Optimized Control Rod of the Research Reactor BR2

BNEN Belgian Nuclear Higher Education Network BNEN

Vrije Universiteit Brussel • Universiteit Gent • Katholieke Universiteit Leuven • Université de Liège • Université Catholique de Louvain • Studiecentrum voor Kernenergie – Centre d’Étude de l’Énergie Nucléaire●Université Libre de Bruxelles

CONFIDENTIAL

BNEN/2006-2007 Xingmin Liu

Promotor: Prof. Dr. Ir. Jean-Marie NOTERDAEME, UGent

Academic year 2006-2007

Thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Nuclear Engineering

Belgian Nuclear Higher Education Network, c/o SCK•CEN, Boeretang 200, BE-2400 Mol, Belgium

BNEN Belgian Nuclear Higher Education Network BNEN

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Master of Science in Nuclear Engineering

Thesis Summary Page

Name of the student:

Xingmin Liu Title:

Optimized nuclear control of the research reactor BR2 Abstract: At the present time the BR2 reactor uses Control Rods with cadmium as neutron absorbing part. The lower section of the Control Rod is a beryllium assembly cooled by light water. A capsule containing about 190 grams of cobalt granules is inserted between the lower part of the cadmium section and the upper part of the beryllium follower. Due to the burn up of the lower end of the cadmium section during the reactor operation, the presently used rods for reactivity control of the BR2 reactor have to be replaced by new ones. Considered are various types Control Rods with full active part of the following materials: cadmium (Cd), hafnium (Hf), europium oxide (Eu2O3) and gadolinium oxide (Gd2O3). Options for decrease of the burn up of the control rod material in the hot spot, such as use of stainless steel in the lower active part of the Control Rod are discussed. Comparison with the characteristics of the presently used Control Rods types is performed. The changing of the characteristics of different types Control Rods and the perturbation effects on the reactor neutronics during the BR2 fuel cycle are investigated. The burn up of the Control Rod absorbing material, total and differential control rods worth, macroscopic and effective microscopic absorption cross sections, fuel and reactivity evolution are evaluated during ~ 30 operating cycles, which is equivalent to ~ 1000 EFPD of reactor operation. The calculations are performed for the full scale 3-D heterogeneous geometry model of BR2 using MCNP&ORIGEN-S combined method. A criterion for choice of the new control rod types is presented. The main procedures for control of the BR2 reactor are revisited and modified to satisfy the new irradiation conditions. Promotor: Prof. Dr. Ir. Jean-Marie Noterdaeme Approval for submission by promotor: Mentors: Ir. Edgar Koonen Dr. Silva Kalcheva Assessors: Prof. Dr. Ir. Pierre-Etienne Labeau Prof. Dr. Ir.Greet Janssens-Maenhout Academic Year 2006-2007

Belgian Nuclear Higher Education Network, c/o SCK•CEN, Boeretang 200, BE-2400 Mol, Belgium

Acknowledgement

I would like to express my gratitude to all those who gave me the possibility to complete this thesis. I want to thank the department of BR2 of SCKyCEN for providing me such a good thesis and a very comfortable office to do this thesis. I would like to express my sincere gratitude to my promoter Prof. Dr. Ir. Jean-Marie Noterdaeme who kept an eye on the progress of my work. I am very grateful to my mentor Edgar Koonen for his important support, help and for all his kindness. It is difficult to overstate my gratitude to another mentor Dr. Silva Kalcheva. With her patience, she taught me the using of code, gave me a lot of good advices at the beginning and helped me solve all the questions throughout the thesis. Without her help, I could not finish my thesis in time. Finally, I would like to give special thanks to my wife. During the study, she took on all family affairs alone, I owe her too much.

Summary

At the present, the BR2 reactor uses Control Rods with cadmium as neutron absorbing part. The lower section of the Control Rod is a beryllium assembly cooled by light water. A capsule containing about 190 grams of cobalt granules is inserted between the lower part of the cadmium section and the upper part of the beryllium follower. Due to the burn up of the lower end of the cadmium section during the reactor operation, the presently used rods for reactivity control of the BR2 reactor have to be replaced by new ones. A modification design for the control rod will be developed. Chapter 1 mainly introduces the importance of control reactivity for a reactor, and then gives a general idea of how to choose a kind of control rod material with good characteristics; finally, according to historical experiences of BR2 operation, shows the importance and the aim of this thesis. The choice of control rod materials is related to the characteristics of BR2, therefore, in Chapter 2 the main description of BR2 reactor will be introduced, which can help us to know how to choose an appropriate control rod type among those candidate materials for BR2. The central topic in the Chapter 3 is the choice of control rod material. Firstly, the general condition of research reactors in the world will be introduced and the basic parameters including control rod material will be listed. Secondly, Chapter 3 describes the detailed characteristics from macroscopic to microscopic of several important absorbing materials such as cadmium (Cd), hafnium (Hf), europium oxide (Eu2O3), gadolinium oxide(Gd2O3) and stellite. The characteristics of these candidate materials will determine the behavior of the rod in the core during irradiation. The calculation methodologies will be given in Chapter 4. The methods of completing the thesis are chiefly numerical methods (application of appropriate computational tools) complemented with analytical considerations (based on reactor kinetics theory) and experimental methods (measurement of the reactor period). The behavior of the control rod is time-dependent, so the reactor kinetics theory is the main principle, which lies in the basis for determination of the reactor period. Therefore, the first part Chapter 4.1 gives some theoretical aspects of the processes of reactor kinetics for the case of a slab reactor combined with the real parameters of BR2. The aim of this chapter is: starting from the one-speed diffusion equation and through the equations of the reactor kinetics to deduce step by step the formula for the reactor period. Several numerical solutions of the kinetics equations are given in Appendices. The reactor period is the main parameter to be measured and to be used in the experimental technique for determination of control rod worth. Therefore, in the second part Chapter 4.2 the experimental method will be also introduced as a kind of validation and certification method to theory analysis and calculation. The basic computational tools used in this thesis for determination of the control rod characteristics are MCNP, SCALE4.4a and MCNP&ORIGEN-S method. The functions of these codes and how to use them to solve the task of the thesis will be discussed in Chapter 4.3.

The first several chapters mostly introduce some knowledge related to the optimal design. With the theoretical support of these chapters, Chapter 5 mainly discusses what should be calculated, that means, we must know what are the factors affecting the behavior of the control rod in the BR2 and what is the criterion of the optimal design. On the basis of those, the calculation will be started. At first, choosing several ideal CR types with different absorbing material, and then modeling using the MCNP Monte Carlo code complemented with time-dependent evaluations by SCALE4.4a, finally, the process of calculation will be given. The content of Chapter 6 is analysis of calculation results which is the most important section in the thesis. The analysis method is the comparison. We can proceed from three main aspects: comparison of CR material characteristics during irradiation, comparison of the reactor neutronics characteristics of BR2 with different type of CR, and comparison of neutron flux distribution with different type of CR. For the first aspect, the comparison includes five terms: macroscopic absorption cross sections, activity, nuclear heating in the lower active part of the control rods, total control rods worth’s and differential control rod worth; for the second aspect, we do the comparison for the keff and reactivity variations during the operating cycle; for the third aspect, we compare the axial distribution of the neutron flux for both the various types of CR at same position and the same type of CR at the different position of the control rod. The final conclusion is given in Chapter 7. The conclusion of each section in the last chapter will be summarized, according to which, a most optimal CR type is chosen as the new control rod type and the detailed reasons are presented. Although the material has been chosen, the ideal dimension of CR has not yet been decided, therefore, the final step is to compare the different dimensions so as to get a most optimal design.

Contents

CHAPTER 1 INTRODUCTION...... 1 CHAPTER 2 THE OVERVIEW OF BR2...... 4 CHAPTER 3 CONTROL ROD MATERIALS...... 7 3.1 Overview of control rod materials of the research reactor in the world[5] ...... 7 3.2 The characteristics of control rod materials...... 9 3.2.1 Cadmium[7] – thermal absorber ...... 9 3.2.2 Hafnium[9] – resonant absorber...... 10 3.2.3 Boron (boron carbide)[10] ...... 11 [11] 3.2.4 Gadolinium oxide (Gd2O3) ...... 11 [12] 3.2.5 Europium oxide (Eu2O3) – both thermal and resonant absorber ...... 12 3.2.6 Stellite[13] ...... 13 CHAPTER 4 CALCULATION METHODOLOGY ...... 20 4.1 Reactor Kinetics for determination of CR worth[14] ...... 20 4.1.1 Kinetic equations without delayed neutrons...... 20 4.1.1.1 Time – dependent slab reactor...... 21 4.1.1.2 Long – time behaviour...... 25 4.1.1.3 Criticality condition...... 27 4.1.2 Kinetic equations with delayed neutrons...... 30 4.2 Experimental methods for determination of CR worth ...... 34 4.2.1 Estimation of differential control rods worth ...... 34 4.2.2 Estimation of total control rods worth...... 35 4.2.3 Experimental data of total control rods worth and burnup of the lower active cadmium part of the Reference control rods ...... 36 4.3 Introduction of the Code...... 38 4.3.1 MCNP-4C...... 38 4.3.2 SCALE4.4a...... 38 4.3.3 MCNP&ORIGEN-S Method...... 40 CHAPTER 5 CALCULATION OF THE OPTIMAL DESIGN ...... 44 5.1 Definitions...... 44 5.2 Impact of various factors on the CR parameters in the reactor BR2 ...... 45 5.3 Criterion for Control Rod Life ...... 47 5.4 Design modifications and the choosing of optimal projects...... 48 5.4.1 General idea for the design modifications...... 48 5.4.2 Choosing of the optimal projects...... 48 5.5 Neutronics modeling of BR2...... 49 5.6 Calculation contents and process...... 49 CHAPTER 6 ANALYSIS OF CALCULATION RESULTS...... 52 6.1 Comparison of various CR material characteristics ...... 52

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6.1.1 Comparison of macroscopic absorption cross sections during 1000 EFPD ...... 52 6.1.2 Comparison of microscopic absorption cross sections...... 54 6.1.2.1 Evaluation of microscopic absorption cross section during T< 600 EFPD ...... 54 6.1.2.2 Comparison of Neutron Spectrum during 1000 EFPD...... 59 6.1.3 Comparison of activity during 1000 EFPD ...... 61 6.1.4 Comparison of nuclear heating in the lower active part of the control rods...... 63 6.1.5 Comparison of total control rods worth's ...... 64 6.1.5.1 Comparison of Total Control Rods Worth at T=0 and T=1000 EFPD...... 64 6.1.5.2 Comparison of Total Control Rods Worth at BOC (T=0) and EOC (T=30 EFPD)...... 66 6.1.6 Comparison of differential control rods worth ...... 67 6.2 Comparison of the reactor neutronics characteristics for different CR types during BR2 fuel cycle 70 6.3 Control rod effects on neutron flux distributions...... 72 6.3.1 Axial distributions as function of CR position ...... 72 6.3.2 Axial distributions of neutron fluxes as function of energy produced during operation cycle 83 6.3.3 Axial distribution of thermal neutron fluxes in axis of a fuel channel for simulated critical cores at 0 power for different Sh (as in BR02)...... 83 CHAPTER 7 CONCLUSIONS ...... 89 7.1 Summary ...... 89 7.2 Proposed new CR type – Hf+AISI304...... 90 7.3 Final optimal design ...... 91 APPENDIX A 95

A.1 Solution of kinetic equations for step – function input in reactivity ρ...... 95

A.2 Solution of kinetic equations for ramp – function input in reactivity ρ...... 96 A.3 Graphical method for the solution of reactor equations with time – varying inputs ...... 99 APPENDIX B 105 B.1 Upper bound of reactor period...... 105 B.2 Lower bound of reactor period ...... 105 B.3 Intermediate value of reactor period...... 108 APPENDIX C 111 C.1 Subcritical level operation...... 111 C.2 Criticality approach ...... 113 C.3 Getting the reactor critical...... 114

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List of Figures

Figure 2.1 the 3D view of Reactor model ...... Error! Bookmark not defined. Figure 3.1 Total cross section of cadmium isotopes...... Error! Bookmark not defined. Figure 3.2 Total cross section of hafnium isotopes ...... Error! Bookmark not defined. Figure 3.3 Total cross section of gadolinium isotopes ...... Error! Bookmark not defined. Figure 3.4 Total cross section of europium isotopes ...... Error! Bookmark not defined. Figure 3.5 Total cross section of isotopes in Stellite...... Error! Bookmark not defined. Figure 4.1 The eigenfunctions for slab geometry...... Error! Bookmark not defined. Figure 4.2 Time decay of higher order spatial modes in slab reactor...... Error! Bookmark not defined. Figure 4.3 Relative neutron level in the BR2 core as a function of time for a step reactivity change of +0.00144, which is equivalent to +0.2 $ ( or 144 pcm) (Eq. 4-82)…………………………………….34 Figure 4.4 Total CR Worth and cadmium burnup in CR with Cd and Co[9]...... 37 Figure 4.5 MCNP model of a standard BR2 fuel element (the fuel plates are divided into azimuth sectors by each 5° in the hot plane)...... Error! Bookmark not defined. Figure 4.6 A scheme of the combined MCNP&ORIGEN-S method for 3D modeling of the isotopic fuel depletion[15-17]...... Error! Bookmark not defined. Figure 5.1 Total Control Rod Effective Worth for 6 CR_Co with “fresh” and “poisoned” beryllium section under the Cd part of the CR...... Error! Bookmark not defined. Figure 5.2 Total Control Rods Effective Worth for different types CR, located “close” or “far” from the

reactor core centre (used “poisoned” Be in CR, ∆hcd ~ 150 – 160 mm)..Error! Bookmark not defined. Figure 5.3 MCNP model representation of a horizontal cut 15 cm below mid-planeError! Bookmark not defined. Figure 5.4 MCNP model of a BR2 control rod, divided into 10 radial sectors by each 0.05 mm.Error! Bookmark not def Figure 6.1 Comparison of effective macroscopic absorption cross sections of dominant and non-dominant isotopes for various CR types...... Error! Bookmark not defined. Figure 6.2 Evolutions of microscopic effective absorption cross and atomic density of dominant isotopes for different absorbing materials ...... Error! Bookmark not defined. Figure 6.3 the changing of the neutron spectrum and the reaction rates in the rods during irradiation for Cd – rod and for Eu – rod ...... Error! Bookmark not defined. Figure 6.4 Neutron spectra in the lower active part of the CR...... Error! Bookmark not defined. Figure 6.5 The activity of each part of CR and dominant nuclides in the CR during 1000 EFPDError! Bookmark not def Figure 6.6 The total control rods worth of different CR at T=0 and T=1000EFPD

( R0 = ρ(0) − ρ(960mm) )...... Error! Bookmark not defined.

Figure 6.7 Comparison of total CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material

for individual CR types ( R0 = ρ(0) − ρ(960mm) )...... Error! Bookmark not defined.

Figure 6.8 Comparison of total CR worth: (a) for various fresh CR materials at BOC of the 1st cycle; and (b) at EOC of the 1st cycle, i.e. after T ~ 30 EFPD; (c) to (e) comparison of total CR worth’s at BOC and EOC during the 1st cycle for individual CR types...... Error! Bookmark not defined. Figure 6.9 Comparison of total CR worth for various CR types (a) at BOC of the 30th cycle (T~1000 EFPD); (b) at EOC of the 30th cycle (T~1030 EFPD)...... Error! Bookmark not defined.

Figure 6.10 Comparison of differential CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for different CR types...... Error! Bookmark not defined. Figure 6.11 Comparison of differential CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for individual CR...... Error! Bookmark not defined. Figure 6.12 Comparison of differential CR worth at BOC (T=0) and EOC (T ~ 30 EFPD) for different CR types ...... Error! Bookmark not defined. st th rd Figure 6.13 Evolutions of keff and reactivity during the 1 , 15 and the 30 operating cycle (all values of keff are normalized to unity at BOC)...... Error! Bookmark not defined. Figure 6.14 Comparison of criticality variation for individual CR types during long time of irradiation (the position Sh at BOC for the corresponding CR is kept constant during irradiation).Error! Bookmark not defined.

Figure 6.15 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel H1/Central...... Error! Bookmark not defined.

Figure 6.16 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel A30 in the central crown...... Error! Bookmark not defined.

Figure 6.17 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, placed in channel C41 close to a Control Rod location.....Error! Bookmark not defined.

Figure 6.18 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in reflector channel E30, which is located near Control Rod channel...... Error! Bookmark not defined.

Figure 6.19 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel H1/Central...... Error! Bookmark not defined.

Figure 6.20 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel C41 close to Control Rod channel...... Error! Bookmark not defined.

Figure 6.21 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in reflector channel E30 close to Control Rod channel.Error! Bookmark not defined.

Figure 6.22 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel H1/Central versus energy produced during typical BR2 cycle.Error! Bookmark not

Figure 6.23 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel A30 in the central crown versus energy produced during typical BR2 cycle...... Error! Bookmark not defined.

Figure 6.24 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel B180 in the central crown versus energy produced during typical BR2 cycle...... Error! Bookmark not defined.

Figure 6.25 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel H1/Central versus energy produced during typical BR2 cycle.Error! Bookmark not

Figure 6.26 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel A30 in the central crown versus energy produced during typical BR2 cycle...... Error! Bookmark not defined.

Figure 6.27 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel B180 in the central crown versus energy produced during typical BR2 cycle...... Error! Bookmark not defined. Figure 6.28 MCNP whole core model of configuration 4 (~ similar to the load for BR02, [19]).Error! Bookmark not defin

Figure 6.29 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in the central crown versus in simulated critical cores, similar to the load of BR02[19]...... Error! Bookmark not defined.

Figure 7.1 Comparison of axial distributions of thermal (a) and fast (b) fluxes in typical fuel channels for different optimization dimensions of the Hf+AISI304 rod...... 92 Figure 7.2 Comparison of differential CR worth for different dimensions of Hf+AISI304 rod...... 92 Figure A.1 Relative neutron level of U-235 reactor versus time for positive step – function reactivity

changes, Λ =10−4 sec. (δk = ρ) ...... Error! Bookmark not defined.

Figure A.2 Front edge shape for a step – function change in reactivity showing results of approximate

formula (A-3), (δk = ρ) ...... Error! Bookmark not defined.

Figure A.3 Relative neutron level versus reactivity remaining in a reactor for various ramp – function

reactivity change rates (δk = ρ) ...... Error! Bookmark not defined.

Figure A.4 Reactor period versus multiplication factor for various ramp – function reactivity change rates

(δk = ρ) ...... Error! Bookmark not defined.

Figure A.5 Relative neutron level versus time for various ramp function reactivity change rates, using

approximate formula (99) (δk = ρ) ...... Error! Bookmark not defined.

Figure A.6 Circuit diagram of variable resistance lumped – inductance series circuit.Error! Bookmark not defined. Figure A.7 Voltage versus current diagram, indicating the graphical construction required to solve the transmission – line problem...... Error! Bookmark not defined. Figure A.8 Construction for series – inductance variable – resistance example.Error! Bookmark not defined. Figure A.9 Transmission – line diagrams for single – group delayed – neutron equations:Error! Bookmark not defined. Figure A.10 Construction diagrams for solution of reactor – kinetic equations having single – group delayed neutrons and ramp input in ρ...... Error! Bookmark not defined. Figure B.1 Period as a function of reactivity indicating upper and lower bounds for a start-up accident.

Prompt critical 7.5×10−3 ...... Error! Bookmark not defined.

Figure B.2 Period attained by a reactor during a start-up accident at a given protection level vs. the number of decades below the protection level reactivity insertion started. Start-up from –13% reactivity

(δk = ρ) ...... Error! Bookmark not defined.

Figure C.1 Subcritical multiplication...... Error! Bookmark not defined. Figure C.2 Relative power level as a function of reactivity in the reactor for infinitely slow reactivity

change, (δk = ρ) ...... Error! Bookmark not defined.

Figure C.3 Plot of 1/c.r. vs. amount of uranium, showing the effect of the instrument placement.Error! Bookmark not de Figure C.4 Power level of a reactor as function of per cent reactivity remaining in the reactor for given linear rates of change of reactivity...... Error! Bookmark not defined. Figure C.5 Period vs. reactivity of a reactor for given rates of change of reactivity.Error! Bookmark not defined. Figure C.6 Period on which a reactor would go through criticality by inserting reactivity at given linear

rates. Λ =10−4 sec. (δk = ρ) ...... Error! Bookmark not defined.

List of Tables

Table 2-1 the main design parameters of BR2 ...... 6 Table 3-1 Some of the main research reactors in the world ...... 8 Table 3-2 the main isotopes of cadmium...... 10 Table 3-3 the main isotopes of hafnium ...... 11 Table 3-4 the main isotopes of gadolinium ...... 12 Table 3-5 the main isotopes of europium ...... 13 Table 4-1 Properties of delayed neutrons and photo neutrons for HEU fuelled (~ 90 % 235U) and beryllium reflected reactor...... 31 Table 4-2 MCNP calculation of effective thermal microscopic cross sections in typical fuel channel of the reactor BR2 ...... 40 Table 4-3 Isotopic composition of the depleted fuel, evaluated with ORIGEN-S and used in the MCNP model...... 42 Table 5-1 Impact of various factors on total worth for the Reference CR ...... 45 Table 6-1 Calculated macroscopic absorption cross sections Σ in fresh CR material (T=0) and burnt CR material (T ~ 600 - 1000 EFPD)...... 54 Table 6-2 Effective microscopic cross sections of 113Cd...... 56 Table 6-3 Effective microscopic cross sections of 151Eu ...... 56 Table 6-4 Nuclear heating (Watt per gram) in the lower part of the Control Rods...... 64 Table 6-5 Comparison of total worth for different control rod types accounting for the axial burnup of the absorbing material during irradiation ...... 64 Table 6-6 Comparison of the positions Sh [mm] at criticality for different CR types at BOC of the 1st operation cycle (loaded fresh CR absorbing material) and for burnt CR absorbing material after ~ 30 operation cycles (i.e., equivalent to about ~ 1000 EFPD of irradiation)...... 70 Table 7-1 The final optimal design of Hf+AISI304 for CR of BR2...... 92

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List of Abbreviations

BOC – Beginning Of Cycle BR2 – Belgium Reactor 2 CR – Control Rod EFPD – Effective Full Power Days EOC – End Of Cycle FE – Fuel Eement HEU – Highly Enriched Uranium (standard BR2 fuel, containing ~ 90% 235U) MTR – Material Test Reactor NTD – Neutron Transmutation Doping

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CHAPTER 1 INTRODUCTION

A nuclear reactor which has to be operated at steady state conditions is initially charged with a significantly larger amount of fuel in order to maintain criticality during long time of operation. The fuel loading and fuel enrichment when inserted into the core must be estimated to have sufficient excess reactivity to allow full power operation for a predetermined period. This excess reactivity will compensate the decrease of the multiplication factor of the system due to the negative reactivity feedback of fuel depletion, fission product poisoning and (eventually) temperature and pressure effects. However, to compensate for the excess reactivity at the beginning of an operation cycle, a certain amount of negative reactivity must be introduced into the core, which one can adjust or control by desire. This control reactivity can be used both to compensate for the excess reactivity necessary for long term core operation and also to adjust the power level of the reactor, and finally to shut down the reactor. The determination of control reactivity requirements and the choice of control rod absorbing materials for the various types of control elements is a very important aspect of nuclear reactor core design. Generally, the material selected for control rods should have a good absorption cross section for neutrons and have a long lifetime as an absorber (not burn out rapidly). Materials with very high absorption cross section may not be preferred because they can disturb strongly the neutron flux and the power in the vicinity of the rod and hence generate big reactivity perturbations in the core. For this reason a special design of the control rod may be required for materials with very high absorption cross section. The same amount of reactivity worth can be achieved by manufacturing the control rod from material with a slightly lower cross section and by loading more of the material. This also results in a rod that does not burn out as rapidly. Another factor for control rod material selection is that materials which resonantly absorb neutrons are often preferred to those that merely have high thermal neutron absorption cross sections. Resonance neutron absorbers absorb neutrons in the epithermal energy range. The path length traveled by the epithermal neutrons in a reactor is greater than the path length traveled by thermal neutrons. Therefore, a resonance absorber absorbs neutrons which have their last collision farther from the control rod than a thermal absorber. This has the effect of making the area of influence around a resonance absorber larger than around a thermal absorber and is useful to maintain a flatter flux profile. The ability of a control rod to absorb neutrons can be adjusted during manufacture. A control rod that is referred to as a "black" absorber absorbs essentially all incident neutrons. A "grey" absorber absorbs only a part of them. Grey rods are sometimes preferred because they cause smaller depressions in the neutron flux and power in the vicinity of the rod. This leads to a flatter neutron flux profile and more even power distribution in the core. Materials with a very high absorption cross section may not be desired for use in a control rod, because it will burn out rapidly due to its high absorption cross section, unless the burning isotopes are transmuted into another ones having also high absorption cross section.

1 CHAPTER 1. INTRODUCTION

The most commonly used elements for reactivity control in research reactors are presented by rods or plates of strong neutron absorbers (such as boron, cadmium, hafnium, gadolinium, europium or combination of these materials with “grey” absorbers), which can be inserted into or withdrawn from the core. Historically, the earliest reactivity control of the BR2 reactor core has been maintained by control rods with full length made of cadmium as absorbing material. The experience has shown that the lower edge of the control rod, which is exposed to highest thermal neutron flux, is burning out under irradiation mainly due to depletion of the dominant cadmium isotope 113Cd. The neutronography analysis has shown that after about 650 EFPD the cadmium length is reduced by about ~ 295 mm for cadmium thickness ~ 2 mm. Therefore the developed control rods have been made with larger cadmium thickness ~ 4 to 5 mm. This allowed prolonging the control life: for the same irradiation period (~ 650 EFPD) the reduction of the cadmium length has been diminished by about 100 mm. The next improvement of the control rods life has been implemented by the use of “grey” material (cobalt), placed in the lower absorbing part of the control rod. Cobalt is not burning out too fast and has high absorption cross section, which is sufficient to depress the thermal neutron flux in the vicinity of the hot edge of the control rod. The implementation of these control rods has allowed depressing the depletion in the lower cadmium edge: for an irradiation period ~ 1000 EFPD the reduction of the cadmium length could be diminished to about 60 mm. At nearest future, the control rods of BR2 will reach the end of their control lives, i.e., these control rods can not be used any more due to the burning up of the lower edge of the active absorption part of the rod, which will be replaced by new ones. The optimization design of control rods will be developed to seek a new type of CR with longer control life which must satisfy the new irradiation conditions at the same time. In order to complete this aim, the following main works need to be developed: y Focus on the choice of main absorbing material for the active part of the control rod among various “black” absorber materials, such as cadmium, hafnium, europium and gadolinium and “grey” materials, such as stellite. y Consider various absorbing material, especially combinations of “black” absorbers with “grey” ones, (e.g. stainless steel) in order to diminish burnup of the “black” absorber and to flatten the neutron flux and power distributions in the core. y Analyze the results and chose a most optimal control rod type. The chosen new control rods must satisfy the following requirements for the reactivity control of the reactor BR2: y They must provide the necessary negative reactivity for adjustment of the power level during long time of operation (~ 30 effective full power days for the BR2 reactor). y They should not disturb too strongly the neutron flux and power distributions in the core. y The chosen main control rod absorbing material should have high thermal and high epithermal absorption cross sections. Too high thermal absorption cross section is not desired, because this may require a significant change of the control rod design

2 CHAPTER 1. INTRODUCTION

(geometry and dimensions). y The control rod material should not burn too fast and should be used at least during 5 calendar years or equivalent to about ~ 25 BR2 operating cycles.

CHAPTER 2 THE OVERVIEW OF BR2

The Belgian Reactor 2 (BR2) (see Fig. 2.1) at the SCK•CEN site at Mol, developed in 1957 and taken into service in 1963, is a high flux nuclear reactor for a variety of materials, engineering and physics experiments. Currently the reactor is used for the internal R&D programs of the SCK•CEN, for international programs such as the fusion program and for the production of radioisotopes and neutron transmutation doping (NTD) silicon[1]. The BR2 is cooled and partly moderated by ordinary water pressurized to about 12bar and consists of a compact core of roughly 1m in diameter and 1m of height with highly enriched uranium fuel elements in a beryllium matrix, which contains 79 cylindrical channels that can be loaded with fuel elements, control rods, experimental setups and beryllium plugs. The cooling equipment has an ultimate capacity of 125MW but usually the reactor operates at 60MW during 5 cycles of 21 days. The maximal thermal neutron flux approaches 1015 n/(cm² s) and the fast neutron flux (>1MeV) inside a fuel element can reach 5x1014 n/(cm² s). The main design parameters of the reactor BR2 are tabulated in Table 2-1.

