Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 39, No. 7, p. 716–728 (July 2002)
Transport Equivalent Diffusion Constants for Reflector Region in PWRs
Yoshihisa TAHARA1, ∗ and Hiroshi SEKIMOTO2
1
Reactor Core Engineering Department, Mitsubishi Heavy Industries, Ltd., Minatomirai 3-chome, Nishi-ku, Yokohama 220-8401
2
Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152-8850
(Received December 27, 2001 and accepted in revised form April 30, 2002)
The diffusion-theory-based nodal method is widely used in PWR core designs for reason of its high computing speed
in three-dimensional calculations. The baffle/reflector (B/R) constants used in nodal calculations are usually calculated
based on a one-dimensional transport calculation. However, to achieve high accuracy of assembly power prediction,
two-dimensional model is needed. For this reason, the method for calculating transport equivalent diffusion constants
of reflector material was developed so that the neutron currents on the material boundaries could be calculated exactly
in diffusion calculations. Two-dimensional B/R constants were calculated using the transport equivalent diffusion
constants in the two-dimensional diffusion calculation whose geometry reflected the actual material configuration in
the reflector region. The two-dimensional B/R constants enabled us to predict assembly power within an error of 1.5%
at hot full power conditions.
KEYWORDS: transport theory, diffusion theory, blackness, diffusion coefficient, homogenization, reflectors,
reflection, accuracy, B/R constants, albedo, PWR type reactors, PHOENIX-P, ANC, CUDISCON, DIFF, MCNP,
ENDF/B-VI, JENDL-3.3
I. Introduction
The diffusion calculation method is widely used for various
kinds of core design, analyses of fuel handling facilities and
critical experiments as an approximate method for calculating the neutron behavior. Especially in the pressurized lightwater reactor (PWR), the two-energy group diffusion calculation method has been used for a long time, because this type
of reactor core is of a big size and the fission occurs mainly
in the thermal energy region.
Recently, according to improvement in the performance of
computers, the two-dimensional transport core calculation, a
so-called large-scale calculation, has been attempted by using
the Characteristics method and so on. Due to the fact that
the core design requires a huge amount of calculation such
as depletion calculation, various kinds of nuclear parameters
with depletion, etc., it will yet take longer time to routinely
adopt the transport calculation in the core design. Hence it
is believed that the main method for the core design is the
diffusion calculation for a time being.
The diffusion calculation requires the diffusion constants
for a fuel cell or fuel assembly and a reflector. For fuel, it
is general to determine the diffusion constants by the energy
condensation and the homogenization with the correction of
the B1 critical spectrum after carrying out the transport calculation by the pin cell model or the single fuel assembly
model.1) However, sufficient research works have not been
performed for the diffusion constants of the reflector. Thus,
a practical calculation method as accurate as transport theory
does not exist yet.
Conventionally, two-group diffusion constants of the reflector region are obtained by collapsing and homogenizing
the diffusion coefficients and cross-sections of each region
by the neutron spectrum obtained from one-dimensional core
∗
Corresponding author, Tel. +81-45-224-9607, Fax. +81-45-2249971, E-mail: [email protected]
model. In that case, the method often used to determine the
diffusion coefficient is to use the transport cross-section:
1
,
DG =
3tr,G
(the first method).2) Another method to determine the diffusion coefficient is to preserve the neutron current on each surface obtained from the transport calculation:
J trg (r, E)d Ed S
Si E
DG,i = − ,
diff
∇φg (r, E)d Ed S
Si
E
where the superscript tr and diff is the abbreviation for transport and diffusion, respectively (the second method).3, 4)
However, there remains a problem how to define the transport cross-section in the first method. Also, in the case the
continuity conditions of the neutron flux and current are imposed on the reflector boundary, it is not assured that the response of the neutron current by the transport calculation can
be preserved. By the second method, the neutron current by
the transport calculation is preserved on each surface. However, this method has a defect that the diffusion coefficient is
different on each surface and varies in the material in question. Furthermore, there is no assurance that the neutron balance in the region is established.
Accordingly, a method to determine transport equivalent
two-group diffusion constants is proposed in this paper. The
constants can reproduce the response of the neutron current
of the transport calculation on the material boundary and are
constant within the material for direct application to the diffusion calculation.
The transport equivalent two-group diffusion constants of
the reflector (diffusion coefficient, absorption cross-section
and removal cross-section) are determined so that the neutron
current on the reflector boundary from the diffusion calcula-
716
717
Transport Equivalent Diffusion Constants for Reflector Region in PWRs
tion may be equal to that from the transport calculation. The
required information was therefore obtained using the transport calculation code; a continuous energy Monte Carlo code
was used in this study.
The response of the transport neutron current on the reflector boundary can be reproduced by using thus obtained transport equivalent two-group diffusion constants, and the high
calculation accuracy of the core reactivity and the fuel rod
power can be achieved in a diffusion core calculation.
II. Transport Equivalent Diffusion Constants
− D1 ∇ 2 φ1 + (a1 + r )φ1 = 0,
(1)
− D2 ∇ 2 φ2 + a2 φ2 = r φ1 ,
(2)
∇ 2 φ1 − κ12 φ1 = 0.
(3)
(4)
Introducing a new non-dimensional variable p,
x − T /2
,
p=
T /2
where T is the reflector thickness, and using the neutron
fluxes as boundary conditions, the solution of Eq. (4) can be
written
sinh[(κ1 T /2)(1 − p)] l
φ1
φ1 ( p) =
sinh(κ1 T )
sinh[(κ1 T /2)(1 + p)] r
φ1 ,
+
(5)
sinh(κ1 T )
where the superscript l or r indicates the left or right boundary
of the reflector. The neutron currents on the boundaries can
J1l − J1r
,
φ1l + φ1r
β=
J1l + J1r
,
φ1l − φ1r
and substituting fluxes and currents into Eq. (6), we have
1
α = D1 κ1 coth(κ1 T ) −
sinh(κ1 T )
1
β = D1 κ1 coth(κ1 T ) +
sinh(κ1 T )
cosh(κ1 T ) =
β +α
,
β −α
D12 κ12 = αβ
are obtained. Solving Eq. (7) for κ1 , we have
β +α
1
.
