Progress in Nuclear Energy 67 (2013) 124e131
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Progress in Nuclear Energy
journal homepage: www.elsevier.com/locate/pnucene
Bondarenko method for obtaining group cross sections in a
multi-region collision probability model
C.L. Dembia 1, G.D. Recktenwald 1, M.R. Deinert*
Department of Mechanical Engineering, The University of Texas at Austin, 1 University Station C2200, Austin, TX 78715, United States
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 27 July 2012
Received in revised form
31 January 2013
Accepted 1 February 2013
Correct multigroup cross sections are essential to modeling the physics of nuclear reactors. In particular,
the presence of resonances leads to the well-known self-shielding effect that complicates any procedure
for obtaining group cross sections. The Bondarenko method for producing self-shielded group cross
sections is widely used. For heterogeneous systems, the approach requires the use of an escape cross
section that captures the probability that a neutron will leave a cell without interaction, and simple
approximations are typically used for this purpose. Here we provide a concise derivation for how to
determine group cross sections using the Bondarenko method with an extension to multi-region collision probability models. The accuracy of the method is demonstrated by comparing group cross sections
derived this way with those produced by Monte Carlo simulations of thermal and fast spectrum reactors.
Ó 2013 Elsevier Ltd. All rights reserved.
Keywords:
Collision probability method
Equivalence theory
Escape cross section
Heterogeneous
Resonance self-shielding
1. Introduction
Hundreds of thousands of energy grid points are required to
fully resolve the energy dependence of neutron interaction cross
sections. Even at the petascale, a brute force resolution of this
dependence in Monte Carlo and discrete ordinate techniques is far
from practical. As a result, reactor physics codes typically use a
‘multigroup’ formulation where the neutron spectrum is represented by a few tens to hundreds of energy groups. Multigroup
cross sections are then required that preserve the correct in-group
reaction rates. However, producing these cross sections is not
straightforward because the presence of in-group resonances affects the flux. Being able to produce the correct group-averaged
cross sections then requires knowing the energy dependent flux
within a group, which is often the quantity being sought.
Furthermore, the attempt to model a heterogeneous system
requires additional considerations. The traditional method for
treating heterogeneity involves applying an equivalence relation to
the background cross section of the Bondarenko method
(Gopalakrishnan and Ganesan, 1998; Joo et al., 2009; Kidman et al.,
1972; Schneider et al., 2006a; Stamm’ler and Abbate, 1983). The
desire to model next generation reactors with more complex
* Corresponding author.
E-mail address: [email protected] (M.R. Deinert).
1
The authors contributed equally to this work.
0149-1970/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.pnucene.2013.02.001
geometries has led to the use of subgroup methods (Chiba, 2003;
Cullen, 1974; Herbert, 1997; Huang et al., 2011). Self-shielding
methods primarily differ in the accuracy with which they attempt
to approximate the neutron flux within a group, and are well
described in the reviews by Hwang (1982) and Herbert (2007).
The subgroup method is used when modeling next-generation
reactors whose complex geometry precludes the use of the Bondarenko method. In this work we are concerned with modeling
simple pin-cell geometries, in which case the Bondarenko method
provides sufficient accuracy. The use of the Bondarenko method for
heterogeneous systems requires the use of an effective escape cross
section that describes the probability that a neutron may escape a
resonance by leaving a region. There are a number of ways by which
the escape cross section can be obtained. The methods vary in
complexity, but most use the Wigner rational approximation and
the mean chord length of a region in order to obtain an expression
for a collision probability (either the first-flight escape probability
or the fuel escape probability). Sometimes this approximation is
adjusted by a Dancoff factor, Bell factor or by replacing the Wigner
rational approximation with an N-term expression such as the one
by Carlvik (Stamm’ler and Abbate, 1983; MacFarlane and Muir,
1994; Herbert and Marleau, 1991; Yamamoto, 2008).
