Nuclear reactor core model for the advanced nuclear fuel cycle simulator FANCSEE. Advanced use of

TVE-F 17 009 juni

Examensarbete 15 hp Juni 2017

Nuclear reactor core model for the advanced nuclear fuel cycle simulator FANCSEE. Advanced use of Monte Carlo methods in nuclear reactor calculations

Alexander Skwarcan-Bidakowski

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Nuclear reactor core model for the advanced nuclear fuel cycle simulator FANCSEE. Advanced use of

Monte Carlo methods in nuclear reactor calculations

Alexander Skwarcan-Bidakowski

A detailed reactor core modeling of the LOVIISA-2 PWR and FORSMARK-3 BWR was performed in the Serpent 2 Continuous Energy Monte-Carlo code.

Both models of the reactors were completed but the approximations of the atomic densities of nuclides present in the core differed

significantly.

In the LOVIISA-2 PWR, the predicted atomic density for the nuclides approximated by Chebyshev Rational Approximation method (CRAM) coincided with the corrected atomic density simulated by the Serpent 2 program. In the case of FORSMARK-3 BWR, the atomic density from CRAM poorly approximated the data returned by the simulation in Serpent 2. Due to boiling of the moderator in the core of FORSMARK-3, the model seemed to encounter problems of fission density, which yielded unusable results.

The results based on the models of the reactor cores are significant to the FANCSEE Nuclear fuel cycle simulator, which will be used as a dataset for the nuclear fuel cycle burnup in the reactors.

ISSN: 1401-5757, TVE-F 17 009 juni Examinator: Martin Sjödin

Ämnesgranskare: Changqing Ruan

Handledare: prof. Waclaw Gudowski, Blazej Chmielarz

1 Populärvetenskaplig sammanfattning av projektet

Kärnkraftverk kan vara svaret till en hållbar och pålitlig energiutvinning som ersätter förbrännin- gen av fossila bränslen. Idag står kolkraftverk för ca 40% av världens elektricitet men samtidigt har det varit en av anledningarna till de höga koldioxidutsläppen. Ersättning av kolkraftverk med exempelvis kärnkraft skulle kräva en del planering och uppskattning av bland annat, hur mycket kärnbränsle som krävs och hur mycket avfall som måste förvaras eller anrikas för vidare operation.

Dagen teknik tillåter användningen av dator-simuleringar för en bra uppskattning av dessa värden.

FANCSEE är en sådan kärnbränsle-simulator som nuvarande utvecklas i Institutionen för Reak- torfysik på Kungliga Tekniska Högskolan och ska ha ett grafisk användargränssnitt till skillnad från andra programvaror. Detta gör det enklare för beslutsfattare att göra informativa beslut. Pro- gramvaran använder sig av värden framtagna från simulationer av detaljerade modeller baserade på verkliga reaktorer.

I detta projekt har två brett använda reaktormodeller undersökts och modelerats i ett program som simulerar reaktionerna i en reaktorkärna. Resultaten från dessa simuleringar används sedan i FANSCEE för att uppskatta kärnavfall, bränslebehov, energinätkapacitet eller budget för en kärn- reaktor. Har man kunskap av hur mycket kärnavfall som produceras, kan man exempelvis i god tid planera hur mycket som ska förvaras eller berikas för fortsatt användning.

Just nu är FANCSEE fortfarande under utveckling. Fler reaktormodeller ska implementeras för bredare data-bibliotek. Med detta verktyg kommer kärnbränsleplaneringen gå fortare, och vara enklare än någonsin tidigare.

Contents

1 Populärvetenskaplig sammanfattning av projektet 1

2 Introduction 3

2.1 Principles of nuclear reactors . . . 3

2.2 Objective . . . 3

2.3 FANCSEE . . . 4

3 Monte Carlo method 4 4 Serpent 2 5 5 Theory 6 5.1 Process of radioactive decay and transmutation through neutron induced reactions 7 5.2 Bateman Equation . . . 8

5.3 Neutron cross-section . . . 9

5.4 Burnup calculation . . . 10

5.5 Chebyshev Rational Approximation method . . . 11

6 Method 12 6.1 LOVIISA-2 . . . 13

6.2 FORSMARK-3 . . . 17

6.3 Implementation of core design to Serpent 2 syntax . . . 21

6.4 Reactor profile . . . 23

7 Results 24 7.1 LOVIISA-2 . . . 24

7.2 FORSMARK-3 . . . 26

8 Conclusions 28

9 Future work 28

10 Acknowledgements 29

2 Introduction

Nuclear energy provides about 11% of the electricity of the world, while burning of coal provides a staggering 40% [1]. The need for nuclear energy will likely increase as the burning of coal decreases due to the global problem of climate change. Though renewable energy sources are popular with the public, they are not as reliable or stable as nuclear power, hence nuclear power is likely to carry the burden of increased energy demand. For a number of reasons, including safety, politics and environmental concerns, nuclear power is still a controversial topic to the public, but only through continuous research, development and communication these concerns can be addressed. Working towards sustainable, safe and economically viable nuclear power, simulations are crucial to finding scenarios of reactor cycles. These reactor cycles include, but are not limited to, simulations of neutron reactions, core criticality or the burnup of nuclear fuel and nuclear transmutation, the latter two whom will be the objective of this paper.

2.1 Principles of nuclear reactors

The energy generated by nuclear reactors come from the fission of certain heavy metals, ie. elements with proton number between 90 and 100. Nuclear reactors mostly use uranium as nuclear fuel, which is extracted from natural uranium ore. Natural uranium is mostly composed of uranium-238 with low enrichments of the fissile isotope uranium-235 (mass fraction 0.72%), which can be found naturally in the environment. The fuel however, is generally an enriched mixture of²³⁸U and²³⁵U . The measure of how much energy is extracted from the nuclear fuel is called Burnup, also known as f uel utilization and is measured in fissions per initial metal atom, %FIMA or the total energy released per ton of initial mass of uranium [M W d/kgU ].