κ1 = cosh−1
T
β −α
∇ 2 φ2 − κ22 φ2 = −γ 2 φ1 .
(9)
(12)
Putting S as
S=
κ22
γ2
,
− κ12
∇ 2 φ̃2 − κ22 φ̃2 = 0.
r
g
x
T
Fig. 1 Fluxes and currents on reflector surfaces
φ and J are the flux and the total neutron current on the reflector boundary. The subscript g indicates energy group and the
superscript l or r indicates left or right boundary of the reflector.
VOL. 39, NO. 7, JULY 2002
(8)
(13)
(14)
where φ̃2 is the solution of the equation
J gr
0
(7)
From Eq. (8), the diffusion coefficient becomes
1
D1 =
αβ.
(10)
κ1
Accordingly, κ1 can be obtained by using Eq. (9) after determining α and β from the neutron flux and the neutron current
obtained from the transport calculation, and thus the diffusion
coefficient of fast group can be obtained from Eq. (10).5–7) In
the case that a neutron does not lose its energy by scattering
and hence any transition to lower energy group does not occur, the diffusion coefficient can be determined by the abovementioned method. However, in general, the neutron loses its
energy by collision and slows down to lower energy group.
Hence the PWR core design calculation is performed in twoenergy groups: fast group and thermal group. Therefore, the
above-mentioned method is extended to the thermal group to
obtain a set of transport equivalent diffusion constants.
Defining
a2
r
, γ2 =
,
(11)
κ22 =
D2
D2
Eq. (2) becomes
φ2 = φ̃2 + Sφ1 ,
g
Jg
(6)
φ2 is given by
Water
Reflector
Core
α=
and further
1. Formulation of Transport Equivalent Diffusion Constants
A set of transport equivalent diffusion constants is determined by using neutron flux and neutron current of the transport calculation. Consider a one-dimensional slab core with
reflectors, e.g. an iron reflector and a water reflector, as shown
in Fig. 1. The neutron fluxes in the reflector can be described
with the equations
where the subscripts 1 and 2 denote energy group.
Putting
a1 + r
κ12 =
D1
gives us the equation of fast group
be obtained from Eq. (5). Introducing parameters
(15)
The flux φ̃2 can be expressed by the following equation
sinh(κ2 T /2)(1 − p) l
φ̃2 ( p) =
φ̃2
sinh(κ2 T )
sinh(κ2 T /2)(1 + p) r
φ̃2 ,
+
(16)
sinh(κ2 T )
718
Y. TAHARA and H. SEKIMOTO
where φ̃2l , φ̃2r can be determined by using the neutron fluxes
of the fast group and the thermal group:
φ̃2l = φ2l − Sφ1l
(17)
φ̃2r = φ2r − Sφ1r .
Here, though the removal cross-section is included in S,
this cross-section is determined as follows. At first, the following neutron balance equation is obtained by integrating
the diffusion equation of the thermal group
(J2r − J2l )A + a2 φ2 V = r φ1 V,
where A and V are surface area and volume of each region,
respectively. Given the absorption cross-section and the average neutron flux, the removal cross-section is obtained from
the above equation in the following manner
−
φ2
A
· + a2 .
(18)
V
φ1
φ1
In the fast group, the sum of absorption cross-section and
removal cross-section (a1 +r ) B can be expressed from
Eq. (3) as
rN =
(J2r
J2l )
(a1 + r ) B = κ12 · D1 ,
(19)
where superscript B means the quantity is obtained based on
the blackness theory. This value can be also obtained from
the neutron balance of the fast group
(J1r − J1l )A + (a1 + r )φ1 V = 0.
Solving this equation for the sum of absorption and removal
cross-sections, we obtain
(a1 + r ) N = −
(J1r − J1l )
·
A
,
V
(20)
φ1
where superscript N means the quantity is obtained based on
the neutron balance. As, in general, this value is different
from that obtained from Eq. (19), we introduce a factor f
defined as
(a1 + r ) B
f =
.
(21)
(a1 + r ) N
Using this factor, the removal cross-section is obtained from
rB = f · rN .
(22)
By employing this removal cross-section, the resonance
escape probability P can be preserved:
rB
rN
P=
=
.
(23)
(a1 + r ) B
(a1 + r ) N
Then, the absorption cross-section of the fast group is calculated from the equation
B
a1
= (a1 + r ) B − rB .
(24)
From Eqs. (14) and (16), the thermal flux is given by the following equation:
1
κ2 T
φ̃ l sinh
(1 − p)
φ2 ( p) =
sinh(κ2 T ) 2
2
κ2 T
r
+ φ̃2 sinh
(1 + p) + Sφ1 ( p).
(25)
2
Differentiating this equation gives us the neutron currents at
left and right surfaces
r
D2 κ 2
D
{φ̃ l cosh(κ2 T ) − φ̃2r } + 2 1 2 J1l
J2l =
sinh(κ2 T ) 2
κ2 − κ1
(26)
r
D2 κ 2
D
J2r = −
{φ̃ r cosh(κ2 T ) − φ̃2l } + 2 1 2 J1r .
sinh(κ2 T ) 2
κ2 − κ1
(27)
By using Eqs. (11), (13) and (17), we can write the neutron
currents as
J2l = D2 A(κ2 ) + B(κ2 )
J2r
= D2 C(κ2 ) + D(κ2 ),
(28)
(29)
where
κ2
{φ l cosh(κ2 T ) − φ2r }
(30a)
sinh(κ2 T ) 2
κ2
r
J1l
r
l
B(κ2 ) = 2
{φ − φ1 cosh(κ2 T )} +
D1
κ2 − κ12 sinh(κ2 T ) 1
(30b)
κ2
{φ l − φ2r cosh(κ2 T )}
C(κ2 ) =
(30c)
sinh(κ2 T ) 2
κ2
r
J1r
r
l
.
{φ
cosh(κ
T
)
−
φ
}
+
D(κ2 ) = 2
2
1
D1
κ2 − κ12 sinh(κ2 T ) 1
(30d)
A(κ2 ) =
Here, these equations are solved for D2 and κ2 . Solving
Eqs. (28) and (29) simultaneously, we obtain
J2l C − J2r A + AD − BC = 0.