In the present contribution we provide a review and simplified
derivation for the escape cross section using a collision probability
model for the transport of neutrons from one reactor region to
another. To demonstrate the accuracy of the approach we use the
Bondarenko method to generate multigroup reaction and kernel
C.L. Dembia et al. / Progress in Nuclear Energy 67 (2013) 124e131
cross sections which we use with an in-house collision probability
spectral solver to obtain a neutron spectrum. We compare the
predictions of neutron spectrum and reaction rates for simulated
fast and thermal spectrum reactors to a published benchmark
(Rowlands et al., 1999) which is commonly used in the analysis of
self-shielding methods (Herbert, 2005) as well as to results produced using MCNPX 2.7.0.
described in the next section. In Sections 3 and 4 we introduce the
background cross section method that allows this integration to be
performed in a problem-independent manner.
2.1. Resonance self-shielding
For a homogeneous system containing M nuclides, the macroscopic total cross section of the mixture is given by:
2. Group cross sections
We define a group structure with G groups that span a range of
energies from EG to E0{eV}. The g-th energy group spans an energy
range from Eg to Eg1. Here we adopt the convention that the groups
are ordered in descending energy so that the highest energy is E0
and the lowest energy is EG, and in general Eg < Eg1. The width of
an energy group is DEg ¼ Eg1 Eg and is different for every
group. The microscopic and macroscopic group cross sections for
any interaction are respectively denoted as sg {barns} and Sg
{cm1}. It will be convenient to define a group flux fg {cm2 sec1}
given by the integral of f(E) {cm2 sec1 eV1} over the energies in
group g:
fg ¼
Z
g
dEfðEÞ
(1)
where the integral over g indicates an integral from Eg to Eg1.
The group cross sections sg must be defined in such a way that
they preserve the reaction rates R(E) ¼ Ns(E)f(E) {cm3 sec1 eV1}
of the interactions that occur in the system, because it is the reaction rates that ultimately dictate the behavior of the reactor. Here
N is the atom density {cm3} of the medium. The group cross
section sg {barns} is given by (Lamarsh, 1972):
Z
sg ¼
Zg
dEsðEÞfðEÞ
g
(2)
dEfðEÞ
where s(E) is the energy dependent cross section {barns}. This
averaging must be applied to all cross sections relevant to the
problem (i.e. absorption, capture, etc). Similarly, the group-togroup scattering cross section ss;g)g’ is defined as:
Z
Z
dE0 ss ðE)E0 ÞfðEÞ
dE
ss;g)g0 ¼
g
g0
fg
(3)
The group cross section ss;g)g’ has units of barns, and the
group-to-group scattering cross section ss (E)E0 ) has units of
barns/eV.
Equations (2) and (3) define the group cross sections, but they
unfortunately depend on the flux f(E) between Eg and Eg1. This
presents a problem, as f(E) is the quantity we seek. If the flux f(E) is
roughly constant through a group then the flux drops out of Eq. (2),
and the flux is not needed to obtain group cross sections. This
assumption is often valid, but cannot be used when the flux varies
rapidly within a group as is the case in groups where the cross
section exhibits resonances. To obtain group cross sections in these
cases, the flux f(E) must be approximated and Eq. (2) must be
solved by integration for the relevant interactions. However, it is
not desirable to work in a spectral code with the point-wise data
this integration requires. It is preferable to precompute this integration with a separate data processing code.
The issue with this procedure is that the flux is necessarily
problem-dependent, largely through the self-shielding effect
125
St ðEÞ ¼
M
X
N m sm
t ðEÞ
(4)
m
where Nm is the number density of the m-th isotope in the mixture
and St is the total macroscopic cross section {cm1}.
In this homogeneous medium, the neutron balance is given by
the following:
St ðEÞfðEÞ ¼ SðEÞ
(5)
where St (E)f(E) {cm3 sec1 eV1} is the interaction rate at energy
E and S(E) is the collision density and can be interpreted as the
corresponding slowing down density if the neutron field is at
equilibrium (Gopalakrishnan and Ganesan, 1998; Lamarsh, 1972).
The idea with the Bondarenko method is to let S(E) be a smooth
function that is functionally equivalent to the flux profile in the
absence of resonance effects and will depend on the type of reactor.
For a thermal spectrum S(E) would follow a MaxwelleBoltzmann
distribution with a fission spectrum peak (MacFarlane, 2000).