2.2 Objective

The objective of this paper is to simulate neutronics of two detailed nuclear reactor core designs in the Monte-Carlo particle transport code [2] - Serpent 2, and to simulate burnup cycle calculations of them. From these burnup cycles, depletion matrices will be derived and used in future work to model realistic fuel cycle scenarios in a fuel cycle simulator, FANCSEE being developed at KTH.

Figure 1 shows the process of transforming the results of the burnup cycles to usable data in the FANCSEE software.

The nuclear reactors under consideration will be the Pressurized Water Reactor LOVIISA-2 (VVER- 440/213) in Finland and the Boiling Water Reactor FORSMARK-3 (ABB-III) in Sweden. Both reactor types were some of the first reactors developed in the 1950s and have since been improved on. They are also the most common reactor types which make them important for the FANCSEE project.

Figure 1: The Figure shows parts of the process, from simulating the Burnup cycle to the final software. Red - Developers side, green - User side, blue - focus of the project.

2.3 FANCSEE

The FANCSEE simulator was initially developed at the Royal Institute of Technology in Stock- holm, KTH, by Torbjörn Bäck and prof. Wacław Gudowski, and during the years progressively developed in cooperation with the University of Tartu. The FANCSEE software is in its final stages of development by Błażej Chmielarz, KTH [3], with a ultimate goal to create a user-friendly, graph- ically controlled software, allowing advanced simulations of a nuclear fuel cycle scenario, even for very complex and diversified scenarios and has the ability of tracking 1307 nuclides at any point in the cycle for up to 1100 years. The target audience would not only include scientists in the field, but also for policymakers as a source of well informed decisions and for educational purposes for students. It could also serve as a way of planning nuclear waste repositories, fuel needs, energy grid capacity or budget in a nuclear park.

3 Monte Carlo method

The Monte Carlo method is a stochastic method of approximating solutions to mathematical prob- lems using computational algorithms that rely on repeated statistical sampling from a probability distribution function. It is often used in problems with probabilistic structures where variables follow a pattern. The randomness of the Monte Carlo method is only applied to the degrees of freedom that are defined and limited by the user, and have great advantages over a deterministic method mainly because it does not require new analytical solutions for every change of the system, is a more flexible method and can be applied to a system of any complexity.

4 Serpent 2

Serpent 2 is a universe based stochastic Monte-Carlo code [2]. It utilizes a user defined geometry (2-3 dimensional) with materials and their densities to simulate neutron reactions and transmu- tations of isotopes in a reactor core. The components that assemble the reactor are defined in separate universes, ie. a space which is filled with a specific geometrical structure with a defined material. What is returned by the simulation are the reaction rates of the neutrons, containing atomic density vectors of all isotopes present in a defined space over time, core criticality and is not limited to keeping track of only the fuel.

The Serpent 2 code has predefined geometrical templates, parameters of which can be modified to suit the desired design. The advantage of the universe based approach is that the overall modeling can be divided into smaller modules, or universes, which can be stacked inside of eachother. The universes are a central part of the Serpent 2 syntax; geometries and designs are created in separate levels, independent of eachother and have the option to be nested within one another. Creating a functionable model requires the scale of the nests to match so they don’t overlap or leave undefined space.

When running the script on a Unix-based system (Ubuntu 14.04), any geometrical errors, whether it be overlaps or undefined regions, the Serpent 2 program automatically throws an error, therefore it is critical to check the geometry before proceeding with the simulation. The Serpent 2 program also checks the atomic densities of the materials used, ensuring that the syntax is correctly defined.

It does not however check if the definitions are reasonable.

An example of how universes are nested within eachother is the design of the core. Usually, the highest universe in a core is occupied by the smallest component; the fuel pin, defining the universe as a single pin. The pin is then nested inside of a lower universe, called a pin lattice, a kind of assembly composed of pins. The assembly is then placed in a assembly lattice making up the core of the reactor, as shown in Figure 2. Within these universes, geometrical structures can be implemented and filled with a desired material. Each material is comprised of a list of nuclides whose properties, namely neutron interaction cross-sections at different temperatures, are defined in the cross-section library JEFF-3.1.

Figure 2: The Figure shows the different universes and how they are nested within eachother.

5 Theory

The theory of the nuclear reactor is based on just a few basic concepts, one of them being nuclear fission. Nuclear fission occurs when fissionable nuclides collide with an incoming neutron splitting into fission fragments and high energy neutrons. The probability of which fission fragments the nuclide is split into is described in a fission yield distribution curve, shown in Figure 3 for²³⁵U . The fragments are shown with their respective mass number. The fission fragments are released at high speeds, and it is the kinetic energy of the fission fragments that stands for the majority of the power generated by a nuclear reactor. The high energy neutrons collide with other nuclides and continue the chain reaction. To control the reaction rate, control rods, moderators, neutron poisons are implemented. The design of the reactor is also of great importance to the reaction rate, as well as many other variables. Neutron poisons are often passive means of reaction control, mostly used when inserting fresh fuel loads into the reactor to lower its high reactivity. In the case of the PWR, neutron poisons are always used with fresh fuel batches, decreasing in concentration throughout the cycle. Some neutron poisons get depleted when acquiring neutrons which could in other cases initiate new reactions of other uranium atoms. Control rods work under the same principle as neutron poison in the sense of catching neutrons, but are used for more finer reaction control or complete reactor shutdown, SCRAM, in which case they are called Safety rods. Reactors have several control rods, where some are kept in the core for an even reaction profile, and other are kept outside of the core or partly inserted, depending on the desired rate of reaction and power.

Figure 3: The Figure shows a fission yield distribution curve of²³⁵U . Graph: Nuclear Data Center, Japan Atomic Energy Agency.

