S D R C SIDEEG SALAH MUSTAFA HASSAN Master of Science Thesis Division of Reactor Physics Stockholm, Sweden June 2011

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S

PATIAL

D

EPENDENCE OF

R

EACTIVITY

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OEFFICIENTS

SIDEEG SALAH MUSTAFA HASSAN

Master of Science Thesis Division of Reactor Physics

Stockholm, Sweden June 2011

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Abstract

The objective of this thesis is to study and understand the behavior of the reactivity coefficients (RCs) in a boiling water reactor (BWR) partially within 25 different segments with different void fractions, with enriched oxide fuel (UOX) core, as well as to evaluate the methodologies exposed in [10]. These two normalization methods (described in chapter 3) are used to analyse the contribution of each segment of the core having different regions (fuel, clad, coolant, moderator and channel box) to the RCs. All the calculations in this work are performed using PSG2 / Serpent – a Continuous-energy Monte Carlo Reactor Physics Burnup Calculation Code.

The overall reactivity coefficient's values are exactly the same for both methodologies, but the partial reactivity coefficients are totally differing from each other. Using the symmetric normalization methodology introduced in chapter 3 section 3.4.3 the partial reactivity coefficients become almost the same for both methodologies. Moreover, the overall reactivity coefficients are exactly the same as well.

Methodologies for reducing the radiotoxicity of the actinides and fission products produced in nuclear fuels are presently under investigation. Therefore, currently the nuclear reactors are designed not only for power generation, but also are used as transmuters as well, by inserting, for example, the minor actinide to a reactor core the safety parameters can get worse. Moreover, it is wiser to study the partial reactivity coefficients of a reactor core.

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Acknowledgments

Initially, I would like to thank much Professor J. Wallenius who was proposed the project idea, for giving me this opportunity to work with this nice group. With all pleasure, I also thank Dr. V.Arzhanov, who was the main guide of my work.

My appreciation goes also to M. Tesinsky for answering my questions and helping about the Serpent plot and to Youpeng Zhang for his helping me about Serpent Code. I convey my special thanks for the corridor group.

I am grateful to my friend N. Harai for his helping about the references, strategy suggestions, etc. Thanks for all my friends and colleagues.

Finally, special thanks to my wife for her patience.

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Abbreviations

LWR Light Ware Reactor BWR Boiling Water Reactor PWR Pressurized Water Reactor NPP Nuclear Power Plant

UOX Uranium Oxide

PSG Probabilistic Scattering Game RCs Reactivity Coefficients

FTC Fuel Temperature coefficient MTC Moderator Temperature coefficient VRC Void reactivity Coefficient

VWC Void Worth

NTE Neutron Transport Equation NDE Neutron Diffusion Equation

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In Europe and Japan, over 30% of the electricity is produced by nuclear energy. In the U.S., the nuclear power share of electricity is about 20%. Currently there are 440 commercial nuclear power reactors operating in 30 countries that produce in total 376 GW of electric power, which constitutes about 15% of the world’s electricity generation. Figure 1.1 shows the share of different sources for the electricity production in the world. About 55 power plants are under construction, to produce about 16% of current capacity, while over 150 are promptly planned, equivalent to 45% of present capacity [1].

The light water reactor's development started after the Second World War in USA. They use normal water (H₂O) as a coolant and neutron moderator, and represent the most common types of nuclear reactors for electricity production [2].

Figure 1.1: world’s sources of electricity generation [1]

The Swedish nuclear reactor park consists only of light water reactors (LWRs). Sweden has a high level of electricity consumption; about 42% of its electricity is produced at nuclear power plants, NPP, and up to half is hydro. Figure

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1.2 below shows the contributions of electricity sources in 2006, 2007 and 2008.

Sweden has ten operational nuclear reactors (1972-) most of them are boiling water reactors (BWRs) and the others are pressurized water reactors (PWRs) as listed in Table 1.1below [1].

Figure 1.2: Swedish sources of electricity generation [1]

1.1 Light water reactors (LWRs)

A vast majority of nuclear power reactors is represented by light water reactors (thermal reactors), which utilize the ordinary (light) water as both moderator and coolant. The normal water (H₂O) has excellent moderating properties; it has well understood thermodynamic properties, economic and is easily available. LWRs can be subdivided into two main categories: pressurized water reactors (PWRs) and boiling water reactors (BWRs) [1, 2].

1.1.1 Pressurized water reactors (PWRs)

In a commercial pressurized water reactor, the fission chain reaction produces heat that warms up the water in the primary coolant loop by thermal conduction through the clad material. The primary coolant is pumped under a high pressure to the reactor core, mostly under a pressure of 15.5 MPa, which prevents the water from boiling in spite of its high temperature. The pressurized water is pumped to steam generators where the steam is produced and then a steam pipeline feeds the turbine plant to rotate the generator for electricity production. The reactor vessel, steam generators, pressurizer and the coolant pumps are enclosed in a containment structure [3, 5].

