Physics and Technology of Nuclear Reactors


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The following presentation was created by me (Paul Callaghan) in order to demonstrate learning on the Physics and Technology of Nuclear Reactors Course I attended from Autumn 2007 to Spring 2008 at The University of Birmingham.

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  • Nuclear reactions involve collisions of a nucleus with a particle c.f chemical reactions where the collisions are between whole atoms or molecules To do this, the particle must overcome the coulombic repulsion of nuclei owing to their positive charge approaching to within 10 -12 cm (diameter of nucleus) before interaction can occur. The coulombic energy is very large in the order of MeV and this must be overcome in order for a reaction to occur. Neutrons are ideal for use as the incident particle for these reactions as they are electrically neutral hence they are unaffected by the charge of the nucleus. Within a reactor this means that low energy slow neutrons (~0.03eV) at room temperature can interact with atomic nuclei In fact the probability of collision between a slow neutron and nucleus is greater than that between a fast neutron and nucleus as a slow neutron will spend more time within the vicinity of a nucleus than a fast neutron. In the Compound Nucleus model - a nuclear reaction occurs in 2 stages: Incident particle absorbed by target nucleus creating a compound nucleus Compound nucleus disintegrates expelling a particle (or photon) leaving a recoil nucleus. Radiative Capture is the process whereby a particle is captured and the excess energy is emitted as radiation
  • Definition: A measure of the probability of occurrence of a particular nuclear reaction under prescribed conditions i.e. the probability of collision Microscopic Cross-Section -  Applies to a particular process on a single nucleus Usually described by beam of light model Macroscopic Cross-Section -  Is volumetric and is for a collection of nuclei Related to  by  = N.  Where N = Number of nuclei per cm 3 Nuclear cross-sections commonly of the order 10 -22 to 10 -26 cm 2 per nucleus Unit of measurement is the barn 1 barn = 10 -24 cm 2 per nucleus Different types of macroscopic cross-section for different nuclear processes Absorbtion Cross-Section (  a ) - neutrons “lost” to the system Fission Cross-Section (  f ) – behaviour of incident particle leads to generation of new particles Scatter Cross-Section (  s) – transfer of energy from one particle to another
  • Hydrogen has a high scatter cross section – a property which makes it a very effective moderator material Deuterium also has a high scatter cross section – a property which makes it effective for use as a moderator Boron has massive absorption cross section – a very useful property for a control rod / reactivity suppressant Zirconium has relatively high absorption and scatter cross-sections – making it a useful cladding material
  • Resonance absorption frequently occurs with neutrons of energy between 1eV and 10 eV i.e. thermal neutrons
  • For nuclides with mass number>100 examination of the absorption cross-section (  a ) with neutron energy shows 3 distinct regions: Low Energy (1/V) Region – Cross-section decreases with increasing neutron energy Resonance Region Fast Neutron Region – Cross-sections decrease steadily with increasing neutron energy Resonance Region Categorised by resonance peaks. Found in regions of low neutron energy with elements of higher mass number Reactions are (n,  ) Resonance absorption cross-sections are high often 10 3 barns c.f geometric cross-section of 2 barns.
  • Types of nuclear interaction As a generalized nuclear process, consider a collision in which an incident particle strikes a previously stationary particle, to produce an unspecified number of final products. If the final products are the same as the two initial particles, the process is called scattering. The scattering is said to be elastic or inelastic, depending on whether some of the kinetic energy of the incident particle is used to raise either of the particles to an excited state. If the product particles are different from the initial pair, the process is referred to as a reaction. The most common type of nuclear reaction, and the one which has been most extensively studied, involves the production of two final products. Such reactions can be observed, for example, when deuterons with a kinetic energy of a few MeV are allowed to strike a carbon nucleus of mass 12. Protons, neutrons, deuterons, and alpha particles are observed to be emitted. The nuclei are indicated by the usual chemical symbols; the subscripts indicate the atomic number (nuclear charge) of the nucleus, and the superscripts the mass number of the particular isotope. These reactions are conventionally written in the compact notation 12 C( d , d ) 12 C, 12 C( d , p ) 13 C, 12 C( d , n ) 13 N, and 12 C( d ,α) 10 B, where d represents deuteron, p proton, n neutron, and α alpha particle. In each of these cases the reaction results in the production of an emitted light particle and a heavy residual nucleus. If the residual nucleus is formed in an excited state, it will subsequently emit this excitation energy in the form of gamma rays or, in special cases, electrons. The residual nucleus may also be a radioactive species, in which case it will undergo further transformation in accordance with its characteristic radioactive decay scheme
  • The hydrogen atoms in the water molecules are very close in mass to a single neutron and thus have a potential for high energy transfer, similar conceptually to the collision of two billiard balls. In addition to being a good moderator, water is also fairly effective at absorbing neutrons.