5.1 Process of radioactive decay and transmutation through neutron induced reactions

The driving force of nuclear energy released in a reactor comes from the transmutation of isotopes through neutron induced chain reactions. What is necessary for a chain reaction to occur is the presence of fissile elements, those being elements that are able to emit neutrons when splitting into fission fragments through a neutron induced fission reaction. When a neutron collides with a nucleus of an element or a isotope it either gets captured, scattered or it splits the target nucleus and releases energy (more detailed explanation and other reactions are discussed in section 5.3 Neutron Cross-section). If the target nucleus is split (fission), it emits neutrons and creates fission fragments which are isotopes of lesser mass number, shown in Figure 3. In the right circumstances the emitted neutrons may lead to a chain reaction. In the other case, if the neutron is captured by the nucleus of the isotope, it can stabilize it, transmute it to another stable nuclide or become unstable and decay with half-lifes stretching from milliseconds to billions of years. It can decay in multiple ways and transmute into other isotopes, some of which are shown in Table 1. If

A ZX,

where A is the mass number, Z is the atomic number and X is the symbol of the isotope. When a isotope decays its isotopic composition is changed, leading to the transmutation of a new isotope.

This new isotope is called the daughter nucleus.

Mode of decay Particles Daughter

Nucleus

Alpha decay (α) An Alpha particle is emitted (A - 4, Z - 2)

Beta minus (β⁻) An Electron and an electron antineutrino is emitted (A, Z + 1) Beta plus (β⁺) A Positron and an electron neutrino is emitted (A, Z - 1)

Gamma decay (γ) A Gamma ray is released (A, Z)

Neutron emission A neutron is emitted from the nucleus (A - 1, Z) Electron capture The nucleus captures a orbiting electron and emits a

neutrino; daughter nucleus is in a excited state (A, Z - 1) Spontaneous fission Nucleus shatters into two or more smaller nuclei and

other particles -

Table 1: Table of common radioactive decays.

The most common isotope used in reactor fuel is²³⁸U , being about 99% of the mass in naturally found uranium. But it is not a fissile isotope and therefore nuclear fuel is usually enriched with the fissile isotope ²³⁵U , ie. the ratio of ²³⁵U to ²³⁸U is increased. ²³⁵U is exchanged between batches of uranium in the enrichment process, resulting in a depleted and an enriched batch of uranium. When the nucleus of ²³⁵U absorbs a neutron, it either undergoes fission, splitting into fissile elements (Equation 1) while emitting 2-3 neutrons or changes its isotopic composition while emitting a Gamma ray (Equation 2). These neutrons can continue the chain reaction by reacting with other fissile elements.

An example of how²³⁵U reacts when absorbing a neutron is shown below, as well as their proba- bilities of happening.

235

92 U +¹₀n →²³⁶₉₂ U −−→ f ission(eg.^{85% 139}₅₆ Ba +⁹⁴₃₆Kr + 3¹₀n) (1)

235

92 U +¹₀n →²³⁶₉₂ U −−→^{15% 236}₉₂ U + γ (2) When the non-fissile isotope²³⁸U captures a neutron, it transmutes into²³⁹U , then, by β⁻ decay, into neptunium-239 (²³⁹N p) and finally into plutonium-239 (²³⁹P u), which is a fissile isotope (Equation 3). This process of decay takes approximately 2.3 days and yields plutonium with a half-life of 24,110 years.

238

92 U +¹₀n →²³⁹₉₂ U ^β

−

−−→²³⁹₉₃ N p ^β

−

−−→²³⁹₉₄ P u (3)

Because ²³⁹P u is a fissile isotope it has a high probability of undergoing fission when absorbing a neutron. If it undergoes fission, it splits into fission fragments, free neutrons, gamma rays and a neutrino, with a released energy of ∼ 200 MeV; one example of how it can split is shown in Equation 4. While there is a 73% probability that the neutron absorption results in fission, there is also a 27% chance that the neutron will be captured by the nucleus and therefore transmute while emitting a γ − ray as shown in Equation 5.

239

94 P u +¹₀n →²⁴⁰₉₄ P u−−→ f ission(eg.^{73% 134}₅₄ Xe +¹⁰³₄₀ Zr + 3¹₀n) (4)

239

94 P u +¹₀n →²⁴⁰₉₄ P u−−→^{27% 240}₉₄ P u + γ (5) The above Equations are examples of fission fragments, but during fission the produced isotopes can vary as long as the mass number N is balanced. Usually, fission generates two fission fragments and a number of neutrons, depending on the energy spectrum and fissioned nuclide.

5.2 Bateman Equation

The Bateman Equation is a basis for all codes dealing with isotope concentration in a nuclear reactor. It is a mathematical model describing the decay chains of isotopes, or in other words, by how much an isotope decays into its daughter isotope as a function of time based on its decay rate and the initial amount of the decaying isotope. The mathematical model is a first order differential Equation, on the form

dn

dt = −λn ⇒ n(t) = e^−λtn(0) (6)

where n is the number of atoms of an isotope per unit volume [atoms/cm³], also known as atomic density and λ is the reaction rate for the change of one isotope to another. Equation 6 describes the decay of a isotope dependent on time. Equation 7 however, is a system of first order differential Equations, which describe the rate at which isotope i decays and at the same time is created from all parent isotopes that transmute into it through nuclear decay or neutron induced fission. If the parent isotopes decay into its daughter isotope at a quicker rate than the daughter isotope decays, the net rate will be positive, increasing the density of the daughter isotope i.

dn_i(t)

dt = −λ_in_i(t) +

k

X

j6=i

λ_j→in_j(t) (7)

While the calculation of the chain can be solved explicitly with k = 2, the Equation quickly becomes nigh impossible to solve analytically in consequence of infinite chains and numerically

identical reaction rate constants. Therefore, the Bateman Equation can be applied with a variety of state-of-the-art numerical methods, one of them being the Chebyshev Rational Approximation Method (CRAM) for solving the burnup matrix exponential, as computed in Serpent 2 and the FANCSEE software.