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1.1.2 Boiling water reactors (BWRs):

In a BWR, the heat produced by nuclear fission in the reactor core will boil the water, which will lead to the steam production. The steam is directly guided through a pipeline to drive a turbine, after which it is cooled down in a condenser and converted back to the liquid phase before entering the reactor core to complete the loop. The cooling water of BWR is usually held at around 7 MPa pressure, and it boils at about 285^°C [3, 5].

Operator Reactor Type MWe net Commercial

operation

OKG Oskarshamn 1 BWR 467 1972

Vattenfall Ringhals 1 BWR 859 1976

Vattenfall Ringhals 2 PWR 866 1975

Vattenfall Forsmark 1 BWR 987 1980

Total 10 7 BWR+3 PWR 9.4 GWe

Table 1.1: Swedish Nuclear Power Park [1]

Since BWRs are widely spread in several industrial countries such as in Sweden, it is very important to evaluate the safety parameters (reactivity coefficients), by understanding the effect of temperature on the reactor core, particularly on the core components such as fuel, clad and coolant. On the other hand, it is a great value to know the contribution of each of these constituents (fuel, clad and coolant) to the reactivity coefficient as described in chapter three.

1.2 Safety parameters

1.2.1 Temperature influence on reactivity coefficients

As any dynamic system, the inborn stability of nuclear reactors can be reached simply by negative feedbacks to ensure the safety of the nuclear reactor.

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Under reactor operation, the energy released from the fission processes in fuel will directly be transferred to the coolant through the clad material. The final temperature distribution in the fuel, clad and coolant is a key point in the reactor design concerning further neutronic analysis of a nuclear reactor. The temperature distribution, which in the most general case is a function of location and time, will affect the values of microscopic and macroscopic cross-sections. Consequently, this will have an influence on various nuclear reactions caused by neutrons.

Accordingly, the reactivity will depend on the temperature changes.

The temperature reactivity coefficients are commonly defined as the change in the reactivity upon temperature change of the i-th component of the reactor core. It might be positive or negative.

i

Ti

  ^

 (1.1)

Where Ti is the temperature of the ith component and the reactivity ρ is defined by 1

k

 k^ (1.2)

Here k is the multiplication factor of the multiplying system in question.

Once we know partial reactivity coefficients, αi we may rapidly evaluate the reactivity change in the i-th component due to the change in temperature, ΔT_i[4, 5]

i Ti

    (1.3)

The main temperature effects in most reactors are the change in resonance absorption of neutron (the Doppler Effect) due to fuel temperature change and the change in the neutron energy spectrum due to change of moderator or coolant density because of temperature, pressure or void fraction changes as in the BWR case.

1.2.2 Doppler reactivity coefficient

The Doppler coefficient (fuel temperature coefficient) plays an important role in the reactor safety. “The fuel temperature coefficient is the change in reactivity per degree change in fuel temperature”.

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1/ Κο fuel

FTC T



  

   (1.4)

FTC is commonly measured in per cent (10^-2), pcm (10^-5), ppm (10^-6) [4].

A negative fuel temperature coefficient is generally considered more important than a negative moderator temperature coefficient. This is because the fuel temperature increases instantly following the increase in reactor power. The two main contributions of the fuel temperature coefficient are the change of the resonance capture in ²³⁸U and the change of fission to the absorption ratio due to the change of fuel temperature [5].

The reactivity is commonly measured in relative units, dollars:

$

 

  (1.5)

Here β is the fraction of delayed neutron. For ²³⁵U fueled reactors, β=0.0065 [6].

1.2.3 Moderator temperature coefficient

In light water reactors, as the name suggests the light water serves as both moderator and coolant. The moderator temperature coefficient (MTC) plays an important role in reactor dynamics. For safe reactors, a negative moderator temperature coefficient is necessary to reach stability during changes in temperature that can be caused, by for example insertion of reactivity. Thus, MTC calculation is a key point in the reactor design process. The MTC coefficient is defined by the change of reactivity per degree change in moderator temperature.

1/ Kο moder

MTC T



  

   (1.6)

Its magnitude and sign (+ or -) is mostly a function of the moderator to fuel ratio [4].

1.2.4 Void coefficient

In BWR reactors the variation of temperature in the core will lead to change of coolant/moderator density, hence the vapor phase of water occurs and the pressure coefficient becomes more important. During the operation of boiling water reactors the more important factor is the void coefficient. The void coefficient is

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due to the creation of steam bubbles in the coolant, and it becomes more important in the reactor when the moderator is saturated or close to saturation conditions. The reactivity void coefficient is defined as the change of reactivity per percent change of void volume [4].