Figure 2.1 the 3D view of Reactor model[2] The methods used to control the reactivity of BR2 are control rods and burnable poisons. The control rods of BR2 are of two types, shim-safety and regulating rods. They are designed to operate from the top of the reactor. Each is mechanically independent of the other control rods

4 CHAPTER 2. THE OVERVIEW OF BR2 and each can be inserted in any standard reactor channel (84mm). There is, therefore, great flexibility in control rod location and choice of the reactor core configuration[3]. a). Shim Rods – safety rods The shim control –safety rods provide both the coarse normal operational control and the safety control. Each mobile rod consists of the two sections. The lower section is a beryllium assembly cooled by water. The upper section is a round cadmium tube clad with aluminum on both sides. The cadmium section is completely inserted in the active core when the rods rest on their shock absorbers. A capsule containing approximately 190g of cobalt particles is inserted between the lower end of the cadmium section and the upper end of the beryllium assembly. The cobalt particles are inside a leak proof capsule. The control rods used at present, are as described in this chapter. It is foreseen to use in the future the original control rods type which has previously been used in BR02 in order to use the available spare parts. The only difference concerns the cobalt capsule which is not present and is replaced by cadmium (the cadmium part is longer). The length of cadmium is 963.5mm. With the rod fully withdrawn, (in the most reactive position), this beryllium section is in the core section of the reactor (height 914.4 mm). Primary cooling water flows through the tubular and annular passage of the beryllium section, the cobalt capsule and the cadmium section. b). Regulating Rods Two new regulation rod mechanisms were installed during the refurbishment. They worked now for nearly ten years without problem and so proved a very reliability. One or two regulating rods can be loaded in the reactor vessel and only one rod can move at a time. Only one rod is required and operational in the regulation loop. The regulating rod is loaded in a reactor channel where its reactivity worth is less than 0.48 $ (equivalent to 345 pcm). The regulating rod provides the following functions: y Automatic start-up of the reactor from 1 to 100% power; y Stabilization of the reactor power at a constant level; y Automatic slow or fast set back of the reactor power. The neutron absorbing part is permanently attached to the drive mechanism with no provision for scram release. It consists of an aluminum cadmium-aluminum co extruded cylinder. The cadmium is cold welded inside the aluminum cylinder. Each end of the cylinder is ended by guide pieces. c). Burnable poisons A standard BR2 fuel element contains HEU (93%~72%) fuel with density 1.3~1.7 gUtot/cm3 235 under the form of UAlx matrix and the mass of U per fuel element is 400 or 330 grams. Burnable poisons B4C and Sm2O3 are homogeneously mixed into the fuel meat of the fresh fuel element[4]. 10B acts similarly to a control rod and reduces the over-all control rod motion during the operating cycle by increasing the absorber material at the beginning of cycle (BOC) and depleting a major portion of it before the end of cycle (EOC), thereby compensating the fissile material consumption. Thus the application of 10B always provides more reactivity at any given

5 CHAPTER 2. THE OVERVIEW OF BR2 time than in a case with no burnable absorbers. 149Sm is used to reduce the control rod motion at the start-up until 135Xe and 149Sm have reached equilibrium concentrations. Table 2-1 the main design parameters of BR2 Reactor type Tank-in-pool Primary flow Water in closed loop Speed between fuel plates, m/s 10.4 Temperature , T of inlet − T of outlet, °C 40 − 57 Fuel plate temperature, °C 150 Power level (MW) 50 to 100 Maximum heat flux (W/cm2) 470 -2 -1 15 Maximum neutron flux (n.cm .s ) 1.2×10 (En<0.5 eV), 14 8.4×10 (En>0.1 MeV) Number of fuel elements in the equilibrium core 32~33, fresh and burnt fuel elements, with varied mean fuel depletion between 0% and 50% Lattice hexagon pitch (cm) 9.64438 Parameters of a standard fuel element (FE) (Common characteristics for all used fuels) Number of Material Test Reactor (MTR) -type 18 plates (6 concentric tubes) fuel plates Channel diameter (cm) 8.42 ± 2 Plate length (cm) 97.0 Fuel length (cm) 76.2 Fuel meat thickness (cm) 0.051 Plate thickness (cm) 0.127 Al clad (cm) 0.038 Water gap between plates (cm) 0.3

Fuel types UAlx UAlx UAlx 235U enrichment (wt%) 93% 72% 73% 235U mass per FE (grams) 400 330 400 Total U mass per FE (grams) 430 458 553 Fuel density, gU/cm3 1.3 1.3 1.7 Burnable poisons, homogeneously mixed into the fuel meat:

Natural boron in B4C (grams) 3.8 1.8 3.2

Natural samarium in Sm2O3 (grams) 1.4 1.3 1.4 Fuel consumption Mean depletion at elimination (235U+239Pu) 50% 42% 55% Cycle length (day) 21~28 21 21~28 Number of batches 4~3 2.8 5~4 Control Rods Type Shim safety and regulating rods Absorber material cadmium Number of safety shim rods 6 operational +2 reserve Number of regulating rods 2 shims (only 1 operational) 6

CHAPTER 3 CONTROL ROD MATERIALS

At present, many materials with high absorption cross section can be used as control rod, such as indium, silver and cadmium, boron, cobalt, hafnium, gadolinium, and europium. The content of this chapter is mainly to introduce condition of using control rod materials of the research reactor in the world and realize the characteristic of every material.

3.1 Overview of control rod materials of the research reactor in the world[5]

Research reactors comprise a wide range of civil and commercial nuclear reactors which are generally not used for power generation. The primary purpose of research reactors is to provide a neutron source for research and other purposes. Their output (neutron beams) can have different characteristics depending on use. They are small relative to power reactors whose primary function is to produce heat to make electricity. Their power is designated in megawatts (or kilowatts) thermal (MWth or MWt), but here we will use simply MW (or kW). Most range up to 100 MW, compared with 3000 MW (i.e. 1000 MWe) for a typical power reactor. In fact the total power of the world's 283 research reactors is little over 3000 MW. Research reactors are simpler than power reactors and operate at lower temperatures. They need far less fuel, and far less fission products build up as the fuel is used. On the other hand, their fuel requires more highly enriched uranium, typically up to 20% U-235, although some older ones use 93% U-235. They also have a very high power density in the core, which requires special design features. Like power reactors, the core needs cooling, and usually a moderator is required to slow down the neutrons and enhance fission. As neutron production is their main function, most research reactors also need a reflector to reduce neutron loss from the core. There is a much wider array of designs in use for research reactors than for power reactors, where 80% of the world's plants are of just two similar types. They also have different operating modes, producing energy which may be steady or pulsed. A common design is the pool type reactor, where the core is a cluster of fuel elements sitting in a large pool of water. Among the fuel elements are control rods and empty channels for experimental materials. Each element comprises several (e.g. 18) curved aluminium-clad fuel plates in a vertical box. The water both moderates and cools the reactor, and graphite or beryllium is generally used for the reflector, although other materials may also be used. Apertures to access the neutron beams are set in the wall of the pool. Tank type research reactors are similar, except that cooling is more active. The control rod materials of this type reactor are mostly In, Ag and Cd, Hf, and Eu. Other designs are moderated by heavy water or graphite. A few are fast reactors, which require no moderator and can use a mixture of uranium and plutonium as fuel. Homogenous type reactors have a core comprising a solution of uranium salts as a liquid, contained in a tank about 300 mm

7 CHAPTER 3. CONTROL ROD MATERIALS diameter. The simple design made them popular early on, but only five are now operating. The control rod materials of this type reactor are mostly B4C, Cd, stainless steel, etc.. The IAEA lists several categories of broadly classified research reactors. They include 60 critical assemblies (usually zero power), 23 test reactors, 37 training facilities, two prototypes and even one producing electricity. But most (160) are largely for research, although some may also produce radioisotopes. As expensive scientific facilities, they tend to be multi-purpose, and many have been operating for more than 30 years. Russia has most research reactors (62), followed by USA (54), Japan (18), France (15), Germany (14) and China (13). Many small and developing countries also have research reactors, including Bangladesh, Algeria, Colombia, Ghana, Jamaica, Libya, Thailand and Vietnam. About 20 more reactors are planned or under construction, and 361 have been shut down or decommissioned, about half of these in USA. Many research reactors were built in the 1960s and 1970s. The peak number operating was in 1975, with 373 in 55 countries. The basic conditions of some main research reactors in the world are listed in Table 3-1[6].

Table 3-1 Some of the main research reactors in the world

Thermal power, Control rods Country Facility name Reactor type status steady (MW) material Algeria ES-SALAM Heavy water 15.00 Cd Operational Argentina RA-3 Pool 5.00 Ag, In, Cd Operational HIFAR Heavy water 10.00 Eu, Cd, SS Operational Australia OPAL Experimental 20.00 Ag, In, Cd Operational BR-1 Graphite 4.00 Cd, C Operational BR-2 Tank 100.00 Cd, Co Operational Belgium THETIS Pool 0.25 Cd, In, Ag Operational RR-BN-1 Canada NRU Heavy water 135.00 Cd,Co Operational CARR Pool 60.00 Hf Under construction CFER Fast breeder 65.00 B4C Under construction China HFETR Tank 125.00 Cd, In, Ag Operational HWRR-II Heavy water 15.00 Cd Operational CABRI Pool 25.00 Hf Operational HFR Heavy water 58.30 Ni,In,Ag,Cd Operational France ORPHEE Pool 14.00 Hf Operational PHENIX Fast breeder 563.00 B4C Operational BER-II Pool 10.00 Hf Operational FRJ-2 Germany Heavy water 23.00 Cd,SS Operational (DIDO) FRM-II Pool 20.00 Hf Operational HTTR HTG 30.00 B4C Operational Japan JMTR Tank 50.00 Hf Operational JRR-3M Pool 20.00 Hf Operational

8 CHAPTER 3. CONTROL ROD MATERIALS

Korea, HANARO Pool 30.00 Hf Operational Republic of BOR-60 Fast breeder 60.00 B4C Operational MIR,M1 Pool/channel 100.00 Dy,Cd Operational Russia PIK Tank 100.00 Eu Under construction SM Press. vessel 100.00 Eu Operational South SAFARI-1 Tank in pool 20.00 Cd,U Operational Africa ATR Tank 250.00 Hf Operational United HFIR Tank 85.00 Eu Operational states NBSR Heavy water 20.00 Cd Operational

3.2 The characteristics of control rod materials Every research reactor has own characteristic, such as power level, neutron flux distribution. However, every control rod material also has own characteristic, such as isotopic composition,

microscopic absorption cross section in function of the neutron energy En. In order to find one or several kinds of material which can satisfy the characteristic of a special reactor type, it is necessary to know the characteristic of each material. The detailed information of each material is shown as followed. 3.2.1 Cadmium[7] – thermal absorber Cadmium is a chemical element in the periodic table that has the symbol Cd and atomic number 48. A relatively rare, soft, transition metal, cadmium is known to cause cancer and occurs with zinc ores. Cadmium is used largely in batteries and pigments, for example for plastic products. Cadmium is also a malleable, ductile, toxic, bluish-white bivalent metal which can be easily cut with a knife. It is similar in many respects to zinc but reacts to form more complex compounds. The most common oxidation state of cadmium is +2, though rare examples of +1 can be found. Naturally occurring cadmium is composed of 8 isotopes. For two of them, natural radioactivity was observed, and three others are predicted to be radioactive but their decays were never observed, due to extremely long half-life times. The two natural radioactive isotopes are 113Cd (beta decay, half-life is 7.7 × 1015 years) and 116Cd (two-neutrino double beta decay, half-life is 2.9 × 1019 years). The other three are 106Cd, 108Cd (double electron capture), and 114Cd (double beta decay); only lower limits on their half-life times have been set. At least three isotopes - 110Cd, 111Cd, and 112Cd - are absolutely stable. Among the isotopes absent in the natural cadmium, the most long-lived are 109Cd with a half-life of 462.6 days, and 115Cd with a half-life of 53.46 hours. All of the remaining radioactive isotopes have half-lives that are less than 2.5 hours and the majority of these have half-lives that are less than 5 minutes. This element also has 8 known meta 113m 115m 117m states with the most stable being Cd (t½ 14.1 years), Cd (t½ 44.6 days) and Cd (t½ 3.36 hours). The known isotopes of cadmium range in atomic mass from 94.950 u (95Cd) to 131.946 u (132Cd). The primary decay mode before the second most abundant stable isotope, 112Cd, is electron capture and the primary modes after are beta emission and electron capture. The primary decay

9 CHAPTER 3. CONTROL ROD MATERIALS

product before 112Cd is element 47 (silver) and the primary product after is element 49 (indium). The main isotopes of Cd are shown in Table 3-2. The total cross sections of the cadmium isotopes are given in Fig. 3.1[8]. Cadmium has one main isotope - 113Cd, which has big thermal absorption cross section ~104-105barns. All of the Cd – isotopes including 113Cd have resonance (~103- 104barns) in the epithermal region: 10-5Mev - 10-2Mev.

Table 3-2 the main isotopes of cadmium.

Decay energy Isotope Natural abundance half-life Decay modes Decay product (MeV) 106Cd 1.25% >9.5×1017 y ε ε 2ν - 106Pd 108Cd 0.89% >6.7×1017 y ε ε 2ν - 108Pd 109Cd Syn 462.6 d ε 0.214 109Ag 110Cd 12.49% Cd is stable with 62 neutrons 111Cd 12.8% Cd is stable with 63 neutrons 112Cd 24.13% Cd is stable with 64 neutrons 113Cd 12.22% 7.7×1015 y β- 0.316 113In β- 0.580 113In 113mCd Syn 14.1 y IT 0.264 113Cd 114Cd 28.73% >9.3×1017 y ββ2ν - 114Sn 116Cd 7.49% 2.9×1019 y ββ2ν - 116Sn

3.2.2 Hafnium[9] – resonant absorber Hafnium is a chemical element in the periodic table that has the symbol Hf and atomic number 72. A lustrous, silvery gray tetravalent transition metal, hafnium resembles zirconium chemically and is found in zirconium minerals. Hafnium is used in tungsten alloys in filaments and electrodes and also acts as a neutron absorber in control rods in nuclear power plants. Hafnium is a shiny silvery, ductile metal that is corrosion resistant and chemically similar to zirconium. The properties of hafnium are markedly affected by zirconium impurities and these two elements are amongst the most difficult to separate. A notable physical difference between them is their density (zirconium is about half as dense as hafnium), but chemically the elements are extremely similar. Nevertheless, separation of them becomes important in the nuclear power industry, as zirconium is a common fuel-rod cladding alloy material, with the desirable properties of low neutron cross section and high chemical stability at high temperatures. However, because of hafnium's neutron absorbing properties, hafnium impurities in zirconium cause it to be far less useful for nuclear reactor materials applications. Hafnium is estimated to make up about 0.00058% of the Earth's upper crust by weight. It is found combined in natural zirconium compounds but it does not exist as a free element in nature. Minerals that contain zirconium, such as alvite (Hf, Th, Zr) SiO4H2O, thortveitite and zircon (ZrSiO4), usually contain between 1 and 5% hafnium. Hafnium and zirconium have nearly identical chemistry, which makes the two difficult to separate. About half of all hafnium metal manufactured is produced as a by-product of zirconium refinement. This is done through reducing hafnium (IV) chloride with magnesium or sodium in the Kroll process. The main isotopes of Hf are presented in Table 3-3. The total cross sections of the different Hf isotopes are 10 CHAPTER 3. CONTROL ROD MATERIALS

given in Fig. 3.2[8], the most important isotopes are 177Hf, 178Hf and 176Hf. The main isotope which is burning out during irradiation is 177Hf. By contrast with Cd, the Hf- isotopes have their absorption mainly in the epithermal region (10-6-10-2Mev), which is characterized with many resonances ~103 – 105 barns, the thermal cross section varies as ~ 102 barns.

Table 3-3 the main isotopes of hafnium.

Natural Decay energy Isotope half-life Decay modes Decay product abundance (MeV) 172Hf syn 1.87 y ε 0.350 172Lu 174Hf 0.162% 2×1015 y α 2.495 170Yb 176Hf 5.206% Hf is stable with 104 neutrons 177Hf 18.606% Hf is stable with 105 neutrons 178Hf 27.297% Hf is stable with 106 neutrons 178m2Hf syn 31 y IT 2.446 178Hf 179Hf 13.629% Hf is stable with 107 neutrons 180Hf 35.1% Hf is stable with 108 neutrons 182Hf syn 9×106 y β 0.373 182Ta

3.2.3 Boron (boron carbide)[10]

Boron carbide (chemical formula B4C) is an extremely hard ceramic material used in tank armor, bullet-proof vests, and numerous industrial applications. With a hardness of 9.3 on the moths scale, it is the fifth hardest material known behind boron nitride, diamond, ultra hard fullerite, and aggregated diamond nanorods. Discovered in the 19th century as a by-product of reactions involving metal borides, it was not until the 1930s that the material was studied scientifically. Boron carbide is now produced

industrially by the carbon-thermal reduction of B2O3 (boron oxide) in an electric arc furnace. Its ability to absorb neutrons without forming long lived radionuclides makes the material attractive as an absorbent for neutron radiation arising in nuclear power plants. Nuclear applications of boron carbide include shielding, control rod and shut down pellets.

[11] 3.2.4 Gadolinium oxide (Gd2O3) Gadolinium is a silvery white, malleable and ductile rare earth metal with a metallic luster. It crystallizes in hexagonal, close-packed alpha form at room temperature; when heated to 1508 K, it transforms into its beta form, which has a body-centered cubic structure. Unlike other rare earth elements, gadolinium is relatively stable in dry air; however, it tarnishes quickly in moist air and forms a loosely adhering oxide that spalls off and exposes more surfaces to oxidation. Gadolinium reacts slowly with water and is soluble in dilute acid. Gadolinium has the highest thermal neutron capture cross-section of any (known) element, 49,000 barns, but it also has a fast burn-out rate, limiting its usefulness as a nuclear control rod material. Gadolinium becomes superconductive below a critical temperature of 1.083 K. It is strongly

11 CHAPTER 3. CONTROL ROD MATERIALS

magnetic at room temperature, and exhibits ferromagnetic properties below room temperature. Gadolinium demonstrates a magneto caloric effect whereby its temperature increases when it enters a magnetic field and decreases when it leaves the magnetic field. The effect is considerably

stronger for the gadolinium alloy Gd5(Si2Ge2) Gadolinium is used in nuclear marine propulsion systems as a burnable poison. The gadolinium slows the initial reaction rate, but as it decays other neutron poisons accumulate, allowing for long-running cores. Gadolinium is also used as a secondary, emergency shut-down measure in some nuclear reactors, particularly of the CANDU type. Naturally occurring gadolinium is composed of 5 stable isotopes, 154Gd, 155Gd, 156Gd, 157Gd and 158Gd, and 2 radioisotopes, 152Gd and 160Gd, with 158Gd being the most abundant (24.84% natural abundance). 30 radioisotopes have been characterized with the most stable being 160Gd with a half-life of more than 1.3×1021 years (the decay is not observed, only the lower limit on the half-life is known), alpha-decaying 152Gd with a half-life of 1.08×1014 years, and 150Gd with a half-life of 1.79×106 years. All of the remaining radioactive isotopes have half-lives that are less than 74.7 years, and the majority of these have half-lives that are less than 24.6 seconds. This 143m 145m element also has 4 meta states with the most stable being Gd (t½ 110 seconds), Gd (t½ 85 141m seconds) and Gd (t½ 24.5 seconds). The primary decay mode before the most abundant stable isotope, 158Gd, is electron capture and the primary mode after is beta minus decay. The primary decay products before 158Gd are element Eu (europium) isotopes and the primary products after are element Tb (Terbium) isotopes. The main isotopes are presented in table 3-4. The total cross sections of the Gd isotopes are shown in Fig. 3.3[8], the Gd isotopes are characterized with high thermal cross section (155Gd, 157Gd ~105barns) and also epithermal, ~ 104barns.

Table 3-4 the main isotopes of gadolinium

Natural Decay energy Isotope half-life Decay modes Decay product abundance (MeV) 152Gd 0.20% 1.08×1014 y α 2.205 148Sm 154Gd 2.18% Gd is stable with 90 neutrons 155Gd 14.80% Gd is stable with 91 neutrons 156Gd 20.47% Gd is stable with 92 neutrons 157Gd 15.65% Gd is stable with 93 neutrons 158Gd 24.84% Gd is stable with 94 neutrons 160Gd 21.86% >1.3×1021y β-β- 1.7 160Dy

[12] 3.2.5 Europium oxide (Eu2O3) – both thermal and resonant absorber Europium is the most reactive of the rare earth elements; it rapidly oxidizes in air, and resembles calcium in its reaction with water; deliveries of the metal element in solid form, even when coated with a protective layer of mineral oil, are rarely shiny. Europium ignites in air at about 150 °C to 180 °C. It is about as hard as lead and quite ductile. There are few commercial applications for europium metal, although it has been used to dope 12 CHAPTER 3. CONTROL ROD MATERIALS

some types of glass to make lasers, as well as being used for screening for Down syndrome and some other genetic diseases. Due to its ability to absorb neutrons, it is also being studied for use in nuclear reactors. Europium is never found in nature as a free element; however, there are many minerals containing europium, with the most important sources being bastnasite and monazite. Europium has also been identified in the spectra of the sun and certain stars. Relative depletion or enrichment of europium in minerals relative to other rare earth elements is known as the europium anomaly. Divalent europium in small amounts happens to be the activator of the bright blue fluorescence of some samples of the mineral fluorite (calcium difluoride). The most outstanding examples of this originated around Weardale, and adjacent parts of northern England, and indeed it was this fluorite that gave its name to the phenomenon of fluorescence, although it was not until much later that europium was discovered or determined to be the cause. Naturally occurring europium is composed of 2 stable isotopes, 151Eu and 153Eu, with 153Eu being the most abundant (52.2% natural abundance). 35 radioisotopes have been characterized, with the most stable being 150Eu with a half-life of 36.9 years, 152Eu with a half-life of 13.516 years, and 154Eu with a half-life of 8.593 years. All of the remaining radioactive isotopes have half-lives that are less than 4.7612 years, and the majority of these have half-lives that are less than 12.2 150m seconds. This element also has 8 meta states, with the most stable being Eu (t½ 12.8 hours), 152m1 152m2 Eu (t½ 9.3116 hours) and Eu (t½ 96 minutes). The primary decay mode before the most abundant stable isotope, 153Eu, is electron capture, and the primary mode after is beta minus decay. The primary decay products before 153Eu are element Sm (samarium) isotopes and the primary products after are element Gd (gadolinium) isotopes. The main isotopes are shown in Table 3-5. The cross sections of Eu – isotopes are given in Fig. 3.4[8], all Eu – isotopes have very big thermal cross section (~ 105barns) and also epithermal cross section (~ 104barns).

Table 3-5 the main isotopes of europium.

Natural Decay energy Isotope half-life Decay modes Decay product abundance (MeV) 150Eu syn 36.9 y ε 2.261 150Sm 151Eu 47.8% Eu is stable with 88 neutrons ε 1.874 152Sm 152Eu syn 13.516 y β- 1.819 152Gd 153Eu 52.2% Eu is stable with 90 neutrons

3.2.6 Stellite[13] Stellite alloy is a range of cobalt-chromium alloys designed for wear resistance. It may also contain tungsten and a small but important amount of carbon. It is a trademarked name of the Deloro Stellite Company and was invented by Elwood Haynes in the early 1900s as a substitute for flatware that stained (or that had to be constantly cleaned)

13 CHAPTER 3. CONTROL ROD MATERIALS

Stellite alloy is a completely non-magnetic and non-corrosive Cobalt alloy. There are a number of Stellite alloys, with various compositions optimized for different uses. Information is available from the manufacturer, Deloro Stellite, outlining the composition of a number of Stellite alloys and their intended applications. The alloy currently most suited for cutting tools, for example, is Stellite 100, due to the fact that this alloy is quite hard, maintains a good cutting edge even at high temperature, and resists hardening and annealing due to heat. Other alloys are formulated to maximize combinations of wear resistance, corrosion resistance, or ability to withstand extreme temperatures. Stellite alloys display astounding hardness and toughness, and are also usually very resistant to corrosion. Stellite alloys are so hard that they are very difficult to machine, and anything made from them is, as a result, very expensive. Typically a Stellite part will be very precisely cast so that only minimal machining will be necessary. Machining of Stellite is more often done by grinding, rather than by cutting. Stellite alloys also tend to have extremely high melting points due to the cobalt and chromium content. Stellite was a major improvement in the production of poppet valves and valve seats in internal combustion engines; by reducing wear in them, the competing slide-valve design was driven from the market. The first third of M60 machine gun barrels (starting from the chamber) are lined with Stellite. Modern jet engine turbine blades are usually made of Stellite alloys, due to their very high melting points and tremendous strength at very high temperatures. In the early 1980s, experiments were done in the United Kingdom to make artificial hip joints and other bone replacements out of precision-cast Stellite alloys. Stellite has also been used in the manufacture of turning tools for lathes. With the introduction and improvements in tipped tools it is not used as often any more, but it was found to have superior cutting properties compared to the early carbon steel tools and even some HSS tools, especially against difficult materials as stainless steel. Care was needed in grinding the blanks and these were marked at one end to show the correct orientation, without which the cutting edge could chip prematurely. While Stellite remains the material of choice for certain internal parts in industrial process valves (valve seat hard facing), its use has been discouraged in nuclear power plants. In piping that can communicate with the reactor, tiny amounts of Stellite would be released into the process fluid and eventually enter the reactor. There the cobalt would be activated by the neutron flux in the reactor and become cobalt-60, a radioisotope with a 5 year half life that releases very energetic gamma rays. While not a hazard to the general public, about a third to a half of nuclear worker exposures could be traced to the use of Stellite and to trace amounts of cobalt in stainless steels. Replacements for Stellite have been developed by the industry, such as the Electric Power Research Institute’s “Norem,” that provide acceptable performance without cobalt. Since the US nuclear power industry has begun to replace the Stellite valve seat hardfacing in the late 70’s and to tighten specifications of cobalt in stainless steels, worker exposures due to cobalt-60 have dropped significantly. The absorption cross sections of Cobalt and Chromium and Iron – isotopes are given in Fig. 3.5[8].

14 CHAPTER 3. CONTROL ROD MATERIALS

Cd106 Cd108

Cd110 Cd111

Cd113 Cd114

Cd115m Cd116

Figure 3.1 Total cross section of cadmium isotopes[8]

15 CHAPTER 3. CONTROL ROD MATERIALS

Hf174 Hf176

Hf178 Hf177

Hf179 Hf180

Figure 3.2 Total cross section of hafnium isotopes[8]

16 CHAPTER 3. CONTROL ROD MATERIALS

Gd152 Gd153

Gd154 Gd155

Gd156 Gd157

Gd158 Gd160

Figure 3.3 Total cross section of gadolinium isotopes[8]

17 CHAPTER 3. CONTROL ROD MATERIALS

Eu151 Eu152

Eu153 Eu154

Eu155 Sm152

Figure 3.4 Total cross section of europium isotopes[8]

18 CHAPTER 3. CONTROL ROD MATERIALS

Co59 Co60

(a) Total cross section of cobalt isotopes

Cr50 Cr52 Cr53 Cr54 Mn55

(b) Total cross section of chromium isotopes

Fe54 Fe56 Fe57 Fe58 Ni58

CHAPTER 4 CALCULATION METHODOLOGY

In this chapter the various methods, used for determination of control rods characteristics are considered in brief: (i) analytical methods (Chapter 4.1, Appendices) – reactor kinetics theory is used to deduce the equation of the reactor period; (ii) experimental & theoretical methods (Chapter 4.2): a. determination of reactivity values and rod worth’s from the measured asymptotic reactor period; b. diffusion and perturbation theory are used for estimation of reactivity values of the partially inserted control rod; (iii) numerical methods (Chapter 4.3): a. MCNP4C – Monte Carlo code is used for development of the realistic 3 – D heterogeneous geometry model of the reactor BR2; used for steady-state flux and spectra calculations, calculations of control rod characteristics (total and differential rod worth). b. SCALE4.4a is used for evaluation of the isotopic fuel depletion; depletion and nuclide inventory of absorbing rod material in different axial positions of the control rod; evolution of the macroscopic absorption cross sections. c. Combined MCNP&ORIGEN-S method is used for the detailed 3 – D space dependent distribution of the isotopic fuel depletion in the reactor core and axial distribution of the nuclide inventory of the depleted absorbing material in the control rods.

4.1 Reactor Kinetics for determination of CR worth[14]

The study of long – term changes in the core composition due to fuel burn up and fission product accumulation involves steady-state analysis or at most a sequence of steady-state criticality calculations of the neutron flux in the reactor based on neutron transport or diffusion model. Only the time-dependence of the slowly varying changes in core composition such as those due to fuel depletion must be explicitly considered. By contrast, the analysis of relatively short-term changes ranging from seconds up to minutes due to normal changes in the reactor power level (e.g., startup or shutdown) or in a transient/accident analysis, requires a different type of methods, which are commonly known as nuclear reactor kinetics. The principle application of the analysis is to predict the time behaviour of the neutron population for a given induced change in the core multiplication. The one-speed diffusion model which usually is used to study the reactor criticality is also capable of describing the time behaviour of a nuclear reactor, if we include the effects of the delayed neutrons. Such a model is frequently too detailed for practical implementation in reactor kinetics analysis due to excessive computation requirements. An

20 CHAPTER 4. CALCULATION METHODOLOGY assumption that the spatial dependence of the flux in the reactor can be described by a single, time-independent spatial mode shape ψ1(r) (the fundamental mode) will allow us to simplify the diffusion equation, involving only description of ordinary differential equations in time. This model is some times known as the point reactor kinetics model, although the model does not treat the reactor as a point, but rather assumes that the spatial flux shape does not change with time. The methods of the reactor kinetics lie in the basis of the experimental techniques for measuring reactor parameters such as reactivity values and related control rod worth’s. A simple type of kinetic measurement can be performed if we make a small perturbation in the core composition of a critical reactor and then to measure the stable or asymptotic period of the resultant core transient. Using the inhour equation, which will be described in the further chapters, one can derive the reactivity “worth” of the perturbation from a measurement of the asymptotic positive period (i.e. when is induced positive reactivity change into the reactor core). For all practical purposes this method applies only for positive periods, since the negative periods are dominated by the longest delayed neutron precursor (fission fragment) decay and hence provide low sensitivity to negative reactivity.

4.1.1 Kinetic equations without delayed neutrons

The equations of the reactor kinetic can be derived from one-speed diffusion equation: 1 ∂φ r r − ∇ ⋅ D(r)∇φ + Σ (r)φ(r,t) =νΣ φ(vr,t) = S (r,t) (4-1) v ∂t a f f

r r Where the fission source is given with: S f (r,t) =νΣ f φ(r,t)

4.1.1.1 Time – dependent slab reactor For the case of a simple bare slab reactor we can write Eq. (4-1) as: 1 ∂φ ∂ 2φ − D + Σ (x)φ(x,t) =νΣ φ(x,t) = S (x,t) (4-2) v ∂t ∂x 2 a f f With initial condition:

φ(x,0) = φ0 (x) = φ0 (−x) (4-3) and boundary conditions: a~ a~ φ( ,t) = φ(− ,t) = 0 (4-4) 2 2 The solution of the partial differential Eq. (2) can be found using separation of variables in the form: φ(x,t) =ψ (x) ⋅T(t) (4-5)

Substituting Eq. (4-5) into Eq. (4-2) and dividing byψ (x) ⋅T(t) , we find:

21 CHAPTER 4. CALCULATION METHODOLOGY

1 dT v ⎡ d 2ψ ⎤ = ⎢D 2 + ()νΣ f − Σ a ψ (x)⎥ = const = −λ (4-6) T dt ψ ⎣ dx ⎦ Hence, the separation of variables given by Eq. (4-5) has reduced the original partial differential equation in two variables to two ordinary differential equations: dT = −λT(t) (4-7) dt

d 2ψ λ D + ()νΣ − Σ ψ (x) = − ψ (x) (4-8) dx 2 f a V The solution of the time – dependent equation (4-7) is simple: T (t) = T (0) ⋅ exp[− λt] (4-9) Where T(0) is an initial value, which will be determined later. The solution of the space – dependent Eq. (4-8) can be found together with the boundary conditions, i.e.: d 2ψ ⎛ λ ⎞ D + ⎜ +νΣ f − Σ a ⎟ψ (x) = 0 (4-10) dx 2 ⎝ v ⎠ With boundary conditions: a~ a~ ψ ( ) =ψ (− ) = 0 (4-11) 2 2 For a homogeneous problem Eqs. (4-10) and (4-11) can be written as: d 2ψ + B 2ψ (x) = 0 (4-12) dx 2

a~ a~ ψ ( ) =ψ (− ) = 0 (4-13) 2 2 Where B is an arbitrary parameter which has to be determined. The general solution of the homogeneous problem, Eq. (4-12) & (4-13) is seeking as an expansion in the set of normal modes (or eigenfunctions), characterizing the geometry of interest.

ψ (x) = A1 cos Bx + A2 sin Bx (4-14)

a~ Ba~ Ba~ ψ (± ) = A cos + A sin (4-15) 2 1 2 2 2 Adding and subtracting Eqs. (4-14) and (4-15) we find that: Ba~ A cos = 0 (4-16) 1 2

Ba~ A sin = 0 (4-17) 2 2

22 CHAPTER 4. CALCULATION METHODOLOGY

In order to satisfy conditions (4-16) and (4-17), we must determine the parameter B. There are many values of B, for which this will occur.

A2=0, then: nπ B = B ≡ , n=1,3,5,…… (4-18) n a~ will get the following solution: nπx ψ (x) = A cos , n=1,3,5,…… (4-19) n n a~ The solution (4-19) can satisfy both the differential Eq. (4-12) and the boundary conditions (Eq. (4-13)).

If A1=0, then: nπ B = B ≡ , n=2,4,6,…… (4-20) n a~ with solution: nπx ψ (x) = A sin , n=2,4,6,…… (4-21) n n a~ The values for B2 for which nontrivial solutions exist to the homogeneous problem are referred as:

2 2 ⎛ nπ ⎞ Eigenvalues: Bn = ⎜ ~ ⎟ , n=1,2,3,…… (4-22) ⎝ a ⎠ The corresponding solutions of the homogeneous problem are referred as eigenfunctions of the problem: ⎧ ⎛ nπx ⎞ ⎪An cos⎜ ~ ⎟,n =1,3,5,.... ⎪ ⎝ a ⎠ Eigenfunctions: ψ n (x) = ⎨ (4-23) ⎪ ⎛ nπx ⎞ An cos⎜ ~ ⎟,n = 2,4,6,.... ⎩⎪ ⎝ a ⎠ Another form of the eigenvalue problem can be found as:

H nψ n = λnψ n (4-24)

Where ψn is eigenfunction corresponding to the eigenvalue λn. Comparing (4-24) with (4-12), d 2 one can easily identify: H = ,λ → −B 2 and ψ →ψ (x). The first few eigenfunctions for dx 2 n n a slab geometry are sketched in Fig. 4.1.

23 CHAPTER 4. CALCULATION METHODOLOGY

Figure 4.1 The eigenfunctions for slab geometry. The eigenfunctions are called also normal modes or natural harmonics of the system. Notice that

An are still undetermined and are in fact arbitrary. Eqs. (4-12) & (4-13) can be re-written as: d 2ψ n + B 2ψ (x) = 0 (4-25) dx 2 n

a~ a~ ψ ( ) =ψ (− ) = 0 (4-26) n 2 n 2 With symmetric solutions,

Eigenfunctions: ψ n (x) = cosB n x , n=1,3,5,…… (4-27)

2 2 ⎛ nπ ⎞ Eigenvalues: Bn = ⎜ ~ ⎟ , n=1,3,5,…… (4-28) ⎝ a ⎠ Comparing Eq. (4-10) with Eq. (4-25), we can find:

2 λ = vΣ a + vDBn − vνΣ f ≡ λn n=1,3,5,…… (4-29)

The values λn are known as time eigenvalues of the equation since they characterize the time decay in Eq. (4-9). Therefore, the general solution of the more general Eq. (4-1) must be in the form:

φ(r,t) = ∑ An exp(− λnt)ψ n (r) (4-30) n i.e., the neutron flux is represented as superposition of spatial modes (eigenvalues) weighted with en exponentially varying time dependence. The coefficients An are determined using orthogonality of the functions ψn(x), in other words, the product of any two of functions ψn(x) will vanish when integrated over the slab unless the functions are identical:

a~ ⎧ 2 0, if m ≠ n ⎪ dxψ (x)ψ (x) = (4-31) ∫ m n ⎨~ a~ a − ⎪ , if m = n 2 ⎩⎪ 2

The second property of the eigenfunctions ψn(x) is that they form a complete set, i.e. one can find

24 CHAPTER 4. CALCULATION METHODOLOGY

a "well-behaved" function φ(x) which can be represented as a linear combination of ψn(x), which can be expressed:

∞ φ(x) = ∑ Anψ n (x) (4-32) n=1 where the coefficients An are determined with Eq. (4-31), and hence: a~ 2 2 A = dxφ (x)ψ (x) (4-33) n ~ ∫ 0 n a a~ − 2 For any symmetric initial distribution we have: a~ 2 2 nπx A = dxφ (x)cos (4-34) n ~ ∫ 0 ~ a a~ a − 2 Thus, the flux is represented as a superposition of modes, weighted by an exponential factor: ~ ⎡ a ⎤ ⎢ 2 2 ⎥ φ(x,t) = dx'φ (x')cos B x' exp − λ t cos B x (4-35) ∑⎢ ~ ∫ 0 n ⎥ ()n n n a a~ odd ⎢ − ⎥ ⎣ 2 ⎦

nπ λ = vΣ + vDB2 − vνΣ , B = (4-36) n a n f n a~ Using separation of variables, we can write: nπx φ(x,t) = ∑Tn (t)cos ~ (4-37) n a odd

dT n = −λ T (t) (4-38) dt n n 4.1.1.2 Long – time behaviour

2 2 2 2 2 ⎛ nπ ⎞ One can order B1 < B3 < B5 < ...... Bn = ⎜ ~ ⎟ . Hence, the time eigenvalues must similarly be ⎝ a ⎠

ordered such that λ1 < λ3 < λ5 < ..... (see Eq. (4-36)). This means, that the modes corresponding to larger n decay out more rapidly in time. If wait long enough, then only the fundamental mode remains:

φ(x,t) = A1 exp(− λ1t)cos B1 x when t → ∞ (4-39)

This implies that regardless of the initial shape of φ0(x) the flux will decay into the fundamental mode shape. The coefficient of the fundamental mode is then:

25 CHAPTER 4. CALCULATION METHODOLOGY

a~ 2 2 πx' A = dx'φ (x')cos (4-40) 1 ~ ∫ 0 ~ a a~ a − 2

Since φ0(x) should be > 0 to represent physically realizable flux, then A1>0. Actually, for large Σf, −λn may be positive corresponding to an exponentially growing flux (see Eq. (4-36)). However, the same argument holds since −λ1> −λ3 > −λ5 > …. Hence regardless of whether the flux grows or decays, it will approach a "persistent" of fundamental cosine distribution (see Fig. 4.2).

Figure 4.2 Time decay of higher order spatial modes in slab reactor.

2 Thus, we define the value Bn characterizing the fundamental mode as geometric buckling:

2 2 ⎛ π ⎞ 2 B1 = ⎜ ~ ⎟ ≡ Bg (4-41) ⎝ a ⎠

2 This terminology is used in accordance with the fact that Bn is a measure of the curvature of the mode shape:

2 2 1 d ψ n Bn = − 2 (4-42) ψ n dx Since there will be a large current density J and hence leakage induced by a mode with larger curvature or buckling, one might expect that the mode with least (minimum) curvature will persist in time the longest. The relation between the neutron current density J and the spatial gradient of the flux is given by the Fick’s law and it is: J(r,t) = −D(r)∇φ(r,t) (4-43)

−1 −1 Where, D(r) = []3Σtr (r) = []3()Σt − µ0Σs is the neutron diffusion coefficient;

Σtr is macroscopic transport cross section;

Σt is macroscopic total cross section;

Σs is macroscopic scattering cross section;

26 CHAPTER 4. CALCULATION METHODOLOGY

µ0 is the average value of the cosine of the scattering angle θ 0 : µ0 = cosθ0 .

4.1.1.3 Criticality condition The situation at which the flux distribution in the reactor is time – independent (in the absence of sources other than fission) is defined as reactor criticality. The condition for this is to make the fission chain reaction steady – state. (A source present in a critical system will give rise to an increasing flux that is linear in time). If we write out the general solution for the flux: ∞ φ(x,t) = A1 exp()− λ1t cos B1 x + ∑ An exp(− λnt)cos Bn x (4-44) n=3 odd Then the requirement for a time – independent flux is just that the fundamental eigenvalue mode vanish:

2 λ1 = 0 = v(Σ a −νΣ f )+ vDB1 (4-45)

Since the higher modes will have negative λn (νΣ f > Σ a ) and decay out in time, i.e.:

φ(x,t) → A1 cos B1 x ≠ Function of time (4-46)

2 2 The criticality conditions Eqs. (4-45) & (4-46) can be re-written using the notation B1 = Bg and:

νΣ − Σ f a ≡ B2 (4-47) D m

2 Where, Bm is material buckling which only depends on the material composition of the core,

2 and Bg depends only on the core geometry. Hence, the criticality condition can be written concisely as:

2 2 Bm = Bg (4-48)

2 Thus, in order to achieve a critical reactor, we must either adjust the size Bg or the core

2 composition Bm . As we know,

2 2 Bm > Bg ⇒ λ1 < 0 ⇒ Supercritical

2 2 Bm = Bg ⇒ λ1 = 0 ⇒ Critical (4-48)

2 2 Bm < Bg ⇒ λ1 > 0 ⇒ Sub-critical

2 When increasing the core size, Bg is decreased (see Eq. (4-41)), but while increasing the

27 CHAPTER 4. CALCULATION METHODOLOGY

2 concentration of the fissile material, Σf and hence Bm are also increased. Both of these modifications would therefore tend to enhance core multiplication. We can write the time eigenvalue (4-45) as: ⎛ νΣ / Σ ⎞ λ =νΣ 1+ L2 B 2 ⎜1− f a ⎟ (4-50) 1 a ()g ⎜ 2 2 ⎟ ⎝ 1+ L Bg ⎠

D −1 Where, L = is the neutron diffusion length, (νΣ) is the mean lifetime for a given neutron Σ a

−1 – nuclear reaction to occur. Hence, ()νΣ a is the mean lifetime of a neutron to absorption in an infinite medium (ignoring leakage). The infinite multiplication factor is defined as:

fuel fuel νΣ f νΣ f Σa k∞ = = fuel ⋅ =ηf (4-51) Σa Σa Σa

2 2 −1 Now we must identify (1+ L Bg ) in Eq. (4-50). Recall that the rate of neutron leakage is given by:

Leakage rate = ∫∫d S ⋅ J = d 3r∇ ⋅ J = −∫d 3rD∇ 2φ (4-52) SV V Where, we have used both Gauss's theorem and the diffusion approximation. (Gauss's divergence theorem: d 3r∇ ⋅ J = dseˆ ⋅ J (4-53) ∫∫s VS

Where, eˆs is the unit vector normal to the surface element dS). Then we can write:

d 3rΣ φ Σ d 3rφ Rate of neutron absorption ∫ a a ∫ 1 = V = V = (4-54) d 3rΣ φ − d 3rD∇ 2φ Σ d 3rφ + DB 2 d 3rφ 1+ L2 B 2 Rate of neutr. absorpt.+ leakage ∫ a ∫ a ∫ g ∫ ()g V V V V

We identify the non-leakage probability PNL and the neutron lifetime l in a finite reactor as:

1 PNL ≡ 2 2 (4-55) 1+ L Bg

⎛ 1 ⎞ 1 l = P ⎜ ⎟ = (4-56) NL ⎜ ⎟ 2 2 ⎝νΣa ⎠ ()νΣa ()1+ L Bg Combining Eq. (4-51) with Eq. (4-55) we find the multiplication factor of finite medium:

νΣ f / Σa k =ηfP NL = 2 2 (4-57) 1+ L Bg 28 CHAPTER 4. CALCULATION METHODOLOGY

According to Eqs. (4-50), (4-56) and (4-57), the fundamental time eigenvalue is just the inverse of the reactor period: k −1 1 − λ = = (4-58) 1 l T From the other side, combining Eqs. (4-28), (4-29), (4-41) and (4-47), we can get:

2 2 λ1 = vD(Bg − Bm ) (4-59) Hence, the various forms of the criticality condition are equivalent:

2 2 λ1 = 0 ⇔ Bg = Bm ⇔ k = 1 (4-60)

νΣ f / Σ a k = 2 2 = 1 (4-61) 1+ L Bg

Now let come back to Eq. (4-36) for which (see Eq. (4-58)): −λ1> −λ2 > …. and therefore for long times the flux approaches an asymptotic form and then we can re-write Eq. (4-30) as: ⎡⎛ k −1⎞ ⎤ φ(r,t) ≈ A1 exp()− λ1t ψ 1 (r) = A1 exp⎢⎜ ⎟t⎥ψ 1 (r) (4-62) ⎣⎝ l ⎠ ⎦ Where, the neutron mean lifetime in the reactor, l, is given with Eq. (4-56), the multiplication factor k is given with Eq. (4-57) and the geometry buckling is given with: νΣ − Σ B 2 = f a (4-63) g D Identically to Eq. (4-5), the solution of the one-speed diffusion equation (4-1) can be written also in the form: r r φ(r,t) = vn(t)ψ 1(r ) (4-64)

Where, ψ 1 (r) is the fundamental mode or eigenfunction of the Helmholtz Equation:

2 2 r r ∇ ψ n + Bnψ n (r ) = 0 ψ n (rs ) = 0 (4-65) Substituting (4-64) into (4-65) we arrive to the following equation for the neutron density: dn k −1 = n(t) (4-66) dt l Where, n(t) is the total number of neutrons in the reactor at time t. The equation (4-66) represents a simplified form of a more general set of equations, commonly referred as point reactor kinetics equations, named like this because of the separation of the spatial dependence by assuming a r time-independent spatial flux φ1()r , i.e.:

⎡⎛ k −1⎞ ⎤ ⎡k −1 ⎤ r r r φ(r,t) = vn(t)ψ 1(r) = vn0 exp⎢⎜ ⎟t⎥ψ1(r) ⇒ n(t) = n0 exp⎢ ⋅t⎥ (4-67) ⎣⎝ l ⎠ ⎦ ⎣ l ⎦

29 CHAPTER 4. CALCULATION METHODOLOGY characterized by time constant, 1 l T = = []sec (4-68) ()1 n ()dn dt k −1 Where, T is the reactor period. The time required for the neutron flux (or neutron density) to change by a factor of e is called reactor period. The period of reactor is a dynamic quantity: when the reactor is operated at a fixed power level, the period is infinite, because keff ≈ 1;δk = k −1 = 0 ; only when the reactor is changing its power (neutron density) level there is a finite measurable period. The quantity being usually measured is the inverse reactor period: 1 = ()1 n ()dn dt (4-69) T It is seen from (4-67) that the neutron flux will rise exponentially with time if the effective multiplication factor is greater than unity. The number of neutrons in the core is proportional to the number of fissions occurring. For 3.2×1010 fissions per second, 1 watt of power is produced. Then the power output of a reactor is proportional to the number of neutrons in the core at any given time, and the symbol n is used to designate neutron level, corresponding to a power level involved.

4.1.2 Kinetic equations with delayed neutrons

All above equations have been written, assuming that all neutrons created in the fission process are prompt with life time l ≈ 10−4 sec. Then it is seen from Eq. (4-67) that with such rapid increase of the neutron flux the reactor will be difficult to control. However, about β ≈ 0.0065 of the total number of neutrons, released during the fission process, are delayed neutrons which appear for an appreciable period of time following the fission act. The delayed neutrons are given in six groups, characterized with different mean life times and different fractional quantity. To these delayed neutrons must be added one more group of the delayed photoneutrons, released after threshold photonuclear reactions of beryllium. All 7 groups of delayed neutrons are presented in Table 4-1.

All delayed neutrons can be treated as one single averaged group with total fraction β = 0.0072 and an average decay constantλ ≈ 0.1sec−1 . A nuclear reactor having multiplication factor equal to 1+β is prompt critical which means the reactor would be capable of sustaining a chain reaction without delayed neutrons. For BR2 an effective multiplication factor of ~ 1.0072 makes the reactor prompt critical. Eq. (4-67) indicates how a reactor behaves with k>1 if all the neutrons were prompt. For example, with prompt neutron life time l ≈ 10−4 sec(as for the BR2 reactor),

30 CHAPTER 4. CALCULATION METHODOLOGY and insertion of positive reactivity change of ρ(t) ≈ +0.00144 into the reactor, the power level after 1 second will have risen by a factor of ~ 1.8E+6. Table 4-1 Properties of delayed neutrons and photo neutrons for HEU fuelled (~ 90 % 235U) and beryllium reflected reactor.

-1 Mean life ti, [sec] Decay constant λi, [sec ] Fraction of neutrons, βi 0.179 5.59 0.00017 (0.0235) 0.496 2.02 0.00083 (0.115) 2.23 0.448 0.00265 (0.367) 6.00 0.167 0.00122 (0.169) 21.84 0.046 0.0014 (0.194) 54.51 0.0183 0.00025 (0.035) > 120.0*) < 0.0083*) 0.00070*) (0.097) *) delayed photo neutrons

The system of partial differential equations describing the neutron flux in the reactor including the delayed neutrons is:

7 1 ∂φ r r r r r r − ∇ ⋅ D(r )∇φ + Σa (r )φ(r,t) = (1− β )νΣ f φ(r,t) + ∑λiCi (4-70) v ∂t i=1

∂C i = −λ C (r,t) + β νΣ φ(r,t) , i=1,…,7 (4-71) ∂t i i i f With the contribution of the delayed neutrons, the fission source can be written as:

7 r r r S f (r,t) = ()1− β νΣ f φ(r,t) + ∑λiCi (r,t) (4-72) i=1 The solution of equations (4-70) & (4-71) can be found by use of separation of variables in space and time, so that the solutions are seeking in the forms: r r φ(r,t) = vn(t)ψ 1(r)

r r Ci (r,t) = Ci (t)ψ1(r) (4-73)

r Where, ψ1(r ) is the fundamental mode of the Helmholtz equation (4-64) and Ci is the concentration of the precursors of the delayed neutrons emitted of group i. The substitution of Eq. (4-73) into the system differential equations (4-70) & (4-71) gives a set of ordinary differential equations for n(t) and Ci(t): dn k(1− β ) −1 7 = n(t) + ∑λiCi (4-74) dt l i=1

31 CHAPTER 4. CALCULATION METHODOLOGY

dC k i = β n(t) − λ C (t) i=1,…,7 (4-75) dt i l i i Equations (4-74) & (4-75) are known as the point reactor kinetic equations and represent generalization of the Eq. (4-66) by including the effect of the delayed neutrons. The equations (4-74) & (4-75) can be re-written in the following form: dn ⎡ ρ(t) − β ⎤ 7 = ⎢ ⎥n(t) + ∑λiCi (t) (4-76) dt ⎣ Λ ⎦ i=1

dC β i = i n(t) − λ C (t) i=1,…,7 (4-77) dt Λ i i

l Where, Λ = is the mean generation time between birth of neutron and subsequent absorption k

k(t) −1 inducing fission and the reactivity is defined as: ρ(t) = . For k ~ 1, Λ=l is the prompt k(t) neutron life time. The solution of the system equations (4-76) & (4-77) is not straightforward as it might first appear, because the reactivity ρ(t) is usually function of time and also depends on the neutron density n(t), so that the equations will be nonlinear. Furthermore, the time constants characterizing the nuclear process vary in wide range from l prompt ≈ 10−4 sec to the lifetime of

delayed the longest lived precursor of the delayed neutrons, ln ≈ 80sec and for the delayed photo

delayed neutrons l ph > 100sec. These widely different time scales complicate even the numerical solution of the system equations (4-76) & (4-77). Therefore complete solutions of the above equations should be considered separately for each specific type of ρ(t) disturbance. The equations (4-76) & (4-77) can be written for one group delayed neutrons, represented by

7 β = ∑ β i and average values λ and C: i=1

dn ⎡ ρ(t) − β ⎤ = n(t) + λC(t) (4-78) dt ⎣⎢ Λ ⎦⎥

dC β = n(t) − λC(t) (4-79) dt Λ The solution of these equations is a summation of two exponential terms in the form: dn(t) b − c c − a = e at + ebt (4-80) dt(0) b − a b − a

32 CHAPTER 4. CALCULATION METHODOLOGY

ρ(t)λ ρ(t) − β ρ(t) Where, a = ; b = ; c = . The product terms containing λ can be λΛ + β − ρ(t) Λ Λ neglected in most of the practical cases and then the Eq. (4-80) can be approximately written as: dn(t) β ρ(t) = exp[λρ(t) (β − ρ(t))]t − exp{}− []()β − ρ(t) / Λ t (4-81) dt(0) β − ρ(t) β − ρ(t) In order to compare with the prompt neutron example previously given (see beginning of this paragraph, let us again insert a positive reactivity change ρ(t) = +0.00144, then,

dn(t) = {}1.25exp[]0.025t − 0.25exp[− 57.6t] (4-82) dt(0) BR2 The increase of the neutron density with time according to Eq. (4-82) is given at Fig. 4.3. As it was mentioned earlier the power level of the reactor with only prompt neutrons increases by a factor of about 1.8E+6 after 1 second when a positive reactivity +0.00144 has been inserted into the core, while with the delayed neutrons, the increase of the power level for the same time is only 1.28. Thus the effect of 0.72 per cent of delayed neutrons makes the entire problem of reactor control feasible. The first term in Eq. (4-81) is dominant while the second term can be neglected. The first term is known as a stable period and the second term is referred as to a transient period. The stable period of Eq. (4-81) is: β − ρ T del = = 40[]sec (4-83) λρ

Λ while for the prompt neutrons only: T prompt = = 0.069[]sec . Eq. (4-83) solved for the reactivity ρ gives the following relationship: β ρ = (4-84) 1+ λT A more exact equation between the reactivity and the stable period including the contributions from the individual groups of delayed neutrons can be written as: Λ 7 β ρ = + ∑ i (4-85) Tkeff i=1 1+ λiT The stable periods are different for positive and negative reactivity changes, but as the periods

1.8

Eq. (4-82) 1.7

1.6 n(0)

/ 1.5

) t ( n

1.4

1.3 33

1.2 0.1 1 10 TIME [SECONDS] CHAPTER 4. CALCULATION METHODOLOGY become larger, the magnitude of the reactivity for both positive and negative periods approaches β λT . Eq. (4-86) is known as inhour formula. The inhour is defined as the reactivity which will make the reactor stable period equal to one hour. Figure 4.3 Relative neutron level in the BR2 core as a function of time for a step reactivity change of +0.00144, which is equivalent to +0.2 $ (or 144 pcm) (Eq. 4-82). It is very difficult to solve the equations of the reactor kinetics, we have to depend on numerical calculation and some approximations. These methods for the solution of the kinetic equations are given in Appendix A.

4.2 Experimental methods for determination of CR worth

4.2.1 Estimation of differential control rods worth The problem is how to determine the period of a reactor if a positive reactivity change is inserted into the reactor by withdrawal of the control rods. Two methods of analysis of reactor power level and period are available if we ignore the temperature and void effects. The first approach is to solve the reactor kinetic equations for various rates of reactivity change starting from various subcritical levels. This method of solution was discussed in details in Appendix A. However this method is tedious and restrictive. Another method that might be used is to solve for the boundary cases of maximum and minimum periods that may be involved in a start-up accident. There will be upper bound on how short the period can get as the reactor becomes greatly supercritical and the second a lower bound at greatly subcritical conditions. The upper bound should be dependent in some way upon the neutron lifetime Λ. On the other hand, for greatly subcritical conditions there must be a certain minimum period involved which will depend upon the rate of insertion of reactivity. The determination of upper and lower bounds of the reactor period is discussed in Appendix B. An alternative, third method for determination of the reactor period is to use MCNP. MCNP can be easily implemented for estimation of the reactor period by: (i) direct calculations of keff, prompt-neutron life time and the average number of neutrons per fission event; (ii) then solving the inhour equation (4-85) for the reactor period (including the fractions of all groups of delayed neutrons and a single group of the photoneutrons). The experimental methods for determination of the differential CR worth are based on measurement of the reactor period. Some theoretical aspects concerning the determination of the period in a subcritical medium are given in Appendix C. The measurements of the asymptotic period of the BR2 reactor are performed usually before start of an operation cycle. The technique involves measurement of the multiplication of the sub-critical reactor, maintained at a low power level by a neutron source at strength S [n/cm3/s] when a small positive reactivity is inserted into the core by slight withdrawal of the control rods. The following calculation procedure is executed: a. Insertion of a positive reactivity by slight withdrawal of the CR ( ∆Sh ≈ +9mm );

34 CHAPTER 4. CALCULATION METHODOLOGY

b. Measurement of the time for which the neutron density increases ~ 2 times for a given step and ramp of reactivity change rate ∆ρ according with Eq. (4-82); c. Determination of the reactivity values for two close positions of the CR using the inhour formula, Eq. (4-85); d. Determination of the differential CR worth ∆ρ / ∆h[$/ mm] for a given critical position Sh of the CR. 4.2.2 Estimation of total control rods worth There are two different approaches for determination of total control rod worth. One of them is the application of the perturbation theory for estimation of the effect of a localized control absorber on core multiplication. This is not a simple task, because of the strong absorption characterizing a control element, which causes a severe local distortion of the flux in the vicinity of control rod location. A more common approach involves the use of one – speed diffusion theory that utilizes transport-corrected boundary conditions at the surface of the control rod. There are many diffusion theory studies of cylindrical rods, partially inserted into bare, homogenous cylindrical reactor cores – one of these studies[15] is used in the technique for determination of control rods worth at the BR2 reactor. As alternative method – the Monte Carlo method can be used for determination of both total and differential CR worth. A routine technique for experimental estimation of total Control Rods worth is based on a relation between differential control rods worth, obtained from period measurements and the buckling of the reactor with the partially inserted rod, which can be derived from perturbation theory or using analytical methods[15]: ⎧ x 1 1 1 ⎫ f (z) = δ ()0 < z < π ⋅⎨ − sin x − sin 2x + sin 3x⎬ (4-86) ⎩π 24 24 72 ⎭

Where, fi(z) is a non-linear regression function, describing the buckling of the reactor with partially inserted control rod inside it. The relation between the reactivity of the partially inserted into the core control rod ρ(z,t) at position z, the total worth of the rod ρ(H,t) (i.e., the difference in reactivity between the control rod fully inserted, z=0, and totally withdrawn, z=H) and the buckling of the reactor with the inserted control rod inside it f(z,t) is given by the diffusion theory. In analogy, a similar relation can be written for the differential and the total control rod worth, obtained from detailed Monte Carlo calculation. This method suits well for determination of total worth of control rods with full active length of one black fresh absorber material. To adapt the method for control rods, composed from combination of black and gray absorbers and also to take into account the axial burnup of the absorbing materials during control rod life a series of corrections should be added into the non-linear regression function, describing the buckling of the reactor with the partially inserted control rod. These corrections, including the blackness and the axial burnup of the absorbing materials, have to be determined – by experiment or by using some theoretical or numerical (e.g. Monte Carlo) method.

35 CHAPTER 4. CALCULATION METHODOLOGY

⎧ ⎫ ρ(z,t) = ρ(H,t) ⋅ ⎨∑α i (z,t) ⋅ f i (z)⎬ (4-87) ⎩ i ⎭

Where,α i (z,t) are the blackness coefficients of the absorber as function of axial variable z and time t. The blackness coefficients αi(z) can be derived from the detailed Monte Carlo plus burn up calculations (ORIGEN-S) for the effective macroscopic cross sections of the absorber, obtained as function of time t and axial variable z during long time of irradiation using the relation: S z J z J (z) eff A () A ( ) ; A (4-88) ∑ A ()z = ⋅ α = VA ()z φ(z) φ(z)

Where, φ is the volume-averaged flux in the cell at z, VA (z) is the control cell volume at axial

position z, and S A (z) is the control element surface. J A (z) is the average neutron current at the surface of the control rod at z. 4.2.3 Experimental data of total control rods worth and burnup of the lower active cadmium part of the Reference control rods

36 CHAPTER 4. CALCULATION METHODOLOGY

After the refurbishment in 1997, when the 2nd beryllium matrix has been replaced by new, 3rd one, CR with fresh cadmium as absorbing material in the active part of the CR and cobalt in the lower rod part have been used since the 1st operating cycle 0197A, which started April 1997. These CR rods have been used during 6 years till May 2003. After that the cadmium in the active part has been replaced by pieces from the previous CR, which were used before 1997. In Fig. 4.4a are presented the values of the total CR worth and the burn up of the lower active cadmium part according to the measurements performed at the BR2 reactor[16]. The CRs were loaded in channels C19, C79, C139, C281 and C341. The total CR worth decreases by about 10% after 300 EFPD of irradiation and by 15% after 650 FPD of irradiation. After 100 FPD, the cadmium burnup of the lower active part is about 10 mm and after 650 FPD – about 15 mm. The value of the total CR worth is influenced by several other effects, which are discussed in the following

Reference CR - Cd with Co 184 15 Dh Cd [mm] R0 [$]

174 ] ] $ 10 m m rth [ o

164 w R l C ta Cd burnup [

to 5 154

0 144 100 200 300 400 500 600 Irradiation time [days]

April 1997 March 2003 chapters: burnup of the loaded fuel elements, Helium-3 poisoning, presence of absorbers in H1/C channel, position of the channel of the CR in the core, etc. Figure 4.4a Total CR Worth and cadmium burnup in CR with Cd and Co[16]; where, ρ ρ R0[]$ = ρ()Sh = 960mm − ρ(Sh = 0mm); $ = BR2 = ; Sh – position in [mm] of the βeff 0.0072 Control Rod in the core; DhCd = dhCo + dhCd : dhCo - length of Cobalt; dhCd - length of reduction of cadmium (=burnt cadmium) (see fig4.4b).

37 CHAPTER 4. CALCULATION METHODOLOGY

Figure 4.4b Reference Control Rod with Cadmium and Cobalt

4.3 Introduction of the Code

4.3.1 MCNP-4C MCNP-4C[17] is a general-purpose Monte Carlo N-Particle code that can be used for neutron, photon, electron, or coupled neutron/photon/electron transport, including the capability to calculate eigenvalues for critical systems. The code treats an arbitrary three-dimensional configuration of materials in geometric cells bounded by first- and second-degree surfaces and fourth-degree elliptical tori. Monte Carlo methods are different from deterministic transport methods. The deterministic methods (e.g., discrete ordinates method) solve the transport equation for average particle behaviour (for example, flux) throughout the phase space of the problem, defined by the phase space variables t, E and Ω for time, energy, direction and for position with incremental volume dV around r. The deterministic methods visualize the phase space to be divided into many small boxes and the particles move from one box to another. In the limit, as the boxes get smaller and smaller, the particles are moving from one box to another, which takes differential amount of time to move differential distance in space. In the limit this approaches the integro – differential transport equation, which has derivates in space and time. By contrast, the Monte Carlo method does not solve an explicit equation but rather obtains answers by simulating particle histories and recording some aspects (tallies) of their average behaviour. Monte Carlo method transports particles between events (collisions) that are separated in space and time. Neither differential space nor time are inherent parameters of Monte Carlo transport. No transport equation need to be written to solve a transport problem by Monte Carlo. However, one can derive an equation

38 CHAPTER 4. CALCULATION METHODOLOGY that describes the probability density of particles in phase space: this equation turns out to be the same as the integral transport equation. This is why often the Monte Carlo method is associated with the integral equation, rather than with the integro-differential equation as in the case of the deterministic methods. Because the Monte Carlo method does not use phase space boxes, there are no averaging approximations required in space, energy and time. This is especially important for detailed representation of any physical data by Monte Carlo and therefore the method is well suited to solving complicated three-dimensional time-dependent problems. Pointwise cross-section data are used. For neutrons, all reactions given in a particular cross-section evaluation (such as ENDF/B-VI) are accounted for. Thermal neutrons are described by both the free gas and S(alpha, beta) models. For photons, the code takes account of incoherent and coherent scattering, the possibility of fluorescent emission after photoelectric absorption, absorption in pair production with local emission of annihilation radiation, and bremsstrahlung. A continuous slowing down model is used for electron transport that includes positrons, k x-rays, and bremsstrahlung but does not include external or self-induced fields. Important standard features that make MCNP very versatile and easy to use include a powerful general source (it can be neutron or gamma external fixed source), criticality source (a Monte

Carlo eigenvalue algorithm, used to determine keff of fissile systems) and surface source; both geometry and output tally plotters; a rich collection of variance reduction techniques; a flexible tally structure; and an extensive collection of cross-section data. The neutron energy interval is from 10-11 MeV to 20 MeV, and photon and electron energy intervals are from 1 keV to100 MeV. 4.3.2 SCALE4.4a The SCALE system is composed from many separate programme codes (modules). Different modules of the SCALE4.4a system can be used for the evaluation of the isotopic fuel depletion, determination of burn up of control rod material and evaluation of the macroscopic absorption cross sections in function of time. These modules are: ORIGEN-S[18], XSDRNPM[19], NITAWL-II[20], COUPLE[21]. The 1-D depletion code – ORIGEN-S can be used in combination with mentioned above modules, utilized by the SAS2H module or used as a stand alone code. The original ORIGEN programme has been developed to perform depletion isotopic analysis and radioactivity analysis from fission products, cladding and fuel materials in LWRs, LMFBRs, MSBRs, and HTGRs reactors. The ORIGEN-S nuclear data libraries contain cross sections and fission yields for four reactor types: HTGR (high temperature gas-cooled reactor), LWR (light-water reactor), LMFBR (liquid metal fast breeder reactor) and MSBR (molten salt breeder reactor). In this thesis, the libraries for LWR were used for the burn up and activity calculations. MCNP is used for calculations of the continuous energy reaction rates and fluxes, which are converted into one – group constants. The MCNP calculated effective microscopic cross sections < σ > eff for the main actinides, dominant and some non dominant fission products of the HEU fuel, weighted in the spectrum of the needed fuel region j, are used to update the existing cross sections for the

39 CHAPTER 4. CALCULATION METHODOLOGY

LWR reactor in the ORIGEN-S libraries (see Table 4-2). The input for ORIGEN-S can be the fission power or the neutron flux, calculated by MCNP in the spatial cells where the burnup calculations are needed (examples of input data for SCALE4.4a are given in Appendix D). ORIGEN-S evaluates the evolution of the isotopic fuel and CR material densities for the desired number depletion time steps. The isotopic fuel composition and the nuclide inventory of the CR material for a given time step is introduced back into the MCNP model and distributed in the core using the detailed 3−D power peaking factors, which are earlier evaluated with MCNP[22-24]. Table 4-2 MCNP calculation of effective thermal microscopic cross sections in typical fuel channel of the reactor BR2

Nuclide ORIGEN −S MCNP Nuclide ORIGEN−S MCNP < σ >therm < σ >therm < σ > therm < σ >therm

235U (n,γ) 98 68 103Rh (n,γ) 150 113

235U (n,f) 520 400 105Rh (n,γ) 1.8E+04 1.2E+04

238U (n,γ) 2.73 2 135Xe (n,γ) 3.6E+06 2.2E+06

238U (n,f) 0 8E–06 147Pm (n,γ) 235 127

237Np (n,γ) 170 153 149Sm (n,γ) 4.15E+04 5.5E+04

237Np (n,f) 0.019 0.013 150Sm (n,γ) 102 72

239Pu (n,γ) 632 360 151Sm (n,γ) 1.5E+03 8.3E+03

239Pu (n,f) 1520 750 152Sm (n,γ) 210 150

4.3.3 MCNP&ORIGEN-S Method The distribution of the fuel burn up in the fuel elements becomes non-uniform during irradiation

due to the high peaking factor in BR2: K Z × K R × K ϕ ≈ 2.1, where KZ , KR and Kϕ are axial, radial and azimuth peaking factor of the fuel element, which is composed from six annular concentric fuel plates. A combination of the 3-D Monte Carlo code MCNP with 1-D depletion code ORIGEN-S is used for modeling of the 3-D space dependent isotopic fuel depletion in the core. The Monte Carlo code MCNP is used for evaluation of the detailed 3−D power distribution which is introduced into the 1−D depletion code ORIGEN−S to evaluate the 3−D isotopic fuel profile [22]. The 3−D

5 space distribution of the isotopic fuel depletion B j in a fuel element j in the fuel zone

(rm ,z n ,ϕl ) at arbitrary time step Ti is calculated using the information for the isotopic fuel

5 j depletion B j (Ti−1 ) and the total power distribution K V (Ti−1 , rm ,z n ,ϕ l ) in the previous time step

[22, 23] Ti-1 :

40 CHAPTER 4. CALCULATION METHODOLOGY

5 5 j C1×Ti Pj (Ti ) j Bj (Ti ,rm ,zn ,ϕl ) = Bj (Ti−1,rm ,zn ,ϕl ) × KV (Ti-1,rm ,zn ,ϕl ) + 5 × × KV (Ti ,rm ,zn ,ϕl ) (4-88) c j (0) Vj

dr dz dϕp (T ,r,z,ϕ) dr dz dϕ ∫∫∫ j i ∫ ∫ ∫ j rzmnϕl rm z n ϕl Where, K V (Ti ,rm ,zn ,ϕl ) = (4-89) Pj (Ti ) Vj

is the total power peaking factor in a fuel zone (rm ,zn ,ϕl ) of the fuel element j during the

j irradiation time Ti; K V (Ti-1, rm , zn ,ϕ) is the total power peaking factor in the previous time step

Ti-1; Zn is the axial coordinate of the fuel zone n in the fuel element; ϕl is the azimuth angle in

azimuth fuel sector l in the fuel element; p j (Ti ,rm ,zn ,ϕl ) , is the fission power in the spatial

segment (rm ,zn ,ϕl ) inside the fuel element; Pj (Ti ) and Vj are the average fuel power and the total fuel meat volume of the fuel element j; the constant C1 is determined with ORIGEN−S[22,23]: 235 -1 -1 235 C1=0.125 [kg Ufresh.MW .days ], that means about 8 MWd per 1 kg U initially charged to

235 5 -3 235 the reactor is necessary for 1% depletion of U; c j (0) [g.cm ] is the initial U density in a

5 fresh fuel. The notation B j ()Ti−1,rm , zn ,ϕl []X% is used to denote an isotopic fuel composition,

235 235 5 in which X atoms U percent are removed from the initial fresh U mass M j (0) at a

depletion time step Ti-1, corresponding to a produced energy Pj ×Ti-1 (see Eqs. 7a, 7b in [23]).

Each one of the 6 fuel plates of all 32 fuel elements is divided into axial zones by each 6 cm with varied isotopic fuel densities. The axial segments of the fuel plates are divided into azimuth sectors with heterogeneous isotopic fuel densities (see Fig. 4.5). The total number of the spatial cells with varied fuel depletion in the model is 4600. ORIGEN-S is used for evaluation of the isotopic fuel depletion versus fuel burn up and preparation of a database (DB) with large number isotopic fuel compositions (each containing 80 depleted isotopes), which are further used in the MCNP model. Schematically the model is shown at Fig. 4.6. Each depleted fuel composition,

5 B j []X% , which is evaluated with ORIGEN-S at the relevant depletion step Ti, contains about 80 fissile, 30 non-fissile (light elements) and fission product inventory of 1000 isotopes, from which the dominant ~ 100 nuclides are selected to be included into the MCNP model (see Table 4-3).

41 CHAPTER 4. CALCULATION METHODOLOGY

Table 4-3 Isotopic composition of the depleted fuel, evaluated with ORIGEN-S and used in the MCNP model. Light elements C, O, Al, 10B, 11B

Actinides 234U, 235U, 236U, 237U, 238U, 237Np, 238Np, 239Np, 238Pu, 239Pu, 240Pu, 241Pu, 242Pu 75As, 81Br, 82Kr, 83Kr, 84Kr, 85Kr, 86Kr, 85Rb, 87Rb, 89Y, 93Zr, 95Mo, 99Tc, 101Ru,

Fission Products 103Ru, 103Rh, 105Rh, 106Pd, 108Pd, 109Ag, 110Cd, 111Cd, 113Cd, 114Cd, 115Cd, 116Cd, 120Sn, 127I, 129I, 135I, 131Xe, 134Xe, 135Xe, 133Cs, 134Cs, 135Cs, 136Cs, 137Cs, 138Ba, 141Pr, 145Nd, 147Nd, 148Nd, 147Pm, 148Pm, 149Pm, 147Sm, 149Sm, 150Sm, 151Sm, 152Sm, 153Eu, 154Eu, 155Eu, 156Gd, 158Gd, etc.

Fuel plates

Figure 4.5 MCNP model of a standard BR2 fuel element (the fuel plates are divided into azimuth

42 CHAPTER 4. CALCULATION METHODOLOGY sectors by each 5° in the hot plane).

3D isotopic fuel depletion in fuel elements

3D Monte Carlo code 1D depletion code MCNP-4C ORIGEN-S

Evaluation of Evaluation Depletion step: 3D relative of isotopic ∆β5 =1% 235U power distribution fuel depletion depletion versus burn up

Radial: in 6 fuel 100 depleted plates isotopes in each fuel composition Axial: Database with 90 19 zones depleted in each fuel Isotopic fuel plate compositions

Azimuth: in 72 sectors in outer fuel plate

Figure 4.6 A scheme of the combined MCNP&ORIGEN-S method for 3D modeling of the isotopic fuel depletion[22-24].

CHAPTER 5 CALCULATION OF THE OPTIMAL DESIGN

In former chapters, we have chosen several candidate materials for the control rod and the methodology of the optimal calculation. What will be done next step is to make the optimal project and do calculations. However, the problem in front of us is how to do the optimal calculation, i.e., what is content of the calculation and what is the criterion of optimization? According to the operation experiences of BR2, many factors affect the control rod total worth. We should determine these factors and make the optimal criterion based on them. Only doing like that, the optimal calculation can be significant.

5.1 Definitions In order to be easily understood the content of this chapter, several definitions characterizing reactivity control will be introduced in advanced, such as excess reactivity, shutdown margin, total and differential control rod worth. Excess reactivity: the core reactivity present with all control elements withdrawn from the core is named excess reactivity, ρex. ρex is a function of both time (due to fuel depletion and nuclide transmutation) and temperature (due to reactivity feedback). Larger values of ρex may generally imply longer core lifetimes, but at the expense of larger control requirements and poorer neutron economy (since with more control reactivity in the core, there will be more neutron absorption). Shutdown margin: The negative reactivity of the core present when all control elements have been fully inserted to achieve minimum core multiplication is called shutdown margin. The shutdown margin ρsm is a function of time and temperature. For example, the shutdown margin for a core loaded with fresh fuel at ambient temperature in which no depletion or fission product buildup has occurred will be quite different from the shutdown margin characterizing a core that has been operating at power for some time. Typically shutdown margins are chosen such that the core multiplication is below critical. The shutdown margin not only characterizes the core multiplication in its shutdown state, but is also related to the rate at which the reactor power level may be reduced in an emergency shutdown or "scram". Total control rod worth: total control rod worth R0 is defined as the difference between the excess reactivity and the minimum reactivity when all control elements are fully inserted. That is,

R0 = ρex − ρsm . For the BR2 reactor, R0 = ρex (Sh = 960mm) − ρsm (Sh = 0mm) ; where, Sh is the position in [mm] of the Control Rod in the core. Differential control rod worth: the differential control element worth is defined as the reactivity change per unit control rod displacement: ∆ρi ∆Shi [$ / mm].

44 CHAPTER 5. CALCULATION OF THE OPTIMAL DESIGN

5.2 Impact of various factors on the CR parameters in the reactor BR2 Through the investigation and analysis to the Reference CR with cadmium and cobalt (Cd+Co) using MCNP, the various factors affecting CR total worth are summarized in the Table 5-1. The reactivity values are expressed in units of BR2 dollars and in pcm. For BR2 one dollar (1$) is

equal to a reactivity of βeff=0.0072, including the fraction of delayed neutrons and photoneutrons. Because the BR2 reacor operates with highly enriched uranium fuel (~90% 235U) the changing of

βeff during typical BR2 operation cycle with duration ~25 days is negligible. Table 5-1 Impact of various factors on total worth R0 for the Reference CR

keff Variation of R0 [$ (pcm)] R0, Sh=0 mm Sh=900 mm ∆R0[$ (pcm)] Burnup of CR Fresh 13.2 (9504) 1.0 (720) material Burnt 12.2 (8784) Fuel depletion High 0.9486±0.0005 1.0464±0.0005 13.7 (9864) 1.0 (720) (6NC*) Low 0.9854±0.0005 1.0827±0.0005 12.7 (9144) Fuel depletion High 0.9504±0.0005 1.0488±0.0005 13.7 (9864) 1.0 (720) (6NG**) Low 0.9799±0.0005 1.0764±0.0005 12.7 (9144) Burnable Present 0.9504±0.0005 1.0488±0.0005 13.7 (9864) 0.7 (504) poisons Absent 1.0298±0.0005 1.1393±0.0005 13.0 (9360) Experiments in Present 0.9608±0.0005 1.0574±0.0005 13.2 (9504) 0.8 (576) H1/Central Absent 0.9834±0.0005 1.0783±0.0005 12.4 (8928) Fresh beryllium 13.9 (10008) Highly poisoned Material of the 12.2 (8784) beryllium 3.5 (2520) follower in CR Aluminium 12.0 (8640) Light water 10.4 (7488) Close to core centre Position of CR Close to core 1.8 (1296) in the core periphery Present Photoneutrons 1.0 (720) Absent Thickness of δAl=1.5 mm 0.9563±0.0005 1.0581±0.0005 14.0 (10080) Al cladding δAl=3.5 mm 0.9601±0.0005 1.0579±0.0005 13.4 (9648) 1.2 (864) ***) δAl=5.5 mm 0.9632±0.0005 1.0573±0.0005 12.8 (9216)

*) NC is the abbreviation a type of standard BR2 fuel, which includes 70% 235U enrichment (1.7 g/cm3 235U); **) NG is the abbreviation another type of standard BR2 fuel, which includes 90% 235U enrichment (1.3 g/cm3 235U); ***) For control rod with full cadmium absorbing length (Cd+Cd)

The detailed explanations are listed as followed: y The burnup of the control rod material (cadmium): - Reduces the total control rod worth by ~ 0.5 $ (360 pcm)

45 CHAPTER 5. CALCULATION OF THE OPTIMAL DESIGN

- Strongly affects the shapes of total and differential CR worth y The poisoning of the beryllium follower of the CR, represented by both helium – 3 and Li – 6 absorption in comparison with fresh beryllium, reduces the total control rod worth up to ~ 1.7 $ (1224 pcm) (see Fig. 5.1). y The presence of strong absorbers in the core increases the total control rod worth: - Presence of experimental samples in H1/Central, having large absorption cross section – increase up to ~ 0.8 ÷ 1.0 $ (576 ÷720 pcm) - Burnable poisons in the fuel (B4C and Sm2O3) – increase up to 0.7 $ (504 pcm) y Increase of fuel depletion (diminish of 235U content) and accumulation of fission products increases the total control rod worth up to about ~ 1.0 $ (720 pcm). y The follower in the CR, which is located below the active cadmium part of the CR, is made by beryllium. In order to study the impact of the material of the follower on the total CR worth, MCNP calculations were performed for evaluation of the total CR worth for different materials of the follower. The following conclusions were made: - maximum control rod worth is for fresh beryllium follower - minimum control rod worth is for light water or Al follower - medium control rod worth is for poisoned beryllium follower y Accounting for photo neutrons reduces the control rod worth by about 10%. y Increase of aluminium cladding around the absorbing control rod material reduces the total control rod worth up to ~ 1.2 $ (864 pcm). y Location of CR close to the core centre increases the total rods worth up to about ~1.8 $ (1296 pcm). The MCNP calculations of the total Control Rods worth for one and the same type CR are performed for different reactor core loads: for location of the CR close to the core centre and for location of CR relatively far from the core centre. The comparison of the total CR effective worth is given at Fig. 5.2.

16 CR close to core center: conf. '7A', '7B' (~1972, ) 14 CR far from core center: cycle 01/2005A.3 (3rd Be matrix)

] 10 $ - y [ 8

tivit c a e r 6

2 MCNP 0 0 100 200 300 400 500 600 700 800 900 Sh [mm] Figure 5.1 Total Control Rod Effective Worth for 6 CR_Co with “fresh” and “poisoned”

16 BR2 Cycle: 01/2005A.3 CR_Co_BR2 ("fresh" Be) 14 CR_Co_BR2 ("poisoned" Be)

] 10 [-$ y it 8

reactiv 6 4 46 2 MCNP 0 0 100 200 300 400 500 600 700 800 900 Sh [mm] CHAPTER 5. CALCULATION OF THE OPTIMAL DESIGN

beryllium section under the Cd part of the CR. Figure 5.2 Total Control Rods Effective Worth for different types CR, located “close” or “far”

from the reactor core centre (used “poisoned” Be in CR, ∆hcd ~ 150 – 160 mm).

5.3 Criterion for Control Rod Life

For a reactor, control rod life is very important, which is relative to the control rod material. A suitable material can make the control rod life as long as possible. How to determine the control rod life? Three criterions are presented as follows: y A criterion for the changing of the absorption properties of the CR is the behaviour of macroscopic absorption cross sections of the CR material during long term of irradiation. y Second criterion for the absorption properties of the different CR and their changing

during the fuel cycle is the behaviour of multiplication effective factor keff of the whole reactor system with a given type of CR.

y Third criterion for the CR life, but again related to the keff of the whole system is the total and differential CR worth's. The first criterion gives an idea for the depletion of the CR material and changing of the local absorption properties of the CR, but it doesn’t draw the actual behaviour of the CR in the union of the whole reactor core. The depletion of the CR material during irradiation may change strongly the absorption properties of the CR itself (e.g. the macroscopic absorption cross sections of the CR material). However, the effect of burn up of absorber on the total absorption and total fission in the reactor is less sensitive. The fraction of the CR absorption at typical critical position (Sh~500 mm) can be about 6-8% from the total absorption in the reactor (this value depends on the reactor load and on the critical height at BOC). The second and third criteria are related to change of the macroscopic absorption and fission processes in the whole reactor core due to CR insertion. If the increase of the keff at the beginning of the next cycles for the same reactor core load and the same power means that the absorption ability of the CR type decreases during the irradiation (as it is for example for Cd rods). If keff at BOC after a certain number of cycles slightly decreases or remains constant for the same reactor core load and power – then this means that the absorption properties of the CR type may slightly increase or remain constant (an example of such behaviour demonstrate europium and hafnium rods, in which the depletion of the main isotopes is compensated with transmutation into another absorbing nuclides). The multiplication factor is a function of the total absorption, fission and (n, xn) reaction rates of all materials in the whole reactor. In MCNP keff is calculated using the following equation:

ν f Σ f Φ ν f F k eff = = (5-1) Σ f Φ + Σ a Φ − Σ n,xn Φ + L F + A − N n,xn + L

Where, the leakage term L ~ 0 for the BR2 reactor; Nn,xn is the multiplicity term; A and F determine the total absorption and total fission processes in the reactor. We can re-write Eq. (5-1) 47 CHAPTER 5. CALCULATION OF THE OPTIMAL DESIGN in the following form:

1 1 k = ν ;k ≈ ν (5-2) eff f A N eff f A 1+ − x,xn 1+ F F F

Therefore, a criterion for the CR life should be not the behaviour of the macroscopic cross sections only, but the changes in the ratio A/F, which determines increase or decrease of keff during the reactor cycles.

5.4 Design modifications and the choosing of optimal projects

5.4.1 General idea for the design modifications For a control rod, what can be changed are material and dimension. Each part of the control rod structure which is mentioned in the chapter 2 can be composed of different material and has various dimensions. Through some comprehensive considerations, some general ideas are presented as followed: y Considered are various control rod candidate black absorber materials, such as: cadmium, hafnium, europium and gadolinium and combination of these materials with “grey” absorbers, such as stainless steel, which can be used to prolong the lower end of the active control rod part. Considered is also a CR with full length made of stellite (grey absorber) y Optimization of the control rod geometry and dimensions are foreseen in case of choice of material with significant higher absorption cross section than cadmium (e.g., europium or hafnium); y Variation of the thickness of Al cladding, such as: variation of the thickness of Al cladding from 1.5 mm to 4.5 mm;

y Variation of the outer diameter and thickness of the Control Rod black material;

y Variation of the outer diameter and thickness of the length of the grey material (for Hf rod). 5.4.2 Choosing of the optimal projects According to the description of former section, there could be hundreds of kinds of combinations. The optimal calculation can not be done one by one due to the time limitation. Compared with other research reactor and according to the self - characteristics of BR2, the following projects will be adopted and be calculated: y Cd+Co: reference Control rod with cadmium and cobalt in the lower active absorbing part of the rod (with Al cladding) y Cd+Cd: control Rod with full length of cadmium (with Al cladding) y Cd+AISI304: control Rod with cadmium and AISI304 in the lower active part (with Al cladding)

48 CHAPTER 5. CALCULATION OF THE OPTIMAL DESIGN

y Hf+Hf: Control Rod with full length of hafnium - With Al cladding - Without cladding y Hf+AISI304: control Rod with hafnium and AISI304 in the lower active part (without cladding)

y Eu2O3: control rod with full length of europium oxide (without cladding)

y Gd2O3: control rod with full length of gadolinium oxide (without cladding)

5.5 Neutronics modeling of BR2

The full-scale 3-D heterogeneous geometry model of BR2 was developed using the Monte Carlo code MCNP and it is presented at Fig. 5.3. The model describes the actual twisted hyperboloid reactor core, formed from skew beryllium prisms with individual orientation of the loaded fuel elements, control rods and engineering devices inside the test holes. The CR is divided into 10 axial segments in which the depletion calculations have been performed by MCNP&ORIGEN-S. The lower part of the absorbing CR material (Cd, Hf…) is divided into 10 radial annular rings. The detailed model of the CR is given in Fig. 5.4. Standard fuel elements are assemblies, composed from 6 annular concentric tubes maintained by 3 aluminum side plates. The hot plane of the fuel plates of each FE are divided into azimuth sectors by each 5° (see Fig. 4.5).

5.6 Calculation contents and process

The burn up of the Control Rod absorbing material, total and differential control rods worth’s, macroscopic and effective microscopic absorption cross sections, fuel and reactivity evolution, activity and nuclear heating are evaluated during ~ 33 consequtive operating cycles, each about 30 days long, which is equivalent to ~ 1000 EFPD of reactor operation. The typical initial mass of loaded 235U (which content is 90% from the total fuel mass) in a BR2 operation cycle is about 10 kg. The burn up at the end of an operation BR2 cycle is ~ 150÷160 MWd/kgU5. The depletion calculations of the fuel composition and CR material are performed by MCNP&ORIGEN-S method for the full 3-D detailed heterogeneous model of BR2 with detailed fuel burn up distribution, which contains about 4600 fuel cells with varied fuel burn up.

Before calculation, two assumptions will be considered as followed:

y A typical BR2 reactor core load, which remains the same in each cycle, is used in the calculations. y The evolution of the CR absorbing material (atomic density, macroscopic and microscopic cross sections, and activity), the fuel depletion and criticality evolution are evaluated during each cycle for about ~ 30 consecutive cycles. The following calculation method is applied:

49 CHAPTER 5. CALCULATION OF THE OPTIMAL DESIGN

Ö BOC (1): 6 CR with fresh absorbing material are loaded into the core of BR2 at beginning of the 1st operating cycle. Ö BOC (2): The densities of the CR material at EOC (1) are used as initial densities at the BOC (2). Ö BOC (3): The densities of the CR material at EOC (2) are used as initial densities at the BOC (3). Ö ……………………………………………………………………………….. Ö BOC (N): The densities of the CR material at EOC (N-1) are used as initial densities at the BOC (N), N=1,…,30. For the time between EOC(N-1) and BOC(N) we assume a shutdown period with duration 30 days, which is taken into account in the calculations.

Figure 5.3 MCNP model representation of a horizontal cut 15 cm below mid-plane

50 CHAPTER 5. CALCULATION OF THE OPTIMAL DESIGN

Figure 5.4 MCNP model of a BR2 control rod, divided into 10 radial sectors by each 0.05 mm.

CHAPTER 6 ANALYSIS OF CALCULATION RESULTS

6.1 Comparison of various CR material characteristics

Within each optimal project, for each CR absorbing material, the productions of various isotopes in CR material are evaluated through neutronic transmutation ((n, γ), (n,p), etc. reactions) and radioactive decay using the combined MCNP&ORIGEN-S method. MCNP is used for calculation of the thermal neutron flux in the corresponding axial segment of the CR. Then this neutron flux is introduced into 1-D depletion code ORIGEN-S, which performs calculations of the nuclide inventory and activities in the CR material for the desired irradiation time steps and shutdown periods. The depleted isotopic composition of the CR material for the relevant time depletion step is introduced back into the MCNP whole core model of BR2 to perform accurate evaluations of the CR characteristics: reactivity worth, macroscopic and microscopic effective cross sections, neutron fluxes and spectra. The CR characteristics needed to be compared include atomic densities, macroscopic and microscopic effective cross sections and activity, which are evaluated in different axial zones over the CR height. 6.1.1 Comparison of macroscopic absorption cross sections during 1000 EFPD The macroscopic absorption cross sections of the dominant and non-dominant nuclide for different CR types are given at Fig. 6.1. According to these graphs, we can qualitatively get the following conclusions:

−1 y The total macroscopic cross sections defined as Σ = N < σ > eff ,cm remain almost

constant for all considered rods – cadmium, hafnium, europium (in Eu2O3) and gadolinium rods during sufficiently long time of irradiation: T ~ 650 EFPD (see Fig 6.1(e,f,g)). y After T ~ 650 EFPD the macroscopic cross sections for cadmium rods drastically decrease due to the rapid burnup of the dominant isotope 113Cd (see Fig 6.1(e,f,g)). y A residual absorption, equivalent to about 30% from the initial value is caused by the other cadmium isotopes – 111Cd and 110Cd. This residual absorption cross section remains constant at least for T ~ 600 to 1000 EFPD (see Fig 6.1(e,f,g)). y The macroscopic cross sections for hafnium and europium rods and combination of these absorbers with stainless steel in the lower part of the control rod, remain almost constant up to ~ 1000 EFPD (Fig. 6.1a,b,c,d,h). The absolute values of the macroscopic cross sections for fresh and burnt CR material are summarized in Table 6-1.Through comparison with each material, the following results can be concluded: 52 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

-1 y Eu2O3 rod has maximum anti-reactivity (l00%): Σ=0.30 cm -1 y Reference rods (Cd+Co) – the anti-reactivity is ~ 66% from Eu2O3 rod: Σ=0.2 cm -1 y Cd+Cd rod – the anti-reactivity is about ~ 77% from Eu2O3 rod: Σ=0.23 cm

y Hf+Hf rod (without Al cladding) – the anti-reactivity is ~ 90% from Eu2O3 rod: Σ=0.27 cm-1 y Hf+Hf rod (with Al cladding, not tabulated in Table 6-1) – the anti-reactivity is ~ 85% -1 from Eu2O3 rod: Σ=0.25 cm

y Hf+AISI304 rod (without Al cladding) – the anti-reactivity is ~ 72% from Eu2O3 rod: Σ=0.22 cm-1 -1 y Gd2O3 rod – the anti-reactivity is ~ 92% from Eu2O3 rod: Σ=0.28 cm

CR with full length o f Hf (no Al cladding) CR with Hf and AISI30 4 in the lower lower part 0.30 0.30 MCNP Hf+Hf MCNP Hf+AISI304 0.25 0.25 total ]

total ] -1 Hf174 -1 Hf174

[cm Hf176 Hf176 n 0.20 n [cm 0.20 o o Hf177 i Hf177 ct

e Hf178 Hf178 ecti s s s s Hf179 s Hf179 s 0.15 0.15 Hf180

Hf180 c cro c cro i i p p co co s s 0.10 0.10 acro acro m m 0.05 0.05

0.00 0.00 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Time [F.P.D.] Time [F.P.D.] (a) (b)

CR with full Hf leng th (with Al cladding) CR with full length of Eu2O3 0.30 0.4 MCNP MCNP Eu O Hf+Hf 2 3 0.25 ] ] -1 -1 m m total 0.3 c c total Hf174 [ n n [ 0.20 o o Sm152 i i Hf176 Eu151 Hf177 ect s s

s sect Eu152 s

s Hf178 0.15 0.2 Eu153

Hf179 cro c c cro Eu154 Hf180 i p pi

o Eu155 c co s s 0.10 Gd152 o O16 acr acro m m 0.1 0.05

0.00 0.0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 Time [F.P.D.] Time [F.P.D.] (c) (d)

53 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

CR with Cd and Co (Reference CR) CR with full Cd length

0.25 0.25 MCNP MCNP Cd+Co Cd+Cd ] ] -1

0.20 -1 0.20 m m c [c [ n n o o i i t total c

total ct e

0.15 e

s 0.15 Cd113 s s Cd113 s s s

o Cd112 o r Cd112 r c

Cd106 c Cd106 c i i p Cd108 0.10 Cd108 p o

co 0.10 c s Cd110 s

o Cd110 o Cd111 cr

acr Cd111 a

m Cd114 Cd114 m 0.05 Cd116 0.05 Cd116

0.00 0.00 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Time [F.P.D.] Time [F.P.D.] (e) (f)

-1 Comparison of CR with (C d+Cd) and CR with (Cd+Co) Comparison of Σ [cm ] of different CR 0.30 0.4 MCNP MCNP total_(Cd+Co) total_(Cd+Cd) ]

] 0.25 -1 -1 Cd113_(Cd+Co) m c

[ Cd113_(Cd+Cd) 0.3 n [cm 0.20 o tion c secti se s oss r 0.15

c 0.2

c cros i opic

cop Cd+Co s osc o r 0.10 r Cd+Cd c c a a Hf+Hf m m 0.1 Hf+AISI304

0.05 Eu2O3 Gd O 2 3 0.00 0.0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 1000 Time [F.P.D.] Time [F.P.D.] (g) (h) Figure 6.1 Comparison of effective macroscopic absorption cross sections of dominant and non-dominant isotopes for various CR types.