Within a group Eq. (5) can then be used to express the behavior of
the flux in terms of the macroscopic total cross section:
fðEÞ ¼
SðEÞ
St ðEÞ
(6)
Equation (6) shows that a resonance peak in St (E) causes f(E) to
dip correspondingly. Equation (6) can be used to approximate the
behavior of the in-group flux (Bell and Glasstone, 1970) and Eq. (2)
becomes:
Z
sm
g ¼
SðEÞ
S
t ðEÞ
g
Z
SðEÞ
dE
St ðEÞ
g
dEsm ðEÞ
(7)
Here, sm
g represents the microscopic group cross section for m-th
nuclide in the mixture. Equation (7) can then be used to generate the
appropriate group cross section for every individual reaction of interest. Note that it is the total cross section that always appears on the
right hand side of Eq. (7) and a resonance in any of the interactions
that contribute to the total cross section causes a depression in the
flux. If the group size is small enough, S(E) is effectively constant and
falls out of Eq. (7) (MacFarlane and Muir, 1994).
2.2. The background cross section
A key contribution of Bondarenko (Bondarenko, 1964) was to
separate the macroscopic total cross section into two terms: one
that depends only on the point-wise cross section of nuclide m and
a second term that encompasses all of the other isotopes in the
mixture. Equation (4) is then written as:
St ¼ N m sm
t þ
X
N n snt
(8)
nsm
The cross sections in Eq. (8) are continuous functions of energy,
but we have omitted the energy argument for brevity. We factor out
126
C.L. Dembia et al. / Progress in Nuclear Energy 67 (2013) 124e131
the number density of nuclide m, for which we desire the group
cross section, from both terms:
m
St ¼ Nm sm
t þ s0
(9)
where,
sm
0 ¼
1 X n n
N st
N m nsm
(10)
where sm
0 is commonly referred to as the ‘background’ cross section. Combining Eqs. (7), (9) and (10) we obtain:
Z
sm
g
g
dEsm
Z
¼
dE
g
S
m
sm
t þ s0
(11)
S
m
sm
t þ s0
m
where sm
t , s0 , and S are functions of energy. It is clear from Eq. (11)
that the group cross section for a specific isotope depends critically
on the background cross section.
2.3. Heterogeneous systems and the equivalence relation
Equations (10) and (11) apply when the medium through
which the neutrons are traveling is homogeneous. However,
nuclear reactors typically have distinct regions with different
material compositions. In cases such as these a neutron at energy
E could leak out of the region in which the flux is being
computed, or it could undergo an interaction which removes it
from the region in which flux is being computed. In either case,
the flux at E would be reduced and Eq. (10) would need to be
modified to capture the effect. An easy way to do this is by adding
an additional term to s0(E) in Eqs. (9)e(11) that takes escape
from a region into consideration. We start by modifying Eq. (5):
h
i
j
j
St ðEÞ þ Se ðEÞ fðEÞ ¼ S j ðEÞ
(12)
1
Sje ðEÞ
Here
{cm } is the macroscopic escape cross section in
the j-th reactor region. Note that S j(E), Sjt(E) now apply specifically to the j-th region. Equation (12) constitutes a so-called
‘equivalence relation’ because it allows us to treat the heterogeneous case identically to how we treated the homogeneous case
by simply adding an effective cross section to the total cross
section (Bell and Glasstone, 1970). Equation (12) can be rearranged to give:
fðEÞ ¼
S j ðEÞ
j
St ðEÞ
(13)
j
þ Se ðEÞ
We can expand the denominator, as we did in Eq. (8), to obtain:
m
Sjt þ Sje ¼ N m sm
t þ s0
(14)
where the background cross section for the m-th isotope in a region, j, is now given by:
sm
0 ¼
1
Nm
X
j
2.4. The escape cross section in collision probability models
Collision probability theory models the movement of neutrons
between homogeneous reactor regions using transmission and
escape probabilities (Lamarsh, 1972; Schneider et al., 2006b). We
define the collision probability Pi)j ðEÞ to be the probability that a
neutron with energy E, born in region j (whether by scattering or
fission), undergoes its next collision in region i. Accordingly, Pi)i is
the probability that a neutron born in region i collides next in region i.
The escape cross section can be related directly to the probability that a neutron will leave a particular region of a reactor before
colliding again:
X
Pi)j ¼
isj
Se
St þ Se
j
(16)
Equation (16) can then be rearranged to provide the escape cross
section in terms of the collision probabilities:
0
P i)j 1
P
1
B
C
isj
j
jB
Se ¼ St @ P i)j C
A
(17)
P
isj
Table 1
Dimensions and composition of the LWR from case 1 of the
Rowlands benchmark. The fuel is enriched uranium dioxide, and
is surrounded by a water moderator. A zirconium fuel rod is
modeled by smearing the appropriate amount of zirconium
across the water.
Pin
Diameter
0.8000 cm
Temperature
294.0 K
Nuclide
Density (#/b/cm2)
U-235
U-238
O-16
0.0007080
0.0226040
0.0466240
Annulus
!
N n snt þ Se
Fig. 1. Self-shielding factor as a function of background cross section for uranium-238
at the 6.67 eV resonance. The raw data is produced by NJOY, and can be fit to the tanh
function in Eq. (20) using the method in (Gopalakrishnan and Ganesan, 1998). Red dots
identify the discrete values of the background cross section at which group cross
sections are obtained from NJOY. These points form what is known as a dilution grid.
(For interpretation of the references to colour in this figure legend, the reader is
referred to the web version of this article.)
(15)
nsm
Here it is understood that snt applies to the j’th region. Equations
(11) and (15) can be used to give the group cross section for any
reaction, any nuclide, and in any region of a reactor provided that
the correct escape cross section from that region can be formulated.
Pitch
1.2000 cm
Temperature
294.0 K
Nuclide
Density (#/b/cm2)
H-1
O-16
Zr-90
0.0574461
0.0286544
0.0061594
C.L. Dembia et al. / Progress in Nuclear Energy 67 (2013) 124e131
This formulation has been used to model the escape cross section in two region models in the past (Schneider et al., 2006b,
2007).
The collision probabilities in Equation (17) depend on group
cross sections. However, it is the group cross sections we are solving
for. Thus, we solve for the group cross sections in an iterative
fashion. First, we guess a value for s0. Then, we compute group
cross sections and collision probabilities. Using these collision
probabilities, we compute the escape cross section and thus s0.
We terminate the iteration when our values for s0 converge. The
iteration typically takes 3 steps or less to converge.
2.5. The self-shielding factor
The infinite dilution (unshielded) cross section is defined as the
limit of the group cross section as the background cross section
goes to infinity It is common to define a self-shielding factor f as
the ratio of the group cross section to the infinite dilution cross
section:
f ðs0 Þ ¼
sg ðs0 Þ
sg ðs0 /NÞ
(19)
Fig. 2. Self-shielded uranium-238 capture cross section in an LWR. The top graph
provides the infinite dilution cross section and compares the self-shielded cross sections produced by our collision probability model with the equivalent group cross
sections produced using MCNPX 2.7.0. The middle graph presents the difference between our cross section and MCNPX’s. The bottom graph shows the background cross
section generated by our code as well as the escape portion of this background cross
section. The relatively small background cross section leads to substantial selfshielding. R2 ¼ 0.99.
127
Accordingly, the quantity sg(s0) ¼ f(s0)sg(s0 / N) is called the
self-shielded cross section. Fig. 1 shows the self-shielding factor as a
function of background cross section for the uranium-238 capture
cross section at 6.67 eV. The figure shows that the self-shielding
factor falls between zero and unity, and approaches unity as the
background cross section increases. The function is commonly fit to
a tanh curve:
f ðs0 Þ ¼ Atanh½Bðlns0 þ CÞ þ D
(20)
where the constants A, B, C, and D are dimensionless fitting parameters. A method for obtaining these for a given set of selfshielding data is described in (Kidman et al., 1972).
3. Accuracy of the Bondarenko method
Cross sections generated using Eq. (17) give excellent results
compared to other computational approaches. To illustrate this, Eq.