5.3 Neutron cross-section

Every material in the reactor consists of different elements whether it is an alloy, the moderator, or the fuel. Material cross-sections are unique for every element and define the ability of the element to absorb a neutron. If the material is neutron poison or a control rod, their material composition needs to contain elements with high neutron capture cross-sections, since they need to capture ex- cess neutrons. The opposite applies to the cladding material around the fuel pellets, it consists of elements with low neutron absorption cross-section, so that neutrons emitted from the fuel passes through the cladding with minimal losses.

For nuclear reactors to work, neutron interactions have to occur in the core. The neutrons can either interact with each other, neutron-neutron interactions, or with the nuclides in matter, the latter being more probable. That is why neutron-neutron interactions are disregarded in the sim- ulations and only neutron-matter interactions are considered. The probability of the reactions happening is described by the neutron cross-section and has the dimension of area [m²]. The standard unit for measuring cross-sections is the barn which is equivalent to 10⁻²⁸ m² [4] and can be conceptualized by area of the target nucleus - the larger the area, the higher the probability of a reaction. The neutron cross-section is given by

σ =µ

n (8)

where n is the atomic density of the target nuclide [atoms/m⁻³] (number of nuclides per unit volume), and µ is the attenuation coefficient [m⁻¹], ie. is characterizes how easily a material can be penetrated by a neutron or energy. If the coefficient is small, the neutron passes through the nuclide as if it was transparent. The cross-section depends on the nuclide as well as on the type of reaction and the kinetic energy of the incident neutron.

There is a large variety of reactions that can occur, but they can be divided into three groups - fission, capture and scattering. Neutron capture is a reaction in which an incident neutron is absorbed by the nucleus and puts the nucleus in a excited state, followed by the nucleus decaying back into its initial ground state, emitting gamma rays. When a Neutron scattering reaction takes place, the target nucleus emits a neutron after a neutron-nucleus interaction. If the energy of the incident neutron is transferred to the nucleus and the emitted neutron, the reaction is called elastic neutron scattering. If the neutron is first absorbed by the nucleus, changing its kinetic energy to the internal energy of the nucleus and remitting the neutron at a significantly lower energy, leaving the nucleus in a excited state, it is called inelastic neutron scattering. Even though the fission reaction is what leads to power generation, it can only do so under proper conditions.

These conditions are partially ensured by neutron scattering events, making them as important as fission. Lastly, the fission reaction is what emits energy that is used to generate the majority of power. The incident neutron collides and enters the fissionable nucleus, creating a compound nucleus and exciting it to energy levels well above the critical energy of the nucleus. This leads to the splitting of the nucleus into two large fission fragments, releasing large amounts of energy in the process and 2 or 3 free neutrons that then interact with other nuclei. A sequence of fission

events is called a fission chain reaction.

If the kinetic energy of the incident neutron is too high (fast neutron), the probability of a fis- sion reaction is typically lower. Neutron cross-sections are highly dependent on incident neutron energies. By scattering reactions the fast neutron decreases its kinetic energy until it reaches the thermal spectrum of energy. The process leads to a higher probability of interaction with a fissile nuclide in a fission reaction.

Figure 4 shows the neutron cross-section, in barn, of ²³⁵U nuclides dependent on the kinetic energy of the incident neutron. If the kinetic energy of the neutron is high (1 MeV), the neutron cross-section is significantly lower, in relation to the neutron cross-section when the neutron has a lower kinetic energy (∼ 10⁻¹⁰ MeV). From the figure it can be seen that the neutron cross-section (in barn) is orders of magnitude greater when the neutron has a lower kinetic energy, than if it would have a high kinetic energy. The region in the middle of the figure is called a resonance region, where the kinetic energy of the neutron and the energy of the target nucleus equal to a excited state of a compound nucleus, creating peaks in the cross-section.

Figure 4: The figure shows the neutron cross-section of ²³⁵U dependent on the energy of the incident neutron (log scale). Graph generated from JANIS NEA with the JEFF-3.1 library.

5.4 Burnup calculation

The main goal of the burnup calculation is to simulate behavior of the reactor during its operation cycle. It is strongly based on the geometry created in the Serpent 2 script and the isotopic com- positions of the materials used. The geometry of the reactor stays the same during the simulation but the material composition changes every step of the cycle, because of the neutron reactions or the spontaneous radioactive decays taking place. Generally, burnup can be defined as a process of nuclear transmutation under neutron flux, which simply means the amount of isotopes (or nuclides) resulting from radioactive decay or neutron-induced reactions.

Before the simulation is run, burnup steps and the amount of neutrons simulated per step are chosen (in our case 10⁷ neutrons per step). These burnup steps can be chosen as the unit time [days] or MegaWatt days per ton of uranium produced [MWd/tU]. The amount of neutrons present in the system will only be a fraction of the amount of neutrons in a real reactor. This is due to the fact that it would be impractical to simulate such a large number of neutrons and, if the chosen number is sufficiently high, it does not cause deviations in the final burnup since Serpent 2 normalizes the reaction rates from thermal power levels of existing reactors, entered by the user.

What Serpent 2 does is it sequentially assigns each simulated neutron energy, direction and distance traveled, using the Monte-Carlo method that stochastically chooses each value from an appropriate distribution. If the neutron enters a material, the code looks at a probability of the neutron reacting with nuclides inside the material and the reaction that would occur, depending on the energy of the neutron and the cross-section of each reaction for every nuclide in the material. This algorithm is called Neutron transport. Every time a reaction occurs, the program takes note of the reaction and how many neutrons are emitted depending on the reaction. From those values it calculates the rate of reaction normalized to either the specific power density of fuel, total power, neutron flux or fission rate in the reactor. In the next step, the material nuclides are corrected, ie the nuclides that have decayed or have been reacted with, are changed and the simulation starts again with these new nuclides for the next burnup step. Each burnup step, Serpent 2 creates a burnup matrix from the Bateman equation and reaction rates yielded by Neutron transport. The burnup matrix is then solved by Chebyshev Rational Approximation Method (CRAM).

