MonteCarlo Calculations Of The Training Reactor Of Budabest University Of Technology and Economics using MCNP code

VIETNAM NATIONAL UNIVERSITY, HANOI
VNU UNIVERSITY OF SCIENCE
FACULTY OF PHYSICS

Nguyễn Hoàng Dũng

Monte – Carlo calculations of the Training Reactor of
Budapest University of Technology and Economics
using MCNP code

Submitted in partial fulfillment of the requirements for the degree of
Bachelor of Science in Nuclear Technology
(Advanced Program)

Hanoi - 2017


VIETNAM NATIONAL UNIVERSITY, HANOI
VNU UNIVERSITY OF SCIENCE
FACULTY OF PHYSICS

Nguyễn Hoàng Dũng

Monte – Carlo calculations of the Training Reactor of
Budapest University of Technology and Economics
using MCNP code

Submitted in partial fulfillment of the requirements for the degree of
Bachelor of Science in Nuclear Technology
(Advanced Program)

Supervisor: Dr. Nguyễn Tiến Cường

Hanoi - 2017


Acknowledgement
It has been a long course since the beginning of this thesis, which was 4 months
ago (Feb 2017). Without the help from my supervisor, my friends, my family, I could
not have made such progress nor have the motivation to finish this thesis. Therefore, I
spend the very first page to send my deepest gratitude toward:
First of all, my supervisor Dr. Nguyen Tien Cuong, Faculty of Physics, VNU
University of Science, who has given me the initial idea of what has to be done. His
invaluable comments throughout this thesis has enlightened, guided me to the very last
word.
Secondly, all of my classmates, thank you for your continuously inspiring and
support.
Finally, I would like to thank my family for their support and for believing in me.
Student,
Nguyen Hoang Dung

i


Abstract
The applicability of the Monte Carlo N-Particle code (MCNP) to evaluate reactor
physics parameters, shielding applications on the Training Reactor of Budapest
University of Technology and Economics (BME). Some of reactor physical calculations
were carried out for simulating the reactor critical state: multiplication factor and its
dependence on the water level, vertical and horizontal neutron flux distributions.
Criticality is the condition where the neutron chain reaction is self-sustaining and the

neutron population is neither increasing nor decreasing.
This study also deals with the analysis of variance reduction methods,
specifically, variance reduction methods applied in MCNP on the determination of dose
rates for neutrons and photons. Calculations for the outer wall contains two different
types of concrete compositions were performed to investigate the impact of the bioshield filling materials on the dose rate estimation.

ii


Table of Contents
Acknowledgement................................................................................. i
Abstract
.......................................................................................... ii
Table of Contents ................................................................................ iii
List of Figures .......................................................................................v
List of Tables ....................................................................................... vi
List of Abbreviations .......................................................................... vi
Introduction ....................................................................................... vii
Chapter 1. The Training Reactor of Budapest University of
Technology and Economics ...........................................1
1.1.

General overview .......................................................................... 1

1.2.

Reactor core geometry and configuration ..................................... 3

Chapter 2. Neutron flux and flux density in the reactor core .......7
2.1.

Diffusion equation in a Finite multiplying system ........................ 7

2.2.

One - group reaction equation ....................................................... 8

2.3.

The BME - Reactor (Parallelepiped Reactor) ............................. 10

Chapter 3. Reactor parameter determinations using MonteCarlo Method ................................................................12
3.1.

Introduction ................................................................................. 12

3.2.

Monte-Carlo and Particle Transport ............................................ 13

3.3.

MCNP Code ................................................................................ 14

3.4.

MCNP model of the BME - Reactor ........................................... 15

3.5.

Neutron flux calculations ............................................................ 16

3.6.

Dose rate determinations ............................................................. 17

3.6.1.

“Weight” of a particle.................................................................. 17

3.6.2.

Geometry Splitting with Russian Roulette .................................. 18
iii


Chapter 4. Results and Discussions ................................................20
4.1.

Reactor-physical calculations ...................................................... 20

4.1.1.

The dependence of 𝑘𝑒𝑓𝑓 on water levels ..................................... 20

4.1.2.

Neutron Flux distribution in reactor core .................................... 21

4.2.

Dose rate calculations in concrete structures .............................. 24

4.3.