(31)
In order to solve this equation for κ2 , the following function
of κ2 is defined:
F(κ2 ) = J2l C − J2r A + AD − BC.
(32)
κ2 can be determined as the root of the equation
F(κ2 ) = 0.
This equation can be solved using the bi-section method,8)
for instance. By substituting κ2 into Eq. (28), the diffusion
coefficient is given by
J2l − B
.
(33)
A
By using this D2 , the absorption cross-section of the thermal
group can be obtained from the equation
D2 =
a2 = D2 κ22 .
(34)
However, when the thickness of an iron reflector becomes
approximately 3 cm or more, the root of F(κ2 ) cannot be determined. This is because thermal neutrons in the iron reflector become to behave differently near the reflector boundaries
when its thickness is thicker than twice the diffusion length
of the thermal neutron.9) In order to include such a case, a
solution should be determined so as to minimize the error of
neutron current of transport and diffusion on the left and right
JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY
719
Transport Equivalent Diffusion Constants for Reflector Region in PWRs
surfaces. For this purpose, the following evaluation function,
W , has been introduced,
2 r
l
J2 − D2 C − D 2
J2 − D2 A − B
+
,
(35)
W =
J2r
J2l
D2 and κ2 are determined by minimizing the function. At
first, from the condition that W takes stationary value against
D2 ,
∂W
=0
(36)
∂ D2
and the diffusion coefficient can be expressed as
D2 =
(J2r )2 (J2l − B)A + (J2l )2 (J2r − D)C
.
(J2r )2 A2 + (J2l )2 C 2
(37)
On substituting this D2 into the function W , we have
1
W = r 2 2
(J l C − J2r A + AD − BC)2 .
(J2 ) A + (J2l )2 C 2 2
(38)
The value of κ2 that minimizes this value is a solution, and the
diffusion coefficient of the thermal group is given by Eq. (37)
using this κ2 .
The condition that W takes the stationary value is
∂W
= 0.
(39)
∂κ2
From this condition we obtain
J2l C − J2r A + AD − BC = 0
or
(40)
dC
dA
dA
[(J2r )2 A2 + (J2l )2 C 2 ] J2l
− J2r
+D
dκ2
dκ2
dκ2
dD
dC
dB
+A
−B
−C
dκ2
dκ2
dκ2
− (J2l C − J2r A + D A − BC)
dA
dC
= 0,
× (J2r )2 A
+ (J2l )2 C
dκ2
dκ2
(41)
(42a)
dC
1
{1 − κ2 T coth(κ2 T )}
=
dκ2
sinh(κ2 T )
VOL. 39, NO. 7, JULY 2002
2. Boundary Conditions
The total neutron current can be written in the diffusion
approximation as
J = −D∇φ,
(43)
where D is the diffusion coefficient.10) There are also following relationships among the neutron flux φ, the partial neutron
currents j + , j − and the total neutron current J ,
φ = 2( j + + j − ),
(44)
J = j + − j −.
(45)
Jg = jgtr+ − jgtr− ,
dB
2r κ2
=− 2
dκ2
(κ2 − κ12 )2
κ2
Jl
{φ1r − φ1l cosh(κ2 T )} + 1
×
sinh(κ2 T )
D1
r
1
+ 2
{φ1r − φ1l cosh(κ2 T )}
2 sinh(κ T )
κ2 − κ1
2
(42b)
× {1 − κ2 T coth(κ2 T )} − κ2 T φ1l ,
× {φ2l − φ2r cosh(κ2 T )} − κ2 T φ2r ,
If Eq. (40) holds, the solution is exact one. In the case that
Eq. (40) does not hold, the κ2 -value must be determined from
Eqs. (41) to (42d) and it gives the minimum of W . By the
above-mentioned method, the exact solution can be obtained
in the case the iron reflector is thin (T <3 cm), and the optimum solution can be obtained even in the case the iron reflector is thick (T ≥3 cm). Though this optimum solution produces the error in the thermal neutron current on the surfaces
of the iron reflector, the amount is only 4% and the influence
to the core characteristics is negligible.
From these relationships, and using the partial neutron current obtained from the transport calculation, we have
where
dA
1
{1 − κ2 T coth(κ2 T )}
=
dκ2
sinh(κ2 T )
× {φ2l cosh(κ2 T ) − φ2r } + κ2 T φ2l ,
2r κ2
dD
=− 2
dκ2
(κ2 − κ12 )2
κ2
Jr
{φ1r cosh(κ2 T ) − φ1l } + 1
×
sinh(κ2 T )
D1
1
r
{φ1r cosh(κ2 T ) − φ1l }
+ 2
2 sinh(κ T )
κ2 − κ1
2
(42d)
× {1 − κ2 T coth(κ2 T )} + κ2 T φ1r .
(42c)
φg = 2( jgtr+ + jgtr− ),
(46)
where the superscript tr is the abbreviation for transport.
If the diffusion equation is solved so that the above net current Jg and boundary flux φg can be met, the partial neutron
currents jgtr+ , jgtr− from the transport calculation can be reproduced by the diffusion equation, in other words, the albedo
obtained by the transport calculation is reproduced. From the
above, in order to determine the transport equivalent diffusion
constants, the boundary conditions to be imposed are as follows:
(1) Total neutron current obtained from the transport calculation:
Jgr = ( jgtr,r+ − jgtr,r− ),
Jgl = ( jgtr,l+ − jgtr,l− ),
(47)
(2) Neutron flux calculated based on the diffusion approximation by using the partial neutron currents from the transport
calculation:
φgr = 2( jgtr,r+ + jgtr,r− ),
φgl = 2( jgtr,l+ + jgtr,l− ). (48)
The difference of fluxes between the transport calculation
and the above-mentioned diffusion approximation is the socalled transport effect, and the relation between them can be
720
Y. TAHARA and H. SEKIMOTO
expressed by
f gk =
φgtr,k
φgk
Table 1 Specifications for TCA fuel rod
=
φgtr,k
,
2( jgtr,k+ + jgtr,k− )
(49)
where k is l or r. This factor may be utilized as a factor to
correct the fuel rod power from the diffusion calculation to an
exact one from the transport calculation for the fuel rods near
the reflector boundary.