Table 6-1 Calculated macroscopic absorption cross sections Σ in fresh CR material (T=0) and burnt CR material (T ~ 600 - 1000 EFPD).

Cd+Co Cd+Cd Hf+AISI Hf Eu2O3 Gd2O3 T=0 0.20 0.23 0.22 0.27 0.30 0.28 T=600 EFPD 0.16 0.08 0.21 0.25 0.31 0.26

T=1000 EFPD 0.08 0.10 0.20 0.24 0.31 −

6.1.2 Comparison of microscopic absorption cross sections 6.1.2.1 Evaluation of microscopic absorption cross section during T< 600 EFPD The macroscopic cross is defined as:

54 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

−1 Σ = N < σ >eff ,[cm ] (6-1) Where, Σ is function of two variables: the atomic density N and the effective microscopic

cross section < σ >eff , which is defined as: ∫σ ()E Φ(E)dE < σ >eff = (6-2) ∫Φ()E dE The evolution of the atomic densities of the dominant nuclides and the microscopic effective cross sections, defined with Eq. (6-2) are given at Fig. 6.2. It is seen from Fig. 6.2 that for 151Eu and 177Hf the effective microscopic cross section remain almost constant during the irradiation, while for gadolinium and especially for cadmium the microscopic cross sections increase during the first ~ 10 to 20 cycles, so that the product of the atomic density and the effective microscopic cross section, i.e. the macroscopic effective cross section Σ remains constant during long irradiation time ~ 600 EFPD.

CR with Cd+Cd CR with Hf+Hf

0.0060 60 0.0090 30 ]

Cd-113 n ] ar

Hf-177 n

density b ar 0.0055 [ 0.0085 b n

o density <σ > [ i

eff 55 n ]

] 25 o -3 i -3 <σ >

m eff 0.0050 c 0.0080 ect .cm ss sect ms. o ms o ss s t cr a o [

c i

y [ato 0.0045 50 0.0075 ity 20 it c cr s i s n p cop e s d o co c den s i cr 0.0040 0.0070 o m timic cr a ato

45 e mi 15 v i e mi t v

0.0035 c 0.0065 i e f ect ef f MCNP MCNP 0.0030 40 0.0060 10 ef 0 50 100 150 200 250 300 0 50 100 150 200

Time [days] Time [days]

CR with Eu O CR with Gd O 2 3 2 3 0.0125 30 0.0055 55 n o Gd-157 ]

Eu-151 n cti

0.0120 0.0050 ar b

density [

density n oss se σ 50 < > o ] r

25 eff i -3 t

0.0115 m] c m

c 0.0045 . /b-c ive c t s se <σ > [a eff s toms 0.0110 y o fec a it r ef y [ 0.0040 45 c it

20 c i i dens p c 0.0105 p o co s atimi o

osc 0.0035 r r c c

0.0100 i atomic dens 15 40 mi e m e v

v 0.0030 i

0.0095 t c cti e e MCNP f eff 0.0090 MCNP ef 10 0.0025 35 0 50 100 150 200 250 0 50 100 150 Time [days] Time [days] Figure 6.2 Evolutions of microscopic effective absorption cross and atomic density of dominant isotopes for different absorbing materials To explain the reason for changing of the microscopic effective cross sections during irradiation, calculations of the neutron spectra and the spectral dependence of the effective microscopic cross sections during irradiation are performed for cadmium and europium rods and the results are presented in Tables 6-2(a), 6-2(b) and Tables 6-3(a), 6-3(b).

55 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Table 6-2 Effective microscopic cross sections of 113Cd (a) at BOC(1) Energy Interval Φ ,[n.cm−2 ] 〈σΦ〉 〈σ 〉,[]barn [MeV] n 1.0000E-08 3.11656E-06 1.06062E-01 34031.8 1.0000E-07 9.91677E-05 2.15052E+00 21685.7 5.0000E-07 1.23862E-04 1.13384E+00 9154.1 1.0000E-06 6.86093E-04 2.52618E-01 368.2 1.0000E-05 4.43831E-03 7.78220E-02 17.5 1.0000E-04 4.58137E-03 1.13696E-02 2.48 1.0000E-03 4.36802E-03 1.50586E-02 3.45 1.0000E+00 2.04142E-02 1.11014E-02 0.544 2.0000E+01 8.24206E-03 2.53679E-04 0.031 Total 4.29562E-02 3.75865E+00 87.5

(b) at BOC(5)

Energy Interval Φ ,[n.cm−2 ] 〈σΦ〉 〈σ 〉,[]barn [MeV] n 1.0000E-08 3.60803E-06 1.24786E-01 34585.6 1.0000E-07 1.23175E-04 2.67114E+00 21685.7 5.0000E-07 1.50971E-04 1.33703E+00 8856.2 1.0000E-06 7.77260E-04 2.83058E-01 364.2 1.0000E-05 4.53269E-03 7.96937E-02 17.6 1.0000E-04 4.46512E-03 1.44074E-02 3.23 1.0000E-03 4.54493E-03 1.62104E-02 3.6 1.0000E+00 2.03839E-02 1.06207E-02 0.52 2.0000E+01 8.04363E-03 2.46959E-04 0.03 Total 4.30253E-02 4.53719E+00 105.4 Table 6-3 Effective microscopic cross sections of 151Eu (a) at BOC(1)

Energy Interval Φ ,[n.cm−2 ] 〈σΦ〉 〈σ 〉,[]barn [MeV] n 1.0000E-08 2.03934E-06 4.68975E-02 22996.4 1.0000E-07 2.12959E-04 9.27599E-01 4355.8 5.0000E-07 1.81148E-04 4.25898E-01 2351.1 1.0000E-06 1.36582E-04 1.22348E-01 895.8 1.0000E-05 1.06478E-03 1.86910E-01 175.5 1.0000E-04 1.64017E-03 1.76904E-01 107.9 1.0000E-03 2.55770E-03 1.57794E-01 61.7 1.0000E+00 1.80888E-02 8.87556E-02 4.9 2.0000E+01 7.79635E-03 1.57104E-03 0.2 Total 3.16806E-02 2.13468E+00 67.4

56 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

(b) at BOC(5)

Energy Interval Φ ,[n.cm−2 ] 〈σΦ〉 〈σ 〉,[]barn [MeV] n 1.0000E-08 1.81455E-06 4.25510E-02 23449.9 1.0000E-07 2.06654E-04 9.16538E-01 4435.1 5.0000E-07 1.78796E-04 4.33808E-01 2426.3 1.0000E-06 1.35954E-04 1.26776E-01 932.5 1.0000E-05 9.30659E-04 1.77717E-01 191 1.0000E-04 1.57445E-03 1.83769E-01 116.7 1.0000E-03 2.45591E-03 1.58116E-01 64.4 1.0000E+00 1.81107E-02 9.32322E-02 5.2 2.0000E+01 7.81719E-03 1.54708E-03 0.2 Total 3.14122E-02 2.13405E+00 67.9

As can be seen from Table 6-2(a), 6-2(b), the reason for increase of the microscopic effective cross sections of 113Cd is the changing of the neutron spectrum in the thermal and epithermal energy regions and as result – increase of the thermal and epithermal effective microscopic cross sections. The depletion of the cadmium density is "compensated" by increase of the microscopic

−1 cross section, so that Σ = N < σ > eff ,cm remains almost constant during ~20 irradiation cycles. On the contrary, this effect is not observed for the 151Eu (see Tables 6-3(a), 6-3(b)), which effective microscopic cross section remains constant and consequently the macroscopic cross

−1 151 151 section Σ = N < σ > eff ,cm of Eu decreases with decreasing of the density of Eu. However, the decrease of the macroscopic cross section of 151Eu is compensated by the produced other Eu – nuclides (152Eu, 154Eu) and 152Sm, so that the total macroscopic absorption cross section of the Eu – rod remains also constant. Additional detailed calculations of the changing of the neutron spectrum and the reaction rates in the rods during irradiation have been performed for Cd – rod and for Eu – rod. The results are given in Fig. 6.3. The changing of the neutron flux defined simply as Φ()E,T = 150d. /Φ(E,T = 0d.) is shown at Fig. 6.3(a). It is seen that for Cd – rod the neutron flux increases after T=150 days in the depleted rod. The explanation of this is simple: the depleted rod absorbs fewer neutrons, the local neutron flux arises and as result – the reaction rates σΦ in 113Cd also increase, which can be seen from Fig. 6.3(b). For Eu – rod, the neutron flux during irradiation remains almost constant as can be seen from Fig. 6.3(a). This has also simple explanation: the depletion of 151Eu is "compensated" by the production of the other Eu, Sm, Gd – isotopes, which absorb neutrons, so that the local neutron flux in the Eu – rod remains almost constant. As result for 151Eu the increase of the reaction rates σΦ after T=150 days is negligible in comparison with 113Cd, which can be seen from Fig. 6.3(c). What do we have finally for the absorption during the irradiation by 113Cd and 151Eu, defined as NσΦ? From Fig. 6.3(d) and Fig. 6.3(e) is seen that the product NσΦ remains constant during irradiation for 113Cd and decreases for 151Eu during the irradiation. However, the decrease of

57 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

NσΦ is compensated by the produced other Eu, Sm, Gd – isotopes. The total absorption of the Cd – rod and of the Eu – rod also remain almost constant during long time of irradiation. All considered materials for the different CR types have one dominant nuclide, which is depleted during the irradiation cycle, i.e. the atomic density of this dominant nuclide decreases with time. For cadmium rod – the dominant nuclide is 113Cd, for europium rod the dominant nuclide is 151Eu, for the hafnium and gadolinium rod the dominant nuclide is 177Hf and 157Gd respectively. The effective microscopic cross sections may change (increase) during irradiation due to the changing of the spectrum in the depleted rod. This is strongly observed for cadmium rod, which microscopic cross section increases during irradiation, so that the macroscopic cross section Σ, determined with Eq. (6-1) remains practically constant during long time of irradiation ~ 600 EFPD. For europium rod – the atomic density of 151Eu decreases, the effective microscopic cross section remains constant, so that the macroscopic cross section decreases proportionally with the density. However this decrease is compensated by the produced other europium isotopes (152Eu, 154Eu), some contributions come also from 152Sm, 154Gd, 155Eu and the resonant absorption cross section of the rod also remains constant during the considered irradiation period. The effective microscopic cross sections of 177Hf and 157Gd are changing slowly, but the final product – the macroscopic cross section Σ also remains constant during long irradiation period. (a) Increase of local neutron flux in the depleted during irradiation

2.5

2.0 CR_Cd: Φ (150 d.)/Φ (0 d.) ~ 1.08 CR_Eu: Φ (150 d.)/Φ (0 d.) ~ 1.02 . u r.

, ) . 1.5 d (0 Φ )/

150 d.

( 1.0 Φ

0.5

MCNP 0.0 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Neutron energy [MeV]

0.10 0.04 113Cd 151Eu

> 0.08 >

σΦ 0.03 σΦ < < , ce n, ce n our 0.06 our r 1 s

r 1 s 0.02 T=0 days s pe s pe te T=150 days te a r n ra 0.04

n io

T=0 days o act T=150 days acti re

re 0.01 0.02

MCNP MCNP 0.00 0.00 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10 10 10 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Neutron energy [MeV] Neutron energy [MeV]

(b) Spectral distribution of the reaction rates in 113Cd during irradiation (c) Spectral distribution of the reaction rates in 151Eu during irradiation

58 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

0.0005 0.0004 151 113Cd Eu T T > > Φ 0.0004 σΦ σ < < 51 3 1 0.0003 1 1 - Eu- T Cd T N

n, , n 0.0003 e c r u o s 1 source

0.0002 T=0 days

r T=150 days e p n per

n 0.0002 o i t T=0 days rp

o T=150 days s b

a 0.0001

ol absorptio l o

a 0.0001 t tota o t

MCNP MCNP 0.0000 0.0000 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10 10 10 10 10 10 10 10 10 10 Neutron energy [MeV] Neutron energy [MeV]

T T 113 T T 151 (d) Total absorption N Cd-113<σφ> in Cd during irradiation (e) Total absorption N Eu-151<σφ> in Eu during irradiation Figure 6.3 the changing of the neutron spectrum and the reaction rates in the rods during irradiation for Cd – rod and for Eu – rod 6.1.2.2 Comparison of Neutron Spectrum during 1000 EFPD The changing of the spectrum in the depleted rod has been investigated during long irradiation period for T ~ 1000 EFPD. Detailed spectral calculations have been performed for the lower part of the absorbing CR rod material. The comparison of the neutron spectra for different types CR at T=0 and T=1000EFPD are given in Fig. 6.4(a), (b). The comparison between the neutron spectra in fresh and burnt CR material for the individual CR types is demonstrated at Fig. 6.4(c), (d), (e), (f). As can be seen from the graph (Fig. 6.4(c), (d), (e)): for CR types: Cd+Co, Cd+Cd, Cd+AISI304, the thermal fluxes increase drastically after ~ 600 EFPD which is related with the complete depletion of 113Cd, which has high thermal absorption cross section (see Fig. 3.1). However, the epithermal and fast neutron spectra almost do not change, because for these energy regions the absorption is caused by the other Cd-isotopes: 110Cd and 111Cd (see Fig. 3.1). For a fresh CR type (with cadmium) the fraction of the thermal flux is less than 2% into the total neutron flux of the rod. After T > 600 EFPD, the thermal flux arises and contributes about 30% into the total n – flux. For this region, after T > 600 EFPD, the effective macroscopic absorption cross sections of 110Cd and 111Cd increase and as can be seen from Fig. 6.1(e), (f), (g). They form a residual absorption which is about 30% from the absorption of the fresh Cd-rod. This residual absorption may remain constant at least for T ~ 600 to 1000 EFPD. From all this we can conclude that in principle, the Cd – rod can be used still long time after the complete depletion of the isotope 113Cd. There is one dangerous time interval moment (not yet determined in this job) between the total disappearance of 113Cd and the following compensation of the absorption by 110Cd and 111Cd. The duration of this time interval has to be determined.

59 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

3.5x1013 4.5x1013 MCNP T=0 MCNP T=1000 EFPD 13 13 4.0x10 Cd+Co 3.0x10 Cd+Co Cd Cd 13 3.5x10 ] Cd+AISI304 ] 1 1 Cd+AISI304 - - 13

.s Hf .s 2.5x10 2 2 -

- Hf 3.0x1013 Hf+AISI304 m Hf+AISI304 .cm .c Eu2O3 n

13 Eu2O3 [n 13 [ n n 2.0x10 2.5x10 Φ

Φ ux

ux 13 l

13 fl 2.0x10 1.5x10 n on f ro r t

ut 13 u 1.5x10 ne ne 1.0x1013 1.0x1013

12 5.0x10 5.0x1012

0.0 0.0 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Neutron energy [MeV] Neutron energy [MeV]

(a) Neutron spectra in the lower absorbing CR part at T=0 (b) Neutron spectra in the lower absorbing CR part at T=1000EFPD

3.0x1013 4.5x1013 MCNP MCNP 13 Cd (T=0) Cd+Co (T=0) 4.0x10 Cd (T=1000 EFPD) 2.5x1013 Cd+Co (T=1000) ]

1 13 - ]

1 3.5x10 - .s 2 - .s 2 - 13 13 m 2.0x10 .cm 3.0x10 [n n [n.c n Φ

13 Φ x 2.5x10 u

13 l ux

l 1.5x10 13 on f n f 2.0x10 o utr e utr

13 n 13 ne 1.0x10 1.5x10

1.0x1013 5.0x1012 5.0x1012

0.0 0.0 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Neutron energy [MeV] Neutron energy [MeV]

13 2.0x10 1.8x1013 MCNP MCNP Cd+AISI304 (T=0) 13 Cd+AISI304 (T=1000 EFPD) 1.6x10 Hf+AISI304 (T=0) Hf+AISI304 (T=1000 EFPD) ] ] 1 1 13 - - 13 1.4x10 .s .s 2 2 - - 1.5x10 m m 13 .c .c n n 1.2x10 [ [ n n Φ Φ

1.0x1013 ux ux 1.0x1013 n fl n fl 12 ro ro 8.0x10 ut ut e e n n 6.0x1012 5.0x1012 4.0x1012

2.0x1012

0.0 0.0 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Neutron energy [MeV] Neutron energy [MeV]

(e) Neutron spectra in the lower absorbing CR part (Cd+AISI304) (f) Neutron spectra in the lower absorbing CR part (Hf+AISI304) Figure 6.4 Neutron spectra in the lower active part of the CR 60 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

6.1.3 Comparison of activity during 1000 EFPD The Activity is defined as: A(t) = λN(t) (6-3)

Where, λ is decay constant, and N(t) is atomic density at time t, which can be given by:

−λt N(t) = N0e (6-4) The time during which the activity falls by a factor is known as the half-life and is given the symbol T1/2, which can be expressed by: ln 2 0.693 T = = (6-5) 1/ 2 λ λ Hence, the activity can be rewritten by:

−0.693t /T1/ 2 A(t) = T1/ 2 N0e (6-6) The evaluation of activity for the different types of CR is shown in Fig. 6.5. The CR is divided into three parts: lower, middle and upper part. Among these CR types, only for Cd+Co and Hf+AISI304, the absorbing material of the lower part is different with that of the middle and upper parts; for the rest, the absorbing material is same for the whole CR.

4.0x105 3.5x105 MCNP Cd+Co MCNP Cd+Co

5 3.5x10 total 3.0x105 60Co in the lower part lower part of CR(Co) 61Co in the lower part 5 3.0x10 midle part of CR(Cd) 107 5 Cd in the middle part upper part of CR(Cd) 2.5x10 115Cd in the middle part 2.5x105 117 5 In in the middle part

] 2.0x10 i] 107 i

C Cd in the upper part 2.0x105 [ 115 ty[C

Cd in the upper part

ity 5 vi 1.5x10 117 ti In in the upper part 5 Ac 1.5x10 Activ

1.0x105 1.0x105

5.0x104 5.0x104

0.0 0.0 0 100 200 300 400 500 0 100 200 300 400 500 Time [days] Time [days]

(a) Cd+Co

5.00x104 MCNP 107Cd in the lower part of CR MCNP Cd+Cd 4 115 Cd+Cd 5 4.50x10 Cd in the lower part of CR 1.0x10 total 117 4 In in the lower part of CR 4.00x10 107 the lower part of CR(Cd) Cd in the middle part of CR 115 the middle part of CR(Cd) 4 4 3.50x10 Cd in the middle part of CR 8.0x10 the upper part of CR(Cd) 117In in the middle part of CR 4 3.00x10 107Cd in the upper part of CR 115 i] 4 Cd in the upper part of CR

C 4 2.50x10 117 Ci]

[ 6.0x10 [ In in the upper part of CR y

it ity 2.00x104 tiv tiv c c 4 A 4 A 4.0x10 1.50x10

1.00x104

4 2.0x10 5.00x103

0.00

0.0 3 -5.00x10 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Time[Days] Time[days]

61 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

3.0x105 153 1.0x106 MCNP Sm in the lower part MCNP Eu2O3 Eu O 152Eu in the lower part 2 3 5 154 total 2.5x10 Eu in the lower part 153 5 lower part of CR Sm in the middle part 8.0x10 middle part of CR 152Eu in the middle part 2.0x105 upper part of CR 154Eu in the middle part 153 5 Sm in the upper part 6.0x10 152 i] 5 Eu in the upper part

C 1.5x10 [ 154 y[Ci] t ty

Eu in the upper part i v i tivi 5 5

Act 1.0x10 Ac 4.0x10

5.0x104 2.0x105

0.0 0.0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Time[days] Time[days]

(b) Cd+Cd

1.4x105 8.0x104 175 MCNP total Hf+Hf MCNP Hf in the lower part Hf+Hf 181 5 lower part of CR Hf in the lower part 1.2x10 middle part of CR 175Hf in the middle part upper part of CR 6.0x104 181Hf in the middle part 5 1.0x10 175Hf in the upper part 181Hf in the upper part 4 i] i) 8.0x10 4

(C 4.0x10 y

viy [C vit 4 i i 6.0x10 Act Act

4 4.0x104 2.0x10

2.0x104 0.0 0.0 0 200 400 600 800 1000 0 200 400 600 800 1000 Time(day) Time [days]

(d) Hf+Hf

4 8.0x104 4.0x10 MCNP Hf+AISI304 MCNP Hf+AISI304 3.5x104

4 6.0x104 3.0x10

total 2.5x104 middle part of CR(Hf) i]

Ci] 175 upper part of CR(Hf) [ 4

4 y 2.0x10 Hf in the middle part

4.0x10 it ty[C 181 Hf in the middle part vi tiv i c t 4 175

c 1.5x10 A Hf in the upper part A 181Hf in the upper part 1.0x104 2.0x104

5.0x103

0.0 0.0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Time[days] Time[days]

(e) Hf+AISI304 Figure 6.5 The activity of each part of CR and dominant nuclides in the CR during 1000 EFPD 62 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

The reaction process is very complicated relative to radioactive nuclide decay and neutron activation. According to the definition, we can explain from two ways: half-life and atomic density. The characteristic of each material including the half-life has been mentioned in chapter 3 except for cobalt. It can be seen from Fig 6.5 that they have not common characters, therefore, we can only explain one by one: For Cd+Co, the main part for the activity is lower part and the dominant nuclide is 60Co. 60Co is a radioactive nuclide with he half-life 5.2714y, so, it provides more contribution to the activity even though small part of which is changing into 61Co by reaction with neutron(see Fig. 6.5(a)). For Cd+Cd, the main part for the activity is lower part and the dominant nuclide is 115Cd, which is the product of the reaction of 113Cd and neutrons.

151 152 For Eu2O3, Eu and Eu have high absorption cross sections, the latter doesn’t exist in the nature which is the product of 151Eu reacting with neutron and has a half-life with 13.516y. Furthermore, 152Eu can decay to 152Sm by EC which can be changed into 153Sm due to neutron activation, the half-life of 153Sm is only 46.284h. Therefore, the dominant nuclides to the activity of Eu-rods are 152Eu and 153Sm. For Hf+Hf, 177Hf has a high absorption cross section and is stable, its product after irradiation is still stable until 181Hf, which has a short half-life. That is the reason why 181Hf does more contributes to the activity. For Hf+AISI304, the explanation is same as for Hf+Hf rod. The only different thing is that the material of lower part is AISI304, which produces negligible radioactive nuclides. 6.1.4 Comparison of nuclear heating in the lower active part of the control rods The nuclear heating in the lower Cd part of the different CR types was calculated using MCNP. The main contributions to the total heating of the cadmium give neutrons (more than 90%). The pr+cap contribution from prompt and captured γ-rays Q γ into the total heating is about 12% to 14% from the heating Qn, caused by neutrons. Additional contributions into the total heating come del from the delayed γ-rays Q γ. MCNP can only solve neutron-induced photon transport, the contribution from the delayed photons from the fission products can not be computed directly by the code. A separate geometry model of BR2 has been developed for evaluation of the heating from the delayed photons. For this purpose the code ORIGEN-S of the SCALE system is used for evaluations of the photon spectra and the photon intensity from fission products accumulated in the fuel elements during irradiation in the BR2 reactor. Using the power peaking factors calculated with MCNP in each fuel element, the axial and radial distributions of the intensity of the delayed photon sources in the core are performed and used as an external source (fixed-source option) in the independent photon transport calculation in the MCNP model of BR2. The contribution from β−− particles (if available) is not included. The neutron and gamma heating in the lower Cd part in the Reference CR and in CR type Cd+AISI304 were evaluated using the MCNP neutron and photon heating tallies. The results of the calculated with MCNP neutron and gamma heating for these CR types are presented in Table 6-4.

63 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Table 6-4 Nuclear heating (Watt per gram) in the lower part of the Control Rods. “Cobalt” part of CR Lower cadmium part of CR CR type prompt+cap delayed prompt+cap delayed Qn Qγ Qγ Qn Qγ Qγ CR_Co 0.039 6.70 1.3 27.5 3.79 0.8 CR_Cd 27.2 3.71 0.7 20.5 2.93 0.6 CR_AISI304 0.047 4.62 0.9 30.5 3.46 0.7

6.1.5 Comparison of total control rods worth's Detailed calculations of the total control rod worth for various types control rods have been performed by MCNP&ORIGEN-S for the beginning of the Control rod life and after long time of irradiation, taking into account the axial burnup of the Control Rod absorbing material. 6.1.5.1 Comparison of Total Control Rods Worth at T=0 and T=1000 EFPD The calculations are performed for the load of cycle of BR2 03/2006A.5[25]. Total CR worth’s for the different types control rods are compared for BOC and EOC during the 1st and during the 30th operation cycle and presented in Table 6-5. It can be seen from Table 6-5 that the burn up of the CR material does not affect strongly the value of total CR worth. Table 6-5 Comparison of total worth (in units of $ and pcm in the brackets) for different control rod types accounting for the axial burnup of the absorbing material during irradiation BOC EOC CR type T=0 T=1000 EFPD T=30 EFPD T=1030 EFPD (1st cycle) (~ 30th cycle) (1st cycle) (~ 30th cycle) Cd+Co 13.4 $ (9648 pcm) 13.0 $ (9360 pcm) 14.5 $ 14.0 $ Cd+Cd 13.6 $ (9792 pcm) 13.3 $ (9576 pcm) 14.7 $ 14.3 $ Cd+AISI304 13.2 $ (9504 pcm) 12.9 $ (9288 pcm) 14.3 $ 13.9 $ Hf+Hf 15.8 $ (11376 pcm) 15.7 $ (11304 pcm) 17.1 $ 17.0 $ Hf+AISI304 15.6 $ (11232 pcm) 15.5 $ (11160 pcm) 16.7 $ 16.7 $

Eu2O3 17.5 $ (12600 pcm) 17.5 $ (12600 pcm) 19.0 $ 18.8 $

The curves of the total control rods worth’s for fresh CR at BOC (T=0) and for burnt CR at BOC of the 30th cycle (T~1000 EFPD) are given in Fig. 6.6. The comparison of the curves of total CR worth during irradiation for the different CR types is given in Fig. 6.7. According to the Fig. 6.7, we can draw some conclusions as followed: y The curve of the total worth of control rods with full cadmium part (Cd+Cd) decreases significantly after ~ 750 EFPD of irradiation see Fig. 6.7(c). y The curves of total rods worth for Cd+AISI304 and for Reference Cd+Co also decrease during irradiation, but less (Fig. 6.7(a), (b)).

y For all other CR types –Hf+AISI304, Hf+Hf, Eu2O3 (Fig. 6.7(d), (e), (f)) the changing of the curves of total CR worth’s during T ~1000 EFPD is negligible.

64 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

18 20 Cd+Co Cd+Co 16 18 Cd+AISI304 Cd+AISI304 Cd 16 14 Cd Hf+AISI304 Hf+AISI304 14 Hf 12 Hf Eu2O3 Eu2O3 $] $] 12 - 10 -

ity [ 10

tivity [ tiv c 8 c a a e e 8 r r

6 6

4 4 MCNP 2 T=0 MCNP 2 T=1000 EFPD 0 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] Figure 6.6 The total control rods worth of different CR at T=0 and T=1000EFPD ( R0 = ρ(0) − ρ(900mm) ).

16 16 MCNPX MCNP Cd+Co (T=0) 14 14 Cd+AISI304 (T=0) Cd+Co (T=1000 EFPD) Cd+AISI304 (T=1000 EFPD) 12 12

] 10 $] $ - [- [ y 8 8 vit i tivity t c a ac e e r r 6 6

4 4

2 2

0 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (a) (b)

16 18 MCNPX MCNP Cd (T=0) 16 Hf+AISI304 (T=0) 14 Cd (T=1000 EFPD) Hf+AISI304 (T=1000 EFPD) 14 12 12

] 10 $] $ - - [ 10 [ ty 8 i v i tivity 8 c a e react r 6 6

4 4

2 2

0 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (c) (d)

65 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

18 20 MCNP Hf (T=0) MCNP Eu2O3 (T=0) 16 18 Hf (T=1000 EFPD) Eu2O3 (T=1000 EFPD) 16 14 14 12 ] ] 12 [-$ [-$ 10 y y t t i i 10 v v i i t t 8 ac ac 8 re re 6 6 4 4

2 2

0 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (e) (f) Figure 6.7 Comparison of total CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for individual CR types ( R0 = ρ(0) − ρ(900mm) ).

6.1.5.2 Comparison of Total Control Rods Worth at BOC (T=0) and EOC (T=30 EFPD) The calculations are performed for the load of cycle 03/2006A.5. The curves of the total control rods worth’s at BOC (T=0) and EOC (T=30 EFPD) are given in Fig. 6.8. Similar calculations were performed for the 30th operation cycle BOC (T~1000 EFPD) and EOC (T~1030 EFPD), which are given at Fig. 6.9. It can be seen from these graphs that at EOC the total control rods worth's increase which is caused by the fuel burnup, i.e. – depletion of 235U and accumulation of fission products.

20 20 st st 18 T=0, BOC( 1 cycle) 18 T ~ 30 EFPD, EOC( 1 cycle) Cd+Cd 16 16 Cd+Cd Cd+AISI304 Cd+AISI304 14 14 Hf+Hf Hf+Hf Hf+AISI304 Hf+AISI304 $] 12 stellite -

$] 12 stellite Cd+Co y [- ity [

it Cd+Co 10 Eu2O3 10

tiv iv

Eu2O3 c Gd2O3 a e

8 Gd2O3 r 8 react

6 6

4 4

2 2 MCNP MCNP 0 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (a) (b)

18 18

st st 16 1 cycle 16 1 cycle

Cd+Cd (BOC) Hf+Hf (BOC) 14 14 Cd+Cd (EOC) Hf+Hf (EOC) Cd+AISI304 (BOC) Hf+AISI304 (BOC) 12 12 Cd+AISI304 (EOC) Hf+AISI304 (EOC) ] ] $ $ 10 - 10 [ [- y t

i ity

v i

tiv 8 8 ac re 6 react 6

4 4

2 2 MCNP MCNP 0 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (c) (d) 66 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

16 1 st cycle

14 Cd+Co (BOC) 12 Cd+Co (EOC)

] Cd+AISI304 (BOC) $ - 10 Cd+AISI304 (EOC)

tivity [ 8 ac e r 6

2 MCNP 0 0 100 200 300 400 500 600 700 800 900 Sh [mm] (e) Figure 6.8 Comparison of total CR worth: (a) for various fresh CR materials at BOC of the 1st cycle; and (b) at EOC of the 1st cycle, i.e. after T ~ 30 EFPD; (c) to (e) comparison of total CR worth’s at BOC and EOC during the 1st cycle for individual CR types.