Conclusions ................................................................................. 28

References .........................................................................................29

iv


List of Figures
Figure 1: Side and upper view of the BME - Reactor ...................................................... 2
Figure 2: Schematic drawing (not to scale) of the EK-10 fuel assembly (dimensions are
given in mm unit) and its various types used in the BME - Reactor ............................... 3
Figure 3: Configuration of the BME - Reactor core ........................................................ 4
Figure 4: Cross sectional diagram of a) Automatic and b) Manual control rod .............. 5
Figure 5: 3d view model of the BME - Reactor ............................................................... 5
Figure 6: Lifecycle of a Neutron in Monte-Carlo simulation ........................................ 13
Figure 7: Cross-section top view of MCNP model of BME - Reactor core .................. 15
Figure 8: MCNP geometric model of the BME - Reactor: a) Side view, b) Cross-section
top view (dimensions are given in cm unit) ................................................................... 16
Figure 9: The Splitting Process ...................................................................................... 18
Figure 10: The Russian Roulette Process ...................................................................... 19
Figure 11: The dependence of 𝑘𝑒𝑓𝑓 on water level ........................................................ 20
Figure 12: Vertical flux distribution in fuel rod ............................................................. 21
Figure 13: Vertical flux distribution in fuel cladding .................................................... 21
Figure 14: Thermal vertical flux distribution in Dy-Al wire, experimental and simulated
result ............................................................................................................................... 22

Figure 15: Horizontal flux distribution in fuel rod ........................................................ 23
Figure 16: Horizontal flux distribution in fuel cladding ................................................ 23
Figure 17: Schematic of split model for calculating dose rate: a) Side view, b) Crosssection top view.............................................................................................................. 24
Figure 18: Neutron dose rate as a function of distance from the core vessel at different
water level (60 to 80 cm) ............................................................................................... 25
Figure 19: Photon dose rate as a function of distance from the core vessel at different
water level (60 to 80 cm) ............................................................................................... 26

v


List of Tables
Table 1: The detailed components and information about materials [2]. ........................ 6
Table 2: Relative errors (Err.) in the split sub-layers of concretes of the MCNP dose rate
calculations for neutrons and photons. ........................................................................... 27

List of Abbreviations
MCNP

Monte Carlo N-Particle

BME

Budapest University of Technology and Economics

vi


Introduction
The Monte Carlo N-Particle (MCNP) code, version 5.19 (MCNP5) and a set of

neutron cross-section data were used to develop an accurate three-dimensional
computational model of the Training Reactor of BME with the geometry of the reactor
core was modelled as closely as possible. The following reactor core physics parameters
were calculated for the low enriched uranium core: multiplication factor, horizontal and
vertical neutron flux distributions.
Shielding analysis also forms a crucial part of reactor design. The precise
calculation of dose rates of neutrons and photons is highly desired to perform neutron
activation analysis, production of radioisotopes, determination of safety in standard
operation circumstance or even in accident situations, criticality calculation or evaluation
of many other processes. Especially, during the planning and the operation of the reactor,
it is a crucial task to make a safety analysis. The public, operating personnel and reactor
components must be protected against sources of radiation. Thermal and biological
shields positioned in front of intense radiation sources are highly absorbent materials to
photons and neutrons. Thermal shields prevent the embitterment of the reactor
components, whereas biological shields protect people from neutrons and gammas.
Typical shielding calculations performed in the industry are the transport of neutrons and
gammas through large regions of shielding material.
The behavior of radiation particles is a stochastic process based on a series of
probabilistic events. These probabilistic events are characterized by random variables
such as location, energy, the particle direction of flight, mean free path of the medium
and type of interaction. The transport phenomena can be solved with the Monte-Carlo
method because radiation particles have a stochastic behavior. However, the
disadvantage associated with this method is that they require long calculation times to
obtain well converged results, especially when dealing with complex systems.

vii


Chapter 1. The Training Reactor of Budapest University of
Technology and Economics

1.1.