(17) was used to generate cross sections for an in-house multi-region collision probability code based on the fully benchmarked
two-region VBUDS code (Schneider et al., 2006b, 2007). Infinite
dilution and self-shielded cross sections were generated using
NJOY (MacFarlane and Muir, 1994) for an infinite lattice light water
reactor running fresh uranium dioxide fuel (Rowlands et al., 1999)
Fig. 3. Self-shielded uranium-235 capture cross section in an LWR. The top graph
provides the infinite dilution cross section and compares the self-shielded cross sections produced by our collision probability model with the equivalent group cross
sections produced using MCNPX 2.7.0. The middle graph presents the difference between our cross section and MCNPX’s. The bottom graph shows the background cross
section generated by our code as well as the escape portion of this background cross
section. Since the background cross section is relatively large, little self-shielding occurs. R2 ¼ 0.99.
128
C.L. Dembia et al. / Progress in Nuclear Energy 67 (2013) 124e131
as well as for an infinite lattice sodium cooled fast reactor fueled
with uranium dioxide. The values of s0 were computed directly
using Eqs. (15) and (17). We also present neutron spectrum, reaction rates, and criticality results for each of the two simulations.
3.1. Thermal cross section comparison
The parameters of the thermal reactor are taken from the
Rowlands benchmark (Rowlands et al., 1999), which is commonly
used in the literature to evaluate resonance self-shielding methods
(Gopalakrishnan and Ganesan, 1998; Herbert, 2005). The benchmark provides 9 different cases; our light water reactor is modeled
after case 1 and its geometry and composition are given in Table 1.
The simulated reactor consists of an enriched uranium dioxide fuel
pin in a square lattice surrounded by water, and both materials are
at 294 K. We perform the transport calculation with 100 energy
groups from 1 meV to 10 MeV. The Rowlands benchmark includes a
zirconium fuel rod; we have treated this by smearing the same
amount of zirconium evenly throughout the water.
The cross sections are compared to their values at infinite
dilution, as well as to the cross sections that MCNPX 2.7.0 provides
for a simulation of this system, Figs. (2) and (3). The MCNPX cross
sections are obtained from the reaction rates provided by a cell tally
Fig. 4. Self-shielded uranium-238 capture cross section in a sodium-cooled fast
reactor. The top graph provides the infinite dilution cross section and compares the
self-shielded cross section produced by our collision probability model with the
equivalent group cross section produced using MCNPX 2.7.0. The middle graph presents the error between our cross section and MCNPX’s. The bottom graph shows the
background cross section generated by our code as well as the escape portion of this
background cross section. The escape cross section is dominated by the sodium-23
resonance present in the coolant. The relatively small value for the background cross
section gives rise to substantial self-shielding. R2 ¼ 0.99.
in conjunction with the appropriate tally multiplier. The MCNPX
cross sections are appropriately assumed to be self-shielded. Results from our method are labeled “CPM” in the figures.
Fig. 2 shows the capture cross section for uranium-238 in the
energy range 1 eVe100 keV. The top graph shows that there is
substantial self-shielding of this cross section. This is expected,
because uranium-238 is present in such a great concentration and
many neutrons are absorbed in its resonances. The bottom graph
shows the background cross section for uranium-238, as well as
the escape portion of this background cross section. Over the
entire energy range, the background cross section has a relatively
small value, as we expect in the case where self-shielding is
substantial. Though the error between MCNPX and our method is
large in a few energy groups, the method is mostly able to
compute the escape probability that yields the correct selfshielded cross sections.
Fig. 3 shows the capture cross section for uranium-235, which is
present in a much smaller concentration than is uranium-238. As a
result, the background cross section is large and the infinite dilution
cross section can be used as the group cross section. By comparing
the bottom graph of Fig. 5 to the unshielded cross section in Fig. 4, it
is evident that the background cross section for uranium-235 is
dominated by the uranium-238 capture cross section.