  • Sb – Antimony Te – Tellurium I – Iodine Xe – Xenon Cs – Caesium Ba - Barium Xenon-135 has a huge absorption cross-section (  a ) and is “Poisonous” to a reactor due to soaking up neutrons  a ( 135 Xe) = 2.7x10 6 barns
  • In the periodic table of elements , the series of light elements from hydrogen up to sodium is observed to exhibit generally increasing binding energy per nucleon as the atomic mass increases. This increase is generated by increasing forces per nucleon in the nucleus, as each additional nucleon is attracted by all of the other nucleons, and thus more tightly bound to the whole. The region of increasing binding energy is followed by a region of relative stability (saturation) in the sequence from magnesium through to xenon. In this region, the nucleus has become large enough that nuclear forces no longer completely extend efficiently across its width. Attractive nuclear forces in this region, as atomic mass increases, are nearly balanced by repellent electromagnetic forces between protons, as atomic number increases. Finally, in elements heavier than xenon, there is a decrease in binding energy per nucleon as atomic number increases. In this region of nuclear size, electromagnetic repulsive forces are beginning to gain against the strong nuclear force.
  • U235 fission with fast neutrons produces around 1.7 neutrons per fission - insufficient to sustain a chain reaction.
  • Delayed neutrons emission increases over a period of minutes Delayed neutrons form 5 (possibly more) groups characterised by a specific half-life Total fraction of delayed neutrons is observed to be 0.0075 The spectrum is virtually the same regardless of the initial energy of the incident neutron
  • These neutron-emitting fission fragments are called delayed neutron precursor atoms.
  • In some cases the available energy in the beta decay is high enough to leave the residual nucleus in such a highly excited state that neutron emission instead of gamma emission occurs.
  • 14 MeV Curve is fast neutron energy – Incident Particle Thermal Curve is slow neutron energy – Incident Particle Sum of Fission products is 200% as each fission results in 2 products Fission products are all radioactive (in excited state) Atomic numbers change with time due to beta decay Mass numbers are unaffected hence the yield is expressed in terms of mass number.
  • Xe – Xenon Sm - Samarium Define Multiplication Factor (K) and reactivity (  )
  • Xe – Xenon Sm - Samarium
  • An equation used to study the nonequilibrium behavior of a collection of particles The rate of change of a function which specifies the probability of finding a particle in a unit volume of phase space is equal to the sum of terms arising from: external forces, diffusion of particles collisions of the particles. Also known as Maxwell-Boltzmann equation.
  • Neutrons that are produced in the reactor but independent of the fission chain.
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  • "CANada Deuterium Uranium ” At the time of its design (1950 ’ s), Canada lacked the heavy industry to cast and machine the large, heavy steel pressure vessel used in most light water reactors . Instead, the pressure is contained in much smaller tubes, 10cm diameter, that contain the fuel bundles. These smaller tubes are easier to fabricate than a large pressure vessel. In order to allow the neutrons to flow freely between the bundles, the tubes are made of a zirconium alloy (zirconium + 2.5% wt niobium), which is highly transparent to neutrons. The zircaloy tubes are surrounded by a much larger low-pressure tank known as a calandria , which contains the majority of the moderator . Canada also lacked access to uranium enrichment facilities, which were then extremely expensive to construct and operate. CANDU was therefore designed to use natural uranium as its fuel. Deuterium is used as a moderator as hydrogen in water has high neutron absorption properties which would preclude a sustainable nuclear chain reaction in fuel with a low U235 content. Source:
  • RBMK is an acronym for the Russian reaktor bolshoy moshchnosti kanalniy: "High Power Channel-type Reactor” An RBMK employs long (7 metre) vertical pressure tubes running through a graphite moderator and cooled by water Water (coolant) allowed to boil in the core at 290 C, much as in a boiling water reactor . Fuel is low- enriched uranium oxide made up into fuel assemblies 3.65 metres long. Moderation largely due to the fixed graphite, excess boiling simply reduces the cooling and neutron absorption without inhibiting the fission reaction Reactor can have a large positive void coefficient , and a positive feedback problem can arise, as with the disaster at Chernobyl . In the case of evaporation of water to steam , the place occupied by water would be occupied by water vapor, which has a density vastly lower than that of liquid water (the exact number depends on pressure and temperature; at standard conditions, steam is about 1/1350th as dense as liquid water). Because of this lower density (of mass, and consequently of atom nuclei able to absorb neutrons), light water's neutron-absorption capability practically disappears when it boils. This allows more neutrons to fission more U-235 nuclei and thereby increase the reactor power, which leads to higher temperatures that boil even more water, creating a thermal feedback loop Source:
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  • Physics and Technology of Nuclear Reactors

    1. 1. Physics and Technology of Nuclear Reactors Paul Callaghan Consultant Engineer
    2. 2. A bit about me <ul><li>(2008 – Present) Consultant Engineer – Atkins (Glasgow/Epsom/Bristol) </li></ul><ul><li>(2006-2008) Stress Engineer – Rolls-Royce Submarines (Derby) </li></ul><ul><li>(2005-2006) Planning/Manufacturing Engineer – Rolls-Royce Submarines (Derby) </li></ul><ul><li>(2004) Undergraduate Engineer – Rolls-Royce AR&O (East Kilbride) </li></ul><ul><li>(2003) Undergraduate Engineer – Rolls-Royce AR&O (East Kilbride) </li></ul><ul><li>(2002) Undergraduate Engineer – Rolls-Royce AR&O (East Kilbride) </li></ul><ul><li>(2000 – 2004) B.Eng (Hons) Aeronautical Engineering (University of Glasgow) </li></ul>
    3. 3. Purpose <ul><li>The following presentation was created by me (Paul Callaghan) in order to demonstrate learning on the Physics and Technology of Nuclear Reactors Course I attended from Autumn 2007 to Spring 2008 at The University of Birmingham. </li></ul><ul><li>I delivered this presentation to a selection of my peers to satisfy the requirements of Further Learning (Engineering and Science Deepening) for the IMechE. </li></ul><ul><li>The presentation was created in order to demonstrate my understanding of nuclear physics and the physics which underpins the operation of Nuclear Reactors. </li></ul>
    4. 4. Contents <ul><li>General Nuclear Physics </li></ul><ul><li>Fission Processes </li></ul><ul><li>Transport Theory </li></ul><ul><li>Point Kinetics Equations </li></ul><ul><li>Reactor Systems </li></ul>
    5. 5. <ul><li>General Nuclear Physics </li></ul><ul><li>Interactions of neutrons with matter </li></ul><ul><li>Cross-Sections </li></ul><ul><li>Resonance Effects </li></ul><ul><li>U 235 Absorption Cross Section vs Energy </li></ul><ul><li>Scattering </li></ul><ul><li>Importance of Xenon transients </li></ul>Learning Outcomes:
    6. 6. Interactions of Neutrons with matter (1) <ul><li>The energy released from a nuclear reaction is much higher than from a chemical reaction e.g. burning coal, oil or gas </li></ul><ul><li>Burning coal releases 4 eV per reaction whereas a (nuclear) fission reaction produces 200 million eV (MeV). </li></ul>Prompt Energies Daughter nuclei of fission fragments ~169 MeV K.E of (2.5) neutrons ~5 MeV Gamma ray photons ~7 MeV Delayed Energies Beta (from decay) ~6.5 MeV Anti-neutrinos ~8.8 MeV Delayed Gamma Emission ~6.3 MeV
    7. 7. Interactions of neutrons with matter (2) <ul><li>Nuclear reactions involve collisions of a nucleus with a particle </li></ul><ul><li>Neutrons are ideal for use as incident particle as they are electrically neutral </li></ul><ul><li>According to the Compound Nucleus model - a nuclear reaction occurs in 2 stages: </li></ul><ul><ul><li>Incident particle absorbed by target nucleus creating a compound nucleus </li></ul></ul><ul><ul><li>Compound nucleus disintegrates expelling a particle (or photon) leaving a recoil nucleus. </li></ul></ul><ul><li>Radiative Capture is the process whereby a particle is captured and the excess energy is emitted as radiation </li></ul>
    8. 8. Cross-sections (1) <ul><li>Definition: A measure of the probability of occurrence of a particular nuclear reaction under prescribed conditions i.e. the probability of collision </li></ul><ul><li>Microscopic Cross-Section -  </li></ul><ul><ul><li>Applies to a particular process on a single nucleus </li></ul></ul><ul><li>Macroscopi c Cross-Section -  </li></ul><ul><ul><li>Is volumetric and is for a collection of nuclei </li></ul></ul><ul><ul><li>Related to  by  = N.  </li></ul></ul><ul><ul><ul><li>Where N = Number of nuclei per cm 2 </li></ul></ul></ul><ul><li>Nuclear cross-sections commonly of the order 10 -22 to 10 -26 cm 2 per nucleus </li></ul><ul><li>Unit of measurement is the barn </li></ul><ul><ul><li>1 barn = 10 -24 cm 2 per nucleus </li></ul></ul><ul><li>Different types of macroscopic cross-section for different nuclear processes </li></ul><ul><ul><li>Absorption Cross-Section (  a ) - neutrons “lost” to the system </li></ul></ul><ul><ul><li>Fission Cross-Section (  f ) – behaviour of incident particle leads to generation of new particles </li></ul></ul><ul><ul><li>Scatter Cross-Section (  s) – transfer of energy from one particle to another </li></ul></ul>
    9. 9. Cross-Sections (2) – Typical Reactor Material Values Source: The Elements of Nuclear Reactor Theory 2 nd Edition - Glasstone and Edlund Element Total -  t (barns) Absorption -  a (barns) Scatter -  s (barns) H 20-80 0.32 20-80 D 2 0 15.3 0.00092 15.3 B 722 718 3.8 Zr 8.4 0.4 8.0
    10. 10. Fast and Slow Neutrons <ul><li>Fast Neutron - a free neutron with a kinetic energy level of about 1 MeV (100 TJ/kg), hence a speed of 14,000 km/s. </li></ul><ul><li>Slow Neutron - a free neutron with a kinetic energy of about 0.03 eV (2.4 MJ/kg) (1/40) hence a speed of 2.2 km/s </li></ul><ul><li>Slow neutrons are often referred to as Thermal Neutrons as their energy corresponds to the most probable velocity at a temperature of 290 K/17°C (Room Temperature) </li></ul><ul><li>Thermal neutrons have a different and often much larger effective neutron absorption cross-section for a given nuclide than fast neutrons, and can therefore often be absorbed more easily by an atomic nucleus </li></ul>
    11. 11. Resonance Effects (1) <ul><li>Experimental studies have shown that bombarding different target elements with projectiles of specific energy values causes a sharp increase in reaction rate. </li></ul><ul><li>For certain energy values the probability that the incident particle will be captured and a compound nucleus formed is exceptionally large. </li></ul><ul><li>This phenomenon is attributed to resonance. </li></ul>
    12. 12. Absorption Cross-Section vs Neutron Energy for U 235 (1)
    13. 13. Scattering (1) <ul><li>Definition: The process in which the overall result is transfer of energy from one particle to another </li></ul><ul><li>Two kinds: </li></ul><ul><ul><li>Elastic Scatter – Kinetic energy and momentum conserved </li></ul></ul><ul><ul><li>Inelastic Scatter – Kinetic energy not conserved, momentum conserved. </li></ul></ul><ul><li>Fast neutrons may be deprived of their kinetic energy and slowed down to become slow neutrons with an energy of ~0.03eV at room temperature. </li></ul><ul><li>The slowing down is performed by inelastic scatter in a process known as moderation </li></ul>
    14. 14. Scattering (2) <ul><li>The medium used in this process is the moderator </li></ul><ul><ul><li>Typically involves atoms of low mass number e.g. H 2 0 or D 2 0 </li></ul></ul><ul><ul><li>Efficient moderators reduce the speed of fast neutrons in as few collisions as possible </li></ul></ul><ul><li>After a number of scattering collisions, the kinetic energy of the neutrons is reduced such that it is similar to the moderator medium. </li></ul><ul><li>The new energy depends on the temperature of the medium and is the thermal energy . </li></ul><ul><li>Neutrons of this energy are thermal neutrons . </li></ul><ul><li>The process is thermalisation . </li></ul>