General overview

The Training Reactor of BME is a swimming pool type reactor located at the
university campus. The reactor was designed and built between 1969 and 1971, by
Hungarian nuclear and technical experts. It first went critical on May 20, 1971. The
maximum power was originally 10 kW. After upgrading, which involved modifications
of the control system and insertion of one more fuel assembly into the core, the power
was increased to 100 kW in 1980 [1].
The main purpose of the reactor is to support education in nuclear engineering
and physics; however, extensive research work is carried out as well. Neutron irradiation
can be performed using 20 vertical irradiation channels, 5 horizontal beam tubes, two
pneumatic rabbit systems and a large irradiation tunnel.
The reactor core is made of 24 EK-10 type fuel assemblies, which altogether
contain 369 fuel rods. The fuel is 10%-enriched uranium dioxide in magnesium matrix.
The pellets are filled into aluminum cladding at a length of 50 cm. The total mass of
uranium in the core is approximately 29.5 kg. The reactivity is controlled by four control
rods, two of them are safety rods with one automatic and one manual rod. To minimize
neutron leakage and thereby conserve neutron economy, the horizontal reflector is made
of graphite and water, while in vertical direction water plays the role of reflector. The
highest thermal neutron flux is 2×1012 𝑛/𝑐𝑚2 /𝑠, measured in one of the vertical
channels [1].
Seven measuring chains are applied for reactivity control and power regulation.
The detectors are ex-core ionization chambers, two of which operate in pulse mode in
the startup range, four operate in current mode and one is a wide range detector. In all
power ranges, doubling time and level signals can invoke automatic scram operations.
The reactor is operated when required for a student laboratory exercise or a
research experiment. Accordingly, operation at 100 kW power for many hours is quite
rare; on the average, it occurs once a week. As a fortunate consequence, burn-up is very

low: only 0.56% of the 235U has been used up and 3.4 g 239Pu and 12.3 g fission products
have accumulated. Therefore, there has been no need to replace any of the fuel
assemblies since 1971 [1].
1


The reactor is used, among others, in the following fields:
• Activation analysis for radiochemistry and archeological research
• Analysis of environmental samples
• Determination of uranium content of rock samples
• Biomedical applications
• Nuclear instrument development and testing
• Experiments in reactor physics and thermohydraulics development and
testing of Neutron tomographic methods for safeguards purposes
• Development of noise diagnostic methods, isotope production and
investigation of radiation damage to instrument/equipment

Figure 1: Side and upper view of the BME - Reactor [1]

2


1.2.

Reactor core geometry and configuration
The BME - Reactor core geometry can be described as having a two-level

hierarchy of lattices; the first level of the hierarchy corresponds to fuel assemblies,
formed as a lattice of cylindrical fuel pins - these pins has an external diameter of 1 cm
and a total length of 60 cm, contain fuel meat of which diameter and height are

respectively 0.7 cm and 50 cm, absorbing material for controlling the nuclear chain
reaction and one layer of aluminum cladding. While in the second level, nine type of
EK-10 fuel assemblies (Figure 2) are arranged in a lattice to form the reactor core, six of
them are cut corner due to the insertion of four control rods and two particularly types
have a cavity for the irradiation channel.

Figure 2: Schematic drawing (not to scale) of the EK-10 fuel assembly (dimensions are
given in mm unit) and its various types used in the BME - Reactor [1]
3


As mentioned above, the layout strategy used in the placement of fuel pin and in
the placement of fuel assemblies of the BME - Reactor is a hexahedrical lattice of fuel,
graphite reflecting block and several irradiation channels; which form a rectangular
prism with the dimensions of 58 by 65 by 60 centimeters. The surrounding materials like
water is either coolant or a neutron energy moderator and the aluminum grid simply is
function as a supporting structure. (Figure 3). With this lattice layout, there are four
formed square located in the center of a group of 2 by 2 fuel assemblies that are designed
for one manual, one automatic, and two safety control rods. For specific, the manual
control rod has its inner part made from boron carbide (1.8 cm of diameter) while the
outer part is iron with 0.1 cm of thickness. The automatic one includes an hollow iron
tube (1.8 cm of inner diameter, 2.0 cm of outer diameter) covered by cadmium with
0.001cm of thickness (Figure 4). The full length of these control rods is 64 cm and the
length of cover materials is 60 cm [2].
a - Automatic control rod
b1, b2 - Safety control rods
c - Fast pneumatic rabbit
system
d - Thermal pneumatic
rabbit system

e - Vertical irradiation
channels in water
f - Fuel assemblies
g - Graphite reflectors
h - Vertical irradiation
channels in graphite
i - Vertical irradiation
channels in fuel assembly
j - Neutron-source

Figure 3: Configuration of the BME - Reactor core
4

k - Manual control rod


Figure 4: Cross sectional diagram of a) Automatic and b) Manual control rod

The active core is enclosed by
water in a cylindrical aluminum (2
cm) - air (3 cm) - iron (1 cm) tank.
Shielding concretes are arranged at
the outer of the reactor core for
radiation protection, which is the
combination of heavy and normal
concrete in case of the Training
Reactor. The accommodation of
two concrete block outside the core
is shown in Figure 5.