III. Verification Calculation
The procedure for generating transport equivalent diffusion
constants in Chap. II was written in a program, which was
named “DIFF”. The adequacy of the diffusion constants from
the DIFF code was tested against some critical experiments
and an actual plant.
1. Analysis of Criticality Experiment
In the analysis of critical experiments, the 70-group library11) based on the ENDF/B-VI file12) was used for twodimensional transport calculations with PHOENIX-P1) and
the continuous energy library DLC-18913) was used for
Monte Carlo calculations with MCNP.14)
(1) Reactivity Effect of Iron Reflectors
To verify the validity of the transport equivalent diffusion
constants, the reactivity effect of iron reflectors9) was analyzed with the two-dimensional finite-difference diffusion
code HIDRA.15) The experimental configuration is shown in
Fig. 2 and the specifications of the fuel rod are described
in Table 1. The core was constructed at the center of the
stainless steel tank of Tank-type Critical Assembly (TCA) of
JAERI. The diameter and the height of the tank are 183 cm
and 214 cm, respectively.
The transport equivalent diffusion constants of the iron reflector and the water reflector were determined by carrying
out MCNP calculation with core geometry shown in Fig. 3.
HIDRA calculations were carried out using the obtained reflector constants. The mesh width of 1/8 cell pitch was
adopted because of the wide pitch of 2.293 cm. Axial neutron
leakage was taken into account by the axial buckling calculated using the measured critical water level of 91.45 cm and
Fig. 2 Core and iron reflector configuration
Items
Fuel material
U enrichment
Pellet density
Pellet outer diameter
Cladding outer diameter
Cladding thickness
Cladding material
235
Dimensions or
material
UO2
2.6 wt%
10.4 g/cm3
12.5 mm
14.17 mm
0.76 mm
Al
the axial extrapolation distance of 11.1 cm. That critical water level corresponds to the core with the water reflector. A
thickness of 30 cm was chosen for the reflector region of the
iron plates and water. The reactivity effect was calculated by
keff (T ) − keff (T = 0)
ρ(%) =
× 100,
keff (T )keff (T = 0)
where T is the thickness of the iron reflectors. The calculated
reactivity effect is shown in Fig. 4 with MCNP results obtained with three-dimensional geometry and the experimental
data. The HIDRA result agrees well with the measured reactivity effect of the iron reflectors. This shows the validity of
the transport equivalent diffusion constants.
(2) A Water Reflected Core
The multiplication factor and power distribution of the
21×21 lattice core of TCA16) were analyzed by the HIDRA
code. This core consists of fuel rods of 2.6 wt% with the rod
pitch of 1.956 cm, the critical water level is 46.01 cm at 20◦ C
and the axial extrapolation distance is 12.2 cm.17–19) The plan
view of the core is shown in Fig. 5. The fuel rod specifications
are presented in Table 1.
At first, this core was analyzed with the two-dimensional
model by PHOENIX-P, and the fuel region was homogenized
and its energy group was collapsed into two groups to obtain
the fuel constants. Next, the infinite multiplication factor and
the power distribution were calculated with HIDRA by using the fuel constants and the transport equivalent diffusion
constants of the water reflector which had been determined
as described in Chap. II. The mesh width was made 1/4 cell
pitch and the axial neutron leakage was taken into account by
the axial buckling in the HIDRA calculation. A thickness of
30 cm was chosen for the water reflector in this analysis.
(a) Comparison of Multiplication Factors
The infinite multiplication factors from PHOENIX-P and
HIDRA are 1.09391 and 1.09354, respectively. The HIDRA
calculation with the transport equivalent diffusion constants
excellently reproduces the infinite multiplication factor of the
PHOENIX-P transport calculation. This means that the transport equivalent diffusion constants for the water reflector are
appropriate.
(b) Comparison of Power Distributions
Figure 6 shows the comparison of fuel rod power among
HIDRA, PHOENIX-P and the measurement. The root-meansquare error of calculated rod power with PHOENIX-P and
HIDRA is 1.8% and 1.6%, respectively. HIDRA gives almost
the same calculation accuracy as PHOENIX-P. Hence, it can
JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY
721
Transport Equivalent Diffusion Constants for Reflector Region in PWRs
Reflector region
Water
Water
Reflective boundary
Iron
Vacuum boundary
Fuel rod
Fig. 3 One-dimensional core model for calculating transport equivalent diffusion constants of the iron reflector and the
water reflector
0.6
0.4
300.00
Water reflector
x
Reactivity effect ( )
0.2
0.0
19.56
2.6 wt
UO rod
410.76
Experimental data(JAERIM83-100)
Experimental data(MHI)
Fuel region (21 21)
Cal. (HIDRA)
Cal. (MCNP error=3 )
Unit : mm
205.38
0
20
40
60
80 100 120 140 160 180
Fig. 5 TCA 21×21 lattice core
Reflector thickness (mm)
Fig. 4 Reactivity effect of iron reflectors
Experimental data (MHI) are cited from Ref. 9).
be said that the transport equivalent diffusion constants are
also appropriate from the viewpoint of fuel rod power prediction.
(3) DIMPLE Core
Two critical experiments were analyzed, which were carried out using DIMPLE reactor of AEA Winfrith Technology
Center in Britain.20) The S06A core with a water reflector
and the S06B core with 25 mm thick stainless steel, which
are shown in Figs. 7 and 8, were constructed in a large aluminum tank (2.6 m diameter and 4 m high) of DIMPLE. The
experimental systems consist of 3,072 fuel rods composed of
12 arrays of 16×16, and the cell pitch is 1.251 cm.
The fuel specifications are shown in Table 2. It was
assumed that theoretical density is 0.93, the ratio (t/D)
of cladding tube thickness (t) to cladding tube outer diameter (D) is 0.03, and the dish ratio is zero. It was
also assumed that the composition of stainless steel is
Fe:Cr:Ni:Mn=72.0:18.0:8.0:2.0 in weight percent (wt%).
