in these analyses:
B
∗
Γ
∗
= q
∗
.
(8.13)
Considering the theoretical background [13–16] and using Eqs. (8.10) through
(8.13), the sensitivity coefficient in Eq. (8.7) is finally expressed by applying the
first-order perturbation approximation [17], as follows:
S =
φ
∗
−
∂B
∂ x
φ
φ ∗ F φ
k e f f .
(8.14)
Finally, applicability of sensitivity analyses to k eff was investigated for a thermal
spectrum core, such as the KUCA core, with the use of SAGEP that had been
originally developed for conducting the sensitivity analyses of fast reactors.
8.2.2.3 Uncertainty
In analyzing the cross-section uncertainty of nuclear data [18], the uncertainty of
reactor physics parameter ν is expressed as follows: