Monte Carlo calculation method can be used for resolving particle transport in matter, and particularly the transport of neutrons in environment of reactor core. The method has become more efficiently because of high accuracy of updated nuclear data and fast development of advanced super-computing system. In this work, we would like to present calculations for kinematical characteristics of neutron transport in a typical configuration of the pressurized water reactor (PWR) fuel assembly based on the Monte-Carlo simulation method. We concentrate in two main results: (1) neutron energy spectrum at fuel rod and (2) optimal thickness of light water reflector

Khai, Nguyen Tuan

Vuong, Phan Quoc

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 25, No. 1 (2015), pp. 91-96
DOI:10.15625/0868-3166/25/1/5661
SIMULATION FOR NEUTRON TRANSPORT IN PWR REACTOR
MODERATOR AND EVALUATION FOR PROPER THICKNESS OF LIGHT
WATER REFLECTOR
NGUYEN TUAN KHAI AND PHAN QUOC VUONG
Institute for Nuclear Science and Technology, 179 Hoang Quoc Viet, Hanoi
E-mail: ntkhai@iop.vast.ac.vn
Received 26 November 2014
Accepted for publication 12 December 2014
Abstract. Monte Carlo calculation method can be used for resolving particle transport in matter, and particularly
the transport of neutrons in the environment of the reactor core. The method has become more efficient because of
high accuracy of updated nuclear data and fast development of advanced super-computing system. In this work, we
would like to present calculations for kinematic characteristics of neutron transport in a typical configuration of the
pressurized water reactor (PWR) fuel assembly based on the Monte-Carlo simulation method. We concentrate in two
main results: (1) neutron energy spectrum at fuel rod and (2) optimal thickness of light water reflector.
Keywords: Monte-Carlo method, neutron transport, pressurized water reactor (PWR), fuel rod, fuel assembly, light
water reflector.
I. INTRODUCTION
Fission reaction takes place when a heavy nucleus as uranium, thorium and plutonium
absorbs neutron, high energy gamma radiation or even charged particles and is splited into two
intermediate-mass fragments with releasing a few new neutrons under prompt and delayed types.
As known, amongst of these the fission of the heavy actinide isotopes as 235U,239U and 241Pu
induced by thermal neutrons is the most important process to generate power in nuclear reactor. It
is obvious that in nuclear reactor the neutron generation from fission and their transport including
slowing down in moderator and absorption in fuel are very essential as basis of reactor physics
and kinetics.
The basis of transport theory can be derived from the particle density balance inside an
infinitesimal element of the six-dimensional phase-space. In the Cartesian geometry space, the
particles are initially located inside an infinitesimal volume element d3r = dxdydz about position
r. The infinitesimal element in the angular space is the differential solid angle dΩ at the direction
of motion Ω. The energies of the particles are simply distributed within energy interval from E to
E +dE. An illustration is given in Fig. 1.
c©2015 Vietnam Academy of Science and Technology
92 SIMULATION FOR NEUTRON TRANSPORT IN PWR REACTOR MODERATOR AND EVALUATION ...
The neutron balance inside the six-dimensional phase-space element is described by the
transport equation [1]:
(1/v)∂ [ψ(r,Ω,E, t)]/∂ t+Ω ·∇ψ(r,Ω,E, t)+Σt(r,E)ψ(r,Ω,E, t) = q(r,Ω,E, t) (1)
 