5.5 Chebyshev Rational Approximation method

The Chebyshev Rational Approximation method (CRAM) is a state-of-the-art method of burnup matrix exponential calculation. It is the most accurate method of approximation using the burnup matrix, A, from the Bateman Equation, shown in Equation 9. Generally, the eigenvalues of the burnup matrix are confined to a region near the negative real axis, which is a condition well-suited for the use of Chebyshev polynomials, used in CRAM, when approximating the atomic density of the nuclides [3] [5].

When the burnup matrix and the initial atomic density of the nuclide is inserted into CRAM (Equation 10) it returns a function dependent on time of the burnup in the system.

n(t) = e^Atn₀(t) (9)

n(t) = a₀n(0) + 2Re





k/2

X

i=1

a_i(At + θ_iI)⁻¹



n(0) (10)

Where n(0) is the initial atomic density of the isotope, A is the burnup matrix, a₀ is a limiting value of the function at −∞, ai is the residue at the poles θi and I is the identity matrix.

CRAM is used in both Serpent 2 and FANCSEE. In Serpent 2, burnup matrices are used and have both a predictor and corrector step, where the program first predicts the final atomic den- sity, followed by a correction based on the predicted value. For FANCSEE, cycle-averaged burnup matrices are used and are assumed to not need further corrections, thus obtaining solutions much quicker. This is done because the atomic densities from FANCSEE should be quickly approximated as opposed to the computationally expensive corrected values returned by Serpent 2.

6 Method

Both reactor types where developed at the early stages of the nuclear age, but have been improved significantly since then. The LOVIISA-2 reactor in Finland is a Pressurized Water Reactor (PWR) which was first designed in the former Soviet Union. The reactors pressure vessel contains the core surrounded by water, which acts as a moderator and a coolant, being at a temperature of approximately 300 degrees centigrade at a pressure of 123 bar. Because of the pressure, the moderator stays in its liquid form throughout the cycle and the system has therefore a primary and a secondary heat transfer circuit. The primary circuit cycles the moderator through the core, cooling it, and then pumps it into the steam generator where it transfers its heat to the water in the secondary circuit, before it enters the condenser. The water in the secondary circuit reaches its saturation point and transforms into steam before entering the turbine, as shown in Figure 5.

Figure 5: The figure shows a diagram of the Pressurized Water Reactor. Photo: United States Nuclear Regulatory Commission.

FORSMARK-3 is a Boiling Water Reactor (BWR) which was developed by ASEA - Atom in Sweden with a different approach to design. FORSMARK-3 has its vessel at a pressure of 69 bar, allowing the moderator to boil when passing through the core, generating steam and drying it, all inside of the reactor vessel, instead of having pressure in the vessel high enough to prevent boiling, like LOVIISA-2. This requires only one circuit, so the steam passes through the turbine after leaving the vessel. A simplified diagram is shown in Figure 6.

Figure 6: The figure shows a diagram of the Boiling Water Reactor. Photo: United States Nuclear Regulatory Commission.

6.1 LOVIISA-2

The design of the core starts at the smallest components of the reactor - the fuel pellets. These need to be designed in detail, considering both its inner and outer diameter of their cylindrical geometry. The inner diameter and the concave geometry of the pellet is there for when the fuel pellets swell up during operation where the core of the pellet would expand the most because of heat. The pellets are kept in a tube consisting of a thin layer of cladding material, in the case of the PWR VVER-440/213 LOVIISA-2 NPP the cladding is made out of a Zirconium-based alloy such as Zr-1%Nb. The usage of Zirconium alloy is central, because it has many properties desirable for cladding in a PWR - it has a good corrosion and stress-corrosion cracking resistance, which is critical to the tube because the fuel rods oscillate under operation and should not crack. The alloy also has low neutron absorption, ie. it does not absorb as many neutrons as other materials, therefore letting them interact with other isotopes and improving neutron economy (a balance of neutrons created and neutrons lost).

The Zr-1%Nb tubes, or pins as they are called, are filled with fuel pellets stacked on top of each other to a certain point. The rest of the tube contains an area called the gas plenum, where exists a spring that applies a force on the pins so that they remain stacked during transport. The gas plenum serves as a margin for when the pressure increases inside of the pin, because of gases released in fission. The rods are then assembled together into fuel assemblies in a hexagonal geom- etry, each assembly containing 126 fuel rods [6] with the same fuel and one central unfuelled rod, serving as a tube for water flow, with a rod pitch of 1.22 cm [6], ie the distance from the center of one fuel rod to an adjacent rod (Figure 7).

Surrounding the rods and the Zr-1%Nb shroud containing the rods is a liquid, that works as a moderator and a coolant; in the PWR - water mixed with Boric acid at an average concentration of 650 ppm. The role of the coolant in the reactor is to cool the core and transport the heat generated by it to the steam generator. The role of a moderator is to control the rate of reaction of fission, which, in the absence of the moderator would cease instantly, and in the absence of coolant cause core meltdown. The way to decrease the rate of reaction is to absorb some of the neutrons so that there is only a necessary amount of neutrons present for chain reaction criticality.

This is done partly by the diluted Boric acid in the moderator, because of its neutron absorbing properties. The moderator also serves as a way to slow down the neutrons because they need to be moving at the right speed for other nuclides to capture them at the right rate, this is called neutron capture (Neutron cross-section 5.3).

The moderator works as a passive reaction control, but for more active control a control rod is implemented. It can be used as a means of increasing/decreasing the reactor activity or as part of a routine reactor shutdown for inspection by lowering or raising them. The geometry of a control rod is nearly identical to that of a fuel assembly, containing rods of the same size, but filled with Borated Stainless Steel consisting of 20% chromium, 16% nickel and 2% natural boron, ie.

a combination of¹⁰B and ¹¹B isotopes. The control rods are approximately twice as long as the fuel rods, with an upper part being the control rod and the lower part being fuel. They are this way so that the fuel can still be inside of the core when the control rods are raised. Throughout the length of the core, 11 spacer grids of INCONEL alloy 690 are placed equidistantly. These are mesh like components of the core that hold the fuel pins in place so they do not oscillate during operation, caused by the moderator flowing through the core.