Figure 5: 3d view model of the BME - Reactor

5


Table 1: The detailed components and information about materials [2].
Fuel: ρ = 5.45719438 g/cm3
235

U 0.076126
C -3.5208e-3

10

B -1.1940e-6

238

U -0.6892785
16
O -0.1064826
11

234

U -0.0006052
Mg -0.1239809

B -4.806e-6

Aluminum: ρ = 2.698900 g/cm3
Al 1.0
Water: ρ = 0.998207 g/cm3
1
16
H 0.06672
O 0.00336

(mt3 LWTR.01)

Iron: ρ = 7.874000 g/cm3
Fe 1.0
Cadmium: ρ = 8.650000 g/cm3
Cd 1.0
Boron Carbide: ρ = 2.52000 g/cm3
10
C +0.217390
B +0.782610
Air: ρ = 2.698900 g/cm3
C -0.000124
Ar -0.012827

14

N -0.755286

16

O -0.231781

Graphite: ρ = 1.70000 g/cm3
C 1.0
(mt8 GRPH.01)
Heavy concrete: ρ = 3.35000 g/cm3
1
16
H -0.003585
O -0.311623
27

Al -0.004183
Ca -0.050194

Si -0.010457
Fe -0.047505

Normal concrete: ρ = 2.25000 g/cm3
1
16
H -0.004530
O -0.512600
27

Al -0.035555
Fe -0.013780

Si -0.360360

Stainless Steel: ρ = 7.860000 g/cm3
C -0.001400

Si -0.009300
S -0.000280
Fe -0.700000

Mg -0.001195

Cr -0.180000
Co -0.090000

6

S -0.107858
138

23

Ba -0.463400

Na -0.015270
Ca -0.057910

31

P -0.000420
55
Mn -0.011860


Chapter 2. Neutron flux and flux density in the reactor core
To design a nuclear reactor properly, it is necessary to predict how the neutrons

will be distributed throughout the system. Unfortunately, determining the neutron
distribution is a difficult problem in general. The neutrons in a reactor move about in
complicated paths as the result of repeated nuclear collisions. However, the overall effect
of these collisions is that the neutrons undergo a kind of diffusion in the reactor medium,
much like the diffusion of one gas in another. The approximate value of the neutron
distribution can then be found by solving the diffusion equation. This procedure, which
is sometimes called the diffusion approximation, was used for the design of most of the
early reactors.

2.1.

Diffusion equation in a Finite multiplying system
Consider an arbitrary volume V within a medium containing neutrons. As time

goes on, the number of neutrons in V may change if there is a net flow of neutrons out
of or into V if some of the neutrons are absorbed within V or if sources are present that
emit neutrons within V. The equation of continuity is the mathematical statement of the
obvious fact [3]. In particular, it follows:
[Rate of change in number of neutrons]
= [rate of production of neutrons] − [rate of absorption of neutrons]
− [rate of leakage of neutrons]
This equation can be written explicitly as:
𝜕𝑛
= 𝑠 − Σ𝑎 𝜙 − 𝑑𝑖𝑣 𝑱
𝜕𝑡
Which:
𝑛: the density of neutrons at any point and time in V

(1)

𝑠: the rate at which neutrons are emitted from sources per cm3 in V
Σ𝑎 𝜙: the rate at which neutrons are lost by absorption per cm3/sec
𝑱: the neutron current density vector on the surface of V
Consider flux is generally a function of three spatial variable, apply Fick’s law:
𝑱 = −𝐷 𝑔𝑟𝑎𝑑 𝜙 = −𝐷𝛻𝜙, into Eq.(1) and assume the diffusion coefficient 𝐷 is not a
function of the spatial variables gives:
𝐷𝛻 2 𝜙 − Σ𝑎 𝜙 + 𝑠 =

7

𝜕𝑛
𝜕𝑡

(2)


Since: 𝜙 = 𝑛𝜐, where 𝜐 is the neutron speed, Eq.(2) can also be written as:
1 𝜕𝜙
(3)
𝐷𝛻 2 𝜙 − Σ𝑎 𝜙 + 𝑠 =
𝜐 𝜕𝑡
This is the diffusion equation [3].