20 20 th th 18 T ~ 1000 EFPD, BOC( 30 cycle) 18 T ~ 1030 EFPD, EOC ( 30 cycle)

16 16 Cd+Cd Cd+Co 14 Cd+AISI304 14 Cd+Cd Hf+Hf Cd+AISI304 ] 12 12 Hf+AISI304 ] Hf+Hf

[-$ Hf+AISI304 [-$ y

t Cd+Co i

ty 10

i v Eu2O3 i Eu2O3 v t i t ac 8 8 re reac 6 6

4 4

2 2 MCNPX MCNPX 0 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] Figure 6.9 Comparison of total CR worth for various CR types (a) at BOC of the 30th cycle (T~1000 EFPD); (b) at EOC of the 30th cycle (T~1030 EFPD) 6.1.6 Comparison of differential control rods worth The calculations are performed for the load of cycle 03/2006A.5. The comparison of the curves of differential worth for the various CR types during irradiation is depicted on Fig. 6.10. At Fig. 6.10(a) are given the differential curves of all CR types, at Fig. 6.10(b) are given only the curves of CR in which the lower part is prolonged by "grey" material. It is seen that prolonging the "black" absorber with a "grey" material shifts the maximum of the differential worth curve to the lower positions Sh of the CR motion. A compromised decision can be found reducing the length of the "grey" material which will be demonstrated in Chapter 7.3. The changing of differential worth curves for the various CR types during long time of irradiation are depicted on Fig. 6.10(c).

67 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

0.050 0.050 MCNP MCNP Cd+Co 0.045 0.045 Cd+Cd Cd+Co Cd+AISI304 Cd+AISI304 0.040 0.040 Hf+Hf Hf+AISI304 Hf+AISI304 0.035 0.035 Eu2O3

0.030 ] 0.030 m m / mm] / $ 0.025

0.025 [ [$

Sh Sh ∆

/ 0.020 ∆

/ 0.020 ∆ρ ∆ρ 0.015 0.015 0.010 0.010 T=0 T=0 0.005 0.005 0.000 0.000 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (a) (b)

0.050 MCNP Cd+Co 0.045 Cd+Cd Cd+AISI304 0.040 Hf+Hf Hf+AISI304 0.035 Eu2O3

] 0.030 m 0.025 [$/m

S (c) ∆

/ 0.020 ∆ρ 0.015

0.010

0.005 T~1000 EFPD

0.000 0 100 200 300 400 500 600 700 800 900 Sh [mm] Figure 6.10 Comparison of differential CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for different CR types. The comparison of the differential CR worth’s for fresh and burnt CR material for individual CR types are given at Fig. 6.11. It can be seen that the burn up of the CR material affects strongly the curves or differential worth for CR types (Cd+Co, Cd+Cd, Cd+AISI304), i.e. - the differential worth for these rods decreases with burn up and the distributions for burnt CR are shifted to the lower positions Sh (see Fig. 6.11a,b,c); the differential CR worth for CR types: Hf+Hf, Hf+AISI304 almost do not change with burn up of the CR material (see Fig. 6.11e,f).

∆ρ ∆ Differential Control Rods Worth: ∆ρ /∆Sh Differential Control Ro ds Worth: i/ Shi i i 0.030 0.030 MCNP MCNP

0.025 0.025

] 0.020 0.020 m m / mm] $ / $ [ i )

[ 0.015

0.015 i ) Sh ∆ Sh / i ∆ / i ∆ρ

0.010 ∆ρ 0.010

Cd+Co (T=0) 0.005 Cd+Co (T=1000 EFPD) 0.005 Cd+AISI304 (T=0) Cd+AISI304 (T=1000 EFPD)

0.000 0.000 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (a) (b)

68 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Differential Control Rods Worth: ∆ρ /∆Sh Differential Control Rods Worth: ∆ρ /∆Sh i i i i 0.030 0.040 MCNP MCNP 0.035 0.025

0.030 0.020 ] ]

m 0.025 m m / m $ /

[ 0.015 i ) [$ 0.020 i ) Sh ∆ Sh / i ∆ / i ∆ρ 0.010 0.015 ∆ρ

0.010 0.005 Cd (T=0) Cd (T=1000 EFPD) 0.005 Eu2O3 (T=0) Eu2O3 (T=1000 EFPD) 0.000 0 100 200 300 400 500 600 700 800 900 0.000 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (c) (d)

∆ρ ∆ Differential Control Rods Worth: ∆ρ /∆Sh Differential Control R ods Worth: i/ Shi i i 0.040 0.035 MCNP MCNP 0.035 0.030

0.030 0.025

] 0.025 m] m

m 0.020 m / / $ [ i

[$ 0.020 i )

Sh) 0.015 Sh ∆ / ∆ i / i 0.015 ∆ρ ∆ρ 0.010 0.010

0.005 Hf+AISI304 (T=0) 0.005 Hf (T=0) Hf (T=1000 EFPD) Hf+AISI304 (T=1000 EFPD)

0.000 0.000 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] (e) (f) Figure 6.11 Comparison of differential CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for individual CR The comparison of the curves of differential worth at BOC and EOC for all rods is shown at Fig. 6.12. At EOC the differential Control Rods worth's increase which is caused by the fuel burnup, i.e. – depletion of 235U and accumulation of fission products.

0.050 0.050 MCNP MCNP Cd+Co 0.045 Cd+Co 0.045 Cd+Cd Cd+Cd Cd+AISI304 Cd+AISI304 0.040 Hf+Hf 0.040 Hf+Hf Hf+AISI304 Hf+AISI304 Eu2O3 stellite 0.035 stellite 0.035 Gd2O3 Eu2O3 Gd2O3 0.030 0.030 m] mm] / /m

0.025 $ 0.025 [ [$

h Sh S ∆ ∆ / / 0.020 0.020 ∆ρ ∆ρ 0.015 0.015

0.010 0.010 T=0, BOC (1st cycle) T=30 E.F.P.D., EOC (1st cycle) 0.005 0.005

0.000 0.000 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Sh [mm] Sh [mm] Figure 6.12 Comparison of differential CR worth at BOC (T=0) and EOC (T ~ 30 EFPD) for different CR types

69 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

6.2 Comparison of the reactor neutronics characteristics for different CR

types during BR2 fuel cycle

There are two important factors in the reactor neutronics, changing of which will both affect directly other reactor parameters such as reactor period and demonstrate indirectly changing of control rod worth during irradiation. Analysis to keff and reactivity with different CR types will provide potent evidence for the optimal design. The CR positions Sh at BOC for fresh (T=0) and burnt (T ~ 1000 EFPD) CR material are calculated and presented in table 6-6. Accurate criticality calculations have been performed by MCNP with loaded different CR type in the reactor core. One and the same fuel load has been used in the calculations. Because the absorption properties of the various CR are different, in order to keep criticality constant in the core (keff=1.0) it is necessary to adjust the positions of the CR for the corresponding CR type (see the 2nd column in Table 6-6). The calculations have been performed for fresh CR material and for burnt CR material using the same fuel load. As expected, the positions of the CR with burnt cadmium will be lower than the position with fresh cadmium due to the burn up of the lower edge of the CR and reduction of the cadmium length. The data in table 6-6 show that the maximum decrease of the position Sh at BOC after long term of irradiation has the cadmium rod (about 70 mm), the decrease in Sh for Cd+Co rod is less (~ 40 mm). For all other rods (hafnium, europium, gadolinium) the position of the rod Sh at BOC remains practically constant during many operating cycles (up to about 30 considered cycles).

st th th The keff and the reactivity evolutions during the 1 , 15 and 30 cycles with the same reactor core load, but with depleted control rods are evaluated for various control rod types and presented at Fig 6.13.

It can be seen from Fig. 6.13, that the tendencies of the keff and the reactivity evolutions are similar for all CR types during the considered operating cycles. The values of keff for all CR types in Fig. 6.13 are normalized to unity at BOC, so to get some more information about the differences among the different CR types, these graphs are re-drawn at Fig. 6.14, which shows the criticality variation for each individual CR type during long term of irradiation. As it is seen from the graphs at Fig. 6.14, after the 15th cycle and especially close to the 30th cycle, the tendencies of the keff and the reactivity evolutions for Cd+Cd rod are quite different from th those for the rest CR types. At end of 30 operating cycle, the values of the keff for Cd+Cd increase very rapidly, that is related to the depletion of 113Cd and thus disappearance of a major portion of the CR material in the lower edge of the CR. The increasing of the keff for Cd+Co rod is smaller than that of Cd+Cd rod. The reason for this is the slower rate of cadmium burn up in the lower rod edge due to the presence of a gray material (cobalt) in the lower active part of the CR.

The values of keff for the CR type for the rest CR types are almost same, which proves the type of Cd+Cd has the shortest control life due to the depletion of the absorbing material – Cd in the lower part.

Table 6-6 Comparison of the positions Sh [mm] at criticality for different CR types at BOC of the 1st operation cycle (loaded fresh CR absorbing material) and for burnt CR absorbing material

70 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS after ~ 30 operation cycles (i.e., equivalent to about ~ 1000 EFPD of irradiation). Calculation by MCNP. T=0 (BOC) T=1000 EFPD (BOC) Cd+Co 440 mm 404 mm Cd+Cd 500 mm 429 mm Hf+Hf 535 mm 522 mm Hf+AISI304 450 mm 450 mm Eu2O3 555 mm 554 mm

Criticality evolution during the 1st cycle Reactivity evolution during the 1st cycle

1.01 1 MCNP MCNP 0 1.00

-1 0.99 -2 [$]

0.98

eff (T) k

ρ -3 Cd+Co Cd+Co 0.97 Cd+Cd Cd+Cd -4 Hf+Hf Hf+Hf Hf+AISI304 Hf+AISI304 0.96 Eu2O3 -5 Eu2O3 Gd2O3 Gd2O3

-6 0.95 0 5 10 15 20 0 5 10 15 20 Time [efpd] Time [efpd] th Reactivity evolution during the 15th cycle Criticality evolutio n during the 15 cycle 1.01 1 MCNP MCNP 0 1.00

-1 0.99

] -2

f 0.98

ef (T) [$ k ρ -3

0.97 Cd+Co -4 Cd+Co Cd+Cd Cd+Cd Hf+Hf Hf+Hf 0.96 -5 Hf+AISI304 Hf+AISI304 Eu2O3 Eu2O3 0.95 -6 0 5 10 15 20 0 5 10 15 20 Time [efpd] Time [efpd] rd rd Criticality evolution during the 30 cycle Reactivity evoluti on during the 30 cycle 1.01 1 MCNP MCNP 0 1.00

-1 0.99 -2 f ] 0.98

ef [$ k

) -3 ρ(Τ 0.97 Cd+Co -4 Cd+Co Cd+Cd Cd+Cd Hf+Hf Hf+Hf 0.96 Hf+AISI304 -5 Hf+AISI304 Eu2O3 Eu2O3 0.95 -6 0 5 10 15 20 0 5 10 15 20 Time [efpd] Time [efpd] st th rd Figure 6.13 Evolutions of keff and reactivity during the 1 , 15 and the 30 operating cycle (all values of keff are normalized to unity at BOC).

71 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

1.02 1.01

Cd+Cd Cd+Co 1.01 1.00

1.00 0.99 f ef f 0.99

k ef k 0.98 0.98

st 1th cycle 0.97 1 cycle th 0.97 th 12 cycle 12 cycle th th 25 cycle 25 cycle MCNP MCNP 0.96 0.96 0 5 10 15 20 0 5 10 15 20 Time [days] Time [days]

1.01 1.01

Hf+Hf Hf+AISI304

1.00 1.00

0.99 0.99

eff k f ef 0.98 k 0.98

th 1st cycle 1 cycle 0.97 th 0.97 12th cycle 12 cycle th 25th cycle 25 cycle MCNP MCNP 0.96 0.96 0 5 10 15 20 0 5 10 15 20 Time [days] Time [days]

1.01 1.01 Gd O Eu O 2 3 2 3 1.00 1.00

0.99 0.99 f

0.98

ef k f ef k 0.98 0.97

0.97 1st cycle 0.96 th th 1 cycle 12 cycle th th 12 cycle MCNP 25 cycle MCNP 0.95 0.96 0 5 10 15 20 0 5 10 15 20 Time [days] Time [days] Figure 6.14 Comparison of criticality variation for individual CR types during long time of irradiation (the position Sh at BOC for the corresponding CR is kept constant during irradiation).

6.3 Control rod effects on neutron flux distributions

6.3.1 Axial distributions as function of CR position The strongly absorbing nature of control elements causes major perturbations in the neutron flux in the vicinity of the control rod and also affects the overall flux and power distribution of the reactor core. At the beginning of core life, the control rods are inserted to a lower position Sh ~ 400 - 500 mm to compensate the excess reactivity of the initial core loading. The areas around the CR location channels experience decreased multiplication and hence lower fluxes. Because the

72 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS overall reactor power is kept constant, there will be a flux-peaking near position Sh of the CR. With depletion of fuel during the reactor operation the CR are gradually withdrawn to compensate for the reduced reactivity of the fuel. During the first stages of operation the fuel near the reactor hot plane is exposed to relatively higher fluxes and hence experience larger burnups. Thus as the control rods are withdrawn, the flux peak will shift toward higher axial positions. In order to investigate the control rods effect on the neutron flux and power distributions in the BR2 reactor core a series of neutronics calculations have been performed by MCNP. The detailed axial distributions of thermal, epi-thermal and fast neutron fluxes in the axis of typical fuel and reflector channels have been calculated for different positions Sh of the CR movement and successively increased produced energy during a typical operation cycle. A loading similar to the one of cycle 02/2003A.6 [25] has been used in the calculations. The reason for the choice of such load was the relatively long way of CR movement during the cycle, i.e. from Sh(0)=443 mm to Sh (1140 MWD)=840 mm at the end of cycle. In order to compare the flux distributions at similar positions Sh for the various CR types, the loadings used in the calculations for all other CR types have been adapted in order to obtain criticality near the same Sh as for the Reference CR in cycle 02/2003A.6. This could be achieved for example by varying the Helium-3 poisoning or varying the content of experimental samples in channel H1/Central. The calculations of the axial distributions of the neutron fluxes have been performed for the various control rod types at positions Sh=400, 500, 600, 700 and 800 mm during the cycle. The thermal fluxes in the axis of FE, located in channels H1/C, A30 and C41 and in the axis of the reflector channel E30 are given at Fig. 6.15 (H1/C), Fig. 6.16 (A30), Fig. 6.17 (C41), Fig. 6.18 (E30) and the fast fluxes in the same channels are given at Fig. 6.19 (H1/C), Fig. 6.20 (C41), Fig. 6.21 (E30). The axial distribution of the epi-thermal neutron fluxes in the axis of a channel, located near the CR location is compared for the different CR types and given in Fig. 6.21a. The main conclusions are: y The strongest perturbations of the axial neutron flux distributions are observed in reflector channels, located near to the channels of CR location (e.g., channel E30). y For fuel channels – maximum disturbed are neutron flux distributions in channels C, located near the CR channels and also in the axis of channels A and B of the central crown,. The perturbation of the flux distributions in the axis of the FE in H1/C is lower. y Maximum perturbation effects on axial flux distributions have rods with full absorbing

length from europium (Eu2O3), hafnium (Hf+Hf) and cadmium (Cd+Cd). y The axial distributions of neutron fluxes for CR rods, composed as a combination of “grey” material (cobalt, stainless steel) and “black” absorber (cadmium, hafnium), such as Cd+AISI304 and Hf+AISI304 are similar to those for the Reference CR (Cd+Co). y Maximum perturbation effect on the axial neutron flux distributions is observed at lower positions of the CR (Sh=400 to 500 mm). For high positions Sh, the differences in the axial neutron flux distributions for the various CR types decrease and practically disappear at Sh > 700 mm.

73 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV in Al sample in the axis of FE in H1/Central in Al sample in the axis of FE in H1/Central

14 6x10 6x1014 Cd+Cd H1/C Cd+AISI304 H1/C

14 5x10 5x1014 ] ] -1 14 -1 .s 14 .s

-2 4x10 -2 4x10 m

14 14

3x10 3x10 l flux [n.cm l flux [n.c a Sh=400 mm a Sh=400 mm 14 Sh=500 mm

therm 14 2x10 therm 2x10 Sh=500 mm Sh=600 mm Sh=600 mm Sh=700 mm Sh=700 mm 14 Sh=800 mm 1x10 1x1014 Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in H1/Central in Al sample in the axis of FE in H1/Central 6x1014 6x1014 Hf+Hf H1/C Hf+AISI304 H1/C

5x1014 5x1014 ] ] -1 14 -1 14 .s .s

-2 4x10 -2 4x10 m m

14 14

3x10 3x10 l flux [n.c l flux [n.c a a m m

14 Sh=400 mm 14 Sh=400 mm ther 2x10 ther 2x10 Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm 1x1014 Sh=700 mm 1x1014 Sh=700 mm Sh=800 mm Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in H1/Central in Al sample in the ax is of FE in H1/Central 6x1014 6x1014 stellite H1/C Cd+Co H1/C

5x1014 5x1014 ] ] -1 -1 14 14 .s .s -2 -2 4x10 4x10 m .cm n [n.c 14 14

3x10 3x10 l flux l flux [ a a m m

er Sh=400 mm 14 Sh=400 mm 14 th ther 2x10 2x10 Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm Sh=700 mm Sh=700 mm 14 1x1014 1x10 Sh=800 mm Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the ax is of FE in H1/Central 6x1014

Eu2O3 H1/C

5x1014 ] -1 14 .s

-2 4x10

14 ux [n.cm

3x10 al fl erm

h 14 t 2x10 Sh=400 mm Sh=500 mm Sh=600 mm 1x1014 Sh=700 mm Sh=800 mm

0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] 74 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the ax is of FE in H1/Central in Al sample in the ax is of FE in H1/Central 6x1014 6x1014 Sh=400 mm H1/C Sh=800 mm H1/C

5x1014 5x1014 ] ] -1 -1 14 14 .s .s -2 -2 4x10 4x10 m m c [n. [n.c x x 14 14

3x10 3x10 Cd+Cd l flu l flu Cd+Cd a a Cd+AISI304 Cd+AISI304 m er erm 14 14 Hf+Hf th th 2x10 Hf+Hf 2x10 Hf+AISI304 Hf+AISI304 stellite stellite 14 14 Cd+Co 1x10 Cd+Co 1x10 Eu2O3 Eu2O3

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Figure 6.15 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel H1/Central.

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in th e axis of FE in A30 in Al sample in th e axis of FE in A30 4x1014 4x1014 Cd+Cd A30 Cd+AISI304 A30

3x1014 3x1014 ] ] 1 - -1 .s .s 2 - -2 m m c c . n [ [n.

2x1014 2x1014 l flux l flux a a m m r er th Sh=400 mm the Sh=400 mm Sh=500 mm 14 14 Sh=500 mm 1x10 Sh=600 mm 1x10 Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux E <0.5 eV n Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in th e axis of FE in A30 in Al sample in th e axis of FE in A30 14 4x10 4x1014 Hf+Hf A30 Hf+AISI304 A30

14 3x1014 3x10 ] ] -1 -1 .s .s -2 -2 m c . 14 14 [n 2x10 ux 2x10 l

l flux [n.cm l f a a m r Sh=400 mm

therm Sh=400 mm

the 14 Sh=500 mm 1x10 Sh=500 mm 14 1x10 Sh=600 mm Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in th e axis of FE in A30 in Al sample in th e axis of FE in A30 4x1014 4x1014 stellite A30 Cd+Co A30

3x1014 3x1014 ] ] -1 -1 .s .s -2 -2 m m c c . . n n [ [

14 14

2x10 2x10 l flux l flux a a m m r r e h the t Sh=400 mm Sh=400 mm Sh=500 mm 14 Sh=500 mm 14 1x10 Sh=600 mm 1x10 Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

75 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in th e axis of FE in A30 4x1014

Eu2O3 A30

3x1014 ] -1 .s -2 m c [n.

x 14

2x10 l flu a m er

th Sh=400 mm Sh=500 mm 1x1014 Sh=600 mm Sh=700 mm Sh=800 mm

0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Axial distribution of thermal neutron flux E <0.5 eV Axial distribution of thermal neutron flux E <0.5 eV n n

in Al sample in th e axis of FE in A30 in Al sample in the axis of FE in A30 14 4x1014 4x10 Sh=400 mm A30 Sh=800 mm A30

14 3x1014 3x10 ] ] -1 -1 s .s . -2 -2 m m c c . n [n. [

x 14 14 ux u l 2x10 2x10 Cd+Cd f fl

l l Cd+Cd a a Cd+AISI304 m m r Cd+AISI304 Hf+Hf e er h h t Hf+Hf t Hf+AISI304 Hf+AISI304 14 stellite 14 1x10 1x10 Eu2O3 Cd+Co stellite Eu2O3 Cd+Co

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Figure 6.16 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel A30 in the central crown.

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in th e axis of FE in C41 in Al sample in th e axis of FE in C41 2,5x1014 2,5x1014 Cd+Cd C41 Cd+AISI304 C41

2,0x1014 2,0x1014 ] ] -1 -1 .s .s -2 -2 m 14 m 14 c c . 1,5x10 . 1,5x10 n [n

[ x u ux l l l f l f a a 14 14 m 1,0x10 m 1,0x10 er Sh=400 mm er Sh=400 mm th th Sh=500 mm Sh=500 mm Sh=600 mm 13 Sh=600 mm 13 5,0x10 Sh=700 mm 5,0x10 Sh=700 mm Sh=800 mm Sh=800

0,0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux E <0.5 eV Axial distribution of thermal neutron flux En<0.5 eV n in Al sample in the axis of FE in C41 in Al sample in th e axis of FE in C41 14 2,5x1014 2.5x10 Hf+Hf C41 Hf+AISI304 C41

14 2,0x1014 2.0x10 ] ] -1 -1 .s .s -2 -2 14 m 14 .cm c 1.5x10 . 1,5x10 n n [ [

x x u u l

fl l l f a a 14 14 m 1.0x10

1,0x10 erm er h t th Sh=400 mm Sh=400 mm Sh=500 mm Sh=500 mm 13 5,0x1013 Sh=600 mm 5.0x10 Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm 0,0 0.0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm] 76 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in th e axis of FE in C41 in Al sample in th e axis of FE in C41 2,5x1014 2,5x1014 stellite C41 Cd+Co C41

2,0x1014 2,0x1014 ] ] -1 -1 .s .s -2 -2 m m 14 14 .c .c 1,5x10 1,5x10 n n [ [

x x

flu flu l l a a 14 14 m m r r 1,0x10 Sh=400 mm 1,0x10 Sh=400 mm e e th th Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm 5,0x1013 Sh=700 mm 5,0x1013 Sh=700 mm Sh=800 mm Sh=800 mm

0,0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in th e axis of FE in C41 2,5x1014

Eu2O3 C41

2,0x1014 ] -1 .s -2 1,5x1014

l flux [n.cm a 14 m r 1,0x10 Sh=400 mm e

th Sh=500 mm Sh=600 mm 5,0x1013 Sh=700 mm Sh=800 mm

0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Comparison of axial distributions of thermal neutron flux E <0.5 eV Comparison of axial distributions of thermal neutron flux E <0.5 eV n n for different Control Rod types at Sh=400 mm for different Control Rod types at Sh=800 mm

2.5x1014 2,5x1014 Sh=400 mm C41 Sh=800 mm C41

2.0x1014 2,0x1014 ] ] -1 -1 .s .s -2 -2

m 14 14 c 1.5x10 1,5x10 [n. x Cd+Co l flu l flux [n.cm a 14 a 14

m Cd+Co m Cd+AISI304 r 1.0x10 r 1,0x10 e Cd+AISI304 e Cd+Cd th th Cd+Cd Hf+Hf Hf+Hf Hf+AISI304 13 Hf+AISI304 13 5.0x10 5,0x10 stellite stellite Eu2O3 Eu2O3

0.0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Figure 6.17 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, placed in channel C41 close to a Control Rod location.

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in E30 in Al sample in the axis of FE in E30 4x1014 5x1014 Cd+Cd E30 Cd+AISI304 E30

4x1014 3x1014 ] ] -1 -1 .s .s -2 -2 m m 14 c c

. 3x10 [n 14 ux

2x10 l flux [n. l l f a a 14 m m

r 2x10 Sh=400 mm the Sh=500 mm ther Sh=400 mm 1x1014 Sh=600 mm Sh=500 mm 14 Sh=700 mm 1x10 Sh=600 mm Sh=800 mm Sh=700 mm Sh=800 mm 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm] 77 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in E30 in Al sample in the axis of FE in E30 14 5x1014 5x10 E30 Hf+Hf E30 Hf+AISI304

14 4x1014 4x10 ] ] -1 -1 .s .s -2 -2 14 m 14 .cm 3x10 .c 3x10 n [n [

l flux l fl a

a 14 14 m

m 2x10

2x10 er er th th Sh=400 mm Sh=500 mm Sh=400 mm 14 1x1014 Sh=600 mm 1x10 Sh=800 mm Sh=700 mm Sh=800 mm 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux En<0.5 eV Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in E30 in Al sample in the axis of FE in E30 5x1014 5x1014 stellite E30 Cd+Co E30

4x1014 4x1014 ] ] -1 -1 .s .s -2 -2 m m 14 14 .c .c 3x10 3x10 n n [ [

l flux l flux a a 2x1014 2x1014 therm therm Sh=400 mm Sh=400 mm Sh=500 mm Sh=500 mm 1x1014 Sh=600 mm 1x1014 Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in E30 5x1014 E30 Eu2O3

4x1014 ] -1 .s -2

m 14

.c 3x10 n [

l fl a 14 m 2x10 er th Sh=400 mm Sh=500 mm 14 1x10 Sh=600 mm Sh=700 mm Sh=800 mm 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Comparison of axial distributions of thermal neutron flux En<0.5 eV Comparison of axial distributions of thermal neutron flux En<0.5 eV

for different Control Rod types at Sh=400 mm for different Control Rod types at Sh=400 mm 4x1014 5x1014 Sh=400 mm E30 Sh=800 mm E30

4x1014 3x1014 ] ] -1 -1 .s .s -2 -2 m m 14 c . .c 3x10 n [n [ x x 14 u

2x10 l fl l flu

a Cd+Co a Cd+Co 14 2x10 Cd+AISI304 erm erm Cd+AISI304 th th Cd+Cd Cd+Cd 14 Hf+Hf 1x10 Hf+Hf Hf+AISI304 1x1014 Hf+AISI304 stellite stellite Eu2O3 Eu2O3 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

78 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Figure 6.18 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in reflector channel E30, which is located near Control Rod channel.

Axial distribution of fast neutron flux En>0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV in Al sample in the axis of FE in H1/Central in Al sample in the axis of FE in H1/Central

14 6x10 6x1014 Cd+Cd H1/C Cd+AISI304 H1/C

14 5x10 5x1014

14 14 ] ] -1 -1 4x10 4x10 .s .s -2 -2 m m c c . 14

[n. 14 [n

3x10

3x10 x u ux l l f f t st fa

fas 14 14 Sh=400 mm 2x10 2x10 Sh=400 mm Sh=500 mm Sh=500 mm Sh=600 mm 14 Sh=600 mm 1x1014 Sh=700 mm 1x10 Sh=800 mm Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of fast neutron flux En>0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV in Al sample in the axis of FE in H1/Central in Al sample in the axis of FE in H1/Central

6x1014 6x1014 Hf+Hf H1/C Hf+AISI304 H1/C

14 5x1014 5x10

14 14 ] ] -1 -1 4x10 4x10 .s .s -2 -2 m m c . 14 14 [n

3x10

3x10 x u l f flux [n.c t st s Sh=400 mm fa

fa 14 2x1014 Sh=400 mm 2x10 Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm Sh=700 mm 14 14 1x10 Sh=700 mm 1x10 Sh=800 mm Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of fast neutron flux En>0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV in Al sample in the axis of FE in H1/Central in Al sample in the axis of FE in H1/Central

6x1014 6x1014 stellite H1/C Cd+Co H1/C

5x1014 5x1014

14 14 ] ]

-1 4x10 -1 4x10 .s .s -2 -2 m m

14 14 [n.c [n.c

3x10 3x10 x x u u fl fl

fast 14 fast 14 Sh=400 mm 2x10 Sh=400 mm 2x10 Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm 14 14 1x10 Sh=700 mm 1x10 Sh=700 mm Sh=800 mm Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Figure 6.19 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel H1/Central.

79 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of fast neutron flux E >0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV n in Al sample in the axis of FE in C41 in Al sample in the axis of FE in C41 14 5x1014 5x10 Cd+Cd C41 Cd+AISI304 C41

14 4x1014 4x10 ] ] -1 -1 .s .s 14 14 -2 3x10 -2 3x10 m m [n.c

[n.c

x u l f 14 flux t 14 s 2x10 2x10 fa Sh=400 mm fast Sh=500 mm Sh=400 mm Sh=600 mm Sh=500 mm 14 1x10 Sh=700 mm 1x1014 Sh=600 mm Sh=800 mm Sh=700 mm Sh=800 mm 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -40-30-20-100 10203040 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of fast neutron flux E >0.1 MeV Axial distribution of fast neutron flux E >0.1 MeV n n

in Al sample in the axis of FE in C41 in Al sample in the axis of FE in C41 14 5x10 5x1014 Hf+Hf C41 Hf+AISI304 C41

14 4x10 4x1014 ] ] -1 -1

.s 14 .s 14 -2 3x10 -2 3x10 m m c .c [n

x u ux [n.

fl 14 14 st 2x10 st fl 2x10 fa Sh=400 mm fa Sh=400 mm Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm 1x1014 1x1014 Sh=700 mm Sh=700 mm Sh=800 boc Sh=800 mm

0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of fast neutron flux E >0.1 MeV Axial distribution of fast neutron flux E >0.1 MeV n n

in Al sample in the axis of FE in C41 in Al sample in the axis of FE in C41 14 5x10 5x1014 Eu O C41 Sh=400 mm C41 2 3

4x1014 4x1014 ] ] -1 -1 .s .s 14 14 -2 3x10 -2 3x10 m .c [n x u ux [n.cm Cd+Cd fl 14 14 st fl st 2x10 2x10 Cd+AISI304 fa fa Hf+Hf Sh=400 mm Hf+AISI304 Sh=500 mm 14 14 stellite 1x10 Sh=600 mm 1x10 Sh=700 mm Cd+Co Sh=800 mm Eu2O3 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Figure 6.20 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel C41 close to Control Rod channel.