The measured criticality water level and the axial buckling are
476 mm and 24.7 m−2 for S06A, and 525 mm and 21.3 m−2
for S06B. The measurement error of ±0.0012 in effective
multiplication factor is due to the error of the dimensions, the
composition of materials and measurement error of the axial
VOL. 39, NO. 7, JULY 2002
Table 2 Specifications for DIMPLE fuel rod
Items
Dimensions or
material
Fuel material
U enrichment
Pellet outer diameter
Cladding outer diameter
Cladding material
UO2
3.0 wt%
10.13 mm
10.94 mm
Stainless steel
235
buckling.
PHOENIX-P was run using the geometry shown in Fig. 7.
The fuel constants were calculated by collapsing into two energy groups and homogenizing into two zones—outer most
fuel rods and the others.
The transport equivalent diffusion constants of the stainless
steel plate and the water reflector were determined by means
of the method of Chap. II using the MCNP results obtained
with 1-D geometry. By using these fuel constants and the
transport equivalent diffusion constants of the stainless steel
and water reflectors, two-dimensional diffusion calculations
were carried out for the two cores of S06A and S06B. The
HIDRA code was used for the diffusion calculations, where
the mesh width was set to a quarter of the cell pitch. A thickness of 30 cm was chosen for the reflector region—water or
722
Y. TAHARA and H. SEKIMOTO
1.292
1.317
1.315
1.315
1.309
1.307
1.304
1.285
1.282
1.244
1.244
1.241
1.191
1.188
1.185
1.108
1.117
1.114
1.038
1.031
1.030
0.942
0.935
0.937
0.859
0.836
0.846
0.805
0.773
0.791
0.929
0.927
0.892
1.298
1.300
1.299
1.261
1.277
1.274
Meas. (JAERI -M 9844)
Cal.(PHOENIX-P), =1.8
Cal.(HIDRA with Transport Equivalent Diffusion Constants),
1.226
1.253
1.250
1.154
1.175
1.172
1.078
1.071
1.068
0.926
0.947
0.944
0.796
0.807
0.808
0.677
0.663
0.669
0.530
0.530
0.546
1.6
0.469
0.451
0.477
0.559
0.565
0.553
Fig. 6 Comparison of rod power
The transport equivalent diffusion constants were used for the water reflector. The σ -value is the standard deviation
of pin-power error. Error is defined as (Meas.-Cal.)/Cal.×100.
Fuel lattice region (16
16 rods)
Water
reflector
Fuel lattice region (16
16 rods)
Water
reflector
Stainless steel 25mm
Rod pitch 12.51mm
Fig. 7 DIMPLE-S06A core
stainless steel plus water. The comparison of kinf -values between HIDRA and PHOENIX-P is shown in Table 3. The
difference of the kinf -values between the two codes for the
S06A and S06B cores is about 0.26%ρ and it can be said
that the transport equivalent diffusion constants give almost
the same calculation accuracy for both of the cores with the
water reflector and the baffle/reflector.
2. Analysis of a PWR Core
A new method for calculating the baffle/reflector (B/R)
region of a PWR core is presented here by integrating the
two-dimensional homogenization method21) and the transport
Rod pitch 12.51mm
Fig. 8 DIMPLE-S06B core
equivalent diffusion constants developed in Chap. II. Its validity has been shown by verification calculations using the actual plant data. A three-loop plant of 2,652 MWt was selected
for this purpose. This core has a large corner part of the B/R
region and is suitable for the verification calculations.
(1) Core Specifications
The plan view of the reactor including the core, baffle
plates and the core barrel is shown in Fig. 9. The fuel assembly is a 17×17 PWR standard fuel assembly, and its effective length is 3.66 m. The initial core of the first cycle (CY1)
consists of 53 assemblies with enrichment of 2.0 wt%, and
52 assemblies with 3.5 wt% and 52 assemblies with 4.1 wt%.
JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY
723
Transport Equivalent Diffusion Constants for Reflector Region in PWRs
Table 3 Multiplication factors for S06A and S06B
Water reflector
(S06A core)
Baffle/Reflector
(S06B core)
1.08511
1.08828
0.268
1.07316
1.07619
0.262
kinf (PHOENIX-P)
kinf (HIDRA)
ρ (%)a)
a)
Reactivity difference between PHOENIX-P and HIDRA
Fig. 9 Plan view of a 3-loop core
The fuel-loading pattern of the first cycle is shown in Fig. 10.
In order to suppress excess reactivity and keep the moderator temperature coefficient negative in the initial core, burnable poison rods composed of pyrex glass (BP) were inserted.
The in-core arrangement of BP rod clusters is also shown in
Fig. 10. Two clusters with 23 BP rods and one primary neu-
tron source rod are loaded in the initial core, but the clusters
were treated as those with 24BP rods to maintain quarter symmetry in the core calculation.
The fuel-loading pattern of the third cycle (CY3) is shown
in Fig. 11. The UO2 –Gd2 O3 fuel rods were used as well as
BP rods. Enrichment of UO2 –Gd2 O3 fuel rods is 2.6 wt%
235
U and the concentration of Gd2 O3 is 6.0 wt%. The sixteen
UO2 –Gd2 O3 fuel rods are included in the Gd assembly and
the in-core arrangement of the Gd fuel assembly is also shown
in Fig. 11.
(2) Neutronic Analysis Method
For the core calculations, the three-dimensional nodal
code, ANC22–24) was used, which is based on diffusion theory. The fuel assembly is divided into 4 nodes and the core is
surrounded with a layer of B/R nodes. The flow of the core
analysis is shown in Fig. 12.
For the fuel nodes, the homogenized constants and neutron
flux discontinuity factors were calculated by PHOENIX-P using a single assembly model.
The B/R constants were obtained using the twodimensional model. At first, the transport equivalent diffusion
constants of water gap, baffle plate and water reflector were
calculated based on the MCNP results of the 1-D core shown
in Fig. 13. Two assemblies with 4.1 wt%235 U fuel rods were
employed as the fuel region and the width of the water region
between the baffle plate and the core barrel was determined
so as to preserve the actual amount of water between them.