The reactor core is assembled by arranging 313 assemblies, where 276 are fuel assemblies and 37 are control rod assemblies [6]. The core is also partly surrounded by shielding assemblies also called reflectors. The reflectors role in the core is to limit neutron leakage, by reflecting the neu- trons back into the core instead of them leaving the core. The core is placed in a 6 cm thick core barrel and then in a pressure vessel of thickness 14 cm, lined with a stainless steel clad on the inside (Figure 8 and 9).

Figure 7: The figure shows the cross-section of the VVER-440 fuel assembly. The distance between the pins is called the pin pitch.

Figure 8: The figure shows the horizontal cross-section of the VVER-440 core.

Figure 9: The figure shows the vertical cross-section of the VVER-440 core, where the top part of the core is the gas plenum and the lines crossing the assemblies are the spacers.

6.2 FORSMARK-3

The FORSMARK-3 NPP has been updated and modernized a number of times to this date. Most importantly, the fuel type has been changed to SVEA-100 which differs from its older counterparts by the 10x10 lattice (100 fuel rods per assembly) and no central tubes, with a larger pitch between the fuel rods (1.24 cm). The pellet formed fuel used in the FORSMARK-3 reactor is uranium dioxide (UO2), largely consisting of²³⁸U with an enrichment of 2.769% ²³⁵U [7].

Surrounding the fuel is the cladding, which consists of Zirconium alloy-2, a low neutron cross- sectional material protecting the fuel from corrosion. The fuel pellets are stacked on top of each other inside of the cladding tube and collected together into a 10x10 square fuel assembly and placed in a sheath of Zirconium alloy-4.

The BWR also distinguishes itself from other reactors by arranging 4 fuel assemblies in a 2x2 manner and having a control blade in the form of a cross between the assemblies (Figure 10). The control blade has 4 symmetrical arms, each containing 21 control rods[8]. The rods are made out of Boron Carbide powder inside of a Type-304 stainless steel tube, then placed inside of the blade frame. When 169 assembly bundles are constructed, they are placed beside each other with a pitch of 15.45 cm, and 24 single assemblies are added symmetrically on the periphery [6]. When the core is assembled it is placed in a 16 cm thick vessel, with an interior cladding of Type-304L stainless steel (Figures 11 and 12). The interior cladding is of great importance because it is usually the vessel that dictates the life of a reactor and the cladding protects the vessel from corrosion. If the cladding looses its integrity, leading to sufficient vessel damage, the reactor has to be decommis- sioned.

The vessel is filled to the top of the core with a moderator composed of demineralized water (as opposed to borated water in the PWR) and kept under a constant pressure of about 69 bar [6].

Just above the core (Figure 12), water is pumped with a temperature of 278C and is then pumped down below the core by 8 equidistant jet pumps. The water then travels through the core, heating up and becoming saturated steam when rising to the top part of the vessel. As the water rises through the core its density decreases as it heats up. The water is defined as layers with different water densities as seen in Figure 12 and Table 2. The steam then reaches the steam separator where the larger droplets get separated from the steam and then the steam dryer before it goes out to the outlet into the turbine at a temperature of 286C. Because the water in the BWR is boiling, its density varies significantly throughout the core.

Figure 10: The figure shows the cross-section of the FORSMARK-3 SVEA-100 fuel assembly bundle with a control blade between the assemblies.

Figure 11: The figure shows the horizontal cross-section of the FORSMARK-3 ABB-III core.

Figure 12: The figure shows the vertical cross-section of the FORSMARK-3 ABB-III core. 1, 2, 3, 4, 5, 6, 7 and 8 are regions of water with decreasing density.

Water ID Axial position in core [cm] Water density [g/cm³]

8 221 - 368 0.2379

7 191 - 221 0.3195

6 162 - 191 0.3662

5 132 - 162 0.4250

4 103 - 132 0.4995

3 74 - 103 0.5910

2 44 - 74 0.6899

1 0 - 44 0.7454

Table 2: Table of moderator density depending on axial position, from top of core to bottom (Core height 368 cm). See Figure 12. [9]

6.3 Implementation of core design to Serpent 2 syntax

Converting the core design into Serpent 2 code needs detailed geometry and correct material definitions to function correctly and provide correct data. All commands used in Serpent 2 are called cards. The Serpent 2 code has a predefined card that describes the fuel rods called PIN.

The PIN card is a universe type that is defined by an string, followed by a material name and the radius of the cross-sectional area that is occupied by the material. Here the PIN is defined as ’pin 1’, followed by the materials and their parameters.

When the necessary PINS are defined, they are assembled together using the LATTICE card, which input is its identification string, lattice type (1 - square, 2 - hex), (x,y) position of center, number of elements in both x and y direction and pitch, which defines the distance between each element. Filler pins, ’pin 3’, are added on the periphery. These are usually a moderator that surrounds the elements of the core. In this case ’lat 10’ is the LATTICE containing 126 ’pin 1’

universes, surrounded by ’pin 3’ which is just defined as water. A single Central tube, ’pin 2’ is placed in the middle of the hexagon.

Universes, either defined as pins or lattices, have the option to add geometrical structures depending on their design. In the VVER-440, the fuel assemblies are placed inside of a hexagonal shroud tube, defined by the SURFACE card. The SURFACE card itself only defines a surface that can be used in multiple universes, but until the CELL card is used, the geometry is not generated. The SURF card defines the boundary of the geometry, its (x,y) position and the radius of the surface, while the CELL fills the surfaces with a universe, in this case - ’lat 10’. It then fills the other surfaces with materials by defining the area between surface 1 and surface 2 with the parameter 1, defining the area outside of surface 1 and −2, the area inside of surface 2.