2.2.

One - group reaction equation
Consider a critical reactor, of which neutrons is supposed to be monoenergetic,

containing a homogeneous mixture of fuel and coolant. It is assumed that the reactor
consists of only one region and has neither a blanket nor a reflector. Such a system is

said to be a bare reactor [3].
This reactor is described in a one - group calculation by the one - group time
dependent diffusion equation - Eq.(3).
In a reactor at a measurable power, the source neutrons are emitted in fission. To
determine 𝑠, let Σ𝑓 be the fission cross-section for the fuel. If there are ν neutrons
produced per fission, then the source is: 𝑠 = νΣ𝑓 𝜙.
If the fission source does not balance the leakage and absorption terms, then the
right-hand side of Eq.(3) is non-zero. To balance the equation, we multiply the source
term by a constant 1/𝑘 where 𝑘 is an unknown constant. If the source is too small, then
𝑘 is less than 1. If it is too large, then 𝑘 is greater than 1. Eq.(3) may now be written as:
1
(4)
𝐷𝛻 2 𝜙 − Σ𝑎 𝜙 + νΣ𝑓 𝜙 = 0
𝑘
The equation may be rewritten as an eigenvalue equation by letting:
1 1
𝐵2 = ( νΣ𝑓 − Σ𝑎 )
𝐷 𝑘
Where 𝐵2 is defined as the material buckling. Then Eq.(4) becomes:
𝛻 2 𝜙 = −𝐵2 𝜙

(5)

Or:
1
(6)
−𝐷𝐵2 𝜙 − Σ𝑎 𝜙 + νΣ𝑓 𝜙 = 0
𝑘
This important result is known as the one - group reactor equation. The one group equation may be solved for the constant 𝑘:
νΣ𝑓 𝜙

νΣ𝑓
𝑘= 2
= 2
𝐵 𝜙 + Σ𝑎 𝜙 𝐵 + Σ𝑎

8


Also, the source term in the one - group equation can be written in terms of the
fuel absorption cross-section - Σ𝑎𝐹 . Let 𝜂 be the average number of fission neutrons
emitted per neutron absorbed in the fuel. The source term is given by:
Σ𝑎𝐹
𝑠 = 𝜂Σ𝑎𝐹 𝜙 = 𝜂
Σ 𝜙 = 𝜂𝑓Σ𝑎 𝜙
Σ𝑎 𝑎
Σ𝑎𝐹
Where:
𝑓=
Σ𝑎

(7)

is called the fuel utilization. Since Σ𝑎 is the cross-section for the mixture of fuel and
coolant, whereas Σ𝑎𝐹 is the cross-section of the fuel only, it follows that 𝑓 is equal to the
fraction of the neutrons absorbed in the reactor that are absorbed by the fuel.
The source term in Eq.(7) can be written in terms of the multiplication factor for
an infinite reactor. Consider an infinite reactor having the same composition as the bare
reactor under discussion. With such a reactor, there can be no escape of neutrons as there
is from the surface of a bare reactor and the neutron flux must be a constant, independent
of position. Thus, the absorption of Σ𝑎 𝜙 neutrons in one generation leads to the

absorption of 𝜂𝑓Σ𝑎 𝜙 in the next. Because the multiplication factor is defined as the
number of fissions in one generation divided by the number in the preceding generation.
So in an infinite reactor, it follows that:
𝑘∞ =

𝜂𝑓Σ𝑎 𝜙
= 𝜂𝑓
Σ𝑎 𝜙

(8)

Introducing Eq.(7) and Eq.(8) into the one-group reactor equation, Eq.(6) gives:
𝑘∞
−𝐷𝐵2 𝜙 + ( − 1) Σ𝑎 𝜙 = 0
𝑘
Dividing by 𝐷 yields:
𝑘
( ∞ − 1)
(9)
−𝐵2 𝜙 + 𝑘 2
𝜙=0
𝐿
𝐷
Where:
𝐿2 =
Σ𝑎
is the one - group diffusion area. For a critical reactor (𝑘 = 1), Eq.(2.9) may be solved
for the buckling:
𝐵2 =

( 𝑘 ∞ − 1)
𝐿2

9


2.3.

