Optimized Control Rods of the Br2 Reactor

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OPEN REPORT SCK•CEN-BLG-1054

OPTIMIZED CONTROL RODS OF THE BR2 REACTOR

Analytical, numerical and experimental methods used for the reactivity control of the reactor BR2

S.Kalcheva and E.Koonen

September, 2007

SCK•CEN Boeretang 200 BE-2400 Mol Belgium

OPEN REPORT OF THE BELGIAN NUCLEAR RESEARCH CENTRE SCK•CEN-BLG-1054

OPTIMIZED CONTROL RODS OF THE BR2 REACTOR

Analytical, numerical and experimental methods used for the reactivity control of the reactor BR2

S.Kalcheva and E.Koonen

September, 2007 Status: Unclassified ISSN 1379-2407

SCK•CEN Boeretang 200 BE-2400 Mol Belgium

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A b s t r a c t

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 (Gd2O3). Options to decrease 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 the Monte Carlo burn up code MCNPX.2.6.E and 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. Table of Content

1. Introduction………………………………………………………………………………..p.06. 2. Function of control elements……………………………………………………………...p.10. 3. Overview of absorbing materials, used for nuclear control……………………………….p.12. 4. Nuclear Control in the BR2 reactor………………….……………………………………p.15. 5. Analytical and experimental methods for determination of CR worth ...... p.17. 6. Calculation methodology…………………...... p.52. 7. Neutronics modelling of BR2……………………………………………………………..p.55. 8. Impact of various factors on the control rod parameters in the reactor BR2……………...p.55. 9. Control rod candidate materials and design modifications………………………………..p.56. 10. Evaluation of control rod characteristics of different control rod types during 1000 EFPD of BR2 fuel cycle……………………………………………………………………………..p.57. 11. Changing of the reactor neutronics characteristics for different control rod types during BR2 fuel cycle………………………………………………………………………………………p.63. 12. Control rod effects on neutron fluxes and core power distributions……………………..p.63. 13. Comparison of activity during 1000 EFPD……………………………………………...p.65. 14. Criterion for choice of new control rods of the BR2 reactor…………………………….p.66. 15. Summary………………………………………………………………………………...p.67. 16. Proposed new control rod type – HF+AISI304 (R0=15.6 $)……………………………p.69. 17. References……………………………………………………………………………….p.70. 18. Tables……………………………………………………………………………………p.72. 19. Figures…………………………………………………………………………………...p.85.

1. Introduction A nuclear reactor which has to be operated at steady state conditions is initially charged with a significantly larger amount of fuel than required to achieve criticality in order to maintain critical during long time of operation. The fuel loading and fuel enrichment must be estimated to insert into the core 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 may strongly disturb the neutron flux and the power in the vicinity of the rod and therefore 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 in control rod material selection is that materials that 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 travelled by the epithermal neutrons in a reactor is greater than the path length travelled by thermal neutrons. Therefore, a resonance absorber absorbs neutrons that 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 in maintaining 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 sometimes may be 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. In Table I are summarized the used control rod materials in different research reactors in the world. It is seen that the most common materials used for control rods are Hafnium and Cadmium.

1.1 Importance of the task The necessity to perform this job has been compelled by the intention to improve the existing reactivity control of the BR2 reactor core. The most commonly used elements for reactivity control in research reactors are presented by rods or plates of strong neutron absorbers (such as

6 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 next 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 a factor of 2 and more (about 100 mm reduction of the cadmium lower edge vs. 295 mm for cadmium thickness 2 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 to depress the depletion in the lower cadmium edge: for an irradiation period ~ 750 EFPD the reduction of the cadmium length could be diminished to about 60 mm. Due to the burning of the lower edge of the active absorption part of the rod, which is exposed to the highest irradiation during the control rod life, the cadmium section of the control rod might be prolonged by another material. This material has to fulfil the following requirements: it must not burn too fast (or not to burn at all) from one side and from the other – it must have sufficiently high absorption macroscopic cross section in order to depress the neutron flux in the vicinity of the control rod edge, i.e. to depress the burn up of the main absorbing part (lower) of the control rod. The role of such “grey” material can play cobalt (as it was during the last 2 – 3 decades of the reactivity control of the reactor BR2), stainless steel or other alternative material.

1.2 Aim The aim is optimization of the control rods used for the reactivity control of the reactor BR2. This task comprises the following sub-tasks: (i) 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, gadolinium and “grey” materials, such as stellite. (ii) Considerations of combinations of various “black” absorbers with “grey” ones, (e.g. stainless steel) both to diminish burn up of the “black” absorber and to flatten the neutron flux and power distributions in the core. The grey absorber is used to prolong the lower absorbing part of the control rod. An option of a control rod with flattened axial burn up distribution over the axial direction is considered (decreasing the cladding around the absorbing CR material). (iii) Improvement of the old control rod design or proposal for a new one, if this is demanded by the choice of the (new) absorbing material. (iv) Improvement of the existing procedures for experimental determination of the reactivity values and related with it control rod worth’s to satisfy the new irradiation conditions.

7 The chosen new control rods must satisfy the following requirements for the reactivity control of the reactor BR2: (i) 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). (ii) They should not disturb too strongly the neutron flux and power distributions in the core. (iii) The chosen main control rod absorbing material should have both high thermal and high epithermal absorption cross sections. (iv) 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.

1.3 Condensed content

This report presents detailed optimization calculations for the choice of main absorbing material for the control rods of the BR2 reactor. Considered are various control rod candidate materials, such as: cadmium, hafnium, europium, gadolinium, stellite and the combination of these materials with “grey” absorbers, such as stainless steel. The “grey” material is used to prolong the lower end of the active control rod part. Diminishing the cladding (aluminium) around the absorbing part of the control rod will also depress the axial burn up of the chosen absorber material due to the higher absorption of thermal neutrons by light water. 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). Improvement of the main procedures used for the reactivity control of the reactor BR2 is presented.

1.4 What is new

Presented is detailed comparative analysis for the control rod life of various absorbing materials such as cadmium, hafnium, europium, gadolinium, stellite during long time of irradiation (~ 1000 EFPD). Comprehensive publications for comparison of burn up of different control rod absorbing materials and estimation of the control rod life during long time of irradiation seem not to be available. Most of the published items deal mainly with specific (own) problems. Applied is a modern Monte Carlo burn up calculation methodology ♦ Detailed depletion calculations of the control rod absorbing material are performed using latest developments of LANL – the burn up Monte Carlo code MCNPX2.6.E. ♦ Automatic depletion and criticality calculations are performed for the full scale 3 – D heterogeneous geometry model of the reactor BR2 in reasonable computation time. ♦ Calculations of continuous energy reaction rates are performed for almost all possible interactions of neutron with the reactor materials – (n,γ), (n,f), (n,2n), (n,3n), (n,α) (n,p), (n, beta), etc. with proper tracking of entire decay chains of more than ~ 3500 nuclides. ♦ Burn up calculations for the evolution of the macroscopic cross sections are validated on measurements for the reduction of the control rod absorption properties (cadmium), exposed under irradiation during long irradiation period.

8 Improvement of the experimental techniques for estimation of the control rod worth’s, which are routinely used in the reactor BR2 is presented. ♦ Proposed is a method which uses a combination of Monte Carlo technique and perturbation theory. The idea is to derive the 'blackness' for various fresh and burnt absorber materials from the macroscopic cross sections, which are obtained from detailed Monte Carlo burn up calculations. ♦ After that, a series of corrections, including the axial distribution of the 'blackness' over the absorber length are introduced into the equation for the relation between the reactivity and the buckling of the partially burnt control rod. ♦ These corrections, including the blackness and the axial burn up of the absorbing materials, have to be determined – by experiment (as it was by now) or by using some theoretical method (proposed here). ♦ Thus, the theoretically determined buckling is used together with the measured relative efficiency of the partially inserted rod to determine the total control rod worth and to recover the curves of the total and differential control rods worth. MCNP can be easily implemented for estimation of the reactor period by direct calculations of keff, prompt-neutron life time and the average number of neutrons per fission event and then solving the equation for the reactor period.

1.5 Calculation methodologies

¾ MCNPX – Monte Carlo code: ♦ used for development of the realistic 3 – D heterogeneous geometry model of the reactor BR2; ♦ used for steady-state flux and spectra calculations; eigenvalue calculations; evaluation of the total control rods worth’s; ¾ Combined MCNPX&ORIGEN-S method; ♦ used for the detailed 3 – D distribution of the isotopic fuel depletion in the reactor core ¾ MCNPX2.6.E – a burnus Monte Carlo code: ♦ used for evaluation of 3 – D fuel depletion and eigenvalue calculations; ♦ evaluation of 3 – D reaction rates, macroscopic and effective microscopic cross sections versus irradiation time; ♦ 3 – D burn up calculations for the absorber material of the control rod ¾ SCALE4.4a ♦ used for the evaluation of the isotopic fuel depletion, evaluation of the macroscopic absorption cross sections and k – infinities ¾ Diffusion and perturbation theory: ♦ used for estimation of reactivity of partially inserted control rod ¾ Reactor kinetics theory ♦ used for estimation of asymptotic reactor period

9 2. Function of control elements 2.1 Reactivity control

The safe and stable reactor operation can be guaranteed implementing a proper reactivity control. This task includes several aspects: (i) the mechanical systems, such as control elements, that provide adjustment of the power level and shut down the reactor; (ii) the processes of fuel depletion and fission product poisoning, that require positive counteraction; (iii) "built-in" a factor, such as burnable poisons, that can ensure a long – term control reactivity. We will introduce several definitions characterizing reactivity control, such as excess reactivity, shutdown margin, total and differential control rods worth.

2.1.1 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).

2.1.2 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." The fractional power level decrease achieved immediately after control insertion is given approximately by:

P β − ρ β 0 = 0 = (1) P1 β − ρ1 ρ sm + β

Where: P0 – power before control rod insertion; P1 – power after control rod insertion; β - effective fraction of the delayed neutrons. For example, for the BR2 reactor β eff = 0.0072 = 1$ .

P0 If ρ sm ≈ 4.5$ , then ≈ 0.18 that means that the power level will drop to 18% of its initial P1 value immediately following a scram.

10 2.1.3 Total control element worth

Total control element 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 (2)

For the BR2 reactor R0 = ρex (Sh = 960mm) + ρsm (Sh = 0mm ) .

2.1.4 Differential control element worth

The differential control element worth is defined as the reactivity change per unit control rod displacement: Δρi ΔShi []$ / mm . In Table II, we have listed typical values of the excess reactivity requirements, shutdown margins, and control element worth for the principal reactor types, including the BR2 reactor.

2.2 Control requirements

One can distinguish several different types of control requirements:

2.2.1 Scram control

The reactor control system must be capable of shutting the reactor down under any credible operating conditions. Elements used for such scram control purposes must be capable of inserting negative reactivity very rapidly and must operate with an extremely high degree of reliability.

2.2.2 Power regulation

Certain control elements are designed to compensate for small reactivity transients caused by changes in load demand, core temperature, and for power-level maneuvering. The regulating element is used to adjust the power level of the reactor and to compensate for short – term changes in reactivity.

2.2.3 Shim control

Shim control elements are designed to cover the excess reactivity necessary to compensate for long-term fuel depletion and fission product buildup, as well as to shape the power distribution in the core in order to obtain better thermal performance and more uniform fuel burn up. The reactivity worth of such elements must be quite large to ensure adjustments over long time periods.

2.3 Control rod types

There are several schemes used for introducing control absorption into a nuclear reactor core. One common method is to insert movable rods of absorbing material into the core. Such movable control elements not only can be used to adjust the core power, but because of their rapid response can also be used for scramming the reactor, as well as for shim and power shaping. Fixed absorbing materials are sometimes fabricated into the core with the intent that such absorption will gradually burn out along with the fuel. These burnable poisons are useful for extending the initial core lifetime of reactors A third very popular control mechanism in LWRs involves dissolving a poison such as boric acid in the coolant itself. Such a soluble poison provides a very uniformly distributed shim control which minimizes spatial power profile perturbations.

3. Overview of absorbing materials, used for nuclear control

Chemical elements with a sufficiently high capture cross section for neutrons include silver, indium and cadmium. Other elements that can be used include boron, cobalt, hafnium, gadolinium, and europium. Because these elements have different capture cross sections for neutrons of varying energies the compositions of the control rods must be designed for the neutron spectrum of the reactor it is supposed to control. Light water reactors (BWR, PWR) operate with "thermal" neutrons, breeder reactors with "fast" neutrons.

3.1 Cadmium – thermal absorber

Cadmium is a soft, 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. 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 X 1015 years) and 116Cd (two-neutrino double beta decay, half-life is 2.9 X 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 states with the most stable being 113mCd (t½ 14.1 years), 115mCd (t½ 44.6 days) and 117mCd (t½ 3.36 hours). The main isotopes of Cd are shown in Table III. The total cross sections of the cadmium isotopes are given in Fig.1. Cadmium has one dominant isotope - 113Cd, which has big thermal absorption cross section ~104-105 barns. All of the Cd – isotopes including 113Cd have resonance ( ~103- 104 barns) in the epithermal region: 10-5 MeV - 10-2 MeV.

3.2 Hafnium – thermal and resonant absorber

12 Hafnium is a shiny silvery, ductile metal that is corrosion resistant and chemically similar to zirconium. 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 shown in Table IV. The total cross sections of the different Hf isotopes are given in Fig. 2, 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 (10-6-10-2 MeV), which is characterized with many resonances ~103 – 105 barns, the thermal cross section varies as ~ 103 barns.

3.3 Boron (boron carbide).

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. 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.

3.4 Gadolinium (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 surface 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 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. 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 element also has 4 meta states with the most stable being 143mGd (t½ 110 seconds), 145mGd (t½ 85 seconds) and 141mGd (t½ 24.5 seconds). 13 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 V. The total cross sections of the Gd isotopes are shown in Fig.3, the Gd isotopes are characterized with high thermal cross section (155Gd, 157Gd~105 barns) and also epithermal, ~ 104 barns.

3.5 Europium (Eu2O3).

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. 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 150-Eu 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 seconds. This element also has 8 meta states, with the most stable being 150mEu (t½ 12.8 hours), 152m1Eu (t½ 9.3116 hours) and 152m2Eu (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 VI. The cross sections of Eu – isotopes are given in Fig. 4, all Eu – isotopes have very big thermal cross section (~ 105 barns) and also epithermal cross section (~ 104 barns).

3.6 Stellite.

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) 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. 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. 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.

14 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. The absorption cross sections of Cobalt, Chromium and Iron – isotopes are given in Fig.5, 6 and 7.

4. Nuclear Control in the BR2 reactor The control rods 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 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.

4.1 Shim Rods – safety rods 4.1.1 Design characteristics 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. A schematic drawing of the shim-safety control rod with a cobalt capsule is given at Fig. 8. The original shim-safety rod mechanical design requirements are realized as follows: - The rod drive is located above the reactor; it extends about 836 mm above the top cover (maximum length); - The control rod assembly fits into any standard reactor channel; - The rod-drive mechanism is submerged in the pool water; - The outside diameter of the rod thimble in the matrix is 83+ 0.2 mm. the outside diameter of the housing above the reactor top over is 117 mm; - Electrical cables connected to the control rod assembly are flexible; - The rod is free to drop by gravity and is not impeded by coolant flow, mechanism or seals; - No connections are provided between the rod assembly are flexible; - The guide tube and the structural wall of the rod thimble have holes to permit water to enter the rod for cooling purposes; - The rod thimble has shock absorbers; - An indication on the control desk shows that the rod travelled a distance of 250 mm after a “scram”, the indicator is operated by the rod being in this position; - Water flow aids the rod to drop in the event of a “scram”. In any event the flow does not oppose the drop; - The rod is attached to a driving mechanism, which can release quickly and is “fail-safe”; - A special sealing device is provided to prevent primary radioactive water leakage from the reactor vessel to the pool; - The position of the indication of the position is ±0,5 mm; - The maximum rod withdrawal speed is 10 cm/min; - There is a warning if leakage occurs at individual seals; 15 - The delay time between flux trip value and initiation of rod motion is less than 50 ms for a single rod. The mean delay time for the array is less than 40 ms; - The single rod drops so that it travels into the reactor a distance of 250 mm during less than 300 ms after scram, for all flow conditions; - The corresponding mean value for the array is ~ 240ms; - The total rod stroke was 960 mm until of 1978. Since the cycle 1/80 it has been reduced to 900 mm.

4.1.2 Shim-safety rods construction The control rods, which are used at present, are as described in this chapter. It was 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. Since end of 2005, studies for optimized new CR of the BR2 reactor have started. The neutron absorbing part can be latched to the holding and drive mechanism, and consists of the following parts: - At the upper end is the latching device, which permits coupling of the assembly to the release and drive mechanism; - At the lower end is a shock absorber for smooth deceleration of the movement at the end of its travel after a scram; - Just under the latching device, and just above the shock absorber are the guide pieces having each three rollers for guiding the assembly inside the rod thimble. The roller are disposed at 120°; - The cadmium section is screwed on the lower end of the upper guide piece. The cadmium section is an aluminium-cadmium-aluminium co extrusion cylinder. The cadmium is cold- welded inside the aluminum cylinder. Outer diameter of the aluminum cylinder: 64 mm Outer diameter of the cadmium: 60 mm Thickness of the cadmium: normally 5 mm ( min. 4 mm) Length of the cadmium (constant thickness) :min. 740 mm Overall length of the cadmium section: max. 870 mm. - The cobalt capsule is screwed on the lower end of the cadmium section. The cobalt particles are vacuum hot pressed between two aluminum layers, which are electron beam-welded at both ends. The cobalt capsule has a total length of 140 mm. the outside diameter is 64 mm. the capsule contains approximately 190g cobalt. After irradiation during 4 calendar years, the specific activity of the 60Co reaches about 250 Ci/g. on a ten cycle’s per year basis. - Between the cobalt capsule and the lower guide piece is an aluminum tubular jacket (outside Ø 64 mm) containing two concentric beryllium cylinders having the following dimensions:

Inner cylinder: Ø inside : 9,52 mm Ø outside : 31,75 mm Length : 965,2 mm

outer cylinder: Ø inside : 34,29 mm Ø outside : 54,5 mm Length : 965,2 mm 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. 16

4.2 Regulating Rods 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$. The regulating rod provides the following functions: - Automatic start-up of the reactor from 1 to 100% power; - Stabilization of the reactor power at a constant level; - 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.

- Outer diameter of the aluminum cylinder: 40,6 mm - Outer diameter of the cadmium: 39 mm - Thickness of the cadmium: 2,5 mm - Minimum length of the cadmium at constant thickness: 880 mm - Overall length of the cadmium-aluminum Cylinder: max. 930 mm

The neutron absorbing part moves inside an aluminum guide tube. A beryllium cylinder is fixed between the inside guide tube and the outer guide tube. The length of the beryllium cylinder equals the length of the beryllium matrix (914 mm).

The stroke of the cadmium (displacement between upper and lower mechanical limits) is 457,5 mm, corresponding to half the beryllium matrix length. Depending on the attachment of the rod to his drive mechanism, the rod stroke can be performed at the upper, median or lower part of the reactor core. In this way an adjustment for cadmium burn-up is possible.

4.3 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. 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 BOC and depleting a major portion of it before EOC, thereby compensating the fissile material consumption. Thus the application of 10B always provides more reactivity at any given 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. The basic parameters of the fuel types used during BR2 operation after 1997 are summarized in Table VII.

5. Analytical and experimental methods for determination of CR worth

17 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., start-up 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. However, an assumption to remove the spatial dependence of the flux will 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 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 in hour 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.

5.1 Reactor period 5.1.1 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) (3) v ∂t a f f r r where the fission source is given with: S f (r,t) =νΣ f φ(r,t) ; v – neutron velocity; ν - average number of fast neutrons, born in fission.

5.1.1.1 Time – dependent slab reactor

For the case of a simple bare slab reactor we can write Eq. (3) as:

1 ∂φ ∂ 2φ − D + Σ (x)φ(x,t) =νΣ φ(x,t) = S (x,t) (4) v ∂t ∂x 2 a f f

18 With initial condition:

φ(x,0) = φ0 (x) = φ0 (−x ) (5) and boundary conditions:

a~ a~ φ( ,t) = φ(− ,t) = 0 (6) 2 2

The solution of the partial differential Eq. (4) can be found using separation of variables in the form:

φ(x,t) =ψ (x) ⋅T (t) (7)

Substituting eq. (7) into Eq. (4) and dividing by ψ (x) ⋅T(t ) , we find:

1 dT v ⎡ d 2ψ ⎤ = ⎢D 2 + ()νΣ f − Σ a ψ (x)⎥ = const = −λ (8) T dt ψ ⎣ dx ⎦

The separation of variables given by Eq. (7) has reduced the original partial differential equation in two variables to two ordinary differential equations:

dT = −λT (t) (9) dt

d 2ψ λ D + ()νΣ − Σ ψ (x) = − ψ (x) (10) dx2 f a v

The solution of the time – dependent equation (9) is simple:

T(t) = T (0) ⋅ exp[− λt] (11)

Where T(0) is an initial value, which will be determined later. The solution of the space – dependent equation (10) can be found together with the boundary conditions, i.e.:

d 2ψ ⎛ λ ⎞ D + ⎜ +νΣ f − Σ a ⎟ψ (x) = 0 (12) dx 2 ⎝ v ⎠

With boundary conditions:

a~ a~ ψ ( ) =ψ (− ) = 0 (13) 2 2

For a homogeneous problem Eq. (12) can be written as:

d 2ψ + B 2ψ (x) = 0 (14) dx 2

19 Where B is an arbitrary parameter which has to be determined. The general solution of the homogeneous problem, Eq. (14) with boundary conditions Eq. 13, is seeking as an expansion in the set of normal modes (or Eigen functions), characterizing the geometry of interest.

ψ (x) = A1 cos Bx + A2 sin Bx (15)

a~ Ba~ Ba~ ψ (± ) = A cos + A sin (16) 2 1 2 2 2

Adding and subtracting Eqs. (15) and (16) we find that:

Ba~ A cos = 0 (17) 1 2

Ba~ A sin = 0 (18) 2 2

In order to satisfy conditions (17) and (18), we must determine the parameter B. There are many values of B, for which this will occur. If we choose A2=0, then:

nπ B = B ≡ , n=1,3,5,…… (19) n a~ will give the following solution:

nπx ψ (x) = A cos , n=1,3,5,…… (20) n n a~

The solution (20) satisfies both the differential Eq. (14) and the boundary conditions (15). If A1=0, then:

nπ B = B ≡ , n=2,4,6,…… (21) n a~ with solution:

nπx ψ (x) = A sin , n=2,4,6,…… (22) n n a~

Therefore, the values for B2 for which nontrivial solutions exist to the homogeneous problem are referred as:

2 2 ⎛ nπ ⎞ Eigen values: Bn = ⎜ ~ ⎟ , n=1,2,3,…… (23) ⎝ a ⎠

The corresponding solutions of the homogeneous problem are referred as Eigen functions of the problem:

20 ⎧ ⎛ nπx ⎞ ⎫ ⎪An cos⎜ ~ ⎟,n = 1,3,5,.... ⎪ ⎪ ⎝ a ⎠ ⎪ Eigen functions:ψ n (x) = ⎨ ⎬ (24) ⎪ ⎛ nπx ⎞ ⎪ An cos⎜ ~ ⎟,n = 2,4,6,.... ⎩⎪ ⎝ a ⎠ ⎭⎪

Another form of the eigenvalue problem can be found as:

H nψ n = λnψ n (25)

Where ψn is Eigen function corresponding to the eigenvalue λn. Comparing (25) with (14), one d 2 can easily identify: H = ,λ → −B 2 and ψ →ψ (x ). The first few Eigen functions for a dx 2 n n slab geometry are sketched in Fig. 9.

The Eigen functions are called also normal modes or natural harmonics of the system. The coefficients An are still undetermined and are in fact arbitrary. Now we can re-write Eq. (14) and Eq. (13) as:

d 2ψ n + B 2ψ (x) = 0 (26) dx 2 n

a~ a~ ψ ( ) =ψ (− ) = 0 (27) n 2 n 2 with symmetric solutions (we are only interested in symmetric solutions since φ0(x) is symmetric, see Eq. 5):

Eigen functions: ψ n (x) = cosB n x , n=1,3,5,…… (28)

2 2 ⎛ nπ ⎞ Eigen values: Bn = ⎜ ~ ⎟ , n=1,3,5,…… ⎝ a ⎠ (29)

Comparing Eq. (12) with Eq. (26), we find:

2 λ = vΣa + vDBn − vνΣf ≡ λn n=1,3,5,…… (30)

The values λn are known as time Eigen values of the equation since they characterize the time decay in Eq. (11). Therefore, the general solution of the more general Eq. (3) must be in the form:

φ(r,t) = ∑ An exp(− λnt)ψ n (r) (31) n i.e., the neutron flux is represented as superposition of spatial modes (Eigen values) weighted with en exponentially varying time dependence. The coefficients An can be determined using 21 orthogonality of the functions ψn(x), i.e. 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) = (32) ∫ m n ⎨~ ⎬ a~ a − ⎪ , if m = n⎪ 2 ⎩⎪ 2 ⎭⎪

The second property of the Eigen functions ψn(x) is that they form a complete set, i.e. one can find a "well-behaved" function φ(x) which can be represented as a linear combination of ψn(x), i.e.:

∞ φ(x) = ∑ Anψ n (x) (33) n=1 where the coefficients An are determined using the orthogonality (32) and hence:

a~ 2 2 A = dxφ (x)ψ (x) (34) n ~ ∫ 0 n a a~ − 2 For any symmetric initial distribution we have:

a~ 2 2 nπx A = dxφ (x)cos (35) 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 (36) ∑⎢ ~ ∫ 0 n ⎥ ()n n n a a~ odd ⎢ − ⎥ ⎣ 2 ⎦

nπ λ = vΣ + vDB2 − vνΣ , B = (37) n a n f n a~

Using separation of variables, we can write:

nπx φ(x,t) = ∑Tn (t)cos ~ (38) n a odd

dT n = −λ T (t) (39) dt n n

22 5.1.1.2 Long – time behavior

2 2 2 2 2 ⎛ nπ ⎞ One can order B1 < B3 < B5 < ...... Bn = ⎜ ~ ⎟ . Hence, the time Eigen values must similarly be ⎝ a ⎠ ordered such that λ1 < λ3 < λ5 < .....(see Eq. 37). 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 as t → ∞ (40)

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:

a~ 2 2 πx' A = dx'φ (x')cos (41) 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. 37). However, the same argument hold since −λ1> −λ3 > −λ5 > ……. Hence regardless of whether the flux grows or decays, it will approach a "persistent" of fundamental cosine distribution (see Fig. 10). 2 Thus, we define the value Bn characterizing the fundamental mode as geometric buckling:

2 2 ⎛ π ⎞ 2 B1 = ⎜ ~ ⎟ ≡ Bg (42) ⎝ 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 (43) 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.

5.1.1.3 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:

23 ∞ φ(x,t) = A1 exp()− λ1t cos B1 x + ∑ An exp ()− λnt cos Bn x (44) n=3 odd

Then the requirement for a time – independent flux is just that the fundamental Eigen value mode vanish:

2 λ1 = 0 = v(Σ a −νΣ f )+ vDB1 (45)

since then 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 (46)

2 2 We can re-write the criticality conditions (45) and (46) using the notation B1 = Bg :

νΣ − Σ f a ≡ B2 (47) D m 2 where Bm is material buckling since it depends only on the material composition of the core, and 2 Bg depends only on the core geometry. Hence, the criticality condition can be written concisely as:

2 2 Bm = Bg (48)

2 2 Thus, to achieve a critical reactor, we must either adjust the size Bg or the core composition Bm . We also note:

2 2 Bm > Bg ⇒ λ1 < 0 ⇒ supercritical 2 2 Bm = Bg ⇒ λ1 = 0 ⇒ critical (49) 2 2 Bm < Bg ⇒ λ1 > 0 ⇒ sub-critical

2 By increasing the core size we decrease Bg (see Eq. 42), but when increasing the concentration 2 of the fissile material, Σf and hence Bm , increase also. Both of these modifications would therefore tend to enhance core multiplication. We can write the time Eigen value (45) as:

⎛ νΣ / Σ ⎞ λ = vΣ 1+ L2 B 2 ⎜1− f a ⎟ (50) 1 a ()g ⎜ 2 2 ⎟ ⎝ 1+ L Bg ⎠

D Where: L = is the diffusion neutron length, (vΣ)−1 is the mean lifetime for a given neutron Σ a −1 – nuclear reaction to occur. Hence, (vΣ a ) is the mean lifetime of a neutron to absorption in an infinite medium (i.e. ignoring leakage). 24 The infinite multiplication factor is defined as:

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

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

Leakage rate = ∫∫d S ⋅ J = d 3r∇ ⋅ J = − ∫d 3rD∇ 2φ (52) SV V

Where we have used both Gauss's theorem and the diffusion approximation. (The Gauss's divergence theorem is:

d 3r∇ ⋅ J = dseˆ ⋅ J (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 = 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

(54)

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

1 PNL ≡ 2 2 (55) 1+ L Bg

⎛ 1 ⎞ 1 l = P ⎜ ⎟ = (56) NL ⎜ ⎟ 2 2 ⎝ vΣa ⎠ ()vΣ a ()1+ L Bg

Combining Eq. (51) with Eq. (55) we find the multiplication factor of finite medium:

νΣ f / Σa k =ηfP NL = 2 2 (57) 1+ L Bg

Then from Eq. (50), Eq. (56) and Eq. (57) the fundamental time Eigen value is just the inverse of the reactor period:

k −1 1 − λ = = (58) 1 l T

From the other side, if we recall Eq. (29), Eq. (30), Eq.(42) and Eq. (47) we can write:

2 2 λ1 = vD(Bg − Bm ) (59)

Then the various forms of the criticality condition are equivalent:

2 2 λ1 = 0 ⇔ Bg = Bm ⇔ k = 1 (60)

νΣ f / Σ a k = 2 2 = 1 (61) 1+ L Bg

Now let come back to Eq. (37) for which (see Eq. (58)): −λ1> −λ2 > …. and therefore for long times the flux approaches an asymptotic form and then we can re-write Eq. (31) as:

⎡⎛ k −1⎞ ⎤ φ(r,t) ≈ A1 exp()− λ1t ψ 1 (r) = A1 exp⎢⎜ ⎟t⎥ψ 1 (r) (62) ⎣⎝ l ⎠ ⎦

Where: the neutron mean lifetime in the reactor, l, is given with Eq. (56), the multiplication factor k is given with Eq. (57) and the geometry buckling is given with:

νΣ − Σ B 2 = f a (63) g D

Analogous to Eq. 7, the solution of the one-speed diffusion equation (3) can be written also in the form:

r r φ(r,t) = vn(t)ψ 1(r) (64)

where ψ 1 (r) is the fundamental mode or Eigen function of the Helmholtz equation:

2 2 r r ∇ ψ n + Bnψ n (r) = 0 ψ n (rs ) = 0 (65)

Substituting (64) into (65) we arrive to the following equation for the neutron density:

dn k −1 = n(t) (66) dt l where n(t) is the total number of neutrons in the reactor at time t. The equation (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.:

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

26 characterized by time constant

1 l T = = []sec (68) ()1 n ()dn dt k −1 which 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 (69) T

It is seen from (67) that the neutron flux (density) 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.

5.1.2 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. (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 photo neutrons, released after threshold photonuclear reactions of beryllium. All 7 groups of delayed neutrons are presented in Table VIII. All delayed neutrons can be treated as one single averaged group with total fraction β = 0.0072 and an average decay constant λ ≈ 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. Equation (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), 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. 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 (70) v ∂t i=1

27 ∂C i = −λ C (r,t) + βνΣ φ(r,t) , i=1,…,7 (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) (72) i=1

The solution of equations (70) and (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)ψ1(r ) r r Ci (r,t) = Ci (t)ψ1(r ) (73)

r where ψ1(r ) is the fundamental mode of the Helmholtz equation (65) and Ci is the concentration of the precursors of the delayed neutrons emitted of group i. The substitution of Eqs. (73) into the system differential equations (70) and (71) gives a set of ordinary differential equations for n(t) and Ci(t):

dn k(1− β ) −1 7 = n(t) + ∑λiCi (74) dt l i=1 dC k i = β n(t) − λ C (t) , i=1,2,...,7 (75) dt i l i i Equations (74) and (75) are known as the point reactor kinetic equations and represent generalization of the Eq. (66) by including the effect of the delayed neutrons. The equations (74) and (75) can be re-written in the following form:

dn ⎡ ρ(t) − β ⎤ 7 = ⎢ ⎥n(t) + ∑λiCi (t) (76) dt ⎣ Λ ⎦ i=1

dC β i = i n(t) − λ C (t) i=1,…,7 (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 (76) – (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 flux 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 the delayed 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 (76) – (77). Therefore complete solutions of the above equations 28 should be considered separately for each specific type of ρ(t) disturbance. The equations (76) – 7 (77) can be written for one group delayed neutrons, represented by β = ∑ β i and average values i=1 λ and C:

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

dC β = n(t) − λC(t) (79) dt Λ

The solutions of these equations is a summation of two exponential terms in the form:

dn(t) b − c c − a = e at + ebt (80) dt(0) b − a b − a

ρ(t)λ ρ(t) − β ρ(t) Where: a = ; b = ; c = . The product terms containing λ (λ~0.1 λΛ + β − ρ(t) Λ Λ sec-1, Λ=l/k~10-4 sec) can be neglected in most of the practical cases and then the equation (80) can be approximately written as:

dn(t) β ρ(t) = exp[]λρ(t) ()β − ρ(t) ⋅t − exp{}− []()β − ρ(t) / Λ ⋅t (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 ] (82) dt(0) BR2

The increase of the neutron density with time according to (82) is given at Fig. 11. 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. (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. (81) is:

β − ρ T del = = 40sec (83) λρ

29 Λ while for the prompt neutrons only: T prompt = = 0.069sec. Equation (83) solved for the ρ reactivity gives the following relationship:

β ρ = (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 (85) Tkeff i=1 1+ λiT

The stable periods are different for positive and negative reactivity changes, but as the periods become larger, the magnitude of the reactivity for both positive and negative periods approaches β λT . Eq. (85) is known as in hour formula. The in hour is defined as the reactivity which will make the reactor stable period equal to one hour.

5.1.3 Solution of kinetic equations 5.1.3.1 Solution of kinetic equations for step – function input in reactivity ρ The solutions of the Eqs. (77) and (78) for step – function are well known [1-2]. 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[P j t] (86) i=1

Where the first exponent P1 has the same sign as ρ (the input disturbance). All the other six exponents are negative. Equations (76) and (77) are a family of differential equations with constant coefficients, and there is a definite relationship among the values of Aj, Pj and ρ. 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 constant source of β strength: n . The initial response of the reactor is described by: Λ 0

dn ρ − β β = ⋅ n + ⋅ n (87) dt Λ Λ 0

The solution of Eq. (87) is:

n(t) β ρ ⎡ ρ − β ⎤ = − + ⋅ exp⎢ ⋅t⎥ (88) n0 ρ − β ρ − β ⎣ Λ ⎦

30 − β The neutron density in Eq. 88 approaches the asymptotic value of: . ρ − β

5.1.3.2 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. 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. (76) takes the same form as approximate equation (87):

dn ρ − β β = ⋅ n + ⋅ n (89) dt Λ Λ 0

Now: ρ = A⋅t (90)

dn A⋅t − β β Then: − ⋅ n = ⋅ n (91) dt Λ Λ 0

The solution for Eq. (91) is of the form:

n = [] exp () ∫ − Pdt ⋅ {} ∫ [] exp () ∫ Pdt Qdt + C (92)

− A⋅t + β β ⋅ n Where: P = ; Q = 0 (93) Λ Λ

n β At 2 − 2βt ⎛ t ⎡− At 2 − 2βt ⎤ C ⎞ ()⎜ ( ) ⎟ Then: = ⋅ exp ⎜ ∫exp⎢ ⎥dt + ⎟ (94) 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. 94 may be written in the form:

⎛ β 2 ⎞ t exp⎜ ⎟ ⋅ exp − αt + γ 2 dt (95) ⎜ 2 AΛ ⎟ ∫ [ ()] ⎝ e ⎠ 0

A β Where: α = and γ = − (96) 2Λ 2AΛ

Substituting αt + γ = μ , we have:

31 t 1 αtt++γ π ⎛ 2 α γ 2γ t ⎞ 2 2 ⎜ 2 2 ⎟ ∫∫exp[]− ()αt + γ dt = exp()− μ dμ = ⎜ ∫∫exp()− μ dμ − exp()− μ dμ ⎟ (97) 0 α γ 2α ⎝ π 00π ⎠

The values of the probability integral are available in mathematical tables [3-4]. 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 ⎥ (98) n0 Λ ⎣⎢ 2α ⎝ π 0 π 0 ⎠ β ⎦⎥

5.1.3.3 Graphical method for the solution of reactor equations with time – varying inputs The analytical methods presented in §5.1.3.1, §5.1.3.2 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 [4] which simplifies the solution into a routine drafting operation and provides good accuracy. The approach is by analogy to the solution of ideal transmission – line equations. Consider the circuit diagram of Fig. 12 which has the familiar differential equation:

di u = L + R(t) ⋅i (99) 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 (100) v

From which the construction angle γ (see Fig. 13) is found as:

l L L tanγ = ±z = ± = ± = ± (101) 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.

5.1.4 Period measurements 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- 32 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.

5.1.4.1 Subcritical level operation Let us insert in a subcritical medium a source of neutrons. 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 ) (102)

Where: n=number of neutrons in multiplying medium; n0=number of neutrons originally present in medium. Then:

n 1− k m = (103) n0 1− k

At the end of sufficiently long interval of time for k<1, Eq. 103 generates into:

n 1 = (104) n0 1− k

This ratio is known as the subcritical multiplication factor, and all reactors exhibit this effect. Kinetically, when a source is suddenly inserted into a subcritical multiplying medium containing no initial neutron population, the medium responds according to the equation:

SΛ ⎧ ⎡ 1− k ⎤ ⎫ n(t) = ⎨1− exp⎢− ⎥ ⋅t⎬ (105) 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. 104. The period that results from this very slow pulling rate can be obtained from the definition of period of Eq. 68:

1− k T = (106) 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.

33 5.1.4.2 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 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 from 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⋅ (107) 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 14 shows such a plot for bringing the reactor to criticality. 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 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.

5.2 Estimation of control rod worth Until now we have been talking about linear rates of change of reactivity. Practically, linear rates exist only under short – term conditions. In general, the control rod reactivity worth as a function of time is nonlinear. The effectiveness of a control rod depends upon its position in the core and the value of the neutron flux at that point of insertion of the rod. An approximate formula can be used that the effectiveness of a control rod varies as the square of the neutron flux in which it is placed. Therefore, a rod is most effective at the center of a reactor and least effective in the periphery. For approximate calculations the worth of a rod, partially inserted to a level z, is proportional to ∫sin 2 zdz .

5.2.1 Estimation of reactor period

The problem is now 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 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 § 5.1.3. However this method is 34 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 startup 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.

5.2.1.1 Upper bound of reactor period The upper bound will be handled in the following manner: Let first re-write the familiar pile kinetic equation:

dn ρ − β 7 = ⋅ n + ∑λiCi + S (108) 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 ⇒ < (109) dt Λ dn dt ρ − β

If the period T is defined as:

n T = (110) dn dt then:

Λ T < (111) ρ − β for ρ ≥ β . For ρ < β the inequality of Eq. 111 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.

5.2.1.2 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 (112) γ

35 as the subcritical relationship. For reactors subcritical by a large amount of negative reactivity it has been shown that Eq. 112 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. 112 represents the lower bound of the reactor period. For simplicity consider the pile kinetic equations (78) and (79) for the lumped-delayed- emitter one group

dn ρ − β = ⋅ n(t) + λ ⋅C(t) + S (113) dt Λ

dC β = ⋅ n(t) − λC(t) (114) dt Λ

Where: n(t) [n/cm3] is the neutron density in function of time 7 β = ∑ β i = 0. 0072 is the effective fraction of all delayed neutrons, i=1 including the photo neutrons C(t) is the density of precursors of delayed neutrons S [n/cm3/s] 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 (115) dt Λ

dC β = ⋅ n − λC = 0 (116) dt Λ

Substituting Eq. (116) into Eq. (115):

dn ρ dC ρ = ⋅ n − + S = ⋅ n + S = 0 (117) dt Λ dt Λ

The equilibrium neutron concentration from Eq. (117) becomes:

SΛ n = − (118) eq ρ

36 Eq. (118) 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,

⎛ 1 1 ⎞ Δn = n − n(0) = −SΛ⎜ − ⎟ (119) ⎝ ρ0 + Δρ ρ0 ⎠

SΛΔρ Δn = (120) ρ0 ()ρ0 + Δρ

The time Δt required for ρ to change by any amount Δρ is:

Δρ Δt = (121) 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:

Δn SΛΔρ γ = ⋅ (122) Δt ρ0 ()ρ0 + Δρ Δρ

Δn SΛγ = (123) Δt ρ0 ()ρ0 + Δρ

If one obtains the time derivative of n in the following manner:

dn Δn SΛγ = lim < (124) dt Δt ()ρ 2 the period lower bound becomes:

n SΛ (ρ)2 ρ T = > − = − (125) 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.

5.2.2 Estimation of differential control rods worth The differential control rod worth is determined from the measured reactor period. The calculation procedure is as follows: a. Insertion of a positive reactivity by withdrawal of the CR; b. Measurement of the time for which the neutron density increases ~ 2 times for a given step and ramp of reactivity change rate Δρ (see Eq. 82); 37 c. Determination of the reactivity values using the in hour Eq. (85); d. Determination of the differential CR worth Δρ / Δh[$/ mm] for a given critical position Sh of the CR.

5.2.3 Estimation of total control rods worth 5.2.3.1 Analytical method The insertion of a control rod in a reactor changes its multiplication factor in two ways – first the rod absorbs neutrons, and second – the rod distorts the flux in such a way that the leakage of neutrons from the system is increased. The curvature or buckling of the flux is greater when the rod is present. The gradient of the flux at the surface of the reactor is also greater, and also is the leakage current. We shall try to derive the rod worth using analytical method. For this purpose let us consider a cylindrical rod of radius a inserted along the axis of a cylindrical reactor of radius R. We assume that the reactor region is localized for r: a ≤ r ≤ R . Using the results from the one-group diffusion theory we can write the following formula (see Eq. 57) for the multiplication factor of the reactor before insertion of the control rod:

νΣ Σ k k = f a = ∞ = 1 (126) 0 2 2 2 2 1+B 0 M 1+ B0 M

The multiplication factor of the reactor after insertion of the rod is:

νΣ Σ k k = f a = ∞ (127) 1+ B2M 2 1+ B2M 2

2 2 Where B0 is the initial buckling of the reactor before insertion of the rod and B − the buckling 1 after the rod insertion; M 2 = L2 +τ is the migration area (τ = < r 2 > is Fermi age, which is 6 equal to 1/6 the average crow – flight distance from the point where a neutron enters a system with zero age to the point where it acquires the age τ). The physical properties of the reactor do 2 2 not change due to the rod motion, therefore k∞ and M are constant. However B will change because of the motion of the rod. Then for the control rod worth we obtain:

2 2 2 1− k k0 − k (B − B0 )M ρ = ρ = = = 2 2 (128) k k 1+ B0 M

If the reactivity equivalent of a rod is small, then B ≈ B0 and Eq. (128) can be simplified:

2 2 ()B − B0 (B + B0 )M 2M B0ΔB ρ = 2 2 ≈ 2 2 (129) 1+ B0 M 1+ B0 M

38 Consider now the case in which the rod is withdrawn slowly to compensate for the reactivity changes. Usually the worth of a such rod is defined in terms of the changes in the properties of the system for which it can compensate. The rod worth can be written as:

k − ()k ρ = ∞ ∞ 0 (130) ()k∞ 0 or:

k − ()k ρ = ∞ ∞ 0 (131) ()k∞

Where k∞ and ()k∞ 0 are the infinite multiplication factors of the core with the rod inserted and with the rod withdrawn, respectively. The Eqs. (130) and (131) are practically equivalent, since the difference between k∞ and ()k∞ 0 is usually small. Denoting the initial and final migration 2 2 areas by M and M 0 , then we obtain:

2 2 k∞ = 1+ B M (132)

2 2 ()k∞ 0 = 1+ B0 M 0 (133)

Then from Eqs. (130), (132) and (133) the rod worth will be:

2 2 2 2 B M − B0 M 0 ρ = 2 2 (134) 1+ B0 M 0

2 2 If the changes in the system compensated by one rod are small so that M ≈ M 0 then Eq. (134) reduces to Eq. 128. Thus in this case the two definitions of the rod worth are equivalent. Furthermore, if the rod worth is small Eq. (134) becomes Eq. (129). Therefore, it is usual practice to use either Eq. (128) or (129) for one-group calculations of rod worth regardless of the way in which the rod is used for control purposes. The one – group theory can not be used for an accurate determination of the rod worth because the in the one – group model it is assumed that the rod absorbs neutrons of all energies at the same rate, whereas the fast neutrons are absorbed at the far lesser extent than thermal neutrons. This can be taken into account considering two – or multigroup diffusion model. However, the one – group model can be used for some rough estimate of the control rod worth. Let us consider a bare homogeneous thermal reactor of extrapolated radius R and height H with a single cylindrical rod of radius a inserted in the center. In view of Eqs. (128) or (129) the rod worth is determined by the buckling of the system, that is the first Eigen values of the reactor equation, with the rod inserted or withdrawn. In both cases the flux satisfies the reactor equation:

1 d dφ r T + B2φ = 0 (135) r dr dr T

With boundary condition: φT (R) = 0 (136) Equation (135) is ordinary Bessel equation whose general solution is: 39

φT = AJ0 (Br) + CY0 (Br ) (137)

The function Y0 (Br ) is singular if r = 0 and so C=0. Then the flux is given by:

φT = AJ0 (Br ) (138)

The function J0 (x ) has an infinite number of zeros at x = x1, x2 , x3 ,..... that is

J0 (x1) = J0 (x2 ) =J 0(x3 ) = ...... = 0 . The boundary condition at r = R will therefore be satisfied provided that B takes on the values of Bn R = xn , where xn is any of the zeros. Thus the reactor equation has the following solutions:

⎛ xnr ⎞ φTn = An J0 ⎜ ⎟ (139) ⎝ R ⎠

The smallest value of xn is x1 = 2.405 . Therefore the buckling of the infinite cylinder is:

2 ⎛ 2.405 ⎞ B2 = ⎜ ⎟ (140) ⎝ R ⎠

And the flux in the critical reactor is:

⎛ 2.405r ⎞ φT = AJ0 ⎜ ⎟ (141) ⎝ R ⎠

The constant A is determined using the formula for the integral of the Bessel functions, i.e.

∫ J1(z)dz = −J0 (z) , ∫ J0 (z)zdz = zJ1(z) , so that:

2.405P 0.738P A = 2 = 2 (142) 2πR γ Σ f ()2.405 R γ Σ f

Where P is the power per unit length of the reactor and it is defined as:

P = γ Σ f ∫φT (r)dV (143) V

Where γ is the recoverable energy per fission and Σ f is the thermal average macroscopic fission cross section. For a finite cylinder of extrapolated height H and extrapolated radius R the flux depends on both r and z and the reactor equation is:

1 ∂ ∂φ ∂2φ r T + T + B2φ = 0 (144) r ∂r ∂r ∂z 2 T 40 and boundary conditions (for origin of the coordinate system in the center of the cylinder):

⎛ H ⎞ φT (R, z) = φT ⎜r,± ⎟ = 0 (145) ⎝ 2 ⎠

Using the separation of variables, then:

φT = X (r)Z(z) (146)

Substituting Eq. (146) into Eq. (144) and dividing by φT we arrive at the following equation:

1 d dX 1 d 2Z r + + B2 = 0 (147) Xr dr dr Z dz 2

The first and the second terms depend only on r and z, respectively, and these can be placed equal to constants:

1 d 2Z = −α 2 (148) Z dz2

1 d dX r = −β 2 (149) Xr dr dr

Where: B2 = α 2 + β 2 (150)

Taking into account the boundary conditions at z = ± H 2 , the separation constant α in Eq. (148) will be: mπ α = , m = 1,3,5,...... (151) m H

And Z(z) has the solutions:

⎛ mπz ⎞ Zm (z) = Am cos⎜ ⎟ (152) ⎝ H ⎠

The constant β in Eq. (149) for the boundary conditions at z = ± H 2 is:

x β = n , (153) n R

Where xn is the nth zero of J0 (x ) and X (r) is:

⎛ xnr ⎞ X (r) = An J0 ⎜ ⎟ (154) ⎝ R ⎠

For the lowest values of m and n the buckling is:

2 2 ⎛ 2.405 ⎞ ⎛ π ⎞ B2 = ⎜ ⎟ + ⎜ ⎟ (155) ⎝ R ⎠ ⎝ H ⎠ and the flux in the critical system is:

⎛ 2.405r ⎞ ⎛ πz ⎞ φT (r, z) = AJ0 ⎜ ⎟cos⎜ ⎟ (156) ⎝ R ⎠ ⎝ H ⎠

Where the constant A is found to be:

2.405πP 3.63P A = = (157) 4Vγ Σ f J1(2.405) Vγ Σ f

The equation (156), describing the flux with the rod out of the system is:

⎛ 2.405r ⎞ ⎛ πz ⎞ φT 0 (r, z) = AJ0 ⎜ ⎟cos⎜ ⎟ (158) ⎝ R ⎠ ⎝ H ⎠

And the buckling is:

2 2 2 ⎛ 2.405 ⎞ ⎛ π ⎞ B0 = ⎜ ⎟ + ⎜ ⎟ (159) ⎝ R ⎠ ⎝ H ⎠ 2 2 ⎛ 2.405 ⎞ Where: α0 = ⎜ ⎟ is the radial buckling, and then Eq. (159) becomes: ⎝ R ⎠ 2 2 2 ⎛ π ⎞ B0 = α0 + ⎜ ⎟ (160) ⎝ H ⎠

The criticality problem with the rod inserted is complicated by the fact that the rod is a strong absorber of neutrons and the diffusion theory is not applicable in the vicinity of the rod. This difficulty can be avoided by simply considering the rod to be a region external to the reactor. Diffusion theory then can be applied in regions up to the edge of the rod by using an appropriate boundary condition at the rod-reactor interface. This method is known as the method of Amouyal, Benoist and Horowitz. Thus, the boundary condition at the rod surface is then:

1 dφ 1 T = (161) dr d φT r =a where a is the physical radius of the rod, and d is the linear extrapolation distance, which in general depends on the radius of the rod, scattering properties and material of the rod. With the rod inserted in the reactor there is no reason to exclude solutions to the reactor equation which are singular at r = 0 , since the region occupied by the rod is considered to be external to the reactor. With the rod present, the flux may then be assumed of the form:

42 ⎡ J0 (αR) ⎤ ⎛ πz ⎞ φT (r) = A⎢J0 (αr) − Y0 (αr)⎥cos⎜ ⎟ (162) ⎣ Y0 (αR) ⎦ ⎝ H ⎠

2 ⎛ π ⎞ Where: B 2 = α 2 + ⎜ ⎟ (163) ⎝ H ⎠

This function satisfies the reactor equation and also vanishes for r = R and z = ± H 2 , that is at the surface of the reactor. Now, applying the boundary condition at the rod-reactor interface, Eq. (161) gives:

α J ' (αa) − J (αR)Y ' (αa) Y (αR) 1 [ 0 0 0 0 ] = (164) J0 (αa) − J0 (αR)Y0 (αa) Y 0(αR) d

' ' We identify that J0 (αa) = −J1(αa ) and Y0 (αa) = −Y1(αa ) , then eq. (164) becomes:

α J (αa) − J (αR)Y (αa) Y (αR) 1 − [ 1 0 1 0 ] = (165) J0 (αa) − J0 (αR)Y0 (αa) Y 0(αR) d

Equation (165) is the critical condition for the reactor with the rod inserted. The smallest positive value of α, which satisfies this equitation, that is, the first Eigen value, must next be determined by numerical or graphical methods. Once the value of α is found, B2 can be computed from Eq. (163), and the rod worth ρ from Eqs. (128) or (129). We note, that it is necessary to solve the critical equation, because this equation determines the buckling, i.e. the first Eigen value of the reactor with the rod present. As it was mentioned earlier (see Eqs. 126 and 127) this reactor may not actually be critical. If the rod and its worth are both small, an approximate solution to Eq. (165) can be found. According to the formulas when αa << 1, then:

J0 (αa) ≈ 1, J1(αa) ≈ 0 (166)

2 ⎡ ⎛ 1 ⎞⎤ 2 ⎡ R ⎤ ⎜ ⎟ ⎛ ⎞ Y0 (αa) ≈ − ⎢0.116 + ln⎜ ⎟⎥ = − ⎢0.116 + ln⎜ ⎟⎥ (167) π ⎣ ⎝α0a ⎠⎦ π ⎣ ⎝ 2.405a ⎠⎦

2 Y (αa) ≈ − (168) 1 παa

Let α = α0 + Δα , then the Bessel functions evaluated at r = R can be expanded as:

' J0 (αR) = J0 (α0R + RΔα) ≈ J0 (α0R) + J0 (α0 R)RΔα = J0 (α0R) − J1(α0R)RΔα (169)

However: J0 (α0R) = J0 (2.405) = 0 and J1(α0R) = J1(2.405) = 0. 519 (170)

And then: J0 (αR) ≈ −0.519RΔα , and Y0 (αR) ≈ Y0 (α0 R) = 0. 510 (171)

43 Substituting Eqs. (166) to (171) into Eq. (165), and solve for Δα , we arrived at the following expression:

−1 1.54 ⎡ ⎛ R ⎞ d ⎤ Δα = ⎢0.116 + ln⎜ ⎟ + ⎥ (172) R ⎣ ⎝ 2.405a ⎠ a ⎦ From Eq. (163) we have:

α0Δα ≈ B0ΔB (173)

And then from Eqs. (172) and (129) we obtain:

−1 7.43M 2 ⎡ ⎛ R ⎞ d ⎤ ρ = 2 2 2 ⎢0.116 + ln⎜ ⎟ + ⎥ (174) ()1+ B0 M R ⎣ ⎝ 2.405a ⎠ a ⎦

2 where B0 is given by Eq. (160). The equation (174) describes the reactivity of a small central cylindrical rod according to the modified one-group diffusion theory. The application of the perturbation theory for estimation of the effect of a localized control absorber on core multiplication 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 [5] is used in the technique for determination of control rods worth at the BR2 reactor.

5.2.3.2 One-group perturbation theory Let us consider a critical reactor described by one-group theory. The flux is then given by the equation:

div D grad φ + ()νΣ f − Σa φ = 0 (175)

where Σ f and Σa are the macroscopic fission and absorption cross sections. The quantities Σ f ,

Σa and D may all be functions of position. The equation (175) can be written as:

Mφ = 0 (176) where M is the operator:

M = divD grad +νΣ f − Σa (177)

' ' Let us imagine that for some reason Σ f and/or Σa are changed, so that Σ f → Σ f and Σa → Σa . Temporarily we assume that D is not changed. Then we can write the following equations:

Σ' f = Σ f + δΣ f (178)

44 Σ'a = Σa + δΣa (179)

As result of these perturbations the reactor becomes either subcritical or supercritical. However, the system can be returned to critical if imagine that we may change the value of ν to an appropriate new value ν'. Then the flux in the perturbed critical reactor satisfies the equation:

div D grad φ'+()ν 'Σ' f −Σ'a φ'= 0 (180) which can be written as:

M 'φ'= 0 (181) where M' is the operator:

M '= div D grad +ν 'Σ' f −Σ'a (182)

Inserting Eqs. (178) and (179) into Eq. (182) and denoting ν '=ν + Δν , M' becomes:

M '= div D grad + ()ν + Δν ()Σ f + δΣ f − ()Σa + δΣa = M +νδΣ f + ΔνΣ f + ΔνδΣ f − δΣa (183)

Perturbation theory can be used only with small changes in a reactor. Therefore, the term ΔνδΣ f in Eq. (183) can be ignored (product of two small quantities). Then we arrive to the following reduced equation:

M '= M +νδΣ f + ΔνΣ f − δΣa (184) which can be written in the form:

M '= M + P (185)

where: P =νδΣ f + ΔνΣ f − δΣa (186) is the perturbation operator. Then Eq. (181) can be written as:

()M + P φ'= 0 (187)

Ignoring for a moment that the operator M is self-adjoint (i.e. M + = M ), the equation adjoint to the Eq. (176) is then:

M +φ = 0 (188)

Then multiplying Eq. (187) by ψ and Eq. (188) by φ', subtracting and integrating over the volume of the reactor we obtain:

45 ∫∫ψ ()M + P φ'dV − φ'M +ψdV = 0 (189) VV or:

∫∫(ψMφ'−φ'M +ψ )dV + ψPφ'dV = 0 (190) VV

From the definition of the adjoint operator, i.e. if the one-group operator is given with the equation: M = div D grad + F (191) where D and F are functions of positions within a reactor volume V, then the adjoint of M is defined as:

∫∫uMvdV = vM+udV (192) VV where u and v vanish on the surface of V. Therefore from the definition of the adjoint operator it follows that the first integral in Eq. (190) vanishes, so that:

∫ψPφ'dV = 0 (193) V

From Eqs. (186) and (193) we obtain:

ψ (νδΣ − δΣ )φ'dV Δν ∫ f a ρ = − = V (194) ν ν ∫ψΣ f φ'dV V

Eq. (194) cannot be solved if we do not know the perturbed flux φ'. However, if the perturbation of the reactor is small, φ' will not differ significantly from φ. Then Eq. (194) becomes:

∫ψ (νδΣ f − δΣa )φdV ρ = V (195) ν ∫ψΣ f φdV V

Now we recall that the one-group operator is self-adjoint so that ψ is proportional to φ. Then Eq. (195) reduces to:

2 ∫(νδΣ f − δΣa )φ dV V ρ = 2 (196) ν ∫Σ f φ dV V

Equation (196) shows that the effect of perturbation in Σ f and Σa is obtained by weighting the perturbation by the square of the flux. As an example, suppose that a small absorber of volume 46 Vp and absorption cross section Σap is inserted into the reactor at the point r0 . This perturbation can be represented approximately by the function:

δΣa = ΣapVpδ (r − r0 ) (197)

where δ (r − r0 ) is the Dirac delta function. There is no perturbation in Σ f , so that δΣ f = 0 . Inserting Eq. (197) into Eq. (196) we obtain:

2 ∫δΣ aφ dV 2 V Σ apV pφ (r0 )dV ρ = − 2 = − 2 (198) ν ∫ Σ f φ dV ν ∫ Σ f φ dV V V

Thus, the effect on reactivity of placing the absorber at r0 is weighted by the square of the flux at that point. Also, it is evident from Eq. (198) that inserting an absorber into a critical reactor introduces negative reactivity, i.e. the reactor becomes subcritical.

Up to now only changes in Σ f and Σa were considered. Let us now suppose that D is changed while Σ f and Σa remain constant. Then:

D → D'= D + δD (199)

And the operator for the perturbed system is:

M '= M + P (200)

Now the perturbation operator is given by:

P = divδD grad + ΔνΣ f (201)

Proceeding in the same manner as before gives:

∫ψPφ'dV ≈∫ψPφdV = 0 (202) V V

Inserting Eq. (201) and noting that the operators are self-adjoint and therefore ψ is proportional to φ, the reactivity is found to be:

∫φ div δD grad φ dV Δν V ρ = − = 2 (203) ν ν ∫Σ f φ dV V

In order to determine the adjoint of the operator D it is convenient to introduce the following vector identity as:

w div W = div wW − W ⋅ grad w (204) 47

Where: w and W are scalar and vector functions, respectively. Then

⎛ ⎞ ∫u divD grad vdV = ∫div ⎜uD grad v⎟dV − ∫ D grad v ⋅ grad udV (205) V V ⎝ ⎠ V

From the divergence theorem, we have:

⎛ ⎞ ∫ div ⎜uD grad v⎟dV = ∫uD grad v⋅ ndA = 0 (206) V ⎝ ⎠ A since u = 0 on the surface of the reactor, so that:

∫u divD grad vdV = −∫ D grad u ⋅grad vdV (207) V V

Using the above identity (this is equivalent of integrating by parts twice, as done earlier), it is seen that:

⎛ ⎞ ∫∫D grad u ⋅grad vdV = div⎜ vD grad u⎟dV − ∫ v div D grad udV = − ∫v div D grad udV VV⎝ ⎠ V V

(208) Substituting Eq. (208) into Eq. (207) gives:

∫∫u div D grad vdV = v div D grad udV (209) VV which shows that the operator in Eq. (191) is self-adjoint. In fact, all one-group diffusion operators are self-adjoint. Therefore the expression in Eq. (203) can be written in a better form using the identity given in Eq. (204). Thus

⎛ ⎞ 2 ∫φ div δD grad φdV = ∫∫div⎜φ δD grad φ ⎟dV − δD ()∇φ dV (210) V VV⎝ ⎠

From the divergence theorem the first term on the right is:

⎛ ⎞ ∫∫div⎜φ δD grad φ ⎟dV = φ δD grad φ ⋅n dA = 0 (211) VA⎝ ⎠

Since φ vanishes on the surface. Then Eq. (203) becomes:

48 ∫ δD ()∇φ 2 dV V ρ = − 2 (212) ν∫Σ f φ dV V

So, it is seen that the changes in D are weighted by (∇φ)2 rather than by ()φ 2 as in the cases for

Σ f and Σa . An increase in D, i.e. a positive δD leads to a negative value of ρ. Physically, this is due to the fact that the neutron current increases with increasing D. A positive δD thus results in additional leakage of neutrons from the reactor, causing the originally critical reactor to become subcritical.

If D, Σ f and Σa are changed simultaneously, the total reactivity is simply the sum of the reactivates given by Eqs. (196) and (212). Then ρ is:

2 2 ∫[(νδΣ f − δΣa )φ − δD(∇φ) ]dV V ρ = − 2 (213) ν∫Σ f φ dV V

For a two – region reactor consisting of a core and reflector, Σ f and δΣ f are both zero in the reflector so that:

1 ⎪⎧ 2 2 ⎪⎫ ρ = νδΣ − δΣ φ 2 − δD ∇φ dV − δΣ φ 2 + δD ∇φ dV (214) 2 ⎨ ∫∫[]()f a () []a () ⎬ ν ∫Σ f φ dV ⎩⎪core reflector ⎭⎪ core

The perturbation theory cannot predict the peaking of the thermal flux in the reflector, therefore the effect of the placing an absorber in the reflector cannot be determined accurately with one- group perturbation theory.

5.2.3.3 Physical interpretation of the adjoint flux

Let us consider the reactivity introduced into an originally critical reactor by the insertion of a small absorber at the point r0 . If the absorber is assumed not to scatter neutrons and has volume

Vp and macroscopic cross section Σap then the perturbation is given by the formula:

δΣa = ΣapVpδ (r − r0 ) (215)

and according with Eq. (195) the reactivity is (notice, that δΣ f = 0 ):

Σ V ψ (r )φ(r ) ρ = − ap p 0 0 (216) ν ∫ψΣ f φdV V

49 The dominator in Eq. (216) is a constant which depends upon the normalization of the functions ψ and φ, but is independent of the nature of the perturbation. Denoting this term by the symbol −1 C and solving Eq. (216) for ψ (r0 ) , the result is:

ρ ψ (r0 )= − (217) CΣapVpφ()r0

The quantity ΣapVpφ(r0 ) is equal to the total number of neutrons absorbed per second in the absorber, and it follows from Eq. (238) that the one-group adjoint function ψ (r0 ) is proportional to the negative change in reactivity of the reactor per neutron absorbed per second at r0 . If for example, the absorber is placed at a point where ψ (r0 ) is small, the reactivity will also be small.

Thus, ψ (r0 ) is a measure for the importance of the point r0 with respect to the reactivity introduced by an absorber located at that point. Therefore the function r0 is called the importance function. The perturbation theory can not be used to determine the rod worth, unless the rod is a weak absorber of neutrons. With strongly absorbing rods, the worth is computed by analytical methods as described in § 5.2.3.1. However, if the rod is partially inserted, the problem becomes too complicated to be handled by ordinary analytical methods. In such cases, perturbation theory can be used for estimation of the worth of the partially inserted rod relative to its worth when fully inserted. We again limit our considerations to a central rod of radius a located in a bare cylindrical reactor of extrapolated radius R and height H. Also, we assume that the rod absorbs, but does not scatter neutrons. For simplicity, one-group diffusion theory is used. It is convenient to take the coordinate system at the top of the cylinder. Then with the rod inserted at distance z, the perturbation is:

⎧ ⎪Σ ap , 0 ≤ z ≤ H, 0 ≤ r ≤ a δΣ a = ⎨ (218) ⎪ ⎩0, elsewhere where Σap is the macroscopic absorption cross section of the rod. In this coordinate system the unperturbed flux is:

⎛ 2.405r ⎞ ⎛ πz ⎞ φ(r, z) = AJ0 ⎜ ⎟sin⎜ ⎟ (219) ⎝ R ⎠ ⎝ H ⎠

Introducing Eqs. (218) and (219) into Eq. (196) and noting that the volume element is 2πrdrdz , the reactivity due to the rod inserted at distance z is:

az 2 2 2 2πA Σ ap ∫∫J 0 ()()2.405r / R rdr sin πz / H dz (z) 00 (220) ρ = − 2 ν ∫ Σ f φ dV V

When the rod is fully inserted the reactivity is:

50 aH 2 2 2 2πA Σap ∫∫J0 ()()2.405r / R rdr sin πz / H dz ρ(H ) = − 00 (221) 2 ν ∫ Σ f φ dV V

Dividing the last two equations gives:

z 2 ∫ sin ()πz / H dz ⎡ z 1 ⎛ 2πz ⎞⎤ ρ(z) = ρ(H ) 0 = ρ(H ) − sin (222) H ⎢ ⎜ ⎟⎥ 2 ⎣ H 2 ⎝ H ⎠⎦ ∫ sin ()πz / H dz 0

In Eq. (222) the fully inserted worth ρ(H) may be commuted using the analytical methods described in § 5.2.3.1 or it is determined by the experiment. Since only relative words appear in Eq. (222), this formula then can be applied even to strongly absorbing rods.

5.2.3.4 Improvement of the experimental techniques for estimation of control rod worths 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 the analytical methods described in § 5.2.3.1. In analogy, a similar equation 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 various black and gray absorbers and also to take into account the axial burn up 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 burn up of the absorbing materials, have to be determined – by experiment or by using some theoretical method. Proposed is a method which uses a combination of Monte Carlo technique and perturbation theory. The idea is to derive the 'blackness' for various fresh and burnt absorber materials from the macroscopic cross sections, which are obtained from detailed Monte Carlo burn up calculations. The determined 'blackness' of different absorbers are used to correct the buckling of the reactor with partially burnt control rod, composed from combination of various black and gray materials. Thus, the theoretically determined buckling is used together with the measured relative efficiency of the partially inserted rod to determine the total control rod worth and to recover the curves of the total and differential control rods worth. This method has been compared with direct measurements of the total Control Rods worth, performed at the Belgian Material Testing Reactor BR2 and also with the detailed Monte Carlo direct calculations of the differential and total control rods worth. Detailed analysis is presented in [6].

5.2.3.5 Estimation of the asymptotic reactor period using the Monte Carlo method

51 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 equation for the reactor period (including the fractions of all groups of delayed neutrons and a single group of the photo neutrons):

Λ 7 β ρ = + ∑ i (223) Tkeff i=1 1+ λiT l Where: Λ = is the mean generation time between birth of neutron and subsequent absorption k inducing fission; for k ~ 1, Λ is the prompt neutron life time.

5.3 Experimental data of total control rods worth and burn up of the lower active part of the Reference control rods. 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. 15 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 [7]. The CR were loaded in channels C19, C79, C139, C281 and C341. The total CR worth decreases by about 10% after 300 F.P.D. of irradiation and by 15% after 650 F.P.D. irradiation. After 100 F.P.D, the cadmium burn up of the lower active part is about 10 mm and after 650 F.P.D – about 15 mm. The value of the total CR worth is influenced by several other effects, which are discussed in the following chapters: burn up 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.

6. Calculation methodology 6.1 MCNP(X)&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 × Kϕ ≈ 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 [8] with 1-D depletion code ORIGEN-S [9] is used for modelling 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 [10]. The 3−D space distribution of the isotopic fuel depletion 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 5 j fuel depletion B j (Ti−1 ) and the total power distribution K V (Ti−1 ,rm ,z n ,ϕl ) in the previous time step Ti-1 [10]:

52 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 ) c j (0) Vj (224)

dr dz dϕp (T ,r,z,ϕ) dr dz dϕ ∫∫∫ j i ∫ ∫ ∫ j rzmnϕl rm z n ϕl where: KV (Ti ,rm ,zn ,ϕl ) = (225) Pj (Ti ) Vj

is the total power peaking factor in a fuel zone (rm ,zn ,ϕl ) of the fuel element j during the irradiation time Ti; 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 235 -1 -1 235 ORIGEN−S [11]: C1=0.125 [kg Ufresh.MW .days ], that means about 8 MW.d per 1 kg U 235 5 -3 initially charged to the reactor is necessary for 1% depletion of U; c j (0) [g.cm ] is the initial 235U density in a fresh fuel. 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. 16). The total number of the spatial cells with varied fuel depletion in the model is 4600 [10-11]. 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. 17 [12]. Each 5 depleted fuel composition, 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 IX).

6.2. MCNPX 2.6

The Monte Carlo burn up code MCNPX2.6 [13]is used for evaluation of the burn up of the control rod material, evaluation of macroscopic cross sections, and evaluation of keff – Eigen values during fuel burn up in operating cycle. The depletion/burn up capability in MCNPX is based on the 1 – D burn up code CINDER90 [14] and Monte Burns [15]. The MCNPX depletion process internally links the steady – state flux calculations in MCNPX with the isotopic depletion calculations in CINDER90. MCNPX runs a steady – state calculation to determine the effective multiplication factor keff, 63 – group fluxes and continuous energy reaction rates for (n,gamma), (n,f), (n,2n), (n,3n), (n,alpha) and (n,p), which are converted into one – group constants and used by CINDER90 to carry out the depletion calculations and to generate new number densities for the next time step. MCNPX takes those new number densities for the corresponding fuel cells and generates another set of fluxes, reaction rates. The process is automatic and repeats itself for each time step until the requested final time step. The calculated with MCNPX 63 – energy group fluxes in combination with the inherent 63 – group cross sections of CINDER90 are used to determine the rest of the interaction rates, which are not calculated by MCNPX. 53 The burn up code MCNPX 2.6 has been tested on the Reactor BR2. Depletion and eigenvalue calculations have been performed for the full scale 3-D heterogeneous geometry model of the reactor, which describes the real reactor core of BR2 in a form of a twisted hyperboloidal bundle.The capabilities for depletion and criticality reactor core analysis of the new burn up Monte Carlo code MCNPX 2.6 have been compared with those of the combined MCNPX&ORIGEN-S method. The evolutions of the isotopic macroscopic, effective microscopic cross sections and atomic densities have been evaluated using CINDER90 and compared with ORIGEN-S [16]. The both methods use the same Monte Carlo code, which is linked with a 1 – D depletion code: CINDER90 in MCNPX 2.6 and ORIGEN-S in the MCNP(X)&ORIGEN-S method. In the both methods the reaction rates are calculated by MCNP(X) and the one – group constants are introduced into the depletion equation. The difference is that in MCNPX 2.6 the whole process is automatic and the steady – state flux calculations by MCNPX in the requested fuel region are internally linked with the depletion calculations by CINDER90. Therefore the reaction rates are updated for each time step in the requested fuel region during the irradiation period. In the MCNP&ORIGEN-S method the reaction rates are calculated by MCNP once – at BOC and introduced into ORIGEN-S, which performs the depletion calculations for all desired time steps. Then the isotopic fuel composition for a given time step is introduced back into the MCNP geometry model and distributed in the core using the calculated earlier 3 – D power peaking factors, and the keff is evaluated. The same procedure is repeated for each time step, so that the different depletion steps can be calculated independently and simultaneously, which saves a lot of computational time. The number of the fuel depletion zones used in the MCNP&ORIGEN-S method is about 4000. Although the number of the fuel cells, in which the material can be burnt is unlimited in the latest version MCNPX 2.6, in practice, for a complex heterogeneous system, the number of the spatial fuel zones, which can be depleted is still limited by the allowed computer memory. The comparison of the depletion methodologies by MCNPX 2.6 and MCNPX&ORIGEN-S method is schematically presented at Fig. 18.

6.3. SCALE 4.4a

The SCALE system is composed from many separate programme codes (modules). Different modules of the SCALE4.4a system can be used for evaluation of the isotopic fuel depletion and evaluation of the macroscopic absorption cross sections. These modules are: XSDRNPM [17], NITAWL-II [18], BONAMI [19], COUPLE [20] and ORIGEN-S [9]. 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 LWR. MCNPX 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 LWR reactor in the ORIGEN-S libraries (see Table X). The input for ORIGEN-S can be the fission power or the neutron flux, calculated by MCNP in the spatial cells where the burn up calculations are needed. ORIGEN-S evaluates the evolution of the isotopic fuel densities for the desired number depletion time steps. The isotopic fuel composition for a given time step

54 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 [10-11].

7. Neutronics modelling of BR2 The Belgian Material Test reactor (MTR) BR2 is strongly heterogeneous high flux engineering test reactor at SCK-CEN in Mol. The reactor was designed in 1957 by Nuclear Development Corporation of America – White Plains (NY-USA). Routine operation of BR2 started in January 1963 [21]. The reactor is cooled and moderated by light water in a compact HEU core, positioned in and reflected by a beryllium matrix. The beryllium matrix is an assembly of a big number of irregular hexagonal prisms, each with a cylindrical test hole which forms the channel in the core region. The central 200 mm hole (H1 channel) containing the central beryllium plugs is vertical. All other beryllium assemblies (84 mm hole or 200 mm hole) are skew and form a twisted hyperboloid bundle around the central H1 channel (Fig. 19a). The main design parameters of BR2 are presented in Table XI. The full-scale 3-D heterogeneous geometry model of BR2 was developed using the Monte Carlo code MCNP and presented at Fig. 19b. 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 MCNPX.2.6. 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. 20. 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 (see Fig. 16). A separate geometry model of BR2 has been developed for evaluation of the heating from the delayed photons. 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. Therefore, 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 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 in the independent photon transport calculation in the MCNP model of BR2.

8. Impact of various factors on the CR parameters in the reactor BR2 The impact of various factors affecting the control rods total have been investigated for the Reference CR with Cd and Co and presented in Table XII. The main conclusions can be made, which in general are valid for all CR types: ♦ The main affecting factor – the choice of the control rod absorbing material – is discussed in the following next sections ♦ The burn up of the control rod material (cadmium): Reduces the total control rod worth by ~ 0.5 $ Strongly affects the shapes of total and differential CR worth ♦ The presence of strong absorbers in the core increases the total control rod worth: 9 Presence of Ir samples in H1/C – increase up to ~ 0.8 ÷ 1.0 $ 9 Burnable poisons in the fuel (B4C and Sm2O3) – increase up to 0.7 $

55 ♦ Increase of fuel depletion (diminish of 235U content) and accumulation of fission products increases the total control rod worth up to about ~1.0 $ for fuel types 6NC and 6NG. ♦ The material of the follower. 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: 9 maximum control rod worth for fresh beryllium follower 9 minimum control rod worth for light water or Al follower 9 medium CR worth for poisoned beryllium follower ♦ 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 $ (see Fig. 21). ♦ Accounting for photo neutrons reduces the control rod worth by about 10%. ♦ Increase of aluminium cladding around the absorbing control rod material reduces the total control rod worth up to ~ 1.2 $. The dependence of the total CR worth on the thickness of the Al – cladding for different CR types is presented in Table XIV. It is seen that with decrease of the Al thickness, the total CR worth increases, because Al is replaced by light water, which is stronger absorber of thermal neutrons. ♦ Location of CR close to the core centre increases the total rods worth up to about ~1.8 $. 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 (channels A90, A270, C19, C161, C199, C341, configuration ‘7A’, ‘7B’, ~1972, [22], see Fig. 22) and for location of CR relatively far from the core centre (channels C19, C79, C139, C221, C281, C341; cycle 01/2005A.3, 3rd Be matrix, [23]). The content of the channels for the loadings 7A and 7B are given in Table XIII. The comparison of the total CR effective worth is given at Fig. 23.

9. Control rod candidate materials and design modifications

9.1. Design modifications ¾ 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. Considered is also a CR with full length made of satellite (grey absorber). ¾ A “grey” material (e.g., stainless steel) is used to prolong the lower end of the active control rod part. ¾ 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). ¾ Variation of the thickness of Al cladding: ♦ Variation of the thickness of Al cladding from 1.5 mm to 4.5 mm ♦ Increase of the thickness of Al cladding decreases the control rod worth ¾ Variation of the outer diameter and thickness of the Control Rod

56 9.2. Considered control rod types

¾ Cd+Co: Reference Control Rod with cadmium and cobalt in the lower active absorbing part of the rod with Al cladding ¾ Cd+Cd: Control Rod with full length of cadmium with Al cladding ♦ Variation of the thickness of Al cladding from 1.5 mm to 4.5 mm ♦ Increase of the thickness of Al cladding decreases the control rod worth ¾ Cd+AISI304: Control Rod with cadmium and AISI304 in the lower active part ¾ Hf+Hf: Control Rod with full length of hafnium ♦ With Al cladding ♦ Without cladding ¾ Hf+AISI304: Control Rod with hafnium and AISI304 in the lower active part without cladding ¾ Control Rod with full length of stellite ¾ Eu2O3: Control Rod with full length of europium oxide without cladding ¾ Gd2O3: Control Rod with full length of gadolinium oxide without cladding

9.3. Control rod geometry and dimensions

¾ Geometry and design – like in the Reference Control Rods ¾ Thickness of absorbing material for all types: 4 to 5 mm; outer diameter 6.1 mm; inner diameter 5.1 mm. ¾ Full length of absorbing part, including the use of “grey” material in the lower part – 895 mm. ¾ Length of the section with “grey” material in the lower part: ~ 140 mm; the thickness is equal to the thickness of the absorbing material. ¾ 6 identical CRs are located in channels: C19, C79, C139, C221, C281 and C341.

10. Evaluation of control rod characteristics of different control rod types during ~ 1000 EFPD of BR2 fuel cycle

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 ~ 30 operating cycles, which is equivalent to ~ 1000 EFPD of reactor operation. The depletion calculations are performed by MCNPX.2.6 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.

57 10.1 Methodology for calculation of changing of control rod characteristics and reactor parameters during BR2 fuel cycle ♦ A typical BR2 reactor core load, which remains the same in each cycle, is used in the calculations. ♦ The evolution of the CR absorbing material (atomic density, macroscopic and microscopic cross sections, activity), the fuel depletion and criticality evolution are evaluated during each cycle for about ~ 30 consecutive cycles . ♦ The following calculation methodology is applied: 9 BOC (1): 6 CR with fresh absorbing material are loaded into the core of BR2 at beginning of the 1st operating cycle. 9 BOC (2): The densities of the CR material at EOC(1) are used as initial densities at the BOC (2). 9 BOC (3): The densities of the CR material at EOC(2) are used as initial densities at the BOC (3). 9 ……………………………………………………………………………….. 9 BOC (N): The densities of the CR material at EOC(N-1) are used as initial densities at the BOC (N), N=1,…,30.

10.2 Comparison of macroscopic absorption cross sections during 1000 EFPD For each CR absorbing material, the productions of various isotopes through (n, γ) reaction is evaluated with MCNPX2.6. ♦ The CR characteristics – atomic densities, macroscopic and microscopic effective cross sections, activity – are evaluated in different axial zones over the CR height. ♦ The macroscopic absorption cross sections of the dominant and non-dominant nuclide for different CR types are given at Fig. 24. −1 ♦ 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 9 After T ~ 650 EFPD the macroscopic cross sections for cadmium rods drastically decreases due to the rapid burn up of the dominant isotope 113Cd. 9 A residual absorption, equivalent to about 30% from the initial value is caused by the other cadmium isotopes – 111Cd and 110Cd. This residual absorption remains constant during long irradiation period (T ~ 1000 EFPD). 9 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

♦ The absolute values of the macroscopic cross sections for fresh and burnt CR material are summarized in Table XV. -1 9 Eu2O3 rod has maximum anti-reactivity (l00%); Σ=0.30 cm

58 - 9 Reference rods (Cd+Co) – the anti-reactivity is ~ 66% from Eu2O3 rod; Σ=0.2 cm 1 -1 9 Cd+Cd rod – the anti-reactivity is about ~ 77% from Eu2O3 rod; Σ=0.23 cm

9 Hf+Hf rod (without Al cladding) – the anti-reactivity is ~ 90% from Eu2O3 rod; Σ=0.27 cm-1

9 Hf+Hf rod (with Al cladding) – the anti-reactivity is ~ 85% from Eu2O3 rod; Σ=0.25 cm-1

9 Hf+AISI304 rod (without Al cladding) – the anti-reactivity is ~ 72% from Eu2O3 rod; Σ=0.22 cm-1 -1 9 Gd2O3 rod – the anti-reactivity is ~ 92% from Eu2O3 rod; Σ=0.28 cm

10.3 Comparison of microscopic absorption cross sections 10.3.1 Evaluation of microscopic absorption cross section during T< 600 EFPD

♦ The macroscopic cross is defined as:

-1 Σ = N < σ > eff , [] cm (226)

♦ The macroscopic cross section 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 = , [] barn (227) ∫Φ(E)dE

♦ The evolution of the atomic densities of the dominant nuclides and the microscopic effective cross sections, defined with Eq. (227) are given at Fig. 25. 9 It is seen from Fig. 25 that for Eu-151 and Hf-177 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 ♦ 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 XVIa, XVIb and Tables XVIIa, XVIIb. 9 As can be seen from Table XVIa, XVIb the reason for increase of the microscopic effective cross sections of cadmium – 113 is the changing of the neutron spectrum in the thermal and epithermal energy regions and as result – increase of the 59 thermal and epithermal effective microscopic cross sections. The depletion of the cadmium density is "compensated" by increase of the microscopic cross section, −1 so that Σ = N < σ >eff ,cm remains almost constant during ~ 20 irradiation cycles. 9 This effect is not observed for the Eu – 151 (see Tables XVIIa, XVIIb), which effective microscopic cross section remains constant and consequently the −1 macroscopic cross section Σ = N < σ >eff ,cm of Eu – 151 decreases with decreasing of the density of Eu – 151. However, the decrease of the macroscopic cross section of Eu – 151 is compensated by the produced other Eu – nuclides (Eu – 152, Eu – 154) and Sm – 152, 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. 26. 9 The changing of the neutron flux defined simply as Φ(E,T = 150d.)(/Φ E,T = 0d. ) is shown at Fig. 26a. 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 Cd – 113 also increase, which can be seen from Fig. 26b. 9 For Eu – rod, the neutron flux during irradiation remains almost constant as can be seen from Fig. 26a. This has also simple explanation: the depletion of Eu – 151 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 Eu – 151 the increase of the reaction rates σΦ after T=150 days is negligible in comparison with Cd – 113, which can be seen from Fig. 26c. ♦ What do we have finally for the absorption during the irradiation by Cd – 113 and Eu – 151, defined as NσΦ? From Fig. 26d and Fig. 26e is seen that the product NσΦ remains constant during irradiation for Cd – 113 and decreases for Eu – 151 during the irradiation. However the decrease of 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 9 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 in time. For cadmium rod – the dominant nuclide is Cd-113, for europium rod the dominant nuclide is Eu-151, for the hafnium rod Hf-177 and for the gadolinium rod Gd-157. 9 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. (155) remains practically constant during long time of irradiation ~ 600 EFPD 9 For europium rod – the atomic density of Eu-151 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 (Eu-152, Eu-154), some contributions come also from Sm-152, Gd-154, E-155 and the resonant absorption cross section of the rod also remains constant during the considered irradiation period.

60 9 The effective microscopic cross sections of Hf-177 and Gd-157 are changing slowly, but the final product – the macroscopic cross section Σ also remains constant during long irradiation period.

10.3.2 Microscopic absorption cross section during T > 600 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=1000 EFPD are given in Fig. 27a, 27b. The comparison between the neutron spectra in fresh and burnt CR material for the individual CR types is demonstrated at Fig. 27c, 27d, 27e, 27f. As can be seen from the graph (Fig. 27c, 27d, 27e) for CR types: Cd+Co, Cd, Cd+AlSi304, the thermal fluxes increase drastically after ~ 600 EFPD which is related with the complete depletion of Cd-113, which has high thermal absorption cross section (see Fig.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: Cd-110 and Cd-111(see Fig.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 Cd – 110 and Cd – 111 increase and as can be seen from Fig. 24a, 24b they form a residual absorption which is about 30% from the absorption of the fresh Cd-rod. This residual absorption may remain long time of irradiation – up to the considered in this job irradiation period: T ~ 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 Cd-113. There is one dangerous time interval moment (not yet determined in this job) between the total disappearance of Cd-113 and the following compensation of the absorption by Cd-110 and Cd-111. The duration of this time interval has to be determined. Also, the “safety” value ∆dCd= 210mm must be revisited and probably increased. For CR types: Hf, Hf+AISI304, Eu2O3 , the changing of the neutron spectrum in the rod during the whole considered irradiation period T ~ 1000 EFPD is negligible.

10.4 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 into the total heating of the cadmium give neutrons (more than 90%). pr+cap The 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 del come from the delayed γ-rays Q γ. The neutron and gamma heating in the lower Cd part in the different CR types were evaluated using the MCNP neutron and photon heating tallies. The results of the calculated with MCNP neutron and gamma heating for the used different CR types are presented in Table XIX.

10.5 Comparison of total control rods worth's Detailed calculations of the total control rod worth for various types control rods have been performed by MCNPX2.6 for the beginning of the Control rod life and after long time of irradiation, taking into account the axial burn up of the Control Rod absorbing material.

61 10.5.1 Comparison of Total Control Rods Worth at T=0 and T=1000 EFPD The calculations are performed for the load of cycle 03/2006A.5. 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 XX. It is seen that the burn up of the CR material does not affect strongly the value of total CR worth; Affected are the shapes of the total CR worth as can be seen from Fig. 28. ♦ The curves of the total control rods worth’s for fresh CR at BOC (T=0) and for burnt CR at B0C of the 30th cycle (T~1000 EFPD) are given in Fig. 28. The comparison of the curves of total CR worth during irradiation for the different CR types is given in Fig. 29. 9 The axial burn up of the control rod absorbing material during T ~ 1000 EFPD (~30 operation cycles) is evaluated with MCNPX 2.6 and taken into account in the calculations of the total CR worth. 9 The curve of the total worth of control rods with full cadmium part (Cd+Cd) decreases significantly after ~ 600 EFPD of irradiation see Fig. 29c. 9 The curves of total rods worth for Cd+AISI304 and for Reference Cd+Co also decrease during irradiation, but less (Fig. 29b, Fig. 29a). 9 For all other CR types –Hf+AISI304, Hf+Hf, Eu2O3 (Fig. 29d, 29e, 29f) the changing of the curves of total CR worth’s during T ~1000 EFPD is negligible.

10.5.2 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 E0C (T=30 EFPD) are given in Fig. 30. ♦ Similar calculations were performed for the 30th operation cycle BOC(T~1000 EFPD) and EOC(T~1030 EFPD), which are given at Fig. 31. ♦ At EOC the total and the differential Control Rods worth's increase which is caused by the fuel burn up, i.e. – depletion of 235U and accumulation of fission products.

10.6 . 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. 32. At Fig. 32a are given the differential curves of all CR types, at Fig. 58b 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 is demonstrated in Fig.

62 32b. The changed differential worth curves for the various CR types during long time of irradiation are depicted on Fig. 32c. ♦ The comparison of the differential CR worth’s for fresh and burnt CR material for individual CR types are given at Fig. 33. ♦ The burn up of the CR material affects strongly the curves or differential worth for CR types: Cd+Co, Cd+Cd, Cd+AISI304 (see Fig. 33a, Fig. 33c, Fig. 33b). The differential worth for these rods decreases with burn up; The distributions for burnt CR are shifted to the lower positions Sh. ♦ The differential CR worth for CR types: Hf+Hf, Hf+AISI304 almost not change with burn up of the CR material. ♦ As can be seen from the graphs (Fig. 32a, Fig. 32b), the maximum of the differential curves for CR made of “black” absorber, which is prolonged by “grey” material in the lower CR part, are shifted to the lower positions Sh of the CR motion. ♦ At EOC the differential Control Rods worth's increase which is caused by the fuel burn up, i.e. – depletion of 235U and accumulation of fission products. The comparison of the curves of differential worth at BOC and EOC for the individual rods is shown at Fig. 34.

11. Changing of the reactor neutronics characteristics for different control rod types during BR2 fuel cycle

¾ Reactivity evolution, CR motion, cycle length

♦ The keff and the reactivity evolutions during few consecutive cycles with the same reactor core load, but with depleted CR are evaluated for various CR types and presented at Fig. 35. ♦ The control rods motions during the 3rd and the 5th cycles for the different CR are given at Fig. 36. ♦ The reactivity excess and CR positions Sh at BOC and EOC for fresh (T=0) and burnt (T ~ 1000 EFPD) CR material are calculated and presented in Table XXI. ♦ For CR types: Cd+Cd, Cd+Co, Cd+AISI304 the positions Sh for burnt CR material (T ~1000 EFPD) will be lower than at T=0 (for fresh CR material), which is related to the depletion of Cadmium – 113 and thus disappearance of a major portion of the CR material in the lower edge of the CR. ♦ For CR types: Hf+Hf, Hf+AISI304, Eu2O3 the positions Sh for fresh CR (T=0) and for burnt CR (T ~ 1000 EFPD) remain the same since the absorbing material is almost not burnt during the irradiation period.

12. Control rod effects on core power and neutron flux distributions 12.1 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 63 the 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 burn up. 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 [24] 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 iridium 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. 37 (H1/C), Fig. 38 (A30), Fig. 39 (C41), Fig. 40 (E30). And the fast fluxes in the same channels are given at Fig. 41 (H1/C), Fig. 42 (C41), Fig. 43 (E30). The main conclusions are: 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). 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. Maximum perturbation effect on axial flux distributions have rods with full absorbing length from europium (Eu2O3), hafnium (Hf+Hf) and cadmium (Cd+Cd). 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). 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.

12.2 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. 44 (H1/C), Fig. 45 (A30), Fig. 46 (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. 47 (H1/C), Fig. 48 (A30), Fig. 49 (B180).

64 12.3 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. 50), 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. 51 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.

13 Comparison of activity during 1000 EFPD

The Activity is defined as:

A(t) = λN(t ) (228)

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

−λt N(t) = N0e (229)

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 = = (230) 1/ 2 λ λ

Hence, the activity can be rewritten by:

−0.693t /T1/ 2 A(t) = T1/ 2 N0e (231)

The evaluation of activity for the different types of CR is shown in Fig. 52. 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. 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 52 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. 52(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.

65 151 152 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.

14 Criterion for Control Rod Life A 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 CR. 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. If keff at BOC after a certain number of cycles decreases or remain 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. 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 = = (232) 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, which are easily calculated by MCNP.. We can re-write (252) in the following form:

1 1 k = ν ;k ≈ ν (233) eff f A N eff f A 1+ − x,xn 1+ F F F

Consequently, a criterion for the changing of the CR properties during irradiation is the ratio between the total absorption and the total fission reaction rates in the system: A/F. The fraction of the CR absorption from the total absorption at 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 depletion of the CR material during irradiation changes the absorption properties not only of the CR itself, but also changes (slightly) the total absorption and total fission in the reactor, i.e. changes the ratio A/F, which determines increase or decrease of keff during the reactor cycles. Another criterion for the CR life, but again related to the keff of the whole system is the total CR worth, which is defined as the difference in the reactivity values between position of the CR at Sh=0 mm and Sh=960 mm:

R0[$] = R0(0) − R0(900),[$] (234)

66 Let summarize the criteria for CR properties: burn up of the CR absorbing material. change of keff and the CR position Sh at BOC after several cycles for the same load and same power in all cycles; change of the total and differential CR worth after several cycles.

15 Summary The main absorption and neutronics data for different CR types have been evaluated during the BR2 Fuel Cycle and they are summarized in Table XXII. About 30 consecutive operating BR2 cycles have been considered, which is equivalent to ~1000 EFPD. The fresh CR 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 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 the Monte Carlo burn up code MCNPX.2.6 with included depletion capability and/or by MCNP&ORIGEN-S combined method.

The following conclusions can be made on the basis of the data in Table XXII: 1) Maximum absolute value of the CR Worth is for the Eu rod (17.5 $), and minimum − for Cd rods (~ 13.5 $) 2) The Total CR Worth increases at EOC for all CR types by about 8%. 3) The maximum value of the differential CR worth is for Eu rod (0.040 $/mm); the minimum differential CR worth is for Cd rod (0.033 $/mm). 4) The differential CR worth increases at EOC for all CR types. 5) The positions of the maximum of the curves of differential CR worth are shifted toward lower positions Sh for CR types, which lower part is prolonged with gray material. 6) Burn – up of absorbing material doesn't affect significantly the value of total CR worth: a. about 3% ÷ 4% decrease of total CR worth for Cd, Cd+Co, Cd+AISI304 rods; b. less than 1% decrease of total CR worth for all other types: Hf, Hf+AISI304, Eu2O3 rods. c. For Reference CR:

i. for ΔhCd = 210 mm the reduction of the total CR worth is ~ 5%;

ii. for ΔhCd = 285 mm the reduction of the total worth is about ~ 10%. 7) Burn – up of absorbing material affects strongly shape of the curves of total/differential CR worth for Cd, Cd+Co, Cd+AISI304 rods. For all other types – Hf, Hf+AISI304, Eu2O3 rods – the total differential curves do not change during irradiation. -1 8) Macroscopic absorption cross section Σa [cm ] for CR types Cd, Cd+Co, Cd+AISI304 remain almost constant during ~ 700 EFPD. For T ~ 700÷1000 EFPD Σa ~ 0.2÷0.3 from 113 the initial value ( Cd is totally burnt, residual Σa is due to other Cd – isotopes). -1 9) Σa [cm ] for all other types – Hf, Hf+AISI304, Eu2O3 rods – remain constant till ~ 1000 EFPD. 10) Changing of positions Sh at BOC during ~ 20 ÷ 30 operating cycles:

67 a. For CR types: Cd+Cd, Cd+Co, Cd+AISI304 the positions Sh for burnt CR material (T > 650 EFPD) will be lower than at T=0 (for fresh CR material).

b. For CR types: Hf+Hf, Hf+AISI304, Eu2O3 the positions Sh for fresh CR (T=0) and for burnt CR (T ~ 1000 EFPD) remain 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 a. This effect is sensitive at Sh ~ 400 to 500 mm. b. For Sh > 650 mm the axial distributions of neutron fluxes for all CR types are practically the same 12) Neutron spectra in CR material during irradiation: a. For CR types: Cd, Cd+Co, Cd+AISI304 thermal fluxes increase strongly after T ~ 650 EFPD (due to burn up of absorbing material)

b. For CR types: Hf, Hf+AISI304, Eu2O3 the change of the neutron spectra in the whole energy region is not significant. 13) Corrections in the standard procedure for estimation of CR worth's: a. Schindler's formula can be used for an approximate estimation of reactivity worth of the rod b. 'Blackness' of burnt Cd part a(Cd-burnt): ♦ In standard procedure is considered a(Cd-burnt)=0 ♦ MCNPX 2.6 calculations: 'blackness' of burnt Cd part is ~ 0.2 of 'blackness' of fresh Cd. c. Not taken into account axial distribution of cadmium 'blackness'. d. 'Blackness' of cobalt during irradiation time is decreasing: ♦ Standard procedure: a(Co)=0.75 from blackness of fresh Cd a(Co) not changed during irradiation time ♦ MCNPX 2.6 calculations: a(Co-fresh)=0.60 from 'blackness' of Cd (fresh) a(Co-burnt, ~ 600epd)=0.46 from 'blackness' of Cd (fresh) e. Normalization of reactivity values: ♦ Effective part of delayed neutron is: β=0.0065 ♦ Correct value is: β=0.0072 (including photo neutrons) 14) For Reference CR:

a. For ΔhCd = 210 mm the reduction of the total CR worth is ~ 5%;

b. For ΔhCd = 285 mm the reduction of the total worth is about ~ 10% 15) CR with Hf+AISI304: a. Length of AISI304 ~ 140 mm 68 b. AISI304 does not burn during at least 900 efpd c. 'Blackness' of AISI304: ♦ 0.42 from 'blackness' of fresh Cd ♦ 0.36 from 'blackness' of fresh Hf

16 Proposed new CR type – Hf+AISI304 (R0=15.6 $)

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 burn up calculations up to ~ 1000 EFPD have been performed by MCNPX 2.6.D for the lower stainless part of the rod and it was obtained that the stainless steel is not burning during long irradiation time. The total and differential control rod worths for these modifications are given in Fig. 53d and Fig. 54d. 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. On the basis of the performed calculations we can conclude that increasing the outer diameter of the rod (not changing the rod thickness) improves it absorption properties, which can be seen from Fig. 53d and Fig. 54d. 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 (see Fig. 55). 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 (Fig. 55). 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=140 mm in the lower part of the CR improves the axial distributions of the neutron fluxes in comparison with CR with full length of hafnium. The reduction of the length of the AISI304 from L=140 mm to L=70 mm significantly improves the curve of the differential CR worth which can be seen from Fig. 54d and slightly worsen the axial distributions of neutron fluxes which is sensitive for z > −5cm . Increasing the thickness of AISI from ΔD=5mm to ΔD=10mm improves the differential CR worth almost for all positions of Sh (see Fig. 54d) and practically does not change significantly the axial distributions of the neutron fluxes (thermal and fast). Therefore, the following optimized dimensions have been chosen for the hafnium rod, which lower part is prolonged with AISI304: Outer diameter of Hafnium: Dout= 6.4 mm Inner Diameter of Hafnium: Din = 5.4 mm Outer diameter of AISI304: Dout= 6.4 mm Inner diameter of AISI304: Din = 5.0 mm Length of AISI304: L =70. mm

69 17 References [1] Isbin H.S. and J.W.Gorman: Applications of Pile – Kinetic Equations, Nucleonics, vol. 10, n. 11, p. 69, November, 1952. [2] Hurwitz H.: Derivation and Integration of the Pile – Kinetic Equations, Nucleonics, vol. 5, n. 1, p. 61, July, 1949. [3] Bonilla, C.F. (ed.): "Nuclear Engineering", McGraw – Hill Book Company, Inc., New York, 1957]. [4] 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. [5] 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 Energy, Vol. 8, pp. 18 to 32. Pergamon Press Ltd., London. [6] S.Kalcheva, E.Koonen, "Improved Monte Carlo – Perturbation Method For Estimation Of Control Rod Worths In A Research Reactor", International Conference on the Physics of Reactors “Nuclear Power: A Sustainable Resource”,Casino-Kursaal Conference Center, Interlaken, Switzerland, September 14-19, 2008. [7]. B.Ponsard, E.Koonen,A.Beeckmans,M.Noel – BR2 data for cadmium burn up of the Reference CR with cadmium and cobalt. [8] MCNP/MCNPX, Monte Carlo N-Particle Transport Code System. Oak Ridge National Laboratory, RSICC Computer Collection, November (2005). [9] ORIGEN-S: SCALE System Module to Calculate Fuel Depletion, Actinide Transmutation, Fission Product Build-up 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. [10] S.Kalcheva, E.Koonen and B.Ponsard, “Accuracy of Monte Carlo Criticality Calculations During BR2 Operation”, Nucl. Technol., 151, 201 (2005). [11] S.Kalcheva, E.Koonen and P.Gubel, “Detailed MCNP Modelling of BR2 Fuel with Azimuthal Variation ”, Nucl. Technol., 158, 36 (2007). [12] S.Kalcheva and E.Koonen, "3-D Fuel Burn Up Modelling With MCNP&ORIGEN-S", Proceedings of the 10th Int. Topical Meeting on Research Reactor Fuel Management, Sofia, Bulgaria, May (2006). [13] MCNPX, Version 26D, John S. Hendricks et al., LANL, LA-UR-06-7991. June 20, 2007. [14] CINDER90, William B. Wilson, LANL, T – 16, 2005. [15] MONTEBURNS, Holly R. Trelue, LANL, X – 1, 2005. [16] S.Kalcheva and E.Koonen, "MCNPX 2.6.C vs. MCNPX & ORIGEN-S: State of the Art for Reactor Core Management", Proceedings of the 11th Int. Topical Meeting on Research Reactor Fuel Management, Lyon, France, March (2007). [17] 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. [18] 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. [19] BONAMI: SCALE System Module for Performing Resonance Self – Shielding by the Bondarenko Method. Oak Ridge National Laboratory, NUREG/CR-0200, Revision 6, v. 2, F1. ORNL/NUREG/CSD- 2/V2/R6, March 2000.

70 [20] 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. [21] P.GUBEL, F.JOPPEN AND E.KOONEN. BR2 Material Test Reactor. 40 Years Operating Experience. IAEA-CN-82. Int. Conference on Topical Issues in Nuclear Safety, Contributed Papers, Vienna, Austria, 3-6 September 2001. [22] A. Beeckmans, H. Lenders, F. Leonard, "Mesures D'Effets Reactifs Dans Le Reacteur BR2", CEN/SCK-EURATOM, Department BR2 , 1973. [23] B. Ponsard, "Expected Irradiation Conditions: BR2 Reactor”. 01/2005A.3. [24] B. Ponsard, "Expected Irradiation Conditions: BR2 Reactor”. 02/2003A.6 [25] B. Ponsard, "Expected Irradiation Conditions: BR2 Reactor”. 04/2005B.1 [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.

Table I.

Facility Thermal power, Control rods Country Reactor type status name steady (MW) material BR-1 Graphite 4.00 Cd,C Operational Belgium BR-2 Tank 100.00 Cd,Co Operational THETIS RR- BN-1 Pool 0.25 Cd,In,Ag Operational Under CARR Pool 60.00 Hf construction Under CFER Fast breeder 65.00 B4C China construction 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 High Temp. HTTR 30.00 B4C Operational gas Japan JMTR Tank 50.00 Hf Operational JRR-3M Pool 20.00 Hf Operational BOR-60 Fast breeder 60.00 B4C Operational MIR,M1 Pool/channel 100.00 Dy,Cd Operational Russia Under PIK Tank 100.00 Eu construction SM Press. vessel 100.00 Eu Operational ATR Tank 250.00 Hf Operational United HFIR Tank 85.00 Eu Operational states NBSR Heavy water 20.00 Cd Operational *remark: data comes from IAEA website: http://www.iaea.org/worldatom/rrdb/

72 Table II. Reactivity Requirement in Major Reactor Types

Reactivity(∆k/k) BWR PWR HTGR LMFBR BR2

Excess reactivity ρex of Clean core at 20°C 0.25 0.293 0.128 0.050 ~ 0.06 at operational T° 0.248 0.037 ~ 0.06 at eq. Xe, Sm 0.181 0.073 Total worth of control, R0 0.29 0.32 0.210 0.074 ~ 0.10 0.09(BOC) Burnable poison worth 0.12 0.08 0.10 0.03(EOC) Chemical shim worth 0.17

Shutdown margin ρsm Cold and fresh (clean) 0.04 0.03 0.082 0.024 > 0.0324 Hot and eq. Xe, Sm 0.14 0.137 0.037

Table III. The main isotopes of Cd isotope NA half-life DM DE (MeV) DP 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

Table IV. The main isotopes of Hf

Isotope NA half-life DM DE (MeV) DP 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

Table V. The main isotopes of Gd

Isotope NA 152Gd 0.2% 154Gd 7.18% 155Gd 19.8% 156Gd 20.47% 157Gd 15.65% 158Gd 24.84% 160Gd 21.86%

Table VI. The main isotopes of Eu

Isotope NA half-life DM DE (MeV) DP 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

Table VII. Basic characteristics of different fuel types used in BR2 standard fuel elements (FE).

Fuel Type VInG VInE VInC

Fuel Material UAlx UAlx UAlx Enrichment, % 93% 72% 72% Fuel density, gUtot/cm3 1.3 1.3 1.7 235U, Mass per FE, 400 330 399.80 totU, Mass per FE, 430 458 552.63 B4C, Mass per FE, 3.8 1.8 3.2

Sm2O3, Mass/FE, grams 1.4 1.3 1.4

Table VIII. Properties of delayed neutrons and photo neutrons for HEU fuelled (~ 90 % 235U) and beryllium reflected reactor.

Mean life ti, Decay constant λi, Fraction of -1 [sec] [sec ] 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

Table IX. 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 Fission Products 75As, 81Br, 82Kr, 83Kr, 84Kr, 85Kr, 86Kr, 85Rb, 87Rb, 89Y, 93Zr, 95Mo, 99Tc, 101Ru, 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.

Table X. MCNPX calculation of effective thermal microscopic cross sections in typical fuel channel of the reactor BR2, which are used to update the existing cross sections in the ORIGEN- S libraries for LWR. The cross sections data from the files ENDF/B-V,VI are used in the calculations of < σ >eff [barn] by MCNPX.

Nuclide ORIGEN−S MCNPX Nuclide ORIGEN−S MCNPX < 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+06 238 U (n,f) 0 8E–06 147Pm (n,γ) 235 127 237 Np (n,γ) 170 153 149Sm (n,γ) 4.15E+04 5.5E+04 237 Np (n,f) 0.019 0.013 150Sm (n,γ) 102 72 239 632 360 1.5E+03 8.3E+03 Pu (n,γ) 151Sm (n,γ) 239 1520 750 210 150 Pu (n,f) 152Sm (n,γ)

76 Table XI. 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 32÷33, fresh and burnt fuel elements, with core 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 MTR-type fuel plates 18 plates (6 concentric tubes) 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 235 U enrichment (wt%) 93% 72% 73% 235 U mass per FE (grams) 400 330 400 Total U mass per FE (grams) 430 458 553 3 Fuel density, gU/cm 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 235 239 Mean depletion at elimination ( U+ Pu) 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)

77 Table XII. Impact of various factors on total CR worth for the Reference Control Rods with fresh cadmium and cobalt (Cd+Co). Calculations by MCNPX.

keff R0 [$] ΔR0 [$] Sh=0 mm Sh=900 mm Burnup of CR Fresh 13.2 1.0 material Burnt 12.2 Fuel depletion High 0.9486±0.0005 1.0464±0.0005 13.7 1.0 (6NC) Low 0.9854±0.0005 1.0827±0.0005 12.7 Fuel depletion High 0.9504±0.0005 1.0488±0.0005 13.7 1.0 (6NG) Low 0.9799±0.0005 1.0764±0.0005 12.7 Present 0.9504±0.0005 1.0488±0.0005 13.7 Burnable poisons 0.7 Absent 1.0298±0.0005 1.1393±0.0005 13.0 HEU (90% 235U) Fuel enrichment LEU (20% 235U) Present 0.9608±0.0005 1.0574±0.0005 13.2 Iridium in H1/C 0.8 Absent 0.9834±0.0005 1.0783±0.0005 12.4 Detailed 3 – D in whole core Fuel modelling ~ 1.0 Homogeneous in FA Fresh beryllium 13.9 Material of the Highly poisoned beryllium 12.2 3.5 follower in CR Aluminium 12.0 Light water 10.4 Position of CR in Close to core centre 1.8 the core Close to core periphery Present Photoneutrons 1.0 Absent δ =1.5 mm 0.9563±0.0005 1.0581±0.0005 14.0 Thickness of Al δ =3.5 mm 0.9601±0.0005 1.0579±0.0005 13.4 1.2 Al cladding*) Al δAl=5.5 mm 0.9632±0.0005 1.0573±0.0005 12.8 *) For control rod with full cadmium absorbing length (Cd+Cd)

Table XIII. Content of the reactor channels for the core loadings used in the calculations.

Configur. 7A (~1972) 7B (~1972) Fuel type VInC VInC A30 20% 20% A90 S2 S2 A150 20% 20% A210 20% 20% A270 S5 S5 A330 20% 20% B0 20% 20% B60 20% 20% B120 20% 20% B180 20% 20% B240 20% 20% B300 20% 20% C19 S1 S1 C41 0% 0% C79 0% 0% C101 0% 0% C139 0% 0% C161 S3 S3 C199 S4 S4 C221 0% 0% C259 0% 0% C281 0% 0% C319 0% 0% C341 S6 S6 D0 30% 30% D60 Be Be D120 30% 30% D180 30% 30% D240 30% 30% D300 Be Be E30 40% 40% E330 40% 40% F14 40% 40% F46 Be Be F106 40% 40% F166 40% 40% F194 40% 40% F254 40% 40% F314 40% 40% F346 40% 40% H1/C Be Be H1-1 36% Be H1-2 36% Be H1-3 36% Be H1-4 36% Be H1-5 36% Be H1-6 36% Be H2 Be Be H3 Be Be H4 Be Be H5 Be Be G0 RS RS G60 Be Be G120 40% 40% G180 Be Be 79 G240 40% 40% G300 40% 40% H23 40% 40% H37 40% 40% H323 40% 40% H337 40% 40% K11 Be Be K49 Be Be K109 Be Be K169 Be Be K191 Be Be K251 Be Be K311 Be Be K349 Be Be L0 Be Be L60 Be Be L120 Be Be L180 Be Be L240 Be Be L300 Be Be

Table XIV. Dependence of the Total CR Worth [$] of different CR types with full absorption part of Cd, Hf or stellite on the thickness δ [mm] of the Al cladding at T=0 and at T=400 F.P.D.

Thickness δ [mm] of the Al cladding

δinner=5.5 δinner=3.5 δinner=1.5 δinner=0 CR type δouter=1.5 δouter=1.5 δouter=1.5 δouter=0 0 400 0 400 0 400 0 400 Reference CR 12.7 − 13.2 − − − − − (CR with Co) CR with full Cd 12.9 12.5 13.4 13.1 14.0 13.8 − − (Cd+Cd) CR with full Hf 14.9 14.4 − − 15.7 15.6 15.5 15.4 (Hf+Hf) CR with full − − − − − − 12.4 − stellite

80 Table XV. Calculated macroscopic absorption cross sections Σ [cm-1] in fresh CR material (T=0) and burnt CR material (T ~ 600 ÷ 1000 EFPD). Calculations by burn up Monte Carlo code MCNPX 2.6.C.

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 −

Table XVIa. Effective microscopic cross sections of 113Cd at BOC(1).

Energy Interval −2 〈σΦ〉 Φ n ,[n.cm ] 〈σ 〉,[]barn [MeV] 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

Table XVIb. Effective microscopic cross sections of 113Cd at BOC(5).

Energy Interval −2 〈σΦ〉 Φ n ,[n.cm ] 〈σ 〉,[]barn [MeV] 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

81 Table XVIIa. Effective microscopic cross sections of 151Eu at BOC(1).

Energy Interval −2 〈σΦ〉 Φ n ,[n.cm ] 〈σ 〉,[]barn [MeV] 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

Table XVIIb. Effective microscopic cross sections of 151Eu at BOC(5).

Energy Interval −2 〈σΦ〉 Φ n ,[n.cm ] 〈σ 〉,[]barn [MeV] 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

Table XVIII. Total Activity [Ci]

T=0 T=300 T=480 T=600 T=870 T=900 T=990 Cd+Cd 0 47110 51610 53393 80479 Cd+Co 0 255600 364800 Eu2O3 0 439000 624300 724400 934900 952400 Hf+Hf 0 88070 96170 100200 114000 121500 Hf+SS 0 59980 65810 65870

82 Table XIX. Nuclear heating (Watt per gram) in the lower part of the Control Rods.

CR type “Cobalt” part of CR Lower Cadmium part of CR 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 CR_Al 0.13 − − 36.5 − −

Table XX. Comparison of total worth 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 st st th (1 cycle) (~ 30th cycle) (1 cycle) (~ 30 cycle) Cd+Co 13.4 $ 13.0 $ 14.5 $ 14.0 $ Cd+Cd 13.6 $ 13.3 $ 14.7 $ 14.3 $ Cd+AISI304 13.2 $ 12.9 $ 14.7 $ 14.3 $ Hf+Hf 15.8 $ 15.7 $ 17.1 $ 17.0 $ Hf+AISI304 15.6 $ 15.5 $ 16.7 $ 16.7 $ stellite 12.5 $ − 13.6 $ −

Eu2O3 17.5 $ 17.5 $ 19.0 $ 18.8 $

Gd2O3 15.8 $ − 17.0 $ −

83 Table XXI. 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 (1st T=1000 EFPD T=30 EFPD T=1030 EFPD 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 $

Table XXII. Comparison of total worth (R0) and macroscopic absorption cross sections (Σa) for different CR rod types accounting for the axial burn up of the absorbing material during irradiation. -1 1) R0 [$] Σa [cm ] ΔhCd-burnt [mm] T=0 T > 650 T=0 T > 650 T > 650 T > 750 Cd+Co 13.4 $ 13.0 $ 0.20 0.06 − 65 (2102) Cd+Cd 13.6 $ 13.3 $ 0.22 0.07 140 (2852) Cd+AISI304 13.2 $ 12.9 $ − − − − Hf+Hf 15.8 $ 15.7 $ 0.26 0.24 − − Hf+AISI304 15.6 $ 15.5 $ 0.21 0.20 − − stellite 12.5 $ − − − − −

Eu2O3 17.5 $ 17.5 $ 0.30 0.31 − −

Gd2O3 15.8 $ − − − − − 1) in lower CR active part, Δh=65 mm 2) "safety" value

84 Cd106 Cd108

Cd110 Cd111

Cd113 Cd114

Cd115m Cd116

Fig. 1 Total cross section of Cadmium isotopes

85 Hf174 Hf176

Hf177 Hf178

Hf179 Hf180

Fig. 2 Total cross section of Hafnium isotopes

Gd152 Gd153

Gd154 Gd155

Gd156 Gd157

Gd160 Gd158

Fig. 3 Total cross section of Gadolinium isotopes

Eu151 Eu152

Eu153 Eu154

Eu155 Sm152

Fig. 4. Total cross section of Europium isotopes

Co59 Co60

Fig. 5 Total cross section of Cobalt isotopes

Cr50 Cr52 Cr53 Cr54

Mn55

Fig. 6 Total cross section of Cromium isotopes

Fe54 Fe56 Fe57 Fe58 Ni58

Fig. 7 Total cross section of Iron isotopes

Figure 9. The eigenfunctions for a slab geometry.

Figure 10. Time decay of higher order spatial modes in slab reactor.

1.8

Eq. (4-82) 1.7

1.6

1.5

n(t)/n(0)

1.4

1.3

1.2 0.1 1 10 TIME [SECONDS]

Figure 11. 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)

Figure 12. Circuit diagram of variable resistance lumped – inductance series circuit.

Figure 13. Voltage versus current diagram, indicating the graphical construction required to solve the transmission – line problem.

Figure 14. Plot of 1/c.r. vs. amount of uranium, showing the effect of the instrument placement.

Reference CR - Cd with Co 184

15 Dh Cd [mm] R0 [$]

174

164

Cd burnup [mm] total CR worth [$] worth CR total 5

154

0 144 100 200 300 400 500 600 Irradiation time [days] April 1997 March 2003

Figure 15. Total CR Worth and Cadmium burnup in CR with Cd and Co [1].

Figure 16. MCNP model of a standard BR2 fuel element – the fuel plates are divided into azimuth sectors by each 5° in the hot plane Zhot= –12 cm÷ –6 cm.

3D isotopic fuel depletion in fuel elements

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

Evaluation Evaluation of of isotopic Depletion step: 3D relative Δβ5 =1% 235U power distribution fuel depletion depletion versus burn up Radial: 100 depleted in 6 fuel 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 17. A scheme of the combined MCNP&ORIGEN-S method for 3D modeling of the isotopic fuel depletion [7].

a) b)

Figure 18. Comparison of depletion methodologies by: a) MCNPX 2.6.C and b) MCNP&ORIGENS.

Figure 19. Inside view of the BR2 reactor (a) and model representation of a horizontal cut 15 cm below mid-plane showing the hexagonal lattice with the various channels and their contents (right).

Fig. 20. MCNP model of the lower active Cd part of CR in channel C281, divided into 10 radial sectors by each 0.05 mm.

Total Control Rods Effective Worth of CR_Co (ΔhCd~150 mm)

16 MCNP, Cycle: 01/2005A.3 [1] CR_Co_BR2 ("fresh" Be) 14 CR_Co_BR2 ("poisoned" Be)

[-$] reactivity 6

0 0 100 200 300 400 500 600 700 800 900 Sh [mm] Figure 21. Total Control Rods Effective Worth for 6 CR_Co with “fresh” and “poisoned” beryllium section under the Cd part of the CR.

7A 7B

Fig. 22. MCNP whole core model for typical BR2 reactor core loads (configurations 7A, 7B) of the 1st Be-matrix (~1972).

Total Control Rods Effective Worth of CR_Co, located in different Fuel Channels (used "poisoned" Be in CR, Δh ~150-160mm) Cd 16 CR close to core center: conf. '7A', '7B' (~1972, [20]) 14 CR far from core center: cycle 01/2005A.3 (3rd Be matrix, [21])

[-$] reactivity 6

2 MCNP

0 0 100 200 300 400 500 600 700 800 900 Sh [mm]

Figure 23. Total Control Rods Effective Worth for different types CR, located “close” or “far” from the reactor core centre.

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

0,25 0,25 MCNPX 2.6.C MCNPX 2.6.C Cd+Co Cd+Cd ] ] -1 0,20 -1 0,20

0,15 total 0,15 total Cd113 Cd113 Cd112 Cd112 0,10 Cd106 0,10 Cd106 Cd108 Cd108 Cd110 Cd110

macroscopic cross section [cm section cross macroscopic Cd111 [cm section cross macroscopic Cd111 0,05 0,05 Cd114 Cd114 Cd116 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.]

CR with full length of Hf (no Al cladding) CR with Hf and AISI304 in the lower lower part 0.30 0.30 MCNPX 2.6.C Hf+Hf MCNPX 2.6.C Hf+AISI304

0.25 0.25 ] ] -1 -1 0.20 0.20

0.15 0.15 total 0.10 0.10 total Hf174 Hf174 Hf176

macroscopic cross section [cm section cross macroscopic Hf176 macroscopic cross section cross [cm macroscopic Hf177 Hf177 0.05 0.05 Hf178 Hf178 Hf179 Hf179 Hf180 0.00 Hf180 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.]

CR with full Hf length (with Al cladding) CR with full length of Eu O 2 3 0,30 0.4 MCNPX 2.6.C MCNPX 2.6.C Eu O Hf+Hf 2 3 0,25 ] ] -1 -1 0.3 0,20

0,15 0.2 total total Sm152 Eu151 0,10 Hf174 Eu152 Hf176 Eu153 macroscopic cross section [cm macroscopic cross section [cm section cross macroscopic 0.1 Hf177 Eu154 0,05 Hf178 Eu155 Hf179 Gd152 Hf180 O16 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.]

100

-1 Comparison of CR with (Cd+Cd) and CR with (Cd+Co) Comparison of Σ [cm ] of different CR 0,30 0,4 MCNPX 2.6.C MCNPX 2.6.C

0,25 ] ] -1 -1 0,3 0,20

0,15 0,2

0,10 Cd+Co macroscopic cross section [cm section cross macroscopic macroscopic cross section [cm cross macroscopic Cd+Cd 0,1 total_(Cd+Co) Hf+Hf 0,05 total_(Cd+Cd) Hf+AISI304 Cd113_(Cd+Co) Eu2O3 Gd O 0,00 Cd113_(Cd+Cd) 2 3 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.]

macroscopic cross sections atomic densities 0,25 0,020 MCNPX MCNPX Cd Cd 0,20 0,015 ] 0,15 Cd106 -3 Cd108 Cd106 ]

-1 0,010 Cd110 Cd108 [atoms.cm [cm 0,10 Cd111 Cd110 Co Σ Cd112 N Cd111 Cd113 Cd112 0,005 Cd113 0,05 Cd114 Cd116 Cd114 Cd116 total Cd 0,00 0,000 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.]

0,25 0,020 MCNPX Co MCNPX Co 0,20 0,015

]

0,15 -3 ] -1

[cm 0,010

0,10 [atoms.cm Cd N Co59 0,005 0,05 Co60 Ni60 Co59 total Co Co60 Ni60 0,00 0,000 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.]

Figure 24. Comparison of effective macroscopic absorption cross sections of dominant and non- dominant isotopes for various CR types.

101

CR with Cd+Cd CR with Hf+Hf

0,0060 60 0.0090 30 Cd-113 density Hf-177 0,0055 0.0085 <σ > density eff 55 ] ] 25 -3 -3 <σ > eff 0,0050 0.0080

0,0045 50 0.0075 20

0,0040 0.0070 atimic density [atoms.cm density atimic atomic density [atoms.cm density atomic 45 15 0,0035 0.0065 effective microscopic cross section [barn] section cross effective microscopic MCNPX MCNPX 0,0030 40 0.0060 10 [barn] section cross effective microscopic 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 Eu-151 Gd-157 0.0120 density 0.0050 density <σ > 50 ] 25 eff -3 0.0115 0.0045 <σ > 0.0110 eff 20 0.0040 45 0.0105

[a/b-cm] density atimic 0.0035 0.0100 atomic density [atoms.cm density atomic 40 15 0.0030 0.0095 MCNPX effective microscopic cross section [barn] section cross effective microscopic 0.0090 MCNPX section cross effective microscopic effective 0.0025 35 10 0 50 100 150 0 50 100 150 200 250 Time [days] Time [days] Figure 25. Evolutions of microscopic effective absorption cross and atomic density of dominant isotopes for different absorbing materials.

102

Increase of the local neutron flux in the depleted CR 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

1,5

(0 d.), r.u. Φ a)

(150 d.)/ 1,0 Φ

0,5

MCNPX 0,0 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Neutron energy [MeV] Spectral distribution of the reaction rates in Cd-113 during irradiation Spectral distribution of the reaction rates in Eu-151 during irradiation 0,10 0,04 Eu-151 Cd-113

> 0,08

> σΦ 0,03 σΦ

0,06

0,02 T=0 days b) c) T=150 days 0,04 T=0 days

reaction rates per 1 source n, < rates per 1 source reaction T=150 days reaction rates per 1 source n, < 0,01 0,02

MCNPX MCNPX 0,00 0,00 10-10 10-9 10 -8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 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] T T T T Total absorption N <σΦ> in Cd-113 during irradiation Total absorption N <σΦ> in Eu-151 during irradiation Cd-113 Eu-151 0,0005 0,0004 Cd-113 Eu-151 T T > > 0,0004 σΦ

σΦ < < 0,0003 Eu-151 Cd-113 T

T 0,0003

0,0002 d) T=0 days e) T=150 days 0,0002 T=0 days T=150 days 0,0001 0,0001 totaol absorption per 1 source n, N totaol absorption per 1 source n, N

MCNPX MCNPX 0,0000 0,0000 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10 10 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10 10 10 Neutron energy [MeV] Neutron energy [MeV]

Figure 26. a) Changing of the neutron spectra in the depleted rods during irradiation; b,c) Spectral dependences of the Reaction Rates in the rods during irradiation; d,e) changing of the total absorption in the depleted rods during irradiation. 103

Neutron spectra in the lower absorbing CR part Neutron spectra in the l ower absorbing CR part 13 3,5x10 4,5x1013 MCNPX 2.6.C T=0 MCNPX 2.6.C T=1000 EFPD 13 3,0x1013 Cd+Co 4,0x10 Cd+Co Cd Cd 13

] 3,5x10 1 ]

- Cd+AISI304 Cd+AISI304 13 1 - .s

2 2,5x10 .s - 2 Hf Hf - 13 Hf+AISI304 3,0x10 Hf+AISI304 13 [n.cm n [n.cm Eu2O3 2,0x10 Eu2O3 n 13

Φ 2,5x10 Φ

13 1,5x1013 2,0x10

13 neutron flux neutron

neutron flux neutron 1,5x10 1,0x1013 1,0x1013 5,0x1012 5,0x1012 0,0 0,0 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 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 10-1 100 101 Neutron energy [MeV] Neutron energy [MeV] a) b)

Neutron spectra in th e lower absorbing CR part Neutron spectra in the l ower absorbing CR part 13 3,0x1013 4,5x10 MCNPX 2.6.C MCNPX 2.6.C Cd (T=0) Cd+Co (T=0) 4,0x1013 Cd (T=1000 EFPD) 2,5x1013 Cd+Co (T=1000) ]

1 13 - ]

1 3,5x10 - .s 2 - .s 2 - 13 2,0x1013 3,0x10

[n.cm n [n.cm n Φ 13 Φ 2,5x10 1,5x1013 2,0x1013

13 neutron flux 13 neutron flux 1,0x10 1,5x10

1,0x1013 5,0x1012 12 5,0x10 0,0 0,0 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 10 10 10 10 10 10 10 10 10 10 10 Neutron energy [MeV] Neutron energy [MeV]

c) d)

Neutron spectra in the l ower absorbing CR part Neutron spectra in the l ower absorbing CR part 13 2,0x10 1,8x1013 MCNPX 2.6.C Cd+AISI304 (T=0) MCNPX 2.6.C 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 -

1,2x1013 [n.cm [n.cm

n n Φ Φ 1,0x1013 1,0x1013 8,0x1012 neutron flux neutron flux 6,0x1012 5,0x1012 4,0x1012

12 2,0x10 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) f)

Figure 27. Neutron spectra in the lower active part of the CR.

104

Total Control Rods Effective Worth of Different CR Total Control Rods Effective Worth of Different CR

18 18 Cd+Co Cd+Co 16 Cd+AISI304 16 Hf Cd Hf+AISI304 14 Hf 14 Hf+AISI (shift=1mm, D=5mm) Hf+AISI304 Hf+AISI (shift=2mm, D=5mm) 12 Hf+AISI (shift=1mm, D=5mm) 12 Hf+AISI (shift=2mm, D=6mm) Hf+AISI (shift=2mm, D=5mm) 10 10 Hf+AISI (shift=2mm, D=6mm)

Eu2O3 8 8 reactivity [-$] reactivity [-$] 6 6

4 4

2 T=0 MCNPX 2 T=0 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]

a) b)

Total Control Rods Worth: R0=ρ(0)-ρ(900)

20 MCNPX Cd+Co 18 Cd+0.5Co 16 Cd+AISI304 Cd 14 Hf+AISI304 Hf 12 Eu2O3 10

reactivity [-$] reactivity 8

4 2 T=1000 EFPD 0 0 100 200 300 400 500 600 700 800 900 Sh [mm]

Figure 28. Comparison of total CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for different CR types.

105

Total Control Rods Worth: R0=ρ(0)-ρ(900) Total Control Rods Worth: R0=ρ(0)-ρ(900)

16 16 MCNPX Cd+Co (T=0) MCNPX Cd+AISI304 (T=0) 14 Cd+Co (T=1000 EFPD) 14 Cd+AISI304 (T=1000 EFPD) Cd+0.75Co (T=1000 EFPD) 12 12

10 10 $] $] - - [ [ ty ty ty i 8 i 8 v v i i

react 6 react 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)

Total Control Rods Worth: R0=ρ(0)-ρ(900) Total Control Rods Worth: R0=ρ(0)-ρ(900)

16 18 Cd (T=0) MCNPX MCNPX Hf+AISI304 (T=0) 14 Cd (T=750 EFPD) 16 Hf+AISI304 (T=750 EFPD)

12 14 12 10 $] - $] [ - [ 10 ty ty y i

8 it v i v ti 8 react 6 reac 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) Total Control Rods Worth: R0=ρ(0)-ρ(900) Total Control Rods Worth: R0=ρ(0)-ρ(900)

18 20 MCNPX Hf (T=0) MCNPX Eu2O3 (T=0) 16 Hf (T=650 EFPD) 18 Eu2O3 (T=1000 EFPD)

14 16 14 12 $] $] 12 - - [ 10 [ ty ty ty ty i i 10 v v i 8 i react react 8 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 29. Comparison of total CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for individual CR types.

106

Total Control Rods Effective Worth of Different CR Total Control Rods Effective Worth of Different CR

20 20 T=0, BOC(1st cycle) st 18 18 T ~ 30 E.F.P.D., EOC(1 cycle) 16 16 14 14

12 12 10 Cd+Cd 10 Cd+SS Cd+Cd

reactivity [-$] reactivity 8 Hf+Hf [-$] reactivity 8 Cd+SS Hf+SS Hf+Hf 6 6 stellite Hf+SS stellite 4 Cd+Co 4 Cd+Co Eu2O3 Eu2O3 MCNPX 2 MCNPX 2 Gd2O3 Gd2O3 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) Total Control Rods Effective Worth Total Control Rods Effective Worth 18 18 1st cycle 1st cycle 16 16

14 14

12 12 10 10 8 8 reactivity [-$] reactivity

6 [-$] reactivity 6

4 Cd+Cd (BOC) 4 Cd+Cd (EOC) Hf+Hf (BOC) 2 Cd+AISI304 (BOC) Hf+Hf (EOC) MCNPX 2 Hf+AISI304 (BOC) Cd+AISI304 (EOC) MCNPX 0 Hf+AISI304 (EOC) 0 100 200 300 400 500 600 700 800 900 0 Sh [mm] 0 100 200 300 400 500 600 700 800 900 Sh [mm] c) d)

Total Control Rods Effective Worth 18 1st cycle 16

14 12 10 e) 8 reactivity [-$] reactivity 6

4 Cd+Co (BOC) Cd+Co (EOC) 2 Cd+AISI304 (BOC) Cd+AISI304 (EOC) MCNPX 0 0 100 200 300 400 500 600 700 800 900 Sh [mm] Figure 30. 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.

107

Total Control Rods Effective Worth of Different CR Total Control Rods Effective Worth of Different CR

20 20 th th 18 T ~ 1000 E.F.P.D., BOC(30 cycle) 18 T ~ 1030 E.F.P.D., EOC (30 cycle)

16 16 14 14 12 12

10 10

8 reactivity [-$] reactivity 8

[-$] reactivity Cd+Cd 6 6 Cd+Cd Cd+SS Cd+SS Hf+Hf 4 4 Hf+Hf Hf+SS Hf+SS 2 Cd+Co 2 Cd+Co Eu2O3 MCNPX Eu2O3 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] a) b)

Figure 31. a) Comparison of total CR worth for various CR types at BOC of the 30th cycle (T~1000 EFPD); b) Comparison of total CR worth for various CR types at EOC of the 30th cycle (T~1030 EFPD)

108 Differential CR worth Differential CR worth 0,050 0,050 Cd+Co Cd+Co MCNPX MCNPX Cd+Cd Hf+Hf 0,045 Cd+AISI304 0,045 Hf+AISI304 Hf+Hf Hf+AISI (shift=1mm, D=5mm) 0,040 Hf+AISI304 0,040 Hf+AISI (shift=2mm, D=5mm) Hf+AISI (shift=1mm, D=5mm) Hf+AISI (shift=2mm, D=5mm) 0,035 Eu2O3 0,035 stellite 0,030 Gd2O3 0,030

0,025 0,025

Sh [$/mm] Sh [$/mm] Δ Δ / / 0,020 0,020 Δρ Δρ 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)

Differential CR worth 0,050 MCNPX Cd+Co 0,045 Cd+Cd Cd+AISI304 0,040 Hf+Hf Hf+AISI304 0,035 Eu2O3 0,030

0,025

Sh [$/mm] Δ / 0,020 Δρ 0,015

0,010 T~1000 EFPD 0,005 0,000 0 100 200 300 400 500 600 700 800 900 Sh [mm]

Figure 32. Comparison of differential CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for different CR types.

109

Differential Control Rods Worth: Δρ /ΔSh Differential Control Rods Worth: Δρ /ΔSh i i i i 0,030 0,030 MCNPX MCNPX 0,025 0,025

] 0,020 0,020 ] mm mm

[$/

i [$/

0,015 i 0,015 Sh) Δ Sh) /

i Δ ρ / i Δ ρ 0,010 Δ 0,010 Cd+Co (T=0) 0,005 Cd+Co (T=750 EFPD) 0,005 Cd+AISI304 (T=0) Cd+0.6Co (T=750 EFPD) Cd+AISI304 (T=1000 E.F.P.D.) 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) Differential Control Rods Worth: Δρ /ΔSh Δρ Δ i i Differential Control Rods Worth: i/ Shi 0,030 0,035 MCNPX MCNPX

0,025 0,030

0,025 0,020 ]

] mm mm 0,020 [$/ [$/

0,015 i

Sh) Sh) 0,015 Δ Δ / /

i i ρ ρ Δ 0,010 Δ 0,010

0,005 Cd (T=0) 0,005 Hf+AISI304 (T=0) Cd (T=1000 E.F.P.D.) Hf+AISI304 (T=1000 E.F.P.D.) 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] c) d) Differential Control Rods Worth: Δρ /ΔSh i i Differential Control Rods Worth: Δρ /ΔSh 0,040 i i 0,040 MCNPX MCNPX 0,035 0,035

0,030 0,030

] 0,025 ] 0,025 mm mm

[$/

i 0,020 [$/

i 0,020 Sh) Δ Sh) / i Δ ρ

0,015 / i Δ

ρ 0,015 Δ 0,010 0,010 Hf (T=0) 0,005 Eu2O3 (T=0) Hf (T=1000 E.F.P.D.) 0,005 Eu2O3 (T=1000 E.F.P.D.) 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] e) f) Figure 33. Comparison of differential CR worth for fresh (T=0) and burnt (T ~ 1000 EFPD) absorbing material for individual CR.

110

Differential CR worth Differential CR worth 0,050 0.050 Cd+Co MCNPX Cd+Co MCNPX Cd+Cd Cd+Cd 0,045 Cd+AISI304 0.045 Hf+Hf Cd+AISI304 0,040 Hf+AISI304 0.040 Hf+Hf Hf+AISI (shift=1mm, D=5mm) Hf+AISI304 Hf+AISI (shift=2mm, D=5mm) stellite 0,035 0.035 Eu2O3 Eu2O3 stellite Gd2O3 0,030 Gd2O3 0.030

0,025 0.025

Sh [$/mm] Sh [$/mm] Δ Δ / 0,020 / 0.020 Δρ Δρ 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]

a) b)

Figure 34. Comparison of differential CR worth at BOC (T=0) and EOC (T ~ 30 EFPD) for different CR types.

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

1,01 9 MCNPX 8 Cd_Co 1,00 Cd 7 Eu Eu2O3 0,99 6 Hf 5 Gd 0,98

(T) [$](T) 4 ρ eff k

0,97 (0)- Cd_Co ρ 3 Cd 2 0,96 Eu Eu2O3 1 0,95 Hf Gd 0 MCNPX 0,94 -1 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time [days] Time [days]

rd rd Criticality evolution during the 3 cycle Reactivity evolution during the 3 cycle

1,01 9 MCNPX 8 Cd_Co 1,00 Cd 7 Eu 0,99 6 Eu2O3 Hf

] 5 $ Gd 0,98 (T) [ (T)

ρ 4 eff

0,97 (0)- Cd_Co ρ 3 Cd 0,96 Eu 2 Eu2O3 1 Hf 0,95 Gd 0 MCNPX 0,94 -1 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time [days] Time [days]

th th Criticality evolution during the 5 cycle Criticality evolution during the 5 cycle 1,01 9 MCNPX 8 Cd_Co 1,00 Cd 7 Eu 0,99 6 Eu2O3 Hf 5 Gd [$] 0,98

(T) 4

ρ eff - k 0,97 (0) ρ 3 Cd_Co Cd 0,96 2 Eu Eu2O3 1 0,95 Hf 0 Gd MCNPX 0,94 -1 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time [days] Time [days] st rd th Figure 35. Evolutions of keff and reactivity during the 1 , 3 and the 5 operating cycle.