These constants were applied to the two-dimensional
HIDRA calculation, where the core, water gap, baffle, barrel and water reflector were simulated as shown in Fig. 14,
and then the neutron flux and current were obtained. The
homogenized constants of B/R nodes were calculated with
CUDISCON code21) using information about the flux and current. Finally the power distribution and the critical boron concentration were calculated with ANC by using thus obtained
Fig. 10 Core loading pattern of the first cycle
Fig. 11 Core loading pattern of the third cycle
VOL. 39, NO. 7, JULY 2002
724
Y. TAHARA and H. SEKIMOTO
PHOENIX-P
Fuel constants
MCNP4B
1D-transport calculation
DIFF
Transport equivalent Diffusion Constants
HIDRA
2D-diffusion core calculation
CUDISCON
2D-B/R nodal constants
Fig. 14 Two-dimensional core model for HIDRA calculation
ANC
3D-nodal calculation
Core Characteristics
Fig. 12 Schematic flow of the core analysis
Fuel rod
Water gap
Core barrel
Neutron pad
x
Water
Baffle plate
Water reflector
Fig. 13 One-dimensional core model for calculating diffusion constants of gap, baffle, reflector, barrel and pad
homogenized constants of the fuel and B/R nodes. Assembly
burnup accumulated by a core management code system25)
was used in the calculation of the third cycle.
The nuclear data of uranium, plutonium, americium and
curium of the preliminary version of JENDL-3.326) were used
in the calculation of fuel constants, and ENDF/B-VI was used
for B/R constants according to the study of the iron reflector.9)
The 70 energy-group cross-section library for PHOENIX-P
was newly generated based on the preliminary version of
JENDL-3.3 using NJOY code,27) where a reduction of 238 U
resonance capture by 3.4%28) was employed as done in the
ENDF/B-VI-based PHOENIX-P library.11) The continuous
energy cross-section library of MCNP was also generated
with AUTONJ.29)
(3) Comparison with Measurement
In-core three-dimensional reaction rate distributions of
235
U are measured using movable detectors (M/D) during the
operation of PWRs. The three-dimensional core power distribution is obtained by INFANT code25) using the M/D data.
Hence the validity of the calculation method can be confirmed
by comparing the measured and calculated power distribu-
tions. The 235 U fission cross-sections of M/D, which are
used in the INFANT code to calculate 235 U fission reaction
rate, were obtained by collapsing the infinite dilution crosssections into two energy groups using the spectrum of the instrumentation thimble cell from the PHOENIX-P assembly
calculation. To take account of the dependence of the M/D
fission cross-sections on the fuel enrichment and the number
of BP rods, the M/D fission cross-sections were calculated for
all the combination of the fuel enrichment and the number of
BP rods.
In addition, at the hot full power conditions, the M/D fission cross-sections depending on fuel burnup were used in
consideration of the change in neutron spectrum with the depletion of fuel and BP.
(a) Comparison of Power Distributions
The reaction rate data, which were measured by M/D at all
control rods out conditions (ARO) in the zero power reactor
physics test, were processed with the INFANT code and the
resultant assembly power distributions were compared with
the calculated ones.
Figure 15 shows the comparison of the power distribution
at the hot zero power conditions (HZP). Although the error of
about 2% is observed around the center of the core, the calculated assembly power agrees well with the measured one on
the core periphery and almost all the area inside the core. The
error of the whole core is 1.3%, and it can be said that the twodimensional B/R constants based on the transport equivalent
diffusion constants give excellent agreement with measurement.
The comparison of core power distribution at the hot
full power conditions (HFP) was also made at the middle of cycle—the cycle burnup of 8,900 MWd/t. The twodimensional B/R constants were interpolated with respect to
boron concentration and then used in the depletion calculation. The burnup dependent M/D fission cross-sections were
used in the INCORE calculation. Figure 16 shows the comparison of the power distributions. The measured and calculated assembly power agrees well each other.
Figure 17 shows the comparison of the power distributions
of the third cycle at HZP. Although the error of over 4%
is observed, the calculated assembly power agrees well with
the measured one on the core periphery and inside the core.
JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY
725
Transport Equivalent Diffusion Constants for Reflector Region in PWRs
1.049
0.7
1.287
4.0
1.197
1.0
0.893
-0.1
1.283
-0.2
1.008
-1.1
1.054
-0.6
1.139
-1.6
1.036
-0.5
0.974
-1.2
1.146
-1.4
0.987
0.3
0.885
0.5
1.068
-2.5
1.022
0.3
0.896
0.3
0.851
0.9
0.982 1.155
0.907
-0.4 -0.6
1.5
0.927
1.000
-0.1
0.5
0.891
1.018 1.271
0.868
1.2
-0.2 -1.1
2.9
0.840
0.877
2.3
1.2
0.824
0.820
1.046
1.086
2.6
2.1
-1.6
-0.8
0.837
1.151 0.976
1.9
-1.0 -0.2
0.817 0.961
0.816
1.7 1.5
1.5
0.937
1.006
1.208
-1.0
1.1
1.9
0.970
1.056 1.023
-1.5
-0.4 0.4
1.042
0.974 0.906
-2.0
1.2 1.4
1.144
1.089
-1.6
-0.7
1.082
-1.1
0.814
Mea.
Error( ) Error=(M-C)*100/C =1.3
1.2
1.190
1.7
0.749
-0.5
1.005
-0.1
1.231
3.4
1.151
-1.4
0.913
-1.2
0.960
-2.2
1.027
-2.8
0.880
-2.4
0.831
2.2
1.032
2.2
1.074
-1.4
0.999
-4.3
0.734
-0.3
0.623
0.1
1.209 0.962
0.761
0.5 0.6
2.6
0.766
1.061
1.0
-1.0
1.051
1.046 0.940
0.644
2.3
-0.1 -2.3
5.1
1.200
1.000
3.0
1.2
1.229
1.222
0.813
1.294
4.0
2.8
-1.3
1.4
1.199
0.946 1.265
1.8
0.6 1.8
1.210 0.978
0.949
1.7 1.6
2.5
0.961
1.012
1.166
0.2
-4.1
1.9
0.832
0.861 1.005
-1.1
-4.8 -3.5
0.811
0.911 0.720
-2.9
-3.4 -1.7
0.915
1.061
-2.7
-3.3
1.246
-3.1
0.937
Mea.