80 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of fast neutron flux En>0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV

in Al sample in the axis of FE in E30 in Al sample in the axis of FE in E30 1,5x1014 1,5x1014 Cd+Cd E30 Cd+AISI304 E30

14 14 ] ] -1 1,0x10 -1 1,0x10 .s .s -2 -2 m m c .c [n. [n

x x u u l fl f st st a a f 13 f 13 5,0x10 Sh=400 mm 5,0x10 Sh=400 mm Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm 0,0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of fast neutron flux En>0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV

in Al sample in the axis of FE in E30 in Al sample in the axis of FE in E30

Hf+Hf E30 Hf+AISI304 E30

1,5x1014 1,5x1014 ] ] -1 -1 .s .s -2 -2 m 14 m 14 .c 1,0x10 .c 1,0x10 [n [n

x x flu flu t t s s a a f f Sh=400 mm 13 13 Sh=400 mm 5,0x10 Sh=500 mm 5,0x10 Sh=500 mm Sh=600 mm Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm 0,0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of fast neutron flux En>0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV

in Al sample in the axis of FE in E30 in Al sample in the axis of FE in E30

stellite E30 Cd+Co E30

1,5x1014 1,5x1014 ] ] -1 -1 .s .s -2 -2 m 14 m 14 c

.c 1,0x10 1,0x10 [n

[n.

x u ux l f fl t st a f fas Sh=400 mm Sh=400 mm 13 13 5,0x10 Sh=500 mm 5,0x10 Sh=500 mm Sh=600 mm Sh=600 mm Sh=700 mm Sh=700 mm Sh=800 mm Sh=800 mm 0,0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of fast neutron flux En>0.1 MeV

in Al sample in the axis of FE in E30

E30 Eu2O3 1,5x1014 ] -1 .s -2

m 14 c

. 1,0x10 n [

ux fl t fas Sh=400 mm 13 5,0x10 Sh=500 mm Sh=600 mm Sh=700 mm Sh=800 mm 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] 81 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of fast neutron flux En>0.1 MeV Axial distribution of fast neutron flux En>0.1 MeV

in Al sample in the axis of FE in E30 in Al sample in the axis of FE in E30

Sh=400 mm E30 Sh=800 mm E30

1,5x1014 1,5x1014 ] ] -1 -1 .s .s -2 -2 m m 14 14 1,0x10 c 1,0x10 [n.c

Cd+Cd flux [n. flux Cd+Cd Cd+AISI304 fast fast Cd+AISI304 Hf+Hf Hf+Hf Hf+AISI304 13 5,0x1013 5,0x10 Hf+AISI304 stellite stellite Cd+Co Cd+Co Eu2O3 Eu2O3 0,0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Figure 6.21 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in reflector channel E30 close to Control Rod channel.

Axial distributions of epi-thermal neutron flux 0.5 eV < En< 0.1 MeV in Al sample in the axis of FE in C41 5x1014 Sh=400 mm C41

] 14 -1 4x10 .s -2

3x1014 flux [n.cm ron

l neut 14 a 2x10 Cd+Co Cd+Cd i-therm

p Cd+AISI304 e 1x1014 Hf+Hf Hf+AISI304 Eu2O3 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Figure 6.21a Axial distributions of epi-thermal neutron fluxes (0.5 eV < En < 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel C41 close to Control Rod channel.

82 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

6.3.2 Axial distributions of neutron fluxes as function of energy produced during operation cycle The axial distributions of the neutron fluxes in Al sample in the axis of typical fuel channels were calculated with MCNP for cycle 04/2005B.1[25]. The axial distributions of the thermal fluxes are presented at Fig. 6.22 (H1/C), Fig. 6.23 (A30), Fig. 6.24 (B180) as function of the energy produced during the cycle and consequently as function of the CR motion during the cycle. It is seen that the thermal flux increases with energy produced, consequently with increase of the height Sh of the CR. This results from the fuel burn up and from the burnout of the burnable poisons - 10B and 149Sm. The axial distributions of the fast fluxes in the same channels are given at Fig. 6.25 (H1/C), Fig. 6.26 (A30), Fig. 6.27 (B180). 6.3.3 Axial distribution of thermal neutron fluxes in axis of a fuel channel for simulated critical cores at 0 power for different Sh (as in BR02) The axial distributions of the neutron fluxes in Al sample in the axis of typical fuel channels were calculated with MCNP for a configuration, which was similar to the load 4 (Fig. 6.28), used in the BR02[26]. In analogy with[26], critical cores at 0 powers were simulated by successive increase of the burnable poisons in the fuel (increase of the boron-10 concentrations). It is seen from Fig. 6.29 that the thermal flux decreases with increase of the height Sh of the CR, which results from the increased poisoning of the core by boron-10. The used fuel type was HEU (93% 235U enrichment) without burnable poisons.

83 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS Axial distribution of the thermal flux in the axis of FE in H1/C Axial distribution of the thermal flux in the axis of FE in H1/C

Control Ro ds with Co Control Rod s with Cd 7x1014 7x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

6x1014 6x1014 ] ] -1 14 -1 14 .s .s -2 5x10 -2 5x10 m c . n [ [n.cm

14 x 14 ux u l 4x10 l 4x10 l f l f a a m m r e 14 14 ther 3x10 th 3x10 Sh=450 mm (E=0 MW.d) Sh=400 mm (E=0 MW.d) Sh=550 mm (E=472 MW.d) 14 Sh=500 mm (E=472 MW.d) 14 2x10 2x10 Sh=680 mm (E=826 MW.d) Sh=620 mm (E=826 MW.d) Sh=810 mm (E=1180 MW.d) Sh=760 mm (E=1180 MW.d) 14 1x10 1x1014 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of the thermal flux in the axis of FE in H1/C Control Rods with AISI304

7x1014 MCNP, cycle 04/2005B.1

6x1014 ] -1 14 .s

-2 5x10 [n.cm 4x1014 l flux a m er 14 th 3x10 Sh=380 mm (E=0 MW.d) Sh=480 mm (E=472 MW.d) 2x1014 Sh=600 mm (E=826 MW.d) Sh=740 mm (E=1180 MW.d)

1x1014 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Figure 6.22 Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel H1/Central versus energy produced during typical BR2 cycle.

Axial distribution of the thermal flux in the axis of FE in A30 Axial distribution of the thermal flux in the axis of FE in A30

Control Ro ds with Co Control Rods with Cd 5x1014 5x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

4x1014 4x1014 ] ] -1 -1 .s .s -2 -2

14 m 14

3x10 .c 3x10 n [n.cm [ ux ux l l fl l f a a 14

m 14 2x10 r 2x10 erm h t the Sh=400 mm (E=0 MW.d) Sh=450 mm (E=0 MW.d)

14 Sh=500 mm (E=472 MW.d) Sh=550 mm (E=472 MW.d) 1x10 1x1014 Sh=620 mm (E=826 MW.d) Sh=680 mm (E=826 MW.d) Sh=760 mm (E=1180 MW.d) Sh=810 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

84 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of the thermal flux in the axis of FE in A30 Control Rods with AISI304

5x1014 MCNP, cycle 04/2005B.1

4x1014 ] -1 .s -2 14 .cm 3x10 n [

ux fl l a 2x1014 erm th Sh=380 mm (E=0 MW.d) Sh=480 mm (E=472 MW.d) 1x1014 Sh=600 mm (E=826 MW.d) Sh=740 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Axial distribution of the thermal flux in the axis of FE in B180 Axial distribution of the thermal flux in the axis of FE in B180

Control Ro ds with Co Control Rod s with Cd 5x1014 5x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

4x1014 4x1014 ] ] -1 -1 .s .s -2 -2 m m c c . 14 14 [n [n. 3x10 3x10 x u ux l fl l f a mal m r e 14 14 th ther 2x10 2x10 Sh=400 mm (E=0 MW.d) Sh=450 mm (E=0 MW.d) Sh=500 mm (E=472 MW.d) Sh=550 mm (E=472 MW.d) Sh=620 mm (E=826 MW.d) Sh=680 mm (E=826 MW.d) 14 14 1x10 Sh=760 mm (E=1180 MW.d) 1x10 Sh=810 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of the thermal flux in the axis of FE in B180 Control Rods with AISI304

5x1014 MCNP, cycle 04/2005B.1

4x1014 ] -1 .s -2 m c .

n 14

[ 3x10 ux l l f a m r 14 the 2x10 Sh=380 mm (E=0 MW.d) Sh=480 mm (E=472 MW.d) Sh=600 mm (E=826 MW.d) 14 1x10 Sh=740 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

85 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of the fast flux in the axis of FE in H1/C Axial distribution of the fast flux in the axis of FE in H1/C Control Rods with Cd Control Rod s with Co 6x1014 6x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

5x1014 5x1014

14 ] ] 14 -1 -1 4x10 4x10 .s .s -2 -2 m m c c .

n 14

[ 14

x 3x10 3x10 u fl t flux [n. t s a fas

14 f 2x10 2x1014 Sh=400 mm (E=0 MW.d) Sh=450 mm (E=0 MW.d) Sh=500 mm (E=472 MW.d) Sh=550 mm (E=472 MW.d) 14 1x10 Sh=620 mm (E=826 MW.d) 1x1014 Sh=680 mm (E=826 MW.d) Sh=760 mm (E=1180 MW.d) Sh=810 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of the fast flux in the axis of FE in H1/C

Control Rods w ith AISI304 6x1014 MCNP, cycle 04/2005B.1

5x1014

14 ]

-1 4x10 .s -2

[n.cm 14 3x10 ux fl fast 2x1014 Sh=380 mm (E=0 MW.d) Sh=480 mm (E=472 MW.d) 1x1014 Sh=600 mm (E=826 MW.d) Sh=740 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Figure 6.25 Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel H1/Central versus energy produced during typical BR2 cycle.

Axial distribution of the fast flux in the axis of FE in A30 Axial distribution of the fast flux in the axis of FE in A30

Control Rod s with Co Control Rod s with Cd 6x1014 6x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

5x1014 5x1014

14 ] 14 ] -1 4x10 -1 4x10 .s .s -2 -2 m m c . .c

14 n 14 [n [ x 3x10 x 3x10 u u fl fl t t s s fa 14 fa 2x10 2x1014 Sh=400 mm (E=0 MW.d) Sh=450 mm (E=0 MW.d) Sh=500 mm (E=472 MW.d) Sh=550 mm (E=472 MW.d) 14 1x10 Sh=620 mm (E=826 MW.d) 1x1014 Sh=680 mm (E=826 MW.d) Sh=760 mm (E=1180 MW.d) Sh=810 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

86 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Axial distribution of the fast flux in the axis of FE in A30

Control Rods w ith AISI304 6x1014 MCNP, cycle 04/2005B.1

5x1014

14 ]

-1 4x10 .s -2 m .c

[n 14

x 3x10 u fl t s fa 2x1014 Sh=380 mm (E=0 MW.d) Sh=480 mm (E=472 MW.d) 1x1014 Sh=600 mm (E=826 MW.d) Sh=740 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Axial distribution of the fast flux in the axis of FE in B180 Axial distribution of the fast flux in the axis of FE in B180

Control Rods with Co Control Rods with Cd 6x1014 6x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

5x1014 5x1014

14 14 ] ] -1 -1 4x10 4x10 .s .s -2 -2 m m .c .c

[n 14 [n 14

x 3x10 x 3x10 u u l l f f t t s s fa fa 2x1014 2x1014 Sh=400 mm (E=0 MW.d) Sh=450 mm (E=0 MW.d) Sh=500 mm (E=472 MW.d) Sh=550 mm (E=472 MW.d) 14 14 1x10 Sh=620 mm (E=826 MW.d) 1x10 Sh=680 mm (E=826 MW.d) Sh=760 mm (E=1180 MW.d) Sh=810 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

Axial distribution of the fast flux in the axis of FE in B180 Control Rods with AISI304

6x1014 MCNP, cycle 04/2005B.1

5x1014

14 ]

-1 4x10 .s -2 m c . 14 [n

x 3x10 u fl t s a f 2x1014 Sh=380 mm (E=0 MW.d) Sh=480 mm (E=472 MW.d) 1x1014 Sh=600 mm (E=826 MW.d) Sh=740 mm (E=1180 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

87 CHAPTER 6. ANALYSIS OF CALCULATION RESULTS

Figure 6.28 MCNP whole core model of configuration 4 (~ similar to the load for BR02, [26]).

Axial distribution of the thermal flux in the axis of a fuel channel Axial distribution of the thermal flux in the axis of a fuel channel Control Ro ds with Co Control Rods with Cd 14 4x1014 4x10 MCNP, simulated critical cores as in BR02 MCNP, simulated critical cores as in BR02

14 3x1014 3x10 ] ] -1 -1 .s .s -2 -2 m m c .c . n n [ [

x x 14 14 u u l l 2x10 2x10 l f l f a a m m r r e the th

1x1014 Sh=300mm (E=0 MW.d)_sub-critical 1x1014 Sh=300mm (E=0 MW.d)_sub-critical Sh=300mm (E=0 MW.d) Sh=300mm (E=0 MW.d) Sh=500mm (E=0 MW.d) Sh=500mm (E=0 MW.d) Sh=800mm (E=0 MW.d) Sh=800mm (E=0 MW.d)

-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

CHAPTER 7 CONCLUSIONS

Through analysis to the results of calculation, some important conclusions have been given in the former chapter. First of all, we will make some simple summaries for the front work and the conclusions in order to demonstrate the characteristics of every absorbing material clearly.

7.1 Summary

About 30 consecutive operating BR2 cycles have been considered, which is equivalent to ~1000 EFPD. The fresh CRs are loaded into the BOC (1) of the 1st cycle. At the EOC (1) the depleted atomic densities of the absorbing material in the CR are used as initial densities of the CR material for the BOC (2) of the 2nd cycle. Then the rods are irradiated till the EOC (2) and the atomic densities at the EOC (2) are used as initial for the BOC (3)…, etc. A typical and same reactor core load and same power is used in each consecutive cycle. The calculations of the fuel depletion and the depletion of the CR absorbing material are performed using MCNP&ORIGEN-S combined method. The conclusions are summarized as followed: 1) Maximum absolute value of the CR Worth is for the Eu - rod (17.5 $ or 12600 pcm) due to the high thermal and epi-thermal microscopic absorption cross sections of europium isotopes, and minimum absolute value for Cd - rods (~ 13.5 $ or 9720 pcm). 2) The Total CR Worth increases at EOC for all CR types by about 8% due to the depletion of 235U and accumulation of fission products. 3) Burn-up of absorbing material doesn't affect significantly the value of total CR worth: - about 3% - 4% decrease of total CR worth for Cd, Cd+Co, Cd+AISI304 rods; - less than 1% decrease of total CR worth for all other types: Hf, Hf+AISI304,

Eu2O3 rods. 4) Burn-up of absorbing material affects strongly shape of the curves of differential CR

worth for Cd, Cd+Co, Cd+AISI304 rods. For all other types – Hf, Hf+AISI304, Eu2O3 rods – the differential curves do not change during irradiation. 5) The maximum value of the differential worth for CR made from one "black" material is for Eu rod (0.038 $/mm or 27 pcm/mm); the minimum differential CR worth is for Cd rod (0.026 $/mm or 19 pcm/mm). 6) The maximum of the differential curves of CR made from a combination of "black" and "gray" absorber is shifted to the lower positions of the CR, because of the reduction of the length of the "black" material. The maximum value for this type of rods is for

89 CHAPTER 7. CONCLUSIONS

Hf+AISI304 rods (0.033$/mm or 24 pcm/mm) and minimum for Cd+Co rod (0.025/mm or 18 pcm/mm). 7) The differential CR worth increases at EOC for all CR types due to the same reason as in 2).

-1 8) Macroscopic absorption cross section Σa [cm ] for CR types Cd, Cd+Co, Cd+AISI304 remain almost constant during ~ 750 EFPD. For T ~ 750 - 1000 EFPD Σa ~ 0.2- 0.3 from 113 -1 the initial value ( Cd is totally burnt, residual Σa is due to other Cd – isotopes); Σa [cm ] remain constant till ~ 1000 EFPD for all other types – Hf, Hf+AISI304, Eu2O3 rods. 9) The maximum value of activity is for Cd+Co and minimum for Hf+AISI304. 10) Changing of positions Sh at BOC during ~ 20 - 30 operating cycles: - For CR types: Cd+Cd, Cd+Co, Cd+AISI304 the positions Sh for burnt CR material (T > 750 EFPD) will be lower than at T=0 (for fresh CR material).

- For CR types: Hf, Hf+AISI304, Eu2O3 the positions Sh for fresh CR (T=0) and for burnt CR (T ~ 1000 EFPD) remains the same since the absorbing material is almost not burnt during the irradiation period.

11) Axial distributions of neutron fluxes: CR types with full length of Eu2O3 and Hf depress more strongly the axial distributions of neutron fluxes: - This effect is sensitive at Sh ~ 400 to 500 mm. - For Sh > 650 mm practically the axial distributions of neutron fluxes for all CR types are the same. 12) Neutron spectra in CR material during irradiation: - For CR types: Cd, Cd+Co, Cd+AISI304 thermal fluxes increase strongly after T ~ 750 EFPD (due to burn up of absorbing material)

- For CR types: Hf, Hf+AISI304, Eu2O3 the change of the neutron spectra in the whole energy region is not significant.

7.2 Proposed new CR type – Hf+AISI304

From the comparison of the characteristics of the considered absorbing materials in BR2 during the irradiation, an optimal design is given and the CR Hf+AISI304 is chosen for the new CR type, which has some reasons as followed: 1). Hf almost does not burn during long time of irradiation (there is one main isotope Hf-177 which is depleted, but this is compensated by production of other Hf-isotopes); 2). The absolute value of the total CR Worth for Hf+AISI304 is 15.6$ (or 11232 pcm) , which is smaller than the Eu - rod (17.5 $ or 12600 pcm), but bigger than for Cd - rods (~ 13.5 $ or 9720 pcm); 3). Hf+AISI304 rod has higher differential worth in comparison with Reference Rod;

90 CHAPTER 7. CONCLUSIONS

4). "Gray" material - AISI304 is used in the lower part of the CR with a purpose to soften the disturbance (caused by the higher neutron absorption) of the axial distributions of the neutron fluxes in the neighboring channels; 5). Hf rod is easier to be fabricated than Cd – rods (Al cladding is not needed in case of Hf rod), so Hf rods will be manufactured without Al cladding; 6). Although Eu – isotopes have the best absorption properties and also Eu rod will not burn during very long time of irradiation, the Eu rod has not been chosen as candidate for the BR2 new type of rods, because of the enormous high costs for fabrication even of a small amount of this absorbing material for use as reactivity control in the reactor BR2; 7). The Hf rod will be heavier than the Cd rod due to the higher atomic mass; however the preliminary simulation tests performed at the BR2 reactor have shown that the time for scram will be equivalent for the both type of rods.

7.3 Final optimal design

Several optimization modifications have been made for the hafnium rod. The primary neutronics evaluations for the hafnium rod were made with geometry model and dimensions as for the Reference CR. Neutronics calculations have been performed for rod with full length of hafnium and for hafnium rod, which lower part is prolonged with stainless steel AISI304. Detailed burnup calculations up to ~ 1000 EFPD have been performed by MCNP&ORIGEN-S for the lower stainless part of the rod and it was obtained that the stainless steel is not burning during long irradiation time. The insertion of rod with full length of hafnium increases significantly the total control rod worth and improves the differential worth in comparison with the Reference Rod. From the point of view of the axial distributions of the neutron fluxes in typical fuel and reflector channels, the insertion of hafnium rod instead of Reference Rod depresses the axial neutron distributions for axial positions z > −20cm . This can be improved if use hafnium rod which is prolonged with stainless steel AISI304, then the depression of the axial distributions (~ −10% compared to the Reference CR) is sensitive for z > −5cm . The application of stainless steel in the lower part of the CR shifts the maximum of the curves of differential CR worth for all rod types toward lower positions Sh, which position is almost the same as for the maximum of the curve of the Reference CR. The application of stainless steel with length L=140mm in the lower part of the CR improves the axial distributions of the neutron fluxes in comparison with CR with full length of hafnium (see Fig. 7.1). The reduction of the length of the AISI304 from L=140mm to L=70mm significantly improves the curve of the differential CR worth which can be seen from Fig. 7.2 and slightly worsen the axial distributions of neutron fluxes which is sensitive for z > −5cm . Increasing the thickness of AISI from δ(AISI)=5mm to δ(AISI)=10mm improves the differential CR worth almost for all positions of Sh (see Fig. 7.2) and practically does not change significantly the axial distributions of the neutron fluxes (thermal and fast).

91 CHAPTER 7. CONCLUSIONS

Axial distribution of fast neutron flux E >0.1 MeV Axial distribution of thermal neutron flux En<0.5 eV n in Al sample in th e axis of FE in A30 in Al sample in the axis of FE in C41 14 4x1014 5x10 Sh=400 mm Sh=400 mm A30 C41

4x1014 3x1014 ] -1 ] .s -1 -2 .s 14 -2 m

c 3x10 m . .c [n

n x

14 [ u 2x10 ux l fl fl a 2x1014 fast erm h t Cd+Co Cd+Co: Dout=61 mm; Din=51 mm 14 Hf+Hf: Dout=61 mm; Din=51 mm 1x10 Hf+Hf: Dout=61 mm; Din=51 mm 14 1x10 Hf+AISI: Dout=61 mm; Din=51 mm; L(AISI)=140mm, δ(AISI)=5mm Hf+AISI: Dout=61 mm; Din=51 mm; L(AISI)=140 mm, δ(AISI)=5 mm Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=140mm; δ(AISI)=5 mm Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=140 mm, δ(AISI)=5 mm Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=70mm; δ(AISI)=5 mm Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=70 mm, δ(AISI)=5 mm Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=70mm; δ(AISI)=10 mm Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=70 mm, δ(AISI)=10 mm 0 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] axis of fuel element [cm]

a) b) Figure 7.1 Comparison of axial distributions of thermal (a) and fast (b) fluxes in typical fuel

Differential CR worth 0.040 MCNP 0.035 T=0

0.030

0.025 m]

m 0.020 / $

Sh [ 0.015 ∆ / ∆ρ 0.010

Cd+Co 0.005 (1) Hf+Hf: Dout=61 mm; Din=51 mm (2) Hf+AISI304: Dout=61 mm; Din=51 mm; L(AISI)=140 mm; δ(AISI)=5 mm (3) Hf+AISI: Dout=63 mm; Din=53 mm; L(AISI)=140 mm; δ(AISI)=5 mm 0.000 (4) Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=140 mm; δ(AISI)=5 mm (5) Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=70 mm, δ(AISI)=5 mm (6) Hf+AISI: Dout=64 mm; Din=54 mm; L(AISI)=70 mm; δ(AISI)=10 mm -0.005 0 100 200 300 400 500 600 700 800 900 Sh [mm] channels for different optimization dimensions of the Hf+AISI304 rod. Figure 7.2 Comparison of differential CR worth for different dimensions of Hf+AISI304 rod. Therefore, the following optimized dimensions have been chosen for the hafnium rod, which lower part is prolonged with AISI304, which are summarized in table 7-1: Table 7-1 The final optimal design of Hf+AISI304 for CR of BR2 Dimension [mm] Material Outer diameter Inner diameter Length As for The upper part of CR hafnium 64 54 Reference Rod The lower part of CR AISI304 64 50 70

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[1]. BR2 Multipurpose Material Testing Reactor, http://www.imr-oarai.jp/about/img/File2.pdf [2]. Drawings: BR2 Reactor, S. Wirix, Drawing N°12306, Dept. BR2, SCK•CEN, [3]. The reactor nuclear control, SCK dossier Vol VI, 72S7680, Reactor BR2. [4]. E.Koonen, A.Beeckmans and P.Gubel, "Fuel Characteristics Needed for Optimal Operation of the BR2 Reactor", 2nd Int. Topical Meeting on Research Reactor Fuel Management, Transactions, March (1998), Bruges, Belgium. [5]. Research Reactors, http://www.uic.com.au/nip66.htm [6]. Nuclear Research Reactors in the World, http://www.iaea.org/worldatom/rrdb/ [7]. http://en.wikipedia.org/wiki/cadmium [8]. MCNPDATA. Standard Neutron, Photon, and Electron Data Libraries for MCNP4C. Oak Ridge National Laboratory, RSICC Computer Collection, DLC-200/MCNPDATA, (March 2001). [9]. http://en.wikipedia.org/wiki/hafnium [10]. http://en.wikipedia.org/wiki/boron [11]. http://en.wikipedia.org/wiki/gadolinium [12]. http://en.wikipedia.org/wiki/europium [13]. http://en.wikipedia.org/wiki/stellite [14]. James J. Duderstadt & Louis J. Hamilton,” Nuclear Reactor Analysis”, JOHN WILEY& SONS, INC, NEW YORK, 1976. [15]. G.M.Schindler, "On the Efficiency of A Concentric Cut-Off Rod of A Thermal Reactor as A Function of the Inserted Length of the Rod", J. Nuclear eneregy, Vol. 8, pp. 18 to 32. Pergamon Press Ltd., London, 1958. [16]. B.Ponsard, E.Koonen,A.Beeckmans,M.Noel – BR2 data for cadmium burnup of the Reference CR with cadmium and cobalt. [17]. MCNP - 4C Manual, Monte Carlo N-Particle Transport Code System,Oak Ridge National Laboratory, April 2000 [18]. ORIGEN-S: SCALE System Module to Calculate Fuel Depletion, Actinide Transmutation, Fission Product Buildup and Decay, and Associated Radiation Terms. Oak Ridge National Laboratory, NUREG/CR-0200, Revision 6, v. 2, F7. ORNL/NUREG/CSD-2/V2/R6, March 2000. [19]. XSDRNPM: A One-Dimensional Discrete-Ordinates Code for Transport Analysis. Oak Ridge National Laboratory, NUREG/CR-0200, Revision 6, v. 2, F7. ORNL/NUREG/CSD-2/V2/R6, March 2000. [20]. NITAWL: SCALE System Module for Performing Resonance Shielding and Working Library Production. Oak Ridge National Laboratory, NUREG/CR-0200, Revision 6, v. 2,

F2. ORNL/NUREG/CSD-2/V2/R6, March 2000. [21]. COUPLE: SCALE system Module to Process Problem-Dependent Cross Sections and Neutron Spectral Data for ORIGEN-S Analyses. Oak Ridge National Laboratory, NUREG/CR-0200, Revision 6, v. 2, F6. ORNL/NUREG/CSD-2/V2/R6, March 2000. [22]. S.Kalcheva, E.Koonen and B.Ponsard, “Accuracy of Monte Carlo Criticality Calculations During BR2 Operation”, Nucl. Technol., 151, 201 (2005). [23]. S.Kalcheva, E.Koonen and P.Gubel, “Detailed MCNP Modeling of BR2 Fuel with Azimuthal Variation ”, Nucl. Technol., 158, 36 (2007). [24]. S.Kalcheva and E.Koonen, "3-D Fuel Burn Up Modeling With MCNP&ORIGEN-S", Proceedings of the 10th Int. Topical Meeting on Research Reactor Fuel Management, Sofia, Bulgaria, May (2006). [25]. B. Ponsard, "Expected Irradiation Conditions: BR2 Reactor. Operation Cycles”. [26]. F. Motte, J. Debrue, H. Lenders, A. Fabry, "Study of the BR2 Nuclear Characteristics by Means of its Mock-up BR02", C.E.N., MOL, 1964. [27]. Isbin H.S. and J.W.Gorman: Applications of Pile – Kinetic Equations, Nucleonics, vol. 10, n. 11, p. 69, November, 1952. [28]. Hurwitz H.: Derivation and Integration of the Pile – Kinetic Equations, Nucleonics, vol. 5, n. 1, p. 61, July, 1949. [29]. Bonilla, C.F. (ed.): "Nuclear Engineering", McGraw – Hill Book Company, Inc., New York, 1957. [30]. Stievenart, P., and P.Erkes: Determination of Reactor Transients and Time Variation of Core Material Concentration and Excess Reactivity by Graphical Methods, A/Conf. 15/P/1896, 2d International Conference on Peaceful Uses of Atomic Energy, Geneva, 1958.

APPENDIX A

A.1 Solution of kinetic equations for step – function input in reactivity ρ

The solutions of the Eqs. (4-76) & (4-77) for step – function are well known[27], [28]. These kinetic equations can be combined to form a single differential equation of the seventh order in n. For a step in reactivity ρ the solution will take the form:

7 n()t = n0 ∑ Aj exp[]Pjt (A-1) i=1

Where, the first exponent P1 has the same sign as ρ (the input disturbance). All the other six exponents are negative. Equations (4-76) and (4-77) are a family of differential equations with constant coefficients, and there is a definite relationship among the values of Aj, Pj and ρ. The neutron – level response for various step ρ inputs is shown in Fig. A.1.

Figure A.1 Relative neutron level of U-235 reactor versus time for positive step – function reactivity changes, Λ =10−4 sec. (δk = ρ) .

The value of Λ=10-4 sec. affects the front edge of the rise. After an initial rise caused mostly by prompt neutrons, the reactor was settled down to a steady period created by the delayed neutrons. It can be seen that for step reactivity ρ ≥ 0.005, extremely rapid and long rises occur. The shape of the initial response of a reactor to a step change in ρ can usually be analyzed by an approximation method. During an initial time of the order of 0.1 sec., the delayed – neutron emitters can be considered as a constant source of neutrons. If the reactor is in equilibrium, the delayed neutron emitters yield β neutrons for each neutron produced. That is, they are acting as a

95 APPENDIX A

β constant source of strength: n . The initial response of the reactor is described by: Λ 0

dn ρ − β β = ⋅n + n (A-2) dt Λ Λ 0 The solution of Eq. (A-2) is: n(t) β ρ ⎡ ρ − β ⎤ = − + ⋅ exp⎢ ⋅t⎥ (A-3) n0 ρ − β ρ − β ⎣ Λ ⎦

− β The neutron density in Eq. (A-3) approaches the asymptotic value of: . The accuracy of ρ − β the approximate solution for the neutron density (see Eq. (A-3)) is compared with the actual case in Fig. A.2.

Figure A.2 Front edge shape for a step – function change in reactivity showing results of approximate formula (A-3), (δk = ρ) .