[1/ %]

C_ 



 

 (1.7)

1.2.5 Void worth

Loss of coolant has two main results on the reactor operation. First, the ability of the heat removal is lost leading to overheat and possibly also loss of reactor core integrity. The loss of heat removal capability was one of the main reasons for a core melting accident at the Three Mile Island nuclear reactor.

Second, it can introduce positive reactivity to the system [7]. The coolant void worth represents one of the most important features of the boiling water reactors. It can be expressed in terms of reactivity change when removing the reactor core coolant; mathematically, it can be written as [8].

( ) 10 [5 ]

v withoutcool withcool

W  k k  pcm (1.8)

k is the system multiplication factor.

In BWR, an increase in the steam to water ratio leads to decrease in the reactivity. There are several events that might cause a change in coolant density or occurrence of the void in the core:

 temperature increase of the coolant pursuant to pump failure, crud deposition, plugging of coolant channels, human errors, and inadequate instrumentation; hence this leads to the change in the coolant density and boiling.

 blocking of the coolant circulation as a consequence of coolant freezing in the steam generator, i.e. overcooling. [7]

1.3 Oxide fuels

The nuclear reactor fuel materials used today are mostly based on uranium and thorium isotopes. Uranium plays an important role due to its availability and usability. It can be utilized as a pure metal, an alloy, carbide, or an oxide and other suitable combinations. Uranium is extracted from the uranium ore, which is condensed in a mill after mining and shipped in the oxide form (yellow cake). After

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yellow cake stage the material is converted to the uranium dioxide (UO₂) as a ceramic form, which is the common nuclear fuel used in the commercial nuclear power plants [9].

Different types of nuclear reactors utilize various kinds of fuel elements. For instance, the light water reactors (LWR), which are the common reactors utilized for commercial power generation in Sweden. The fuel pin consists of circular pellets of arranged above each other inside cladding tube made of zirconium alloy.

Its diameter is about one centimeter and height 2.8 to 3.85 meters. These fuel pins are arranged together as a bundle into the fuel assembly to form the square lattice.

The enrichment in ²³⁵U is typically 2.5 to 4 %.

The main disadvantage of uranium dioxide is its poor thermal conductivity, but it has a very high melting temperature, more than 2700ºC, which partially compensates for that disadvantage. Oxide fuels are chemically compatible with the surroundings; they resist corrosion better than most materials. They are more stable under gamma radiation, neutrons and wide temperature variations [5, 9].

1.4 Zirconium alloy cladding

The cladding materials for thermal reactors must have small absorption cross-sections. There are only four elements that can form alloys with high melting points and reasonably low thermal neutron absorption cross-section. They are zirconium, beryllium, magnesium and aluminium. All of these elements have been utilized as fuel cladding. Zircaloy-4 and zircaloy-2 are commonly used in cladding materials; both of them have sufficient mechanical properties and excellent corrosion resistance compared to pure zirconium.

The mechanical strength is improved by the presence of tin and oxygen while the resistance to water corrosion is increased by iron. Presence of nickel allows for the absorption of hydrogen, which is more likely to occur in PWRs than BWRs. Zircaloy-4, which does not contain nickel, is more usable in PWRs whereas in BWRs, Zircaloy-2 is used [9].

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Chapter 2

Cross sections, neutron flux and reaction rates

2.1 Neutron density, neutron beams and collision rates

The neutron population in a reactor is characterized by the neutron density, i.e. the number of neutron in a unit volume around the point of interest, r, at the time t, n(r,t). Mathematically speaking, the neutron density is defined as.

 

^, ³ number of neutrons in ³ about point

n r t dr d r r (2.1)

In either case, the neutron density is measured in cm^-3. This definition accounts for all neutrons irrespective of their speed v or equivalently kinetic energy E = mv²/2.

In what follows, we will restrict ourselves to a stationary situation when the neutron density does not change in time, n(r).

Often we must distinguish neutrons by their kinetic energy. In this case, we characterise the neutron population by the number of neutron in a unit volume around the point in question having energies in a unit energy interval about the energy of interest, n(r,E). More exactly, it is defined as

 

^, ³ number of neutrons in ³ about with energy in about

n r E d dEr  dr r dE E (2.2)

It should be noted that units of n(r,E) are cm^-3J^-1; they are different from those of n(r). Electron-volts are frequently used instead of joules.