Fig. 1. A moving particle located inside an infinitesi-
mal volume element d3r, where the energy of the par-
ticle lies within interval dE at a given value E
where ψ(r,Ω,E, t) is the neutron flux, Σt(r,E)
the total macroscopic cross section of the
medium and q(r,Ω,E, t) a generalized source
term. The first term on the left-hand side is
the time rate of change in the neutron density.
The second term describes the rate at which
neutrons travelling in the direction of motion
Ω leave or enter the differential volume ele-
ment d3r. The third term consists of all inter-
actions removing neutrons either by absorption
or scattering away from the differential angular
and energy space. The source term q(r,Ω,E, t)
consists of contribution from the external, scat-
tering and fission sources. If the source and
loss terms are in balance, the time dependence
disappears, and the result is the steady-state
transport equation.
In this work we have used the Monte Carlo (MC) calculations for resolving neutron trans-
port problem in the light water moderator for the PWR fuel assembly based on the physical pro-
cesses described by Eq. (1). The MC method is being widely used in nuclear physics, radiation
physics and shielding calculations. The basic principle of the method is not so complicated. The
life of a single neutron or any other particle type is simulated from its initial emission until the
eventual death by absorption in the system or escape from the defined boundaries of the system.
The frequency and outcome of the various interactions that may occur during the particle’s life
are randomly sampled and simulated according to interaction laws derived from particle physics.
When the procedure is repeated for a large number of particles, the obtained result shows a de-
tailed simulation of the transport process. In reactor physics applications, the method has been
developed to solve the neutron transport problem, consider neutronic characteristics and analyze
criticality safety in the reactor core. These researches play role as physics core in the present MC
computation codes such as MCNP, MVP/GMVP. The main advantage of the MC method is a good
adaptability in modeling geometry and interaction physics without any approximations. In prac-
tice, the method is possible to include with relative ease all desired variables, which may be three
spatial coordinates, energy, two directional variables and time, and to represent accurately almost
any physical model for particle scattering, cross section data and geometric detail. Recently, the
application of the Monte Carlo method has become more efficient because of high accuracy of
updated nuclear data and fast development of advanced super-computing system.
We have constructed the MC calculation algorithms to simulate the neutron transport in a
typical configuration of the PWR fuel assembly, where it is assumed that the neutrons are gen-
erated from the fission of 235U in fuel rods and considered until they are either absorbed in the
moderator, or hit on the fuel rods. The interactions of neutron inside the fuel are not considered
NGUYEN TUAN KHAI AND PHAN QUOC VUONG 93
here. In addition, it is also assumed that the reactor core is surrounded by a light water reflector,
then the trajectory of the neutrons hitting boundary between the core and the reflector are contin-
ued to be followed. Based on considering a quantity called “backscattering fraction”, i.e. the ratio
between the neutrons coming back the core from the reflector to the total number of the neutrons
hitting the boundary, as a function of the reflector thickness, the optimization for the thickness of
the light water reflector is evaluated.
II. CALCULATION PROCEDURE
Fig. 2 shows the typical configuration of the PWR fuel assembly. The dimension of the
assembly is 21.4 × 21.4 cm2 containing 289 rods (17×17) including 264 fuel rods, 24 control
rods and one instrumental guide tube. A coordinate XYZ is established with the origin O at center
of the assembly. This made us possible to define the position of the rods in the assembly and to
track the neutron trajectory during transport.
The fission energy spectrum of 235U is shown in Fig. 3 and expressed by [3]:
Pn(E) = 0.453e−1.036E sinh
√
2.29E (2)
where Pn(E) is probability for generation of neutrons with energy from E to E +dE.
 
Fig. 2. Typical configuration of the PWR fuel as-
sembly [2]
 
Fig. 3. Neutron fission spectrum of 235U
The MC simulation calculations have been
carried out by tracking the transport of neutron
in the moderator (light water) and the position at
which the neutron hits the fuel rods. The neu-
trons are assumed to be emitted from the fission
source as described by expression (2). Then, the
life of these neutrons from their birth to the death,
which is referred as the particle history, is consid-
ered based on the statistical nature of three main
interaction processes: elastic and inelastic scatter-
ing, and the radiation capture reactions. In our cal-
culations a set of neutron histories is generated by
following individual neutrons through successive
collisions. The locations and results of collisions
are determined from the range of probabilities by
sets of random numbers. The simulation proce-
dure is summarized as follows:
- As mentioned above, the fuel rods are
assumed to be the sources to generate the fis-
sion neutrons with the energy spectrum shown in
Fig. 3. Three initial parameters related to the neu-
tron emission which are the direction, energy and
position are generated as random variables.
- Then, the distance the neutron travels
from the emitted position on the source to its first
94 SIMULATION FOR NEUTRON TRANSPORT IN PWR REACTOR MODERATOR AND EVALUATION ...
interaction is defined as free path by random sampling:
λ =− ln(r)/ΣT (3)
where ΣT is total macroscopic cross section,
ΣT = ΣT−H +ΣT−O (4)
with ΣT−H and ΣT−O being the total macroscopic cross sections for hydrogen and oxygen, re-
spectively, in which the main interaction processes mentioned above, i.e. elastic and inelastic
scattering, and radiation capture (absorption), are taken into account: p(n,n’)p, 16O(n,n’)16O,
16O(n,n’)16O* with producing 6.1 MeV γ-ray and radiation capture p(n,γ)d with producing 2.2
MeV γ-ray.
- At a given interaction position, the atom of the material with which the neutron interacts
and the type of interaction are considered via random sampling based on the normalized probabil-
ity distributions built from the cross section data of the interactions of interest. If the interaction
leads to absorption, the life of neutron is terminated. If the interaction is elastic scattering or in-
elastic scattering, the direction of scattered neutron is selected from the distribution of scattering
angles, i.e. differential cross section, and the energy of the scattered neutron is determined using
energy and momentum conservations.
- To follow the neutron trajectory in the original coordinate a series of frame rotations has to
be done after each collision in order to convert the neutron’s position from the existing coordinate
into the original one. This is necessary to determine the physical parameters characteristics for the
slowing down process such as moderation length, number of collisions and etc.
In this work we have used a large library of nuclear data on cross section and angular dis-
tribution from National Nuclear Data Center, Brookhaven Laboratory, US [4]. They are necessary
to build the normalized probability distributions for the MC random sampling algorithm.
III. RESULTS AND DISCUSSION
First of all, to validate the simulation algorithms we have performed two calculations for
comparison:
Firstly, the conventional calculations related to the neutron transport and moderation in-
cluding: (1) the energy decrease of 2 MeV neutrons as a function of the number of collisions in
hydrogen and oxygen (the calculation results are shown in Fig. 4), and (2) the collision-number
distribution for neutron slowing down from 2 MeV energy to the thermal one in Hydrogen (the
obtained distribution is shown in Fig. 5). Our obtained results from these calculations are well
consistent with the well-known references [5, 6].
Secondly, based on the configuration of PWR fuel assembly (shown in Fig. 2) it is assumed
that a neutron source placed at the center of the assembly generates the mono-energy neutrons of
2 MeV we have compared the results obtained by our calculation program and MCNP code for
both the neutron energy and flux spectra along the assembly. Figs. 6&7 show a good consistence
for these comparisons.
Thus, we can conclude on validation for the used nuclear data and simulation algorithms.
Following, the calculations were used to consider the neutron transport in the PWR fuel assembly,
where the fission neutron source is generated at the fuel rods.
NGUYEN TUAN KHAI AND PHAN QUOC VUONG 95
 