112 rd Reactivity evolution and CR motion during the 3 cycle th Reactivity evolution and CR motion during the 5 cycle 8 1000 8 1000 7 Cd 7 Cd 900 6 900 6 5 800 5 800 ] [$] 4 [$] )

T 4 [ (

ρ 700 (T) -

ρ 700

- 3 Sh

(0) 3 ρ (0) ρ 2 600 2 600 1 1 500 0 500 0 MCNPX MCNPX -1 400 0 5 10 15 20 -1 400 0 5 10 15 20 Time [days] Time [days] Reactivity evolution and CR motion during the 3rd cycle Reactivity evolution and CR motion during the 5th cycle

8 1200 8 1200 Eu2O3 7 Eu O 7 2 3 1100 1100 6 6 1000 1000 5 5 ] [$] ] 4 900 4 900 mm (T) [$](T) [ (T) mm ρ ρ [ -

3 Sh 3 (0)- (0) 800 Sh ρ 800 ρ 2 2 700 700 1 1 600 600 0 0 MCNPX MCNPX -1 500 -1 500 0 5 10 15 20 0 5 10 15 20 Time [days] Time [days]

th th Reactivity evolution and CR motion during the 5 cycle Reactivity evolution and CR motion during the 5 cycle 8 1100 9 1100 Hf 7 8 Gd 1000 1000 6 7 6 5 900 900

] 5

] [$] [$]

4 mm [ (T) (T) mm

4 [ ρ 800 ρ 800

- -

3 Sh (0) (0) Sh Sh ρ ρ 3 2 700 700 2 1 1 600 600 0 0 MCNPX MCNPX -1 500 -1 500 0 5 10 15 20 0 5 10 15 20 Time [days] Time [days]

Figure 36. CR motion of the different CR during the operating cycle.

113 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 the axis of FE in H1/Central in Al sample in the axis of FE in H1/Central

6x1014 6x1014 Cd+Cd H1/C Cd+AISI304 H1/C 5x1014 5x1014 ] ] -1 14 -1 14 .s .s -2 4x10 -2 4x10

3x1014 3x1014 Sh=500 mm Sh=420 mm Sh=514 mm 14 Sh=595.5 mm 14 thermal flux [n.cm flux thermal 2x10 Sh=691 mm [n.cm flux thermal 2x10 Sh=608 mm Sh=786.5 mm Sh=702 mm Sh=882 mm Sh=796 mm 1x1014 1x1014

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)

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 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

3x1014 3x1014

14 Sh=535 mm 14 thermal flux [n.cmthermal flux 2x10 [n.cmthermal flux 2x10 Sh=639 mm Sh=450 mm Sh=744 mm Sh=555 mm Sh=660 mm 14 Sh=848 mm 14 1x10 Sh=952 mm 1x10 Sh=765 mm Sh=870 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]

c) d)

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 the axis of FE in H1/Central in Al sample in the axis of FE in H1/Central

6x10 14 6x1014 stellite H1/C Cd+Co H1/C 5x10 14 5x1014 ] ] -1 -1 14 14 .s .s -2 -2 4x10 4x10

14 14 3x10 3x10

14 14 thermal flux [n.cm flux thermal thermal [n.cm flux 2x10 Sh=493 mm 2x10 Sh=440 mm Sh=614 mm Sh=540 mm Sh=736 mm Sh=640 mm 14 1x1014 Sh=858 mm 1x10 Sh=740 mm Sh=979 mm Sh=840 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] e) f)

114 Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in H1/Central 6x1014

Eu2O3 H1/C

5x1014 ] -1 14 .s

-2 4x10

14 3x10

14 flux [n.cm thermal 2x10 Sh=555 mm Sh=961 mm Sh=656 mm 14 1x10 Sh=758 mm Sh=860 mm

0 -50 -40 -30 -20 -10 0 10 20 30 40 50 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 6x10 14 6x1014 Sh=400 mm H1/C Sh=800 mm H1/C

5x10 14 5x1014 ] ] -1 14 -1 14 .s .s -2 4x10 -2 4x10

14 14

3x10 3x10 Cd+Cd Cd+Cd Cd+AISI304 Cd+AISI304 14 14 Hf+Hf thermal flux [n.cm flux thermal 2x10 Hf+Hf [n.cm flux thermal 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]

h) i)

Figure 37. Axial distributions of thermal neutron fluxes (En < 0.5 eV) in Aluminium sample in the axis of fuel element, located in channel H1/Central.

115

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 the axis of FE in A30 in Al sample in the axis of FE in A30 14 14 4x10 4x10 Cd+Cd A30 Cd+AISI304 A30

14 3x10 3x1014 ] ] 1 - -1 .s .s 2

- -2 m

14 2x10 2x1014

thermal flux [n.c thermal Sh=400 mm thermal fluxthermal [n.cm 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]

a) b)

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

14 3x1014 3x10

] ] -1 -1 .s .s -2 -2

14 14 2x10

2x10

Sh=400 mm

thermal flux [n.cm thermal Sh=400 mm thermal flux [n.cm 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] c) d)

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

3x1014 3x1014 ] ] -1 -1 .s .s -2 -2

14 14 2x10 2x10

thermal flux [n.cm Sh=400 mm thermal flux [n.cm Sh=400 mm Sh=500 mm Sh=500 mm 14 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]

e) f)

116

Axial distribution of thermal neutron flux E <0.5 eV n in Al sample in the axis of FE in A30 14 4x10 Eu2O3 A30

3x1014 ] -1 .s -2 14

2x10

thermal flux [n.cm flux thermal 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] g)

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 A30 in Al sample in the axis of FE in A30 4x1014 4x1014 Sh=400 mm A30 Sh=800 mm A30

3x1014 3x1014 ] ] -1 -1 .s .s -2 -2

14 14

2x10 2x10 Cd+Cd Cd+Cd Cd+AISI304 Cd+AISI304 Hf+Hf thermal flux [n.cm thermal thermal flux [n.cm flux thermal Hf+Hf Hf+AISI304 Hf+AISI304 14 14 stellite 1x10 Eu2O3 1x10 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]

h) i)

Figure 38. 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.

117 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 the axis of FE in C41 14 2,5x10 14 2,5x10 Cd+Cd C41 Cd+AISI304 C41

14 2,0x1014 2,0x10 ] ] -1 -1 .s .s -2 -2 14 1,5x1014 1,5x10

14 1,0x1014 1,0x10 Sh=400 mm

Sh=400 mm fluxthermal [n.cm thermal flux [n.cm flux thermal Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm 13 5,0x1013 5,0x10 Sh=700 mm 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]

a) b)

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 the axis of FE in C41 in Al sample in the axis of FE in C41 14 2,5x1014 2.5x10 Hf+Hf C41 Hf+AISI304 C41

14 2,0x10 14 2.0x10 ] ] -1 -1 .s .s -2 -2 14 1,5x10 14 1.5x10

14 1,0x10 14 1.0x10 thermal flux [n.cm flux thermal thermal flux [n.cm flux thermal Sh=400 mm Sh=400 mm Sh=500 mm Sh=500 mm 13 5,0x10 13 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] c) d)

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 C41 in Al sample in the axis of FE in C41 14 14 2,5x10 2,5x10 stellite C41 Cd+Co C41

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

-2 -2 1,5x1014 1,5x1014

14 14 1,0x10 Sh=400 mm 1,0x10 Sh=400 mm thermal flux [n.cm flux thermal Sh=500 mm [n.cm flux thermal 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]

e) f)

118

Axial distribution of thermal neutron flux En<0.5 eV

in Al sample in the axis of FE in C41 2,5x1014 Eu O C41 2 3 2,0x1014 ] -1 .s

-2 1,5x1014

14 1,0x10 Sh=400 mm [n.cm flux thermal 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

2.5x10 14 2,5x1014 Sh=400 mm C41 Sh=800 mm C41

2.0x10 14 2,0x1014 ] ] -1 -1 .s .s -2 -2 1.5x1014 1,5x1014

Cd+Co 1.0x1014 Cd+Co 1,0x1014 Cd+AISI304 Cd+AISI304 Cd+Cd thermal flux [n.cm flux thermal thermal flux [n.cm flux thermal 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]

h) i)

Figure 39. 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.

119 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 3x1014

2x1014

14 2x10 Sh=400 mm thermal flux [n.cm Sh=500 mm fluxthermal [n.cm 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] a) b)

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 5x10 5x1014 Hf+Hf E30 Hf+AISI304 E30

14 4x10 4x1014 ] ] -1 -1 .s .s -2 -2 14 3x10 3x1014

14 2x10 2x1014 thermal flux [n.cm Sh=400 mm fluxthermal [n.cm Sh=500 mm 14 Sh=400 mm Sh=600 mm 14 1x10 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] c) d)

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 E30 E30 stellite Cd+Co 4x1014 4x1014 ] ] -1 -1

.s .s -2 -2 3x1014 3x1014

2x1014 2x1014 thermal fluxthermal [n.cm Sh=400 mm fluxthermal [n.cm 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]

e) f)

120 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 3x1014

2x1014 thermal fluxthermal [n.cm 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] g)

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 14 3x10

2x10 Cd+Co Cd+Co 14 2x10 Cd+AISI304 Cd+AISI304 thermal flux [n.cm thermal

thermal flux [n.cm Cd+Cd Cd+Cd Hf+Hf 1x1014 Hf+Hf 14 Hf+AISI304 Hf+AISI304 1x10 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]

h) i)

Figure 40. Axial distributions of thermal neutron fluxes (En < 0.5 eV) in reflector channel E30, which is located near Control Rod channel.

121 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 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

14 14

3x10 3x10 fast flux [n.cm

fast flux [n.cm flux fast 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] a) b)

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

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

5x10 14 5x1014

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

14 14 3x10 3x10

Sh=400 mm fast flux [n.cm fast flux [n.cm flux fast 14 14 2x10 Sh=400 mm 2x10 Sh=500 mm Sh=500 mm Sh=600 mm Sh=600 mm 14 14 Sh=700 mm 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] c) d)

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 14 14

3x10 3x10

fast flux [n.cm 14 fast flux [n.cm 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] e) f)

Figure 41. Axial distributions of fast neutron fluxes (En > 0.1 MeV) in Aluminium sample in the axis of fuel element, located in channel H1/Central.

122 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 5x1014 5x10 Cd+Cd C41 Cd+AISI304 C41

4x1014 4x1014 ] ] -1 -1 .s 14 .s

-2 14 3x10 -2 3x10

14 2x10 2x1014 fast flux [n.cm flux fast Sh=400 mm fast flux [n.cm 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] a) b)

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

14 2x10 2x1014 fast flux [n.cm Sh=400 mm fast flux [n.cm 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] c) d)

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 5x1014 5x1014 stellite C41 Cd+Co C41 4x1014 4x1014 ] ] -1 -1 .s .s 14 14 -2 -2 3x10 3x10

2x1014 2x1014 fast flux [n.cm flux fast fast flux [n.cm flux fast Sh=400 mm Sh=400 mm Sh=500 mm Sh=500 mm Sh=600 mm 14 14 Sh=600 mm 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] e) f)

123 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 4x10 14 4x1014 ] ] -1 -1 .s .s 14 14 -2 3x10 -2 3x10

Cd+Cd 14 14 2x10 2x10 Cd+AISI304 fast flux [n.cm fast flux [n.cm 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] g) h)

Figure 42. 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.

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 1,5x10 14 1,5x1014 Cd+Cd E30 Cd+AISI304 E30

14 ] 14 ] -1 1,0x10 -1 1,0x10

.s .s -2 -2

fast flux flux [n.cm fast 13 Sh=400 mm [n.cm flux fast 13 5,0x10 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]

a) b)

124

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 1,5x10 14 1,5x1014 ] ] -1 -1

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

fast flux [n.cm flux fast [n.cm flux fast 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]

c) d)

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 E30 in Al sample in the axis of FE in E30

stellite E30 Cd+Co E30

14 14 1,5x10 1,5x10 ] ] -1 -1 .s .s -2 -2 14 1,0x10 1,0x1014

fast flux [n.cm flux fast Sh=400 mm fast flux [n.cm Sh=400 mm 13 Sh=500 mm 13 5,0x10 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]

e) f)

125

Axial distribution of fast neutron flux En>0.1 MeV in Al sample in the axis of FE in E30

Eu O E30 2 3 1,5x1014

] -1 .s -2 1,0x1014

fast flux [n.cm flux fast 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]

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 1,0x1014 1,0x1014

Cd+Cd Cd+Cd Cd+AISI304 fast flux [n.cm fast flux [n.cm Cd+AISI304 Hf+Hf

13 Hf+Hf 13 Hf+AISI304 5,0x10 Hf+AISI304 5,0x10 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]

h) i)

Figure 43. 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.

126 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 Rods with Co Control Rods with Cd 7x1014 7x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

6x10 14 6x1014 ] ] -1 -1 14 14 .s .s -2 -2 5x10 5x10

14 4x1014 4x10

14 14 thermal flux [n.cm thermal flux [n.cm 3x10 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 1x1014 1x10 -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)

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

7x1014 MCNP, cycle 04/2005B.1 14 6x10 ] -1 14 .s

-2 5x10

14 4x10

14 thermal flux [n.cm flux thermal 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 44. 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.

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

4x10 14 4x1014 ] ] -1 -1 .s .s -2 -2 14 3x1014 3x10

2x10 14 2x1014 thermal flux [n.cm thermal flux [n.cm Sh=400 mm (E=0 MW.d) Sh=450 mm (E=0 MW.d) 14 Sh=500 mm (E=472 MW.d) 14 Sh=550 mm (E=472 MW.d) 1x10 1x10 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]

a) b)

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 3x1014

2x1014 thermal flux [n.cm flux thermal 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 45. 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.

128

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 Rods with Co Control Rods with Cd

5x1014 5x1014 MCNP, cycle 04/2005B.1 MCNP, cycle 04/2005B.1

4x10 14 4x1014 ] ] -1 -1 .s .s -2 -2 3x1014 3x1014

14 14 thermal flux [n.cm thermal flux [n.cm 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) 1x1014 1x1014 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]

a) b)

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 3x1014

14 thermal flux [n.cm flux thermal 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] c)

Figure 46. 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.

129

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 Co Control Rods 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

3x1014 3x1014

fast flux [n.cm fast flux [n.cm 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) 1x1014 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]

a) b)

Axial distribution of the fast flux in the axis of FE in H1/C Control Rods with AISI304 6x1014 MCNP, cycle 04/2005B.1

5x1014

14 ] -1 4x10 .s -2

3x1014

fast flux [n.cm 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 47. 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.

130

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 Rods with Co Control Rods 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

3x1014 3x1014

a) b)

Axial distribution of the fast flux in the axis of FE in A30 Control Rods with AISI304 6x1014 MCNP, cycle 04/2005B.1

5x1014

14 ] -1 4x10 .s -2

3x1014

fast flux [n.cm 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 48. 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.

131

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 4x10 -1 4x10

.s .s -2 -2

3x1014 3x1014

fast flux [n.cm fast flux [n.cm 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]

a) b)

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

3x1014

fast flux [n.cm 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 49. 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.

132

Fig. 50. MCNP whole core model for typical BR2 reactor core loads (configurations 7A, 7B) of the 1st Be-matrix (~1972) and of configuration 4 (~ similar to the load for BR02).

133

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

3x1014 ] -1 .s

-2

2x1014

thermal flux [n.cm flux thermal

14 1x10 Sh=300mm (E=0 MW.d)_sub-critical Sh=300mm (E=0 MW.d) Sh=500mm (E=0 MW.d) Sh=800mm (E=0 MW.d) -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm] a)

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

3x1014 ] -1 .s

-2

2x1014

[n.cm flux thermal 1x1014 Sh=300mm (E=0 MW.d)_sub-critical Sh=300mm (E=0 MW.d) Sh=500mm (E=0 MW.d) Sh=800mm (E=0 MW.d) -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Figure 51. 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.

134

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 Sm in the middle part 8.0x10 lower part of CR 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 1.5x105 Eu in the upper part 154

Eu in the upper part

5 5

Activity[Ci] 1.0x10 Activity[Ci] 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]

(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 115Cd in the upper part of CR 4 2.50x104 6.0x10 117In in the upper part of CR

2.00x104

4 Activity[Ci] 4 Activity[ Ci] 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]

(b) Cd+Cd

Eu in the upper part

5 5

Activity[Ci] 1.0x10 Activity[Ci] 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]

135

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 8.0x104 4.0x104

6.0x104 Activiy [Ci] Activity(Ci)

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 MCNP Hf+AISI304 Hf+AISI304 3.5x104

4 6.0x104 3.0x10

total 2.5x104 middle part of CR(Hf)

upper part of CR(Hf) 4 175 4 2.0x10 Hf in the middle part 4.0x10 181 Hf in the middle part 1.5x104 175 Activity[Ci] Hf in the upper part Activity[Ci] 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 52. The activity of each part of CR and dominant nuclides in the CR during 1000 EFPD

136

Total Control Rods Effective Worth of Different CR 18 MCNPX 16 T=0

8 [-$] reactivity 6 Cd+Co Hf 4 Hf+AISI304 Hf+AISI: shift=1mm, D=5mm, L(AISI)=140 mm 2 Hf+AISI: shift=2mm, D=5mm, L(AISI)=140 mm Hf+AISI: shift=2mm, D=6mm, L(AISI)=140 mm Hf+AISI: shift=2mm, D=5mm, L(AISI)=70 mm 0 0 100 200 300 400 500 600 700 800 900 Sh [mm]

Figure 53. Comparison of total worth for different dimensions of Hf+AISI304 rod.

137

Differential CR worth 0,040 MCNPX 0,035 T=0

0,030

0,025

0,020

Sh [$/mm]

Δ 0,015 /

Δρ 0,010 Cd+Co 0,005 (1) Hf+Hf (2) Hf+AISI304: D(Hf)=5mm, L(AISI)=140 mm (3) Hf+AISI: shift=1mm, D(Hf)=5mm, L(AISI)=140 mm 0,000 (4) Hf+AISI: shift=2mm, D(Hf)=5mm, L(AISI)=140 mm (5) Hf+AISI: shift=2mm, D(Hf)=5mm, L(AISI)=70 mm, D(AISI)=5mm (6) Hf+AISI: shift=2mm, D(Hf)=5mm, L(AISI)=70 mm, D(AISI)=10mm -0,005 0 100 200 300 400 500 600 700 800 900 Sh [mm]

Differential CR worth 0,040 MCNPX

0,035 T=0

0,030

0,025

0,020

Sh [$/mm]

Δ 0,015

/ Δρ 0,010

0,005 Cd+Co Hf+Hf 0,000 Hf+AISI: shift=2mm, D(Hf)=5mm, L(AISI)=70 mm, D(AISI)=5mm Hf+AISI: shift=2mm, D(Hf)=5mm, L(AISI)=70 mm, D(AISI)=10mm -0,005 0 100 200 300 400 500 600 700 800 900 Sh [mm]

Figure 54. Comparison of differential worth for different dimensions of Hf+AISI304 rod.

138

Axial distribution of thermal neutron flux E <0.5 eV n in Al sample in the axis of FE in A30 4x1014 Sh=400 mm A30

3x1014

] -1 .s -2

2x10

Cd+Co thermal flux [n.cm thermal Hf+Hf 14 1x10 Hf+AISI: L(AISI)=140mm, D(AISI)=5mm Hf+AISI: shift 2 mm; L(AISI)=140mm, D(AISI)=5mm Hf+AISI: shift 2 mm; L(AISI)=70mm, D(AISI)=5mm Hf+AISI: shift 2mm, L(AISI)=70mm, D(AISI)=10mm 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Axial distribution of fast neutron flux En>0.1 MeV

in Al sample in the axis of FE in C41 4,5x1014 Sh=400 mm C41 4,0x1014

3,5x1014

14 ]

-1 3,0x10

.s -2 14 2,5x10

2,0x1014

fast [n.cm flux 1,5x1014

1,0x1014 Cd+Co Hf+Hf 13 5,0x10 Hf+AISI: shift 2 mm; L(AISI)=70mm, D(AISI)=5mm Hf+AISI: shift 2mm, L(AISI)=70mm, D(AISI)=10mm 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 axis of fuel element [cm]

Figure 55. Comparison of axial distributions of thermal and fast fluxes in typical fuel channels for different optimization dimensions of the Hf+AISI304 rod.

139