Every material used in the geometry has a defined nuclide composition. This is done by the MATE- RIAL card, containing parameters of nuclide names and their densities. The material identification is on the form ZZAAA.XXc, where ’ZZ’ is the integer proton number (1-99), ’AAA’ is the mass number and ’XX’ is the temperature estimate at which the isotope will be present in the material, expressed in hundreds of Kelvin. On the same line, the density of the isotope is expressed, either in atomic density (10²⁴/cm³) or mass density (g/cm³). Combining the isotopes, the material ’fuel1’

defines one of the fuels in the core.

With these basic cards, the whole geometry of the reactor core can be generated. For more complicated designs, surface cards are stacked, creating desired geometries.

6.4 Reactor profile

For accurate reaction rates and a balanced core activity, the fuel should not be approximated to be identical in the core, but instead have the most fresh fuel at the periphery, and the most burned in the center of the core, with intermediate burned fuel between the two. After a full burnup cycle, the central, most burned fuel is taken out and replaced by the second to oldest fuel and so on. The previously fresh fuel on the periphery is moved towards the center of the core and its place it taken by new fresh fuel, as shown in Figure 13. If the core would have identical fuel, there would be more reactions occurring in the center of the core, which is preferably avoided because a flattened core power profile is desired.

Figure 13: The blue arrows signify the movement of the nuclear fuel at the end of each fuel cycle.

Different colors of the fuel indicate each zone (FORSMARK-3).

For the LOVIISA-2 reactor, the core is divided into 3 radial zones containing differently burned nuclear fuels. The fuels are moved at the end of each 12-month burnup cycle. In the case of FORSMARK-3, the reactor has 5 radial zones and a burnup cycle of 12 months.

Determining the level at which the nuclear fuels are burned for the different zones is done by simulating the cycle with identical fuel for all zones followed by a calculation of how much of each zone has been depleted. The fuel is then redefined in the Serpent 2 code and placed in appropriate zones for a realistic power profile. The profile is essential for acquiring accurate reaction rates throughout the core.

The reactor core models presented were made to resemble the actual reactor cores active today.

Components that were either confidential or protected by Patents were approximated, as well as components the exact geometry of which did not affect the reactions directly.

7 Results

After modeling the core and running it through burnup simulation, the Serpent 2 code returns several files. It returns files with depletion matrices for all fuel batches and control rods at each burnup step, using a debugging function of the code. It also returns a set of mesh figures showing neutron reactions at every burnup step in the reactor core. The depletion file consists of a set of matrices with rows corresponding to different isotopes and columns to burnup steps. These matrices keep data of atomic density, mass density, fission rates, cross-sections and other useful data.

For each reactor the simulations have been done for 10 different power levels ranging from 60%

to 105% with interval steps of 5%, where 100% power level corresponds to the Thermal Capacity of each reactor [M Wt]. In some circumstances the power grid can only take a certain amount of power from the reactor or the reactivity in the core is too low to sustain the thermal capacity so the power level is adjusted accordingly.

The results below show the atomic densities of some of the most interesting nuclides in one fuel batch. The figures follow the life cycle of one fuel batch, from the point at which it entered the reactor core as fresh fuel (on the periphery) and was taken out as burned fuel, each year replaced closer to the center of the core (6.4 Reactor profile). This explains the sudden discontinuities every 365 days for the SERPENT data in the following figures.

7.1 LOVIISA-2

The Thermal Capacity of the LOVIISA-2 reactor is 1500M Wt [10]. The Figures below show the change in atomic density for some of the most important isotopes as a function of time in days for one full reactor cycle operating at 100% Thermal Capacity.

Figure 14: The figure shows a comparison of data calculated by Serpent 2 (corrected) and FANC- SEE (predicted) of how the isotope concentration for²³⁵U and ²³⁹P u changes through transmu- tation in the LOVIISA-2 reactor at 100% power.

Figure 14 depicts the atomic density of²³⁵U and²³⁹P u as a function of time (in days) for a full life cycle of one fuel batch generated by Serpent 2 (shown by the discontinuous line) and FANCSEE

(continuous line). It shows how the atomic density of²³⁵U decreases throughout the cycle, whilst the atomic density of²³⁹P u increases. The reason is that the²³⁵U nuclides undergo fission through neutron induced reactions, causing a nuclear chain reaction and creating nuclides of lower atomic mass. During a chain reaction, neutrons collide with²³⁸U starting a process of transmutation and since its transmutation chain leads to²³⁹P u, as shown in section 5.1 (Equation 3), the atomic den- sity of²³⁹P u increases substantially in the early days of the cycle. Eventually, the curve of ²³⁹P u balances out as a cause of it transmuting into other nuclides through both fission and capture and the gradual depletion of²³⁵U (5.1 Equation 4 and 5).

Figure 15 shows additional nuclides of ²⁴⁰P u and ²⁴¹Am on a semi-logarithmic graph. Here,

240P u is a nuclide created as a result of the transmutation of ²³⁹P u among others. The only nu- clides that are diminishing are²³⁵U and²³⁸U (not shown), being the initial nuclides in the fuel. It can also be noted that there is a similarity between the curves of²⁴⁰P u,²⁴¹Am and their substrate isotope²³⁹P u, the transmutation rate of which they are highly dependent on.

Comparing the Serpent 2 generated data with the FANCSEE data reveals similarities between the two. In Serpent 2, the CRAM uses the burnup matrix exponential, both when predicting and correcting the reaction rates of the isotopes. This yields a precise graph of the atomic densities but has a long computational time (approximately 8 hours, depending on the number of neutrons in the system). In FANCSEE, the CRAM uses a cycle-averaged burnup matrix exponential and only computes the predicted value. It assumes that the prediction does not need corrections, thus obtaining solutions in the matter of seconds. If the prediction through FANCSEE is well approximated to the corrected data from Serpent 2, the model of the reactor can be used in the FANCSEE program. As seen in Figures 14 and 15 for the LOVIISA-2 reactor, the prediction from the cycle-averaged burnup matrix exponential in FANCSEE is well approximated to the corrected graph from Serpent 2, especially at both the point of origin and the end value.