The BME - Reactor (Parallelepiped Reactor)
For a parallelepiped, the eigenvalue equation - Eq.(5): 𝛻 2 𝜙 = −𝐵2 𝜙 becomes:
𝑑2 𝜙 𝑑2 𝜙 𝑑2 𝜙
+
+
= −𝐵2 𝜙
𝑑𝑥 2 𝑑𝑦 2 𝑑𝑧 2

(10)

To solve this, we use a separable form of 𝜙 [4]:
𝜙(𝑥, 𝑦, 𝑧) = 𝑓(𝑥)𝑔(𝑦)ℎ(𝑧)
Substituting this form into Eq.(10) and divide both side by 𝑓 (𝑥)𝑔(𝑦)ℎ(𝑧) gives:
1 𝑑 2 𝑓 (𝑥 )
1 𝑑 2 𝑔 (𝑦 )
1 𝑑 2 ℎ (𝑧 )
+
+
= −𝐵2
𝑓 (𝑥) 𝑑𝑥 2
𝑔(𝑦) 𝑑𝑦 2
ℎ(𝑧) 𝑑𝑧 2

Since the three terms on the left-hand side are functions of x only, y only, and z
only respectively, and the sum is a constant, then each term must be a constant:
1 𝑑 2 𝑓 (𝑥 )
1 𝑑 2 𝑔 (𝑦 )
1 𝑑 2 ℎ (𝑧 )
2
2
(11)
= −𝐵𝑥 ;
= −𝐵𝑦 ;
= −𝐵𝑧 2
2
2
2
𝑓 (𝑥) 𝑑𝑥
𝑔(𝑦) 𝑑𝑦
ℎ(𝑧) 𝑑𝑧
and the three partial bucklings must add to the total buckling
𝐵𝑥 2 + 𝐵𝑦 2 + 𝐵𝑧 2 = 𝐵2
To solve Eq.(11), we evaluate the first part of it:
1 𝑑 2 𝑓 (𝑥 )
= −𝐵𝑥 2
2
𝑓 (𝑥) 𝑑𝑥

(12)

Without loss of generality, the slab is placed symmetrically about 𝑥 = 0, in the
interval [−𝑎/2, 𝑎/2]. Because of that, the flux must be symmetric about 𝑥 = 0 and
vanishes at the extrapolated boundaries, which is called ±𝑎𝑒𝑥 /2.

The general solution to Eq.(12) is:
𝑓 (𝑥) = 𝐴𝑐𝑜𝑠(𝐵𝑥 𝑥) + 𝐶𝑠𝑖𝑛(𝐵𝑥 𝑥)

(13)

Placing the derivative of Eq.(13) equal to zero at 𝑥 = 0 gives: 𝐶 = 0.
Next, introducing the boundary conditions gives:
𝑎𝑒𝑥
𝐵𝑥 𝑎𝑒𝑥
)=0
𝑓 ( ) = 𝐴𝑐𝑜𝑠 (
2
2
𝑛𝜋
⟹ 𝐵𝑛𝑥 =
𝑤𝑖𝑡ℎ 𝑛 𝑜𝑑𝑑
𝑎𝑒𝑥
The various constants 𝐵𝑛𝑥 are known as eigenvalues, and the corresponding
functions 𝑐𝑜𝑠(𝐵𝑛𝑥 𝑥) are called eigenfunctions. It can be shown that if the reactor under
consideration is not critical, the flux is the sum of all such eigenfunctions, each
multiplied by a function that depends on the time. However, if the reactor is critical, all
10


of these functions except the first die out in time. So, the one - group flux on the x-axis
becomes:
𝑓 (𝑥) = 𝐴𝑐𝑜𝑠 (

𝜋𝑥
)

𝑎𝑒𝑥

Thus, the total flux of the reactor can then be written:
𝜋𝑥
𝜋𝑦
𝜋𝑧
𝜙(𝑥, 𝑦, 𝑧) = 𝐴𝑐𝑜𝑠 ( ) 𝑐𝑜𝑠 ( ) 𝑐𝑜𝑠 ( )
𝑎𝑒𝑥
𝑏𝑒𝑥
𝑐𝑒𝑥

(14)