0.2
Error( ) Error=(M-C)*100/C =2.4
Fig. 15 Comparison of assembly power (CY1, 0 MWd/t, HZP,
ARO)
Fig. 17 Comparison of assembly power (CY3, 0 MWd/t, HZP,
ARO)
0.788
-1.3
0.891
0.4
1.016
0.5
0.951
-0.3
0.743
-1.0
1.022
-1.4
0.789
-1.2
0.971
-0.1
1.143
-0.4
1.166
-0.5
1.013
-0.5
1.184
0.0
1.199
0.6
1.036
0.8
1.068
-0.8
1.025
0.7
0.747
-0.4
0.697
0.0
0.975 1.148
0.744
0.3 0.1
-0.8
1.023
1.025
0.7
1.3
1.048
1.026 1.169
0.698
2.0
0.8 -0.2
0.3
1.064
1.037
2.6
1.9
1.065
1.064
1.159
0.984
2.5
2.4
-1.1
-0.7
1.057
1.139 0.865
1.9
-0.7 -0.6
1.055 1.212
0.653
1.6 1.4
0.1
1.186
1.023
0.950
-0.8
1.0
-0.3
1.180
1.182 1.023
-1.0
-0.2 0.5
1.155
0.861 0.747
-1.4
0.0 -0.4
1.121
1.059
-2.4
-1.6
0.971
-2.0
0.650
Mea.
-0.4
Error( ) Error=(M-C)*100/C =1.2
0.807
0.1
1.129
1.4
0.886
0.3
1.221
-1.0
0.912
-1.1
1.034
-0.4
1.251
-1.0
0.989
-0.9
0.923
1.8
1.224
2.0
1.048
-1.2
1.252
-0.7
0.812
1.0
0.643
0.4
1.229 0.920
0.821
-0.9 -1.0
2.1
0.853
1.113
0.2
-0.1
1.232
1.257 1.027
0.646
1.6
-0.3 -0.6
1.5
1.156
1.040
2.0
0.1
1.114
1.108
0.900
1.040
1.5
0.6
-1.6
-1.1
1.121
0.922 0.999
0.3
-1.2 -0.4
1.103 0.972
0.732
0.2 0.0
0.6
0.980
1.108
1.020
-0.1
-0.2
1.0
0.929
0.997 1.256
-0.8
-0.8 -0.4
0.909
1.006 0.811
-1.6
0.4 1.0
0.915
1.060
-1.7
-0.4
1.045
-0.9
0.739
Mea.
1.5
Error( ) Error=(M-C)*100/C =1.0
Fig. 16 Comparison of assembly power (CY1, 8,900 MWd/t, HFP,
ARO)
Fig. 18 Comparison of assembly power (CY3, 7,550 MWd/t, HFP,
ARO)
The error of the whole core is 2.4%. The comparison of core
power distributions at HFP was also made at the cycle burnup of 7,550 MWd/t. Figure 18 shows the comparison of the
power distributions. The error of the whole core is 1.0%. The
measured and calculated values agree well with each other.
From the above comparisons, it can be said that the validity
of the two-dimensional B/R constants has been verified from
the viewpoint of power distribution prediction.
(b) Comparison of Critical Boron Concentrations
The validity of the calculation method and the nuclear data
was verified by comparing the measured and calculated critical boron concentrations at HZP and HFP. Table 4 shows
the comparison between the measured and calculated boron
concentrations of the first and third cycles at HZP for all
the control rods out and control bank D fully inserted cases.
Figures 19 and 20 show the comparisons of critical boron
concentration between measurement and calculation at HFP
for the first cycle and for the third cycle, respectively. The
prediction error of critical boron concentration is smaller than
30 ppm for both cases at HZP and HFP. This shows that transport equivalent diffusion constants enables the evaluation of
neutron leakage from the core correctly and the prediction of
critical boron concentration with high accuracy.
VOL. 39, NO. 7, JULY 2002
726
Y. TAHARA and H. SEKIMOTO
Table 4 Comparison of critical boron concentrations at HZP
Cycle 1
Control
rod
AROa)
Db)
b)
c)
DG,i
= DG (B1 )
Meas.
(ppm)
Cal.
(ppm)
Meas.
(ppm)
Cal.
(ppm)
1,729
1,604
1,714(15)c)
1,591(13)
1,726
1,564
1,728(−2)
1,567(−3)
All control rods out.
Control bank D inserted.
The value in the parenthesis represents error: (Meas.-Cal.).
1800
50
Meas.
Boron concentration (ppm)
1600
Cal.
1400
40
30
Error (Meas.-Cal.)
20
1200
10
1000
0
800
-10
600
-20
400
-30
200
-40
0
0
Prediction error (ppm)
a)
Cycle 2
-50
2000 4000 6000 8000 10000 12000 14000 16000 18000
Cycle burnup (MWd/t)
Fig. 19 Critical boron concentration vs. burnup for the first cycle
(HFP, ARO)
50
Meas.
Boron concentration (ppm)
1600
Cal.
1400
40
20
1200
10
1000
0
800
-10
600
-20
1.300
0.6
1.292 1.284 Cal.(HIDRA with Eq.(50))
=2.0
1.8
1.1 Error ( ),
1.268 1.260 1.237
0.9
2.9
0.1
-30
200
-40
1.229
1.2
-50
2000 4000 6000 8000 10000 12000 14000 16000 18000
1.175
1.4
0
(50)
In order to show the superiority of the transport equivalent
diffusion constants, the experimental core and the 3-loop actual core (See Sections III-1 and III-2) were reanalyzed using
the diffusion coefficients from Eq. (50).
First, the multiplication factor and fuel rod power of the
21×21 lattice core were calculated. The results are shown
in Fig. 21 and Table 5. The calculated multiplication factor is overestimated by 0.57%ρ compared with PHOENIXP, whereas the transport equivalent diffusion constants give
a good agreement with the PHOENIX-P result. Figure 21
shows that the diffusion coefficients by Eq. (50) produce the
power error that exceeds 5% for the fuel rods on the diagonal
axis. On the other hand, when transport equivalent diffusion
constants are used, the error is about 3% for the fuel rods on
the diagonal axis (See Fig. 6). The outer core region on the diagonal axis is the area where the neutron flux is directly influenced by the water reflector facing both sides of the core. The
fact that the fuel rod power on this axis is evaluated correctly
means that the neutron leakage from the core is evaluated correctly and, therefore, it can be said that the core reactivity is
also evaluated accurately.