Figure 15: The figure shows a comparison of data calculated by Serpent 2 (corrected) and FANC- SEE (predicted) of how the isotope concentration for ²³⁵U , ²³⁹P u, ²⁴⁰P u and ²⁴¹Am changes through transmutation in the LOVIISA-2 reactor at 100% power.

Figure 16 is a mesh depicting the fission density of the LOVIISA-2 core at burnup step 9.80 MWd/kgU, ie. where the process of fission occurs. As expected, the fission density is most dense in the central part of the core, signified by the red and yellow colors. The blue scale is neutron

thermalization, caused by neutron scattering in the moderator, where the neutrons loose kinetic energy and enter the thermal spectrum, provided that they are not absorbed. These figures are generated for each burnup step and are convenient when analyzing the core model. If there are any discrepancies in the result, they can provide useful information on where the problem may reside.

Figure 16: The figure shows neutron thermalization (blue scale) and fission density (yellow scale) in a cross-section of the LOVIISA-2 reactor core at 100% power.

7.2 FORSMARK-3

The FORSMARK-3 has a Thermal Capacity of 3300M Wt[11] and the following figures depict the change in atomic density of some of the most important isotopes as a function of time in days in the reactor operating at 100% Thermal Capacity.

Figure 17 and 18 show the atomic densities for ²³⁵U , ²³⁹P u and²³⁵U , ²³⁹P u, ²⁴⁰P u and²⁴¹Am respectively. ²³⁸U is less relevant, so it is left out, since it mostly becomes²³⁹P u. The graph is also limited between 50 days and 1825 days, for clearer results. Here, it can be seen that the average atomic density from FANCSEE is poorly approximated to the data generated from Serpent 2. Not only is the approximation poor, but also the overall result of the atomic density. What the figures seem to show, is that in the middle of the fuel cycle the²³⁵U and²³⁹P u isotopes cross, increasing the content of²³⁹P u to levels higher than²³⁵U . The overall increase of fissile nuclides, observable in Figure 17, is typical to a breeder reactor, which breeds more fissile isotopes than it consumes (ie. having a conversion ratio > 1). The FORSMARK-3 reactor is not a breeder reactor and the atomic densities of²³⁵U and²³⁹P u should approach each other at a later point, towards the end of the cycle.

Figure 17: The figure shows a comparison of data calculated by Serpent 2 and FANCSEE of how the isotope concentration for²³⁵U and²³⁹P u changes through transmutation in the FORSMARK-3 reactor at 100% power.

Figure 18: The figure shows a comparison of data calculated by Serpent 2 and FANCSEE of how the isotope concentration for²³⁵U , ²³⁹P u, ²⁴⁰P u and ²⁴¹Am changes through transmutation in the FORSMARK-3 reactor at 100% power.

Looking at Figure 19, the fission density of a axial cross-section of the FORSMARK-3, it can be noted that the top part of the core has relatively low reactivity. This is caused mainly by the relatively low density of the moderator at the top part of the core. In a BWR, the moderator (water) boils, when passing through the core. This was implemented in the model by dividing the moderator into different levels with densities varying from high density at the lower part of the core and low density at the top part of the core (see Table 2, 6.2 FORSMARK-3). The moderator not only serves as a way to transport heat away from the core, but also as a medium for fast neutrons to scatter against, losing kinetic energy and entering the thermal spectrum. When the neutrons are in the thermal spectrum, the fission cross-section of²³⁵U , which is dependent on the energy

of the incident neutron, is high enough to the point that a fission reaction can take place. But because of the low density of the moderator in the top part of the core, these scattering reactions occur at significantly lower rates compared to the rest of the core. Low scattering reactions lead to few thermal neutrons created, which then results in even fewer fission reactions.

The position of the control blades were also altered to increase the rate of reaction in the core, but so far it too yielded flawed results. The produced figures came from the simulation of the FORSMARK-3 reactor core with fully withdrawn control blades.

Figure 19: The figure shows neutron thermalization (blue scale) and fission density (yellow scale) in a cross-section of the FORSMARK-3 reactor core at 100% power. (Axial cross-section)

8 Conclusions

The process of modeling reactors and simulating them in the Monte-Carlo code Serpent 2, showed to be an appropriate way to approximate the atomic densities of the nuclides present in the core.

From the results, it could be seen that the LOVIISA-2 core model generated appropriate re- sults when it was used to simulate fuel burnup. The simple geometry of the reactor core and a homogeneous moderator throughout the core yielded an even reaction profile. In the case of FORSMARK-3, the moderator had different densities depending on its axial position. This turned out to be problematic because of the approximations made when defining the densities of the moderator, which resulted in incorrect reaction rates throughout the core. The problem may also lay in the definition of the initial fuel composition in the Serpent 2 code - the enrichment of²³⁵U may have been too low, resulting in less reactions occurring in the core. A detailed revision of all defined materials in the syntax will be made.

The fully functional models will prove to be important to the FANCSEE program when simu- lating the fuel cycle, since they can be used to estimate and calibrate the burnup of the nuclides in the nuclear fuel.

Tracking nuclear fuel through its cycle should become easy with the help of a graphical user interface in the FANCSEE software.

9 Future work

Future work for the FANCSEE project will include the modeling of more reactor cores in the Ser- pent 2 code, creating a database of burnup matrices for the software. Some of the major remaining reactor types to model are the Pressurized Heavy water Reactor, Gas Cooled Reactor, Light Water Graphite Reactor and a Fast Breeder Reactor.

A correction of the FORSMARK-3 BWR code is also a priority, since it is a reactor of high interest - a reactor of this type (BWR) is used widely throughout the word and is the dominant reactor type in Sweden.

10 Acknowledgements

I want to thank Prof. Wacław Gudowski for giving me the opportunity to be a part of the FANC- SEE project and for the feedback that I received throughout the duration of my work.

I also want to give huge thanks to Błażej Chmielarz for aiding me in any difficulties that came up during the development of the models as well as the feedback and guidance that I received.

References

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