A.2 Solution of kinetic equations for ramp – function input in reactivity ρ

When a control rod is pulled out of the reactor, a ramp function in the reactivity change of the form ρ = α + γ ⋅ t can be considered. In general control rod effectiveness is such that the reactivity does not change linearly as a function of rod position. Nevertheless, useful information can be obtained by considering linear rates of reactivity change and modifying the slope of these linear reactivity rates of change in discrete intervals. The exact solution of the reactor kinetic equation for this type of ramp is a very complex one. The effect of different linear reactivity change rates may be seen from Fig. A.3. The curves are plotted as a function of the reactivity remaining in the reactor. The subcritical multiplication curve is

96 APPENDIX A furnished as a reference curve, being the case of infinitely slow reactivity change. It can be seen that the reactivity is inserted into the reactor at higher and higher rates, the critical point comes at lower and lower neutron levels. The reactor period as a function of linear reactivity changes is given in Fig. A.4. It is seen that when the multiplication factor of the reactor comes closer and closer to unity as linear rates of change. At high reactivity change rates, a short period, easily detectable by measuring instruments, is available at quite low multiplication factors. If the rate of change of rate of reactivity is small, the multiplication factor must be very close to unity to obtain a reactor period of 20 sec. Fig. A.4 can be interpreted as meaning that one should extract the rods rapidly in order to see quickly a measurable period.

Figure A.3 Relative neutron level versus reactivity remaining in a reactor for various ramp – function reactivity change rates (δk = ρ) .

97 APPENDIX A

Figure A.4 Reactor period versus multiplication factor for various ramp – function reactivity change rates

(δk = ρ) .

An approximate solution of kinetic equations for a ramp – function for a critical reactor can be developed, assuming that the reactor is operated at a steady state before the application of the disturbance and the disturbance at t=0 is ρ = A⋅t . Then Eq. (4-76) takes the same form as approximate equation (A-2): dn ρ − β β = ⋅ n + ⋅ n (A-4) dt Λ Λ 0

Now: ρ = A⋅t (A-5)

dn A⋅t − β β Then: − ⋅ n = ⋅ n (A-6) dt Λ Λ 0

The solution for Eq. (A-6) is of the form:

n = [exp(∫ − Pdt)]⋅{∫[exp(∫ Pdt)]Qdt + C} (A-7)

− A⋅t + β β ⋅ n Where, P = ; Q = 0 (A-8) Λ Λ Then: n At 2 2 t ⎛ t ⎡ At 2 2 t ⎤ C ⎞ β ()− β ⎜ − ( − β ) ⎟ = ⋅ exp ⎜ ∫exp⎢ ⎥dt + ⎟ (A-9) n0 Λ 2Λ ⎝ 0 ⎣ 2Λ ⎦ n0 ⎠

Since at t=0, n n0 =1, C n0 must be equal to Λ β , the integral term in Eq. (A-9) may be written in the form:

98 APPENDIX A

⎛ β 2 ⎞ t exp⎜ ⎟ ⋅ exp − αt + γ 2 dt (A-10) ⎜ 2 AΛ ⎟ ∫ []() ⎝ e ⎠ 0

A β Where, α = and γ = − (A-11) 2Λ 2AΛ

Substituting αt + γ = µ , we can get:

t 1 αt+γ π ⎛ 2 αt+γ 2γ t ⎞ 2 2 ⎜ 2 2 ⎟ ∫exp[]− ()αt + γ dt = ∫exp()− µ dµ = ⎜ ∫exp()− µ dµ − ∫exp()− µ dµ ⎟ (A-12) 0 α γ 2α ⎝ π 0π 0⎠

The values of the probability integral are available in mathematical tables[29]. The complete solution then becomes:

αt +γ γ n β 2 ⎡ π ⎛ 2 2 2 2 ⎞ Λ 2 ⎤ µ ⎜ −µ −µ ⎟ −β 2 AΛ = ⋅ e ⋅ ⎢ ⎜ ∫ e dµ − ∫e dµ ⎟ + e ⎥ (A-13) n0 Λ ⎣⎢ 2α ⎝ π 0 π 0 ⎠ β ⎦⎥

The Eq. (A-13) plotted in Fig. A.5 shows the results of suddenly applying rates of change of reactivity ρ to a critical reactor. It is easily seen that putting in reactivity at finite rates into a reactor, instead of in step – function fashion, results in considerably slower rates of initial rise of neutron level.

Figure A.5 Relative neutron level versus time for various ramp function reactivity change rates, using approximate formula (99) (δk = ρ) .

A.3 Graphical method for the solution of reactor equations with time –

varying inputs

The analytical methods presented in §4.1.3.1, §4.1.3.2 for solving the reactor kinetic equations for step – and ramp – function inputs are tedious and time – consuming. A graphical method has been developed by Stievenart and Erkes[30] which simplifies the solution into a routine drafting operation and provides good accuracy.

99 APPENDIX A

The approach is by analogy to the solution of ideal transmission – line equations. Consider the circuit diagram of Fig. A.6 which has the familiar differential equation: di u = L + R(t) ⋅i (A-14) dt Where: u – the voltage, L – inductance, R – resistance. The length of the transmission line is a, then l is the distributed induction per unit length, i.e. L ⋅ a = l . The capacitance of the transmission line is very small (almost zero). We can select a capacitance such that the time interval T is small compared with the time constant L R(t) ; the smaller the time interval, the more accurate the solution. Then: a T = = a lc = Lac (A-15) v From which the construction angle γ (see Fig. 4.10) is found as: l L L tanγ = ±z = ± = ± = ± (A-16) c ac T The selection of a value of c and hence T is a compromise between a too small time interval requiring too many lines of graphical construction and a too large time interval leading to an inaccurate solution. Experience indicates that time intervals should usually be so selected that values of γ between 60 and 85° result. The stipulation T < LR(t) requires that the angle α must be smaller than γ.

Figure A.6 Circuit diagram of variable resistance lumped – inductance series circuit.

100 APPENDIX A

Figure A.7 Voltage versus current diagram, indicating the graphical construction required to solve the transmission – line problem. The method may be now tried on an example. In the circuit of Fig. A.7, let L=1 henry, u=10 volts, and R(t) = ()1− t ohms for 0.5 sec. and then remains constant at 0.5 ohms. We arbitrarily select a time interval T=0.1 sec. for which tanγ = L T = 1 0.1 = 10 . Fig A.8 shows the graphical construction. From the graph and from obvious physical considerations the voltage ultimately returns to its time zero value of 10 volts and the current ultimately becomes 20 amp.

Figure A.8 Construction for series – inductance variable – resistance example. Now we infer from the above problem that any equation having the form: dx a + bx + c = 0 (A-17) dt Where, a, b, c may be arbitrary functions of time, can theoretically be solved graphically by analogy with the inductance – resistance problem just presented. As an elementary illustration of this graphical method of solution for reactor transients, we may use the kinetic equations that represent a reactor with one lumped group of delayed neutrons: dn ρ − β = ⋅ n + λC (A-18) dt Λ

101 APPENDIX A

dC β = ⋅ n − λC (A-19) dt Λ which can be written in the form: dn β − ρ λC = + ⋅ n (A-20) dt Λ

di Analogous to: u = L 1 + R i (A-21) 1 1 dt 1 1

β dC And: n = + λC (A-22) Λ dt

di Analogous to: u = L 2 + R i (A-23) 2 2 dt 2 2

β Then: u = λi and u = i (A-24) 1 2 2 Λ 1 The transmission – line diagrams are given in Fig. A.9. The solution is obtained by working out two simultaneous graphical constructions. First one finds the value of n in the construction for Eq.

(104) at the end of the first time interval. This value multiplied by β Λ becomes the value of u2 at the end of the first time interval for the construction of Eq. (A-22). This second construction yields a value for c at the end of a time interval, and this value in turn multiplied by λ gives i, for use in the first equation. By working back and forth between the two graphs one quickly arrives at a solution for n.

Figure A.9 Transmission – line diagrams for single – group delayed – neutron equations: (a) Eq. (104); (b) Eq. (106). (δk = ρ) .

As an example, let consider a 235U – fueled thermal reactor, having the constants: Λ=10-4, β=0.0064, λ=0.08. Let the reactor is critical at t=0. Let insert the ramp function ρ(t) = 0.01t into the reactor. For the graph, let n0=1. At t=0, ρ=0, and for the first construction: β 0.0064 ()u = λC = n = = 64 (A-25) 1 0 0 Λ 10−4

102 APPENDIX A

()i1 0 = n0 =1 (A-26) i = n (A-27) β (tanα ) = ()R = = 64 (A-28) 1 0 1 0 Λ

β − ρ(t) 0.0064 − 0.01t (tanα ) = ()R = = (A-29) 1 t 1 t Λ 10−4

The time interval T < 1 R(t) can arbitrarily be chosen as 0.1 sec. Then:

tanγ 1= 10 (A-30) For the construction for the solution of the second equation: β ()u = n = λC = 64 (A-31) 2 0 Λ 0

β n ()i = C = 0 = 800 (A-32) 2 0 0 Λ λ

()tanα 2 0 = R2 = λ = 0.08 (A-33) We use the same time interval T=0.1, so that:

tanγ 2= 10 (A-34) The form of the graphical solution is given in Fig. A.10. The starting point for the delayed – neutron equation in the graph (b) is at point B rather than in point A in order to obtain a value of β n from the graph (a). Λ The method of solution for six delayed neutron groups is now also obvious. Each equation has a separate graph, and the sum of the delayed – neutron – equation contributions is added into the primary – neutron equation.

103 APPENDIX A

Figure A.10 Construction diagrams for solution of reactor – kinetic equations having single – group delayed neutrons and ramp input in ρ.

104

APPENDIX B

B.1 Upper bound of reactor period

The upper bound will be handled in the following manner: Let fisrt re-write the familiar pile kinetic equation: dn ρ − β 7 = ⋅ n + ∑λiCi + S (B-1) dt Λ i=1

The last two terms on the right-hand side of Eq. 129 are always positive. Then: dn ρ − β n Λ > ⋅ n ⇒ < (B-2) dt Λ dn dt ρ − β

If the period T is defined as: n T = (B-3) dn dt then: Λ T < (B-4) ρ − β for ρ ≥ β . For ρ < β , the inequality of Eq. (B-4) loses its significance with regard to fixing the magnitude of the period. However, this inequality establishes a real upper bound for the period when the reactor is above prompt critical. We have then established a maximum limit on the period a reactor can possess by saying that within a given upper criticality range the period cannot be larger than a specified value.

B.2 Lower bound of reactor period

A lower bound to the period in the subcritical range may also be established. It is seen from the ρ subcritical multiplication formula: T = − that this lower bound will depend on the rate d()ρ dt of change of reactivity. Defining this rate of reactivity change by symbol γ []ρ sec ,

ρ T = − ,[]$/sec (B-5) γ as the subcritical relationship.

105 APPENDIX B

For reactors subcritical by a large amount of negative reactivity it has been shown that Eq. (B-5) is exact, but as the reactor approaches criticality, the formula does not hold, because it assumes that all neutrons are effectively prompt neutrons. As criticality is approached in a start-up, the delayed neutrons become more and more important, in that the rates of change of level become comparable with delayed – neutron – emission times. So, from a control point of view the effect of the delayed neutrons is to slow down any level changes, thus increasing the reactor period. Therefore, Eq. (B-5) represents the lower bound of the reactor period. For simplicity consider the pile kinetic equations (4-76) and (4-77) for the lumped-delayed-emitter one group, dn ρ − β = ⋅ n(t) + λ ⋅C(t) + S (B-6) dt Λ

dC β = ⋅ n(t) − λC(t) (B-7) dt Λ

7 3 Where, n(t) [n/cm ] is the neutron density in function of time; β = ∑ β i = 0.0072 is the i=1 effective fraction of all delayed neutrons, including the photoneutrons; C(t) is the density of precursors of delayed neutrons; S[n/cm3/s] is the strength of source of neutrons For sub-critical condition the reactor must reach some equilibrium, which can be defined as when the concentration of the delayed-neutrons emitters C(t) becomes constant, i.e. the neutrons are lost at the same rate they are being produced. Then, at equilibrium we have the following equations: dn ρ − β = ⋅ n + λ ⋅C + S = 0 (B-8) dt Λ

dC β = ⋅ n − λC = 0 (B-9) dt Λ

Substituting Eq. (B-9) into Eq. (B-8): dn ρ dC ρ = ⋅ n − + S = ⋅ n + S = 0 (B-10) dt Λ dt Λ

The equilibrium neutron concentration from Eq. (B-10) becomes: SΛ n = − (B-11) eq ρ

We notice that Eq. (B-11) is independent of whether we use one lumped delayed emitter or seven (or more) individual ones. If now make a small change in δk = ρ about this equilibrium state,

106 APPENDIX B

⎛ 1 1 ⎞ ∆n = n − n(0) = −SΛ⎜ − ⎟ (B-12) ⎝ ρ0 + ∆ρ ρ0 ⎠

SΛ∆ρ ∆n = (B-13) ρ0 ()ρ0 + ∆ρ

The time ∆t required for ρ to change by any amount ∆ρ is: ∆ρ ∆t = (B-14) γ

Where, γ [$/sec] is the previously defined rate of change of reactivity. Then if we consider inserting a positive reactivity change, the equilibrium neutron flux level must be greater than the level before equilibrium is established for any subcritical ρ. The greatest possible value for the time rate of change of the neutron flux level then would be: ∆n SΛ∆ρ γ = ⋅ (B-15) ∆t ρ0 (ρ0 + ∆ρ)∆ρ

∆n SΛγ = (B-16) ∆t ρ0 ()ρ0 + ∆ρ

If one obtains the time derivative of n in the following manner: dn ∆n SΛγ = lim < (B-17) dt ∆t ()ρ 2

The period lower bound becomes: n SΛ ()ρ 2 ρ T = > − = − (B-18) dn dt ρ SΛγ γ An upper and a lower bound for the period of a reactor involved in a start-up – type accident have thus been established. Fig. B.1 illustrates these bounds for several conditions. The upper bound depends only on Λ and the amount of reactivity present in the core at a given time. The lower bound, however, is a function of the rate of insertion of reactivity. Fig. C.1 also indicates a possible start-up accident when reactivity is being inserted at the linear rate of 1.2×10−4[ρ sec] into a reactor having Λ =10−4 sec. At large sub criticalities the reactor follows the lower period bound. Above prompt critical the period closely follows the upper bound. From safety point of view the upper bound is really in the wrong direction, so it would be desirable to know that a reactor could never get on a period shorter than a given amount. However, above prompt critical, the period follows the upper bound as developed so closely that this bound may also be used as a practical value of minimum period. It is to be stressed that the temperature coefficient is not taken

107 APPENDIX B into account in the approach presented. The lower bound would be probably not affected, but the upper bound would be drastically changed.

Figure B.1 Period as a function of reactivity indicating upper and lower bounds for a start-up accident. Prompt critical 7.5×10−3 .

B.3 Intermediate value of reactor period

The period attained in the intermediate region between the bounds depends upon how subcritical the reactor is at the start and what is the reactivity change rate. The period, attained at any level above the start-up level is a function of from how far below that level the initial reactivity insertion started. Let us assume that we have established by the safety system a protection level above which a signal is given for a scram or some emergency means to reverse the rod motion. For example, if the reactor power level exceeded 200% of full power, a signal would be provided to the scramming system to insert all control rods as quickly as possible. The problem then exists of how dangerous level is noted by an instrument and the rods actually start moving in. This available time depends upon the period that the reactor is on at this level. In turn, the period that may be attained depends upon how far below this protection level the start-up accident began. Fig. B.2 indicates the period attained at a given protection level as function of the decades of rise in reactor power level before the protection level is reached. The curves are for the case that the reactor was originally – 13% subcritical. It can be seen from Fig. B.2 that if only a few decades of power – level change are involved between the start-up point

108 APPENDIX B and the protection level, the periods attained are quite modest and a comparatively long amount of time is available to initiate some protection device. On the other hand, if an accident occurs when the reactor is started ~ 10 decades below the protection level, very fast periods can occur, particularly at high reactivity insertion rates.

Once a period has been established in this manner or from boundary conditions, then the amount of time to do something about protection is directly established. Fig. B.3 indicates the relative power level that a reactor would attain above the protection excursion level as a function of the time delay in doing something about the rising power level. Hence, while waiting for the protection signal to act, following a trip signal, the delay time in the protection system is most important, as the power could become very high during this delayed interval. Conversely, Fig. B.3 indicates that if a 200% absolute power level is needed to protect a reactor, than if a protection signal is provided at 100% full power, the protection system must reverse the reactivity and cause the power level to start coming down with a delay of less than 0.7 sec., provided that the reactor is on a 1 – sec period.

109 APPENDIX B

Figure B.3. Relative power level attained by a reactor on a given period as a function of the time delay in doing something about it.

110

APPENDIX C

C.1 Subcritical level operation

Let us insert in a subcritical medium a source of neutrons. Such sources exist naturally from cosmic rays, or neutrons may be artificially provided from radioactive isotopic mixtures such as Ra-Be or Po-Be. Under the condition of sub criticality the number of neutrons which exist in this multiplying medium at the end of a sufficiently long interval of time is:

2 m−1 n = n0 (1+ k + k + ...... + k ) (C-1)

Where, n=number of neutrons in multiplying medium; n0=number of neutrons originally present in medium. Then: n 1− k m = (C-2) n0 1− k

At the end of sufficiently long interval of time for k<1, Eq. (C-2) generates into: n 1 = (C-3) n0 1− k This ratio is known as the subcritical multiplication factor, and all reactors exhibit this effect. The result of Eq. (C-3) is shown graphically in Fig. C.1 for a source suddenly inserted into a multiplying medium. With k ~ 0.5, the number of neutrons saturates after several neutron lifetime n l to a value = 2. The subcritical multiplication factor M of this reactor is then M=2. If we n0 change k to 0.9 by removing part of the CR from the medium, then the subcritical multiplication n factor M=10, etc. As k approaches 1, M = → ∞ and the number of neutrons in the medium n0 rises in straight line with time. As the subcritical multiplication factor M becomes higher and higher, more time is always taken for the medium to settle out at a given level. And finally it will not settle out at all but continues to rise. This situation holds only if there is an external source present. Without a source, the neutron level in any subcritical medium dies down to zero. Kinetically, when a source is suddenly inserted into a subcritical multiplying medium containing no initial neutron population, the medium responds according to the equation:

111

SΛ ⎧ ⎡ 1− k ⎤ ⎫ n(t) = ⎨1− exp⎢− ⎥ ⋅t⎬ (C-4) 1− k ⎩ ⎣ Λ ⎦ ⎭

Let us now examine how the power level changes in this subcritical reactor as we increase slowly k → 1 by removing the control rods at infinitely slow rate. In this case the total multiplication is always the subcritical multiplication. Because the withdrawal rate is so slow, the decay times of even the longest – lived delayed emitters are short in comparison with the time for a noticeable reactivity change. Therefore, all the delayed neutrons have ample time to be emitted before the power changes appreciably. The number of neutrons present is described with the subcritical multiplication formula of Eq. (C-3). Figure C.2 shows the neutron – level build – up under this condition. The curve is a hyperbola and approaches criticality asymptotically. The curve is plotted so that when k=0.9, the power level is 1. The period that results from this very slow pulling rate can be obtained from the definition of period of Eq.: 1− k T = (C-5) dk dt

So, if the rate of change of k is constant with time, then the period decreases directly as the negative reactivity remaining in the reactor becomes smaller. For our example of extremely slow rod pulling at a constant time of change, the period approaches zero as the medium approaches criticality. If the power level of the medium is constant with k=1, however the source neutrons continue to add in and create a rising power level. The usual reactor source strength may vary from a few neutrons per second to a few millions per second, which is only a negligible percentage of the number of neutrons (billions) involved in a power operation.

Figure C.1 Subcritical multiplication.

112

Figure C.2 Relative power level as a function of reactivity in the reactor for infinitely slow reactivity change, (δk = ρ) .

C.2 Criticality approach

Bringing the reactor to initial criticality can be done in a number of ways depending upon its design. Some reactors such as the water boiler, add liquid uranium fuel mixture until the reactor becomes critical. Other reactors, such as NRX reactor add moderator in the form of heavy water to such level that the system becomes critical. Other heterogeneous reactors have fixed fuel and moderator and remove control rods so that the reactor becomes critical. In the BR2 reactor, the criticality is achieved at fixed core load by adding 6 fuel elements in channels A and B, located in the central crown. In any of these cases, the effect is the same. A multiplication factor is brought from some fractional value to unity. As the initial operation is apt to be a very slow, the reactor condition is under subcritical multiplication. That is the instruments will be reading a signal proportional to l (1− k) , or the counting rate from an instrument channel will be: l c.r. = A⋅ (C-6) 1− k Where, c.r. – counting rate, A – instrument constant. As k approaches unity very slowly, the counting rate approaches infinity, or in other words, when 1/c.r. approaches zero, the reactor approaches criticality. Figure C.3 shows such a plot for bringing the BR2 reactor to criticality by adding consecutively 6 fuel elements in the channels A and B, located in the central crown. As the uranium (i.e., the fuel elements) is being added, the curve can be constantly extrapolated to zero to predict where the reactor will go critical. The shape of the curve as the reactor approaches criticality will depend on the position and type of detecting instruments. If the chamber is located very close to the reactor, it will effectively see the source for a long period of time and the shape of the curve will be concave downward. As criticality is approached in this manner, the reactor will take longer and longer to settle out at a

113

fixed neutron level or counting rate. At criticality the reactor level will continue to rise indefinitely because of additive neutrons from the source. Actually, criticality will be reached at such a high level that the source is contributing only a per cent of the over – all counts by the instruments. When the reactor is brought close to criticality, the source can be removed from the reactor.

Figure C.3 Plot of 1/c.r. vs. amount of uranium, showing the effect of the instrument placement.

C.3 Getting the reactor critical

Figure C.4 shows the reactor period as function of reactivity for given rates of change of reactivity. It is seen that at infinitely slow rates of change in reactivity the reactor never becomes critical and follows a hyperbolic curve approaching criticality. At finite rates of reactivity change the reactor becomes critical at lower and lower neutron levels as the reactivity change rate is increased. At large values of sub criticality all the curves merge and the rates of change of reactivity do not matter.

Figure C.4 Power level of a reactor as function of per cent reactivity remaining in the reactor for given linear rates of change of reactivity.

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Fig. C.5 indicates the period that might be obtained by a given reactor as a function of the reactivity remaining in the reactor, as this reactivity is inserted linearly up to the critical point. The period is: − ρ T = (C-7) dρ dt

Eq. (C-7) indicates that the period changes linearly with the reactivity as long as the reactivity is being inserted at a given linear rate. This equation holds only for large values of sub criticality, as when the reactor approaches critical at a finite rate the curves start to bend over as is seen in Fig. C.4.

Figure C.5 Period vs. reactivity of a reactor for given rates of change of reactivity. Eq. (C-7) may be combined with the subcritical multiplication factor equation, giving the level: 1 L = (C-8) − ρ

The relationship between period and level is: n 1 L = = (C-9) n0 []dρ dt T i.e., any subcritical period may be obtained at any subcritical level depending upon how fast reactivity is inserted into the reactor. The formula (C-9) does not hold when the level gets close to critical, but for operation at T < 60 sec. the formula is still valid. Fig. C.6 indicates the type of the curve that can be obtained showing reactor period at criticality as function of reactivity insertion at various rates.

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Figure C.6 Period on which a reactor would go through criticality by inserting reactivity at given linear rates. Λ =10−4 sec. (δk = ρ)

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APPENDIX D

Examples of input data for SCALE4.4a

=COUPLE - LIBRARY MADE IN TEST CASE 1 -

0$$ A2 28 A6 34 E 1$$ 1 1 A11 1010 E 2** A2 0.632 0.333 2 1T 3T 35$$ 0 5T

0$$ A4 34 E 1$$ A5 1 A13 -1 E 1T DONE END =NITAWL 0$$ 82 E 1$$ 0 11 A8 2 E T 2$$ 92234 92235 92236 92238 5010 5011 4009 13027 1001 8016 3006 3** 92235 293. 2 5.04 5.04 0.0 3.09519-3 1 16. 49.783 1 238.125 53.743 1 1 92238 293. 2 5.04 5.04 0.0 1.12864-3 1 16. 136.42 1 235.117 4.546 1 1 4** F293 2T T END =XSDRNPM PWR TEST CASE 1$$ 2 29 145 1 3 29 68 8 3 1 10 10 0 0 0 E 2$$ -1 3Z E 3$$ 1 E 4$$ 1 3 0 -1 E 5** A3 6.7+14 E T 13$$ 1R1 2R2 1R3 2R4 1R5 6R6 1R7 2R8 1R9 6R10 1R11 2R12

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1R13 6R14 1R15 2R16 1R17 6R18 1R19 2R20 1R21 6R22 1R23 2R24 1R25 6R26 1R27 2R28 2R29 14$$ 13027 1001 8016 13027 1001 8016 13027 92234 92235 92236 92238 5010 5011 13027 1001 8016 13027 92234 92235 92236 92238 5010 5011 13027 1001 8016 13027 92234 92235 92236 92238 5010 5011 13027 1001 8016 13027 92234 92235 92236 92238 5010 5011 13027 1001 8016 13027 92234 92235 92236 92238 5010 5011 13027 1001 8016 13027 92234 92235 92236 92238 5010 5011 13027 1001 8016 4009 3006 15** 0.0602 0.066778 0.033389 0.0602 0.066778 0.033389 0.0602 4.01150E-05 3.09519E-03 2.13000E-07 1.12864E-03 1.15273E-04 4.24467E-04 0.0602 0.066778 0.033389 0.0602

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4.01150E-05 3.09519E-03 2.13000E-07 1.12864E-03 1.15273E-04 4.24467E-04 0.0602 0.066778 0.033389 0.0602 4.01150E-05 3.09519E-03 2.13000E-07 1.12864E-03 1.15273E-04 4.24467E-04 0.0602 0.066778 0.033389 0.0602 4.01150E-05 3.09519E-03 2.13000E-07 1.12864E-03 1.15273E-04 4.24467E-04 0.0602 0.066778 0.033389 0.0602 4.01150E-05 3.09519E-03 2.13000E-07 1.12864E-03 1.15273E-04 4.24467E-04 0.0602 0.066778 0.033389 0.0602 4.01150E-05 3.09519E-03 2.13000E-07 1.12864E-03 1.15273E-04 4.24467E-04 0.0602 0.066778 0.033389 0.123443606 0.0000045 T 33** F1.0 T 35** 4I0.0 4I0.7912 4I0.97 4I1.3015 4I1.5977 4I1.64996926 4I1.6866 4I1.725 4I2.0244 4I2.073843 4I2.1133 4I2.151 4I2.4511 4I2.4985496 4I2.54 4I2.578 4I2.8778 4I2.923960925 4I2.9667 4I3.005 4I3.3045 4I3.3495405 4I3.3934 4I3.431 4I3.7313 4I3.7755812 4I3.8202 4I3.858 4I4.21 5.040361 36$$ 5R1 5R2 5R3 5R4 5R5 5R6 5R7 5R8 5R9 5R10 5R11 5R12 5R13 5R14 5R15 5R16 5R17 5R18 5R19 5R20 5R21 5R22 5R23 5R24 5R25 5R26 5R27 5R28 5R29 39$$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 40$$ 29R3

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51$$ 5R1 15R2 7R3 T END =COUPLE *** PWR TEST CASE LIBRARY *** * (PROBLEM DEPENDENT NEUTRON SPECTRUM FACTORS) USED 27 GROUP SCALE LIBRARY AND CONVERTED......

0$$ A4 34 A6 15 E 1$$ A12 922350 A16 3 1 1 1 E 2** 303.3 E T EDIT

0$$ A4 15 E 1$$ A2 1 A5 1 A13 -1 E T DONE END =ORIGENS 0$$ A5 28 E 1$$ 1 T SAMPLE CASE 2 3$$ 15 A3 1 A10 1 1 A16 2 A33 12 E 2T 35$$ 0 4T 56$$ 10 A6 1 A13 17 4 3 0 2 1 1 0 57** 0 0 1-14 E T SAMPLE CASE 2 MT OF HEAVY METAL CHARGED TO REACTOR 58** F2.00 60** 8I0 21 66$$ 1 A5 1 A9 1 E 73$$ 10010 10030 20030 30060 40090 50100 50110 80160 130270 621470 621490 621500 621520 922340 922350 922360 922380 74** 289.6 0.33 0.00500 0.10 3496.6 0.6336 2.5664 2293.84 2157.051 0.211 0.19376 0.10458 0.37282 5.15953 399.8 0.02763 147.6453 75$$ 9R1 4R3 4R2 T 56$$ 0 10 A10 10 A14 4 0 0 4 E 5T 60** 0 10 15 20 40 50 60 70 80 90 61** F1-30 65$$ A1 1 A22 1 A43 1 E 81$$ 2 0 26 1 E 82$$ 6 6 6 6 6 6 6 6 6 6 83** 10+6 2+6 1.5+6 1+6 0.8+6 0.6+6 0.4+6 0.2+6 0.1+6 0.05+6 0.03+6 0.01+6 0 6T

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GRAM U-235, NEUTRONS /(SEC-CM**2), 0 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 1 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 5 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 10 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 25 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 50 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 100 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 150 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 200 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 300 SEC 56$$ F0 T END

=ORIGENS 0$$ A5 28 E 1$$ 1 T SAMPLE CASE 2 3$$ 15 A3 1 A10 1 1 A16 2 A33 12 E 2T 35$$ 0 4T 56$$ 10 A3 1 A6 1 A13 8 4 3 0 2 1 1 0 57** 0 0 1-14 E T SAMPLE CASE 2 MT OF HEAVY METAL CHARGED TO REACTOR 59** F1E12 60** 8I0 20 66$$ 1 A5 1 A9 1 E 73$$ 481060 481080 481100 481110 481120 481130 481140 481160 74** 12.25 8.894 129.01 133.97 255.19 131.14 311.44 83.24 75$$ 8R1 T 56$$ 0 10 A3 1 A10 10 A14 4 0 0 4 E 5T 60** 0 10 15 20 40 50 60 70 80 90 61** F1-30 65$$ A1 1 A22 1 A43 1 E 81$$ 2 0 26 1 E 82$$ 6 6 6 6 6 6 6 6 6 6 83** 10+6 2+6 1.5+6 1+6 0.8+6 0.6+6 0.4+6 0.2+6 0.1+6 0.05+6 0.03+6 0.01+6 0 6T GRAM U-235, NEUTRONS /(SEC-CM**2), 0 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 1 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 5 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 10 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 25 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 50 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 100 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 150 SEC GRAM U-235, NEUTRONS /(SEC-CM**2), 200 SEC

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GRAM U-235, NEUTRONS /(SEC-CM**2), 300 SEC 56$$ F0 T END

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