    15. 15. Importance of Xenon Transients (1) <ul><li>Xenon-135 is a fission product poison produced during fission of U 235 and U 238 </li></ul><ul><li>Xenon-135 is formed from successive beta decays of it’s fission product precursors </li></ul><ul><ul><li>51 Sb 135 52 Te 135 53 I 135 54 Xe 135 55 Cs 135 56 Ba 135 </li></ul></ul><ul><li>135 Xe is of particular concern in a reactor as it has a half-life of 9.1 hrs compared with a 6.6 hr half-life of its precursor 53 I 135 </li></ul><ul><ul><li>Thus 53 I 135 decays quicker to 54 Xe 135 than 54 Xe 135 can decay </li></ul></ul><ul><ul><li>Leads to increased concentration of 54 Xe 135 </li></ul></ul>
    16. 16. Importance of Xenon Transients (2) <ul><li>On restart after a shutdown, the Xenon transient becomes important as the reactivity must be greater than the absorbing effect of the Xenon to establish criticality. </li></ul><ul><li>Increases to reactivity are achieved by withdrawing the control rods </li></ul><ul><li>If the absorbing effect of Xenon concentration in core is greater than the reactivity that can be achieved by withdrawing the control rods – criticality cannot be achieved! </li></ul>
    17. 17. <ul><li>Fission Processes </li></ul><ul><li>Binding energy curve </li></ul><ul><li>Number of neutrons per fission </li></ul><ul><li>Prompt and delayed neutrons </li></ul><ul><li>Delayed neutrons from fission products </li></ul><ul><li>Fission yield curve </li></ul><ul><li>Importance of reactor poisons </li></ul>Learning Outcomes:
    18. 18. Binding Energy Curve
    19. 19. Number of Neutrons per fission <ul><li>U 235 undergoes fission with thermal neutrons as well as those of higher energies. </li></ul><ul><li>It has been observed that fission of U 235 with slow neutrons produces 2.5 ±0.1 neutrons per fission </li></ul><ul><li>Not an integer as U nucleus splits in a number of different ways </li></ul><ul><ul><li>Individually discrete </li></ul></ul><ul><ul><li>Mean may not be whole number </li></ul></ul>
    20. 20. Prompt and Delayed Neutrons (1) <ul><li>Two categories of neutron emitted from fission: prompt and delayed . </li></ul><ul><li>Prompt neutrons </li></ul><ul><ul><li>released in 10 -14 sec </li></ul></ul><ul><ul><li>account for 99% of total fission neutrons </li></ul></ul><ul><ul><li>energies cover considerable range c.f Watt Spectrum </li></ul></ul><ul><li>Delayed neutrons are emitted by one of the fission products anytime from a few milliseconds to a few minutes later </li></ul>
    21. 21. Prompt and Delayed Neutrons (2) <ul><li>Delayed neutrons make it possible to run a reactor subcritically (in terms of prompt neutrons) </li></ul><ul><li>Delayed neutrons come a moment later, just in time to sustain the chain reaction when it is going to die out </li></ul><ul><li>Consequently, neutron production overall still grows exponentially, but on a time scale slow enough to be controlled </li></ul>
    22. 22. Prompt and Delayed Neutrons (2) Fission (Watt) Spectrum Majority 1-2 MeV Range: 0.03eV- 10MeV S(E)=0.484e -E sinh  2E
    23. 23. Delayed Neutrons from Fission Products (1) <ul><li>In the case of U-235, the nucleus absorbs thermal neutrons </li></ul><ul><li>The immediate mass products of a fission event are two large fission fragments. </li></ul><ul><li>These fragments emit, on average, two or three free neutrons (2.5±0.1), prompt neutrons. </li></ul><ul><li>A subsequent fission fragment occasionally undergoes a stage of radioactive decay that yields an additional neutron, called a delayed neutron. </li></ul>
    24. 24. Delayed Neutrons from Fission Products (2) <ul><li>Delayed neutrons are associated with the beta decay of the fission products. </li></ul><ul><li>After prompt fission neutron emission the residual fragments are still neutron rich and undergo a beta decay chain. </li></ul><ul><li>The more neutron rich the fragment, the more energetic and faster the beta decay. </li></ul>