The most accurate description of the neutron distribution in a nuclear reactor is given by the angular neutron density that is defined by



^{, ,}



³ number of neutrons in ³ about with energy in about traveling within solid angle about direction



r Ω r Ω r r

Ω Ω

n E d dEd d dE E

d (2.3)

The concept of a neutron beam turns out to be very useful. It is a collimated beam of neutrons all having the same energy and traveling in the same direction. A neutron beam is characterized by its intensity, I, i.e. the number of neutrons passing through a unit area per unit time. The intensity is given by the product of the neutron's speed, v, and the neutron density, n:

I v n (2.4)

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The intensity of a neutron beam is measured in cm^-2s^-1 when we speak about the total neutron density, n(r), otherwise the intensity has units of cm^-2s^-1J^-1 when we deal with the energy dependent neutron density, n(r,E).

Reaction rates are quantities of paramount importance in nuclear engineering. They tell us how many interactions of a certain kind are occurring each second in a cubic centimetre. We will be interested most exclusively in the following kinds of neutron interactions: fission, radioactive capture, absorption, elastic or inelastic scattering. The reaction rate of a certain type, let’s say x, is defined as the number of neutron interactions of the specified kind happening in a unit volume per unit time. Here x is one of the interactions listed above. It is denoted as Rx(r) and measured in cm^-3s^-1. Similarly we may sometimes wish to single out only neutron interactions caused by neutrons of a specified energy, E.

Then we speak about the energy dependent reaction rate, R_x(r,E), that has units of cm^-3 s^-1J^-1. In what follows, we will call any instance of a neutron interaction with matter as a collision [6, 11].

2.2 Cross sections

The interaction of neutrons with nuclei can be described by quantities known as cross-sections. It was found experimentally that the number of collisions of any type, x, in a small volume V, R_coll,x(V), irradiated by a collimated neutron beam of intensity I is given by.

,

 

coll x x B

R V   I N V (2.5)

Here σ_x is just a coefficient of proportionality, but it plays an exceptionally important role in nuclear engineering. We label this coefficient with the subscript x to stress it refers to the specified interaction. Because of its physical dimension, cm², this coefficient was named as the microscopic cross section. The microscopic cross section, in turn, is proportional to the probability of neutron interaction with a single nucleus; it depends on the kind of the nucleus itself, the incident neutron energy, E, and the type of interaction concerned, x. Because of this, we often write the microscopic cross section as σ_x(E).

A closely related concept is that of macroscopic cross section. It stems also from experiments showing that the decrease in intensity, –dI, due to an interaction x in a target is proportional to the intensity itself, I, and the path length, dz.

 

x B

 

dI z  N I z dz

     (2.6)

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Here z is a point in question along the neutron beam path and N_B is the atom density of the background material with unit of a/cm³. The coefficient of proportionality, σ_xNB, is named as the macroscopic cross section, thus by definition.



 _x _xN_B (2.7)

Obviously, the macroscopic cross section is measured in cm^-1.

Various cross-sections can be divided into several categories according to the interaction they describe, e.g. cross-sections of scattering, absorption, fission, etc.

The scattering cross-section itself, σ_s, can be further divided into two subcategories, elastic and inelastic:

s e i

    (2.8)

The absorption cross-section, σ_a, describes the probability that a neutron can be absorbed by a specific nucleus, the absorption can cause a fission, capture or any other reactions such as (n, p), (n, d) etc. The absorption cross-section can be written as.

a f  p 

      (2.9)

The total cross-section, σ_t, is the sum of the absorption and scattering cross- sections; it characterizes the probability of any interaction (or collision as we agreed earlier):

t a s

    (2.10)

The fission cross section of heavy isotopes is very important in the design of a nuclear reactor [4, 6, 11].

The heavy nuclei that can undergo fission by absorption of a thermal neutron are called fissile isotopes; the most important fissile isotopes are ²³³U, ²³⁵U and

239Pu, fission and capture cross-section for ²³⁵U are shown in Figure 2.1. In UOX fuel, ²³⁸U has a lower fission probability by thermal neutrons than ²³⁵U, but it may eventually be transmuted to ²³⁹Pu, which is a fissile material. Figures 2.2 and 2.3 show the fission and capture cross-sections of ²³⁵U compared to ²³⁸U respectively.

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Figure 2.1 The capture and fission cross-sections for ²³⁵U data /ENDF/B-VII.0 cross-section [12].

Figure 2.2 The total fission cross-section for ²³⁸U and ²³⁵U data /ENDF/B-VII.0 cross-section [12].

Incident neutron energy (MeV) Cross-section (b) Cross-section (b)

Incident neutron energy (MeV)

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Figure 2.3 The capture cross-section for ²³⁸U and ²³⁵U data /ENDF/B-VII.0 cross- section [12].