Fig. 5. Self-shielded uranium-235 capture cross section in a sodium-cooled fast
reactor. The top graph provides the infinite dilution cross section and compares the
self-shielded cross section produced by our collision probability model with the
equivalent group cross section produced using MCNPX 2.7.0. The middle graph presents the error between our cross section and MCNPX’s. The bottom graph shows the
background cross section generated by our code as well as the escape portion of this
background cross section. The escape cross section is dominated, as in Fig. (4), by the
sodium-23 resonance present in the coolant. However, uranium-235 is present in a
smaller concentration than uranium-238 and so its background cross section is not as
large. R2 ¼ 0.99.
C.L. Dembia et al. / Progress in Nuclear Energy 67 (2013) 124e131
Table 2
Dimensions and composition of a sodium fast reactor. The fuel is
enriched uranium dioxide and is surrounded by a sodium
moderator. A steel fuel rod is modeled by smearing chromium,
iron, and nickel across the sodium.
Pin
Diameter
1.3727 cm
Temperature
900.0 K
Nuclide
Density (#/b/cm2)
U-235
U-238
O-16
0.0062999
0.0183091
0.0492181
129
Table 3
Comparison to Rowlands results of 3-group reaction rates in the LWR from case 1 of
the Rowlands benchmark. The absorption rates for the uranium isotopes are provided. The collision probability results using group cross sections obtained with the
Bondarenko methods are denoted by “CPM”. The numbers are normalized to
100,000 total absorptions in the system. The errors are slightly higher than in Table 4
as a result of how we have approximated the presence of the fuel rod.
Nuclide
Reaction
MCNPX
CPM
Error (%)
U-235
Fission
Capture
Fission
Capture
56,438
14,802
5126
22,848
56,325
14,694
5340
22,870
0.20
0.73
4.17
0.10
U-238
Annulus
Pitch
2.0000 cm
Temperature
600.0 K
Nuclide
Density (#/b/cm2)
Na-23
Cr-52
Fe-56
Ni-58
0.0210618
0.0016310
0.0053902
0.0007107
3.2. Fast cross section comparison
The fast reactor, whose geometry and composition is given in
Table 2, consists of an enriched uranium fuel pin surrounded by a
sodium coolant. A fuel rod is modeled by smearing a 0.5 cm thick
steel rod (using only chromium, iron, and nickel) across the coolant.
For this system, we use 42 energy groups from 400 eV to 10 MeV.
The capture cross section for uranium-238 and uranium-235 are
given respectively in Figs. (4) and (5). The shape of the escape cross
section for both of these nuclides is dominated by the sodium cross
section. Again, as expected, the background cross section for
uranium-238 is much smaller than it is for uranium-235 because it
is present in a higher concentration. As a result, the uranium-238
cross section is substantially self-shielded.
3.3. Thermal spectrum and reaction rates
The flux obtained by our method for the thermal reactor, shown
in Fig. (6), correctly captures all essential features of the MCNPX
flux, including the three resonance dips in the epithermal region.
The coefficient of determination R2 between the two results, at
0.99, indicates a high level of accuracy of our method across the
energy groups.
Table 3 compares 3-group reaction rates from our method to
those given in the Rowlands benchmark. The results are normalized
to 100,000 absorptions in the system (in both the fuel and
moderator). Table 4 compares our results to those obtained by an
MCNPX simulation in which the zirconium fuel rod has been
smeared evenly. Table 4 indicates a high level of accuracy of our
method in computing reaction rates, as all errors are below 5%.
However, it is clear from Table 3 that smearing the zirconium
throughout the moderator introduces substantial error.
3.4. Fast spectrum and reaction rates
The neutron spectrum in the fuel is compared in Fig. 7 to results
obtained by MCNPX for the same system. The relative error between the two results is shown in the lower graph. The coefficient
of determination R2 between the two results is 0.99, which indicates a good correlation of the result across the energy groups.
Table 4
Comparison to MCNPX of 3-group reaction rates in the LWR modified from case 1 of
the Rowlands benchmark. The absorption rates for the uranium isotopes are provided. The collision probability results using group cross sections obtained with the
Bondarenko methods are denoted by “CPM”. The numbers are normalized to
100,000 total absorptions in the system. All errors are below 5%, and are slightly
smaller than in Table 3 because the fuel rod has been smeared in MCNPX just as it
has been in our model.