The flux “amplitude” 𝐴 can be determined only by “anchoring” the flux to some
measured or desired quantity. Usually this is the total power 𝑃. Mathematically, this is
because the reactor equation (Eq.(5) or (10)) is homogeneous, and 𝜙 multiplied by any
constant is still the solution. If 𝑃 is used, integrate Eq.(14) over the volume and
neglecting the extrapolation distance gives:
𝐴=
Which:

𝜋3𝑃
8𝑎𝑏𝑐𝐸𝑅 Σ𝑓

Σ𝑓 : the macroscopic fission cross-sextion
𝐸𝑅 : the effective energy released per fission event

The equation above is an approximation, as it is supposed that production, leakage
and absorption occur at a single energy. In reality, neutrons have a range of energies that
is constantly changing as a result of collisions; high energy neutrons slow down through

scattering collisions with atomic nuclei until they are thermalized, while thermal
neutrons can exchange energy with moderator atoms and gain energy. The most
noteworthy simplification in these equations is the assumption that all neutrons are
produced instantaneously at the time of fission. In reality a small fraction of fission
neutrons are delayed because they are emitted as a result of the decay of certain fission
products. These delayed neutrons have profound effects on the behavior of chain
reaction.

11


Chapter 3.

Reactor parameter determinations using
Monte-Carlo Method

This chapter gives a general review, also some background and application on the
Monte-Carlo method; Monte-Carlo applied to particle transport; Monte-Carlo N-Particle
Transport (MCNP) code and variance reduction method. Monte-Carlo methods are often
used when simulating physical and mathematical systems. They are a class of
computational algorithms that rely on repeated random sampling to compute their
results.

3.1.

Introduction

Scientific computing in general and, more specifically, Monte-Carlo methods
date back as far as the 1940s. Von Neumann, Fermi, Ulam and Metropolis played major
roles in research to define the basis of Monte-Carlo methods.

With the constant improvement in computing power, Monte-Carlo simulation
soon became main-stream, and gained popularity in several fields of research. The ability
to sample from a large number of possible scenarios and predict the outcome quickly
found use in finance, operations research and risk analysis. Monte-Carlo methods are
currently used in a large and diverse number of fields. Statistical physics and molecular
modelling make extensive use of Monte-Carlo, as do finance and stock market analysis.
Risk and reliability assessment are well suited to stochastic approximations, while
computer science algorithms, artificial intelligence and game theory inherit its
methodology. Finally, particle transport is a natural target field for Monte-Carlo, whether
that includes ray-tracing for graphics, γ-rays for radiation and biological studies or
neutron transport for nuclear and reactor physics, which is the subject of this thesis.
It was shown that Monte Carlo methods are highly accurate but expensive (in
terms of calculation time). However, the nature of the Monte Carlo method allowed
multiple independent samples can be calculated simultaneously. In fact, ,the only
limiting factor is the data input/output speed between the systems performing the
calculations and the system collecting the results. This allows Monte Carlo codes utilize
effectively on all types of computer architectures, especially on a parallel,
supercomputers, or a clusters.

12


3.2.

Monte-Carlo and Particle Transport
Monte-Carlo methods make use of a sequences of random numbers that can be

reproduced reliably - a pseudo random number generator to simulate particle histories.
Random numbers are generated with each particle history and used to sample probability
distributions, e.g. scattering angles, track lengths distances between collisions.

The Monte Carlo sampling process is summarized below [5]:
Consider a fixed source in a non-multiplying medium with only capture and
elastic scattering. Each history begins by sampling the source distribution in order to
determine the particle’s initial energy, position and direction. After stochastically
determining the distance that the particle will travel before colliding, the material region
and point of collision are determined. By sampling cross-section data, it is determined
with which nuclide the particle collided and whether the collision is a capture or a
scattering reaction. If it is a scattering reaction, the distribution of scattering angles must
be sampled to give a new direction. In the case of elastic scattering, a new energy is
determined by the conservation of energy and momentum. Once the energy, position and
direction have been determined after a collision, the above procedure is repeated for
successive collisions until the particle is absorbed or escapes from a system. Figure 6
shows the lifecycle of the neutron based on the analog Monte Carlo calculation model.
The analog model makes use of natural probabilities that events such as collisions,
fission and capture may occur. Thus, this model is directly analogous to the way in which
the transport occurs naturally.

Figure 6: Lifecycle of a Neutron in Monte-Carlo simulation [6]
13


3.3.