Next, in order to evaluate the influence to the PWR
core analysis, critical boron concentration and fuel assembly
power were recalculated at HZP for the first cycle. The core
calculation was done by ANC using the two-dimensional B/R
constants based on the diffusion constants from Eq. (50). The
results are shown in Table 6. The calculated critical boron
concentration is 1,725 ppm and is lower than the measured
value only by 4 ppm, but the power of the assemblies on the
400
0
tr,G (whole core)
.
tr,G,i
30
Error (Meas.-Cal.)
Prediction error (ppm)
1800
of region i for energy group G is calculated from the B1 diffusion coefficient DG (B1 ) and transport cross-sections:
Cycle burnup (MWd/t)
Fig. 20 Critical boron concentration vs. burnup for the third cycle
(HFP, ARO)
IV. Discussions
In the past, diffusion constants have been defined as a set of
cross-sections and diffusion coefficients obtained in a various
way from transport cross-sections. A simple and easy method
for calculating diffusion coefficients is to preserve neutron
leakage in a critical state from the homogeneous medium obtained by smearing a whole system. The diffusion coefficient
1.107
0.1
1.026
1.1
0.938
0.5
0.852
0.7
0.809
0.5
0.941
1.3
1.162
0.7
1.062
1.5
0.942
1.8
0.810
1.8
0.676
0.2
0.558
4.9
0.495
5.1
0.579
3.4
Fig. 21 Comparison of rod power
The diffusion coefficients from Eq. (50) were used for the water
reflector. Error is defined as (Meas.-Cal.)/Cal.×100.
JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY
727
Transport Equivalent Diffusion Constants for Reflector Region in PWRs
Table 5 Calculation accuracy for the 21×21 lattice core of TCA
Diffusion coefficients
from Eq. (50)
Transport equivalent
diffusion constants
PHOENIX-P
1.10077
2.0
1.09354
1.6
1.09391
1.8
kinf
Error of rod power (%)
Table 6 Calculation accuracy for the 3-loop core at HZP
Diffusion coefficients
from Eq. (50)
Transport equivalent
diffusion constants
1,725(4)a)
2.9
1,714(15)
1.3
Critical boron (ppm)
Error of assembly power (%)
a)
The value in the parenthesis represents error: (Meas.-Cal.).
core periphery is overestimated by about 2–3% and the assembly power around the center of the core is underestimated
by 6–7% as shown in Fig. 22. This in-out power tilt results in
a relatively large error of 2.9%. On the other hand, the twodimensional B/R constants based on the transport equivalent
diffusion constants produce a better assembly power error of
1.3%, although the calculated critical boron concentration is
lower than the measured one by 15 ppm. This excellent agreement of power distribution with the measurement demonstrates the neutron leakage from the core is correctly evaluated by using the transport equivalent diffusion constants.
The transport equivalent diffusion constants give lower boron
concentration than the diffusion coefficients from Eq. (50),
but this error is due to uncertainty of nuclear data. However,
the calculated boron concentration is influenced by the core
power distribution. Therefore, high calculation accuracy of
-3.4
-1.8
-1.6
0.0 -0.2
-0.6
-3.0
0.0
-4.5
-1.5
0.9
1.5
6.2
4.1
4.0
1.1
6.9
-1.3
-1.2
-0.8
-1.0
-2.2
6.0
3.0
6.0
3.3
-3.3
1.2 -0.5
5.8
-1.6 -2.3
5.5
-1.7
2.2
1.6
2.2
1.1
1.2
-0.7
-2.4
1.3
0.4 -0.5
-1.9
-2.7
-1.8
-2.0
-2.1
-3.4
Error( )Error=(M-C)*100/C
=2.9
Fig. 22 Error of assembly power (CY1, 0 MWd/t, HZP, ARO)
The 2D B/R constants based on Eq. (50) were used in the core
calculation.
VOL. 39, NO. 7, JULY 2002
core power distribution is needed for boron prediction as well.
The transport equivalent diffusion constants enable the accurate core calculation by producing correct neutron currents.
V. Conclusions
The characteristics of the core, especially, the radial power
distribution largely depend on the treatment of the radial reflector region. The diffusion calculation code is generally
used in the core design of PWRs. Therefore, the use of diffusion constants that preserve the current response of transport
calculation is desired for the reflector.
In this paper, the method for calculating the transport
equivalent two-group diffusion constants in this sense was developed and validated against the Monte Carlo transport calculations and critical experiments. The effectiveness was also
confirmed by applying them to the core analysis of the actual
plant.
The transport equivalent two-group diffusion constants
were defined as the diffusion coefficients and cross-sections
that minimize the difference of the transport and diffusion
neutron currents on the reflector boundaries.
In the analyses of the critical experiments, the three critical
experiments were analyzed: reactivity effect of iron reflectors; the 21×21 lattice core with the water reflector; cores
with the water reflector and with the stainless steel baffle
plate. As the results, the transport equivalent two-group diffusion constants of the reflectors reproduced the experimental
results very well, and the validity of the transport equivalent
two-group diffusion constants was verified.
By applying the transport equivalent diffusion constants to
the two-dimensional diffusion calculation, a new method for
calculating the two-dimensional B/R constants was developed
for PWR core calculations with a nodal code. The analysis of
the actual 3-loop core using the B/R constants showed that the
critical boron concentration and the power distribution were
well agreed with the measurements. These verification calculations show the validity of the two-dimensional B/R constants.
Moreover, the superiority of the transport equivalent diffusion constants was demonstrated by comparing with the re-
728
sults obtained by the diffusion coefficients based on the B1
method.
From the above results, it is concluded that the method for
calculating transport equivalent diffusion constants is valid
and that the two-dimensional B/R constants based on the diffusion constants enable the prediction of the power distribution and critical boron concentration of PWR cores with high
accuracy.
Acknowledgments
One of the authors (Y. T.) would like to thank Ms. Yoko
Hitosugi, Mr. Hideyuki Noda, and Mr. Kazuhiro Tani of Engineering Development Co., Ltd. for their help in the analysis.
He is also grateful to Nuclear Data Center of Japan Atomic
Energy Research Institute for the use of the latest version of
JENDL.
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