Fig. 4. Energy decrease of 2 MeV neutrons as a
function of collisions in Hydrogen and Oxygen
 
Fig. 5. Collision number distribution for 2
MeV neutrons slowing down to thermal energy
in Hydrogen
 
Fig. 6. Neutron energy spectrum at the fuel rod
using a mono-energy neutron source of 2 MeV.
 
Fig. 7. Neutron flux as a function of distance
from the source to the fuel rod.
The first result we have obtained is the energy spectrum of the neutrons hitting the fuel rods
and core boundaries. Fig. 8 shows typical spectrum at one rod. In principle, the registered neutron
amount with energies from E to E+dE depends on the position at which neutron hits the rod and
also the position of the rods in the assembly.
A gross spectrum at each assembly can be obtained by summing the spectral data from all
the rods. This makes us possible to determine two essential distributions on the neutron flux, par-
ticularly for thermal neutrons: neutron distribution along each assembly and neutron distribution
between the assemblies in the core. Physically, two these results will give image of power distri-
bution in the reactor core. However, in frame of this work we only concentrate in evaluating the
optimal value for thickness of light water reflector using the neutron energy spectrum obtained at
the core boundary. This is performed by simulating the neutron trajectory in the reflector in order
to see its life end which is defined by one of the three following events:
- Neutron is backscattered from the reflector,
- Neutron is absorbed in the reflector,
- Neutron is leaked from the reflector.
Fig. 9 shows the obtained calculation results on neutron backscattering, absorption and
leakage fractions as a function of the reflector thickness. In our consideration the optimal thickness
of the reflector is selected from two evaluations: the first one is to determine the maximum value
96 SIMULATION FOR NEUTRON TRANSPORT IN PWR REACTOR MODERATOR AND EVALUATION ...
 
Fig. 8. Neutron energy spectrum at the fuel rod
using the fission neutron source.
 
Fig. 9. Simulation results on neutron backscat-
tering, absorption and leakage fractions as a
function of the reflector thickness.
of the backscattering fraction, and the second one is based on the neutron shielding from leakage.
Thus, from Fig. 9 we can see that for the first evaluation the reflector thickness satisfies at the
value of more less 12 cm, where the backscatter fraction becomes saturated. However, from the
neutron shielding requirement based on evaluating the neutron leakage from the reflector to the
containment the reflector thickness should be selected about 38- 40 cm, at which the neutron
leakage is quite negligible. This evaluation result is well consistent with the design of AP-1000
reactor, a PWR type [7].
IV. CONCLUSION
We have built a MC calculation program to consider the neutron transport in a typical
configuration of the PWR fuel assembly, where the fission neutrons were assumed to generate at
the fuel rods.
The validity of the simulation algorithms is confirmed by comparing our calculations with
literature and MPNP code for the kinematic characteristics of the neutron transport.
The energy spectra of the neutrons hitting the fuel rod and the core boundary were obtained.
The latter result was used to determine the optimum thickness of the light water reflector based on
the concept of the backscattering fraction obtained by simulating the effects on neutron backscat-
tering, absorption and leakage. The conclusion on the optimum thickness of about 38-40 cm is
well consistent with the design of AP-1000 reactor.
REFERENCES
[1] John R. Lamarsh and Anthony J. Baratta, Introduction to Nuclear Engineering, Third Edition, Prince Hall 2001.
[2] HUST-Misubishi Collaboration program, Nuclear Power Plant Engineering -Text Book- Mitsubishi heavy indus-
tries, Ltd, 2011.
[3] Paul Reuss, Neutron Physics, EDP SCIENCES 2008.
[4] National Nuclear Data Center (NNDC), Brookhaven National Laboratory, http://www.nndc.bnl.gov.
[5] K. Bekurs and K. Virts, Neutron Physics, ATOMIZDAT 1968.
[6] T. Ueki, Lecture on Reactor Physics (JAEA-2013), Private publication.
[7] AP1000 European Design Control Document, EPS-GW-GL-700, Revision 1, Westinghouse.


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