2.3 Neutron flux and reaction rates

For monoenergetic neutrons having speed v, one may show that the collision density of a certain type x is given as

x x

R   nv (2.11)

It gives grounds to define a new quantity, neutron flux, as the product of the neutron density n and the neutron speed v, thus by definition

 nv (2.12)

Thus the collision density, R_x, can be written in terms of neutron flux  as [6]

x x

R    (2.13)

As was pointed out earlier, the reaction rate gives us the number of interactions of the specified kind x occurring in unit volume per unit time. We can easily relax the assumption about neutrons being monoenergetic. Let n(E)dE be the number of neutrons per cm³ with energies between E and E+dE in a neutron field

Incident neutron energy (MeV)

Cross-section (b)

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within a target. Thus the number of interactions due to such neutrons evaluates to ( ) ( ) ( )

x x

dR   E v E n E dE (2.14)

Thus the total collision density is found as

0 0

( ) ( ) ( ) ( ) ( )

x x x

R  ^



E v E n E dE ^



E E dE ^(2.15) 2.4 Multiplication factor

Conceptually, controlling a nuclear reactor means manipulating the neutron population in the reactor. In particular, it is very important to keep the ratio between the number of neutrons in one generation and the number of neutrons in the next generation in a desired range; this ratio is called the multiplication factor, k, and is defined as:

Number of neutrons in one generation Number of neutrons in the preceding generation



k (2.16)

A nuclear reactor is said to be critical if the multiplication factor is unity, k = 1.

Clearly, the fission chain reaction in a critical reactor continues at a constant level.

In the case when the neutron population is decreasing in successive generations of neutrons, the multiplication factor, k, is smaller than unity. Obviously, the reactor power is decreasing also. This mode of operation is called subcritical, k < 1. On contrary, when the neutron population is increasing in successive generations of neutrons, the multiplication factor is greater than unity. This mode of operation is called supercritical, k > 1. [6, 11]

2.5 Mathematical model of a nuclear reactor 2.5.1 Neutron transport equation

The most general and exact description of the neutron population in a reactor is given by the neutron transport equation, NTE, which may also be classified as a linear (we ignore neutron to neutron interaction) Boltzmann equation:

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       

   

     

0 4

, , ,

1 , , , , , , ,

; , , , , ,

( ) , , , , , , ,

4

t

s

f

E t E t E E t

v t

E E E t d dE

E E E t d dE Q E t



  



  





      



     

   

    

  

 

r Ω Ω r Ω r r Ω

r Ω Ω r Ω Ω

r r Ω Ω r Ω

(2.17)

Here Σ_t is the total cross-section and the fission interaction is separated from all other nuclear reactions that are represented by the scattering cross-section, Σ_s. In addition, the delayed neutrons are disregarded in (2.17) for the sake of simplicity.

[11, 18]

2.5.2 Neutron diffusion equation

In many practical situations, the knowledge about the neutron density distribution, n(r,E,t) or even n(r,t), would be enough to design and describe a nuclear reactor. Unfortunately there is no rigorous mathematical procedure to reduce the neutron transport equation (2.17) to a purely differential equation in terms of the neutron density. However, the neutron population in a reactor may be approximately treated as a neutron gas diffusing in the reactor medium. In this case, the energy dependent neutron diffusion equation, NDE, reads as [11, 18]

         

   

     

0

0 4

1 , ,

, , , , , ,

; , ,

( ) , , , , ,

t

s

f

E t D E E t E E t

v t

E E E t dE

E E E t dE Q E t



  



  



       

  

   

  

  



 

r r r r r

r r

r r r

(2.18)

Here, D(r,E) is the so called energy dependent diffusion coefficient.

2.5.3 One group diffusion equation

We can suppress the neutron energy dependence by postulating that all neutrons have the same kinetic energy or equivalently speed. Such a one-group (or one-speed) approximation greatly simplifies the mathematical study of nuclear reactor behavior. Surprisingly enough, this one-speed model may give a reasonably accurate quantitative description of a nuclear reactor if one chooses appropriately

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averaged one-group cross sections. The one-speed neutron diffusion equation reads as [11, 18]

 

^,

             

1 t , _a , _f , ,

D t t t Q t

v t

    

         

r r r r r r r r (2.19)

Here D(r) is the one-group diffusion coefficient.