Nuclide
Reaction
Group
MCNPX
CPM
Error (%)
U-235
Fission
Fast
Res.
Thermal
Fast
Res.
Thermal
Fast
Res.
Fast
Res.
Thermal
675
4093
49,424
110
2371
8762
2765
1
2171
15,218
8395
674
4038
49,555
110
2365
8795
2785
1
2161
15,303
8420
0.15
1.34
0.27
0
0.25
0.38
0.72
0
0.46
0.56
0.30
Capture
U-238
Fig. 6. Comparison to MCNPX of spectral flux in the fuel of the LWR from case 1 of the
Rowlands benchmark. The top graph compares the fuel flux from our method ("CPM")
to MCNPX results. The bottom graph provides the error between our method and
MCNPX. R2 ¼ 0.99.
Fission
Capture
130
C.L. Dembia et al. / Progress in Nuclear Energy 67 (2013) 124e131
Table 6
Comparison to MCNPX of multiplication factor for both the fast and thermal reactor.
The collision probability results using group cross sections obtained with the Bondarenko methods are denoted by “CPM”. Our model provides very accurate results in
comparison to MCNPX simulations. There is greater error between our model and
the results provided by the Rowlands benchmark because we have smeared the
zirconium fuel rod in the benchmark across the moderator.
Run
Comparison
CPM
Error (mk)
Fast
Rowlands
Rowlands in MCNPX
1.536340
1.390110
1.397500
1.539315
1.401140
1.401140
2.975
11.030
3.640
4. Conclusions
We have provided a review and simplified derivation for the
implementation of the Bondarenko method for obtaining group
cross sections in multi-region collision probability models. We have
used the results in an in-house collision probability model to show
how the group cross sections obtained in this way compare to those
generated from infinite lattice Monte Carlo simulations of thermal
and fast spectrum reactors as well as with the Rowlands benchmark
for a thermal spectrum system. The results confirm that the Bondarenko method, while simple, can yield excellent results.
Fig. 7. Comparison to MCNPX of spectral flux in the fuel of the sodium fast reactor in
the results. The top graph provides the infinite dilution cross section and compares the
self-shielded cross section produced by our collision probability model with the
equivalent group cross section produced using MCNPX 2.7.0. Our method is able to
capture the flux dips corresponding to resonances. The bottom graph provides the
error between our method and MCNPX. R2 ¼ 0.99.
Table 5 compares one-group reaction rates computed by our
method to those obtained from MCNPX. The numbers are again
normalized to a total of 100,000 absorptions in the entire system.
The results indicate good agreement with MCNPX.
3.5. Criticality
A summary of the multiplication factors for both the fast and
thermal reactors is provided in Table 6. The thermal results are
compared to both MCNPX and the results given by Rowlands
benchmark. The error is less than 5 mk for comparisons with
MCNPX, and the larger error with respect to the Rowlands results
can be attributed to the fact that in our model we have smeared the
zirconium fuel rod through the water.
Table 5
Comparison to MCNPX of one-group reaction rates in a sodium fast reactor. The
absorption rates for the uranium isotopes are provided. The collision probability
results using group cross sections obtained with the Bondarenko methods are
denoted by “CPM”. These reaction rates encompass the entire energy range that is
modeled, from 400 eV to 10 MeV. The numbers are normalized to 100,000 total
absorptions in the system. All errors are below 5%.
Nuclide
Reaction
Group
MCNPX
CPM
Error (%)
U-235
Fission
Fast
Res.
Thermal
Fast
Res.
Thermal
Fast
Res.
Fast
Res.
Thermal
728
4363
51,926
117
2386
9051
2963
1
2262
16,148
8771
674
4038
49,555
110
2365
8795
2785
1
2161
15,303
8420
7.42
7.45
4.57
5.98
0.88
2.83
6.01
0
4.47
5.23
4.00
Capture
U-238
Fission
Capture
Acknowledgments
We would like to thank the United States Nuclear Regulatory
Commission for grant NRC-38-08-946 which helped to support this
work.
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