MCNP Code
MCNP version 5.19 and VisualEditor is the main geometry plotting, flux

calculating, shielding and criticality analysis tool used in this study.
MCNP is useful for complex geometry problems that often cannot be modelled
efficiently with computer codes that, use deterministic methods, for example. The code
has the capability of dealing with continuous energies, generalized geometries and timedependent problems. The code deals with the transport of neutrons, photons, electrons,

combined neutrons and photons where the photons are produced by neutron interactions,
or combined photons and electrons, etc.
Important standard features that make MCNP very versatile and easy to use
include a powerful general source, criticality source, and surface source; both geometry
and output tally plotters; a rich collection of variance reduction techniques; a flexible
tally structure; and an extensive collection of cross-section data.
The MCNP input file, created with any generic editor such as Notepad, contains
the geometrical description of the model system, the description of materials for the
system and a selection of cross-sections. The location and the characteristics of the
neutron, photon or electron source, the type of answers or tallies and variance reduction
methods that are needed to improve the efficiency of the calculation, are also specified
in the input file. MCNP has the functionality of providing information such as the
population of particles in a cell, the weight balance of each cell. The code also does
extensive internal checking to find input errors. Furthermore, the geometry plotting
capability in MCNP helps the user check for geometry errors [5].
MCNP can be instructed to make various tallies related to particle current, particle
flux, and energy deposition. MCNP tallies are normalized to be per starting particle
except for a few special cases with criticality sources. Fluxes across any set of surfaces,
surface segments, sum of surfaces, and in cells, cell segments, or sum of cells are
available. Similarly, the fluxes at designated detectors (points or rings) are standard
tallies, as well as radiography detector tallies. A pulse height tally provides the energy
distribution of pulses created in a detector by radiation. Tallies such as the number of
fissions, the number of absorptions, or any aspect of the flux could be calculated with
any of the MCNP tallies. The type of tallies used in this thesis is the track length estimate
of cell flux - tally F4.
14


Besides the tally information, the output file contains tables of standard summary
information to summerize how the problem ran. This information can give insight into

the physics of the problem and the adequacy of the Monte Carlo simulation. If errors
occur during the running of a problem, detailed diagnostic prints for debugging are
given. Printed with each tally is also its statistical relative error corresponding to one
standard deviation. Following the tally is a detailed analysis to aid in determining
confidence in the results.

3.4.

MCNP model of the BME - Reactor
Based on the real structure of the BME - Reactor, a simplified MCNP geometry

model was created in a three-dimensional, Cartesian coordinate system. The center of
the graphite reflector cell in position B7 was taken as the origin (0 0 0) in the x- and y plane and the core’s bottom in the z- plane. In the criticality benchmark of the reactor,
simplification of the geometry is done by neglecting some details of the surroundings of
the core to an extent that does not affect the critical condition significantly. Although the
omitted structures do not have a significant effect, they do affect neutron flux
distribution. Therefore the benchmark model serves only as the basis speculation.

Figure 7: Cross-section top view of MCNP model of BME - Reactor core
15


Figure 8: MCNP geometric model of the BME - Reactor: a) Side view,
b) Cross-section top view (dimensions are given in cm unit)

3.5.

Neutron flux calculations
In MCNP, the most common way to calculate neutron flux in a reactor core is

through the use of KCODE option. Flux and dose rate have been calculated using a cell
flux tally (tally F4) along with DE, DF cards of MCNP. Dose function DF and dose
energy DE cards are used for energy to dose conversion using ANSI/ ANS - 6.1.1 - 1977
data. Since MCNP results are normalized to one source particle, the result has to be
properly scaled to compare with the measured quantities such as flux and dose rate. The
following formula has been used to scale the calculated results.
𝑛𝑒𝑢𝑡𝑟𝑜𝑛
[
]
[
]
𝑃
𝑊
𝜈̅
𝑛𝑒𝑢𝑡𝑟𝑜𝑛
1
1
𝑓𝑖𝑠𝑠𝑖𝑜𝑛
]=
𝜙[
×
×𝜙𝐹4 [ 2 ]
(15)
2
𝐽
𝑀𝑒𝑉
𝑐𝑚 . 𝑠
𝑘
𝑐𝑚
−13

𝑒𝑓𝑓
(1.6022×10
)𝐸 [
]
𝑀𝑒𝑉 𝑅 𝑓𝑖𝑠𝑠𝑖𝑜𝑛

16