2.5.4 General neutron flux equation

We can introduce the concept of the effective multiplication factor independently of a particular neutron equation. To this end, we rewrite equations (2.17), (2.18) and (2.19) in a symbolic operator form

 

1 L S Fˆ ˆ ˆ Q v t

 

     

 (2.20)

Here ˆL is the streaming operator; _ˆS is the scattering operator; and ˆF is the fission operator. For example, the streaming operator is [18]

     

       

, , , , , , , : NTE

ˆ , , , , , , : NDE

, , : One‐speed

t

a

E t E E t

L D E E t E E t

D t t

 

  

 

   

  

      

    

  



Ω r Ω r r Ω

r r r r

(2.21)

There is no scattering in one-group model (all neutrons have the same speed) thus we have

   

0 4

0

; , , , , , : NTE

ˆ

; , , :NDE

s

E E E t d dE

S

E E E t dE







       



     



 



r Ω Ω r Ω Ω

r r

(2.22)

Finally, the fission operator is defined as

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   

0 4

( ) , , , , : NTE

4

ˆ ( ) , , , : NDE

, : One‐speed

f

E E E t d dE

F E E E t dE

t



  



   

 



      



   

 

 



 

r r Ω Ω

r r

(2.23)

2.5.5 Effective multiplication factor

We are primarily interested in stationary solutions of eq. (2.20) moreover a nuclear reactor has typically no external source, Q = 0, thus we are seeking solutions to:



^ˆ ^ˆ ^ˆ



0   L S F  (2.24)

This equation says basically that in the stationary (critical) state there is a perfect balance between the major competing processes in a nuclear reactor, namely streaming/leakage, scattering and fission. Generally, eq. (2.24) has no non-trivial stationary solutions. To ensure such solutions and to characterise how far from criticality the reactor is, it is customary to introduce a critical parameter, k, that divides the fission term in eq. (2.24) artificially increasing or decreasing the number of neutrons produced in fission

ˆ 1

ˆ ˆ

0 L S F

k 

 

    

  (2.25)

On physical grounds, it is easy to show existence of a physically meaningful (positive everywhere) solution to eq.(2.25) for some positive k. The introduction of this critical parameter transforms eq. (2.24) into an eigenvalue problem (2.25), which is clearly seen if we rewrite eq. (2.25) as

 

^{L S}^ˆ^^ˆ ^¹^F^ˆ^^^k^ ^(2.26)

Obviously, any eigenfunction,  , is determined up to an arbitrary scalar factor, more important, similar to the matrix eigenvalue problem, there may be many

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17

solutions (countably or even uncountably many). However, it has been proven that under suitable (physically meaningful) conditions there exists a unique positive k and a unique (up to a scalar factor) positive everywhere function  that obey eigenvalue problem (2.26), in addition, this eigenvalue is characterized as being the greatest among other eigenvalues. It is exactly this dominant eigenvalue that is called the effective multiplication factor (constant), k_eff, moreover it is directly linked to the physical definition of the multiplication factor given earlier in (2.16).

Finally, this eigenfunction  is called the fundamental mode or equally often it is referred to as the critical flux.

2.6 Averaged macroscopic cross-sections 2.6.1 Notation

For future reference, let us introduce the following notations:

 Rx [nx /(cm³.s)]: specific reaction rate for a reaction of kind x, (x = a, f, s, …).

 Rx(V) [n_x/s]: total reaction rate in volume V

 Ra [na /(cm³.s)]: reaction rate for absorption.

 Rf [ nf /(cm³.s)]: reaction rate for fission.

 Rv [n/(cm³.s)]: reaction rate for fission neutrons (fission neutron production).

 ^^{( , )}^r ^E [n/J.cm².s]: space and energy dependent (detailed) neutron flux.

 ( )r [n/cm².s]: energy integrated flux.

 ( )V [n.cm/s]: volume integrated flux.

 V [cm³]: volume of a cell.

2.6.2 Energy integrated neutron flux

The energy integrated neutron flux is defined as

0

( ) ( , )E dE

 ^r ^



 ^r ^(2.27)

It is worthwhile to note a notation introduced here, the detailed and the energy integrated quantities are distinguished by the number of their arguments:

 ( , )r E is the detailed flux;

 ( )r is the total (energy integrated) flux;

In the same fashion, we introduce the energy integrated reaction rate

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       

0 0

, , ,

x x x

R ^r ^



R ^r E dE ^



^r E  ^r E dE ^(2.28) As contrast to equation(2.15), we emphasise here that both the macroscopic cross- section and neutron flux may be space dependent.

2.6.3 Energy averaged macroscopic cross-sections

The main idea behind the concept of the energy averaged macroscopic cross sections is to ensure a relationship similar to the definition of reaction rate for a monoenergetic neutron flux(2.13), thus we demand

     

x x

R r   r  r (2.29)

It leads to the following definition of the energy averaged cross-section

0

( , ) ( , ) ( ) ( )

( ) ( , )

x x

x

E E dE R

E dE



 





  



r r

(2.30)

In each region i, we define a total, energy and space integrated, flux as

3 3

0

( ) ( , )

i i

i

V V

d E dEd

 ^

 



^r ^r

 

^r ^r ^(2.31)

It should be noted here, we use capital Φ to denote the volume integrated flux.