    25. 25. Delayed Neutrons from Fission Products (3) Properties of Delayed Neutrons in Slow Neutron Fission of U 235 Source: The Elements of Nuclear Reactor Theory 2 nd Edition - Glasstone and Edlund Half Life (Sec) Mean Life (Sec) Decay Constant (Sec -1 ) Fraction  I Energy (MeV) 0.43 0.62 1.61 0.00084 0.42 1.52 2.19 0.456 0.0024 0.62 4.51 6.5 0.151 0.0021 0.43 22.0 31.7 0.0315 0.0017 0.56 55.6 80.2 0.0124 0.00026 0.25
    26. 26. Fission Yield Curve – U 235 Light Group Heavy Group
    27. 27. Importance of Reactor Poisons (1) <ul><li>Nuclear (or Neutron) poisons are substances with large neutron absorption cross-sections which retard +ve reactivity </li></ul><ul><li>Two main types of poison can be found within the core: </li></ul><ul><ul><li>Transient fission product poisons </li></ul></ul><ul><ul><li>Control poisons </li></ul></ul><ul><li>Transient fission product poisons generated by fission of U 235 and subsequent beta decay </li></ul><ul><li>Common fission product poisons are 135 Xe and 149 Sm </li></ul><ul><ul><li> a ( 135 Xe) = 2.7x10 6 barns </li></ul></ul><ul><ul><li> a ( 149 Sm ) = 4.1x10 4 barns </li></ul></ul><ul><li>Poisoning of a reactor core by fission product poison may become significant when the –ve reactivity of the poison > +ve reactivity of the fuel </li></ul><ul><ul><li>Stops chain reaction and criticality is lost. </li></ul></ul>
    28. 28. Importance of Reactor Poisons (2) <ul><li>Fuel loading in reactor cores is often greater than that required for exact criticality in order to prolong reactor life. </li></ul><ul><li>The +ve reactivity of the excess fuel must be balanced by –ve reactivity of an added neutron absorbing material. </li></ul><ul><li>The control poison employed to absorb neutrons may be 1 of 3 kinds: </li></ul><ul><ul><li>Burnable poison </li></ul></ul><ul><ul><li>Non-burnable poison </li></ul></ul><ul><ul><li>Soluble poison </li></ul></ul>
    29. 29. Importance of Reactor Poisons (3) <ul><li>Burnable poisons are materials of high neutron absorption cross-section and are incorporated into the core structure as rods, pins or plates dependant on reactor construction. </li></ul><ul><li>Typical materials used as burnable poisons are Boron alloys and Gadolinium alloys </li></ul><ul><li>Burnable poisons are depleted by absorption of neutrons from fuel and converted to material of lesser neutron absorption cross-section. </li></ul><ul><ul><ul><li>5 B 10 + 0 n 1 3 Li 7 + 2 He 4 </li></ul></ul></ul><ul><li>An ideal burnable poison will deplete (burn-up) at the same rate as reactivity is lost from the fuel. </li></ul>
    30. 30. Importance of Reactor Poisons (4) <ul><li>Non-Burnable poisons maintain a constant negative reactivity worth throughout core life. </li></ul><ul><ul><li>A typical non-burnable poison material is Hafnium. </li></ul></ul><ul><ul><li>The removal of one isotope of Hafnium by neutron absorption leads to the formation of another isotope of equivalent absorption cross-section a process which continues through a chain of 5 absorbers. </li></ul></ul><ul><ul><li>The result of this is a long-life burnable poison approximating non-burnable characteristics. </li></ul></ul><ul><li>Soluble poisons produce spatially uniform absorption when dissolved in water coolant. </li></ul><ul><ul><li>A common soluble poison used in commercial PWR plant is Boric Acid (H 3 BO 3 ) </li></ul></ul><ul><ul><li>The Boron concentration in the water may be controlled by adding more water (dilution) or adding more Boron. </li></ul></ul><ul><li>The benefit of maintaining control in this way is a flatter flux profile more so than could be obtained by control rod insertion. </li></ul>
    31. 31. <ul><li>Transport Theory </li></ul><ul><li>Neutron Distribution </li></ul><ul><li>Boltzmann Transport Equation </li></ul><ul><li>Importance of terms </li></ul><ul><li>Strategies for solving </li></ul>Learning Outcomes:
    32. 32. Neutron Distribution <ul><li>Reactor Physics deals with the determination of neutron distribution in: </li></ul><ul><li>Space </li></ul><ul><li>Energy </li></ul><ul><li>Time </li></ul><ul><li>Neutron Transport Theory is used to determine neutron distribution by solution of the Boltzmann Transport Equation </li></ul><ul><li>Upon determining neutron distribution, we can then