A region integrated reaction rate of kind x is defined in a similar way

 

³ ³ ³

, ,

0

( ) ( ) ( ) ( , ) ( , )

i i i

x i x i x i x x

V V V

R R V 



R ^r d ^r 



^r  ^r d ^r

 

^ ^r E  ^r E dEd ^r ^(2.32)

We stress here that it is exactly these quantities, Φ_i and R_x,i, that are readily reported by many Monte Carlo codes, e.g. MCNP, Serpent etc.

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2.6.4 Region averaged macroscopic cross-sections

Volume V of a cell is assumed to be a union of smaller volumes (regions).

1 2 ... _N

V V V   V (2.33)

Very often, any such sub-volume, Vi, will be assumed to be homogeneous.

Similarly, we define a region averaged macroscopic cross section, Σx,i,such that it holds

, ,

x i x i i

R    (2.34)

Hence we arrive at the following definition

3 3

, 0

, 3

3 0

( ) ( ) ( , ) ( , )

( ) ( , )

i i

i

x x

V V

x i x i

i

V V

d E E dEd

R

d E dEd

 



 

   



  

r r r r r r

r r

(2.35)

In the similar manner, a cell of volume V defines the cell integrated flux as

3 3

0 1

( ) ( , )

 ^



 ^V



^r 

 

^r ^r



^N ⁱ

V V i

d r E dEd (2.36)

Correspondingly, the cell integrated reaction rate is given by

 

³

, , ,

1 1

N N

x V x x i x i i

i i

V

R R d R

 





^r ^r







  ^(2.37)

Similarly, we define a cell averaged cross-section by requiring

, ,

x V x V V

R    (2.38)

It follows then

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, 1 ,

,

1 N

x i i

x V i

x V N

V i

i

R _



 

  

 



^(2.39)

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Chapter 3 Methodology 3.1 Serpent Code

All the calculations for this study have been done using PSG2 / Serpent – a Continuous-energy Monte Carlo Reactor Physics burnup Calculation Code. The code development started at the VTT Technical Research Centre of Finland in 2004, with the name Probabilistic Scattering Game “PSG”. The first version (Serpent 1.0.0) was released in October 2008 [13, 14]. The code is still under development.

The serpent code reads continuous-energy interaction data from ACE format cross section libraries, based on JEF-2.2, JEFF-3.1, JEFF-3.1.1, ENDF/B-VI.8 and ENDF/B-VII evaluated nuclear data files. Different integral reaction rates (e.g fission and capture reaction rate) can be calculated using the detectors capability of Serpent and the integral flux in specific region as well, various response functions are available to calculate different reaction rates in different materials and isotopes.

A region could be defined by a cell, an universes, a lattices and a materials, or one can use a three-dimensional super imposed mesh.[15]

3.2 Physical model

A BWR one pin and sub-assembly lattice models were used for all calculations; the sub-assembly consists of one hundred thirty five fuel pins with a moderator channel at the centre of the core. All the pins have the same fuel materials, which is 2.573 enriched UOX fuel surrounded by zirconium alloys as a cladding material and coolant around the clad.

3.2.1 Pin cell model

In view of the fact that the Serpent code is basically a lattice code, the geometry was first simplified into fuel pins consisting of fuel pellets, void and clad as nested annular layers bounded by coolant. The fuel is the uranium dioxide with zircaloy clad.

The simple pin model is 3.85 meters long with a half a meter of water below and one and a half meter above. It has been divided vertically into 25 segments each segment has a different water density and a different void fraction, (see Table 3.1), the idea was recommended by Janne Wallenius. The properties of water below and above were the same as the first and the last segments respectively.

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Figure 3.1: Cross-cut of pin cell model.

The fuel pin dimensions are shown in Table 3.2. Figure 3.1 and Figure 3.2 show the cross-section and axial view of the pin cell model.

From bottom to top the pressure was fixed at 70 bars with temperature 285.15^°C. This pin has been used to define the BWR fuel assembly as in the next section.

Figure 3.2: Axial view of pin cell model

Void

Coolant Clad

Fuel

Void

Clad

0.5 meters of water

25 segments with different

water densities and void fractions 1.5 meters

of water

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Distance from top to bottom [cm]

Void fractions Coolant densities [g/cm³]

535.0-385.0 0.39665 0.4618

385.0-369.6 0.7933 0.1822 369.6-354.2 0.7915 0.1835 354.2-338.8 0.7873 0.1865 338.8-323.4 0.7807 0.1911 323.4-308.0 0.7724 0.1970 308.0-292.6 0.7625 0.2039 292.6-277.2 0.7511 0.2120 277.2-261.8 0.7383 0.2210 261.8-246.4 0.7298 0.2270 246.4-231.0 0.7123 0.2393 231.0-215.6 0.6929 0.2530 215.6-200.2 0.6705 0.2688 200.2-184.8 0.6452 0.2866 184.8-169.4 0.6158 0.3073 169.4-154.0 0.5816 0.3314 154.0-138.6 0.5411 0.3600 138.6-123.2 0.4972 0.3909 123.2-107.8 0.4432 0.4290 107.8-92.4 0.3822 0.4720

92.4-77.0 0.3136 0.5204 77.0-61.6 0.2376 0.5739

61.6-46.2 0.1580 0.6300

46.2-30.8 0.0825 0.6833 30.8-15.4 0.0234 0.7249 15.4-0.0 0.0000 0.7414

0.0 to -50.0 0.0000 0.7414

Table 3.1: Void fractions and coolant densities for the pin cell model.

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3.2.2 BWR sub-assembly model

Figure 3.3 shows the top view cross section of a BWR fuel sub-assembly of a typical 12 by 12 pin model. A large square moderator channel is located at the centre of the fuel sub-assembly. Table 3.2 shows the dimensions of the sub- assembly model.

The moderator channel is also divided vertically into twenty five segments with different void fractions and different water densities, the void fraction values in the moderator channel and the water outside the channel box has been approximated to be typically a half of void fractions values in the coolant channels, recommended by Henryk Anglart, compare Table 3.3 below with Table 3.1.

Pin pitch [cm] 1.2950 Outer radius of fuel [cm] 0.4335 Inner radius of clad [cm] 0.4420 Outer radius of clad [cm] 0.5025 Inner radius of moderator channel box [cm] 1.6742 Outer radius of moderator channel box [cm] 1.7445 Assembly pitch [cm] 15.375 Inner radius of outside channel box [cm] 7.9400 Outer radius of outside channel box [cm] 7.7100 Table 3.2: Dimensions of BWR sub-assembly model [13].

The water properties of the lower and upper plenums were the same as the first and last segment’s moderator properties respectively. The box of moderator channel and the outside box material have the same properties; the density and isotopes compositions are same as clad.

Each segment, i, is split into several regions as explained in Figure 3.4:

1- Fuel 2- Void 3- Clad 4- Coolant

5- Moderator channel walls 6- Moderator in the channel box 7- Outside channel box walls 8- Water outside channel box

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Distance from bottom [cm]

Void fraction Moderator densities [g/cm³]

535.0-385.0 0.39665 0.4618

385.0-369.6 0.39665 0.4618 369.6-354.2 0.39575 0.4624 354.2-338.8 0.39365 0.4639 338.8-323.4 0.39035 0.4663 323.4-308.0 0.38620 0.4692 308.0-292.6 0.38125 0.4727 292.6-277.2 0.37555 0.4767 277.2-261.8 0.36915 0.4812 261.8-246.4 0.36490 0.4842 246.4-231.0 0.35615 0.4904 231.0-215.6 0.34645 0.4972 215.6-200.2 0.33525 0.5051 200.2-184.8 0.32260 0.5140 184.8-169.4 0.30790 0.5244 169.4-154.0 0.29080 0.5364 154.0-138.6 0.27055 0.5507 138.6-123.2 0.24680 0.5662 123.2-107.8 0.22160 0.5852 107.8-92.4 0.19110 0.6067

92.4-77.0 0.15680 0.6309 77.0-61.6 0.11880 0.6577 61.6-46.2 0.07900 0.6857 46.2-30.8 0.04125 0.7123 30.8-15.4 0.01170 0.7332 15.4-0.0 0.00000 0.7414

0.0 to -50.0 0.00000 0.7414

Table 3.3: Void fractions and moderator densities.

As we can see in Figure 3.4 the very first and the last segments consists of only one region, water. In any case, we may write

i ij

j

V 



V ^(3.1)

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26

Here V_i is the volume of the ith segment and V_ij is the jth sub volume of segment i.

3.2.3 Material specifications

In spite of BWR fuel assembly being more complicated than PWR, all the fuel pellets were simply made of uranium dioxide surrounded by a zircalloy-2 cladding. As shown in Figure 3.3 all fuel rods have been made of only one kind of material, 2.573 wt% ²³⁵U enriched UO₂ rod. Table 3.4 shows isotopes composition (normalized) per percentage and fuel density. These for cladding/channel box and coolant/moderator are shown in Table 3.5 and Table 3.6 for respectively.

Note that waters densities was calculated using equation 3.2.

15.375 cm

Figure 3.3: Cross-cut of the BWR sub assembly model

Reflective boundary condition

Reflective boundary condition

Reflective boundary condition

Reflective boundary condition

Moderator and outside

Channel box

Moderator

Coolant channel

Fuel

S D R C SIDEEG SALAH MUSTAFA HASSAN Master of Science Thesis Division of Reactor Physics Stockholm, Sweden June 2011

S

D

R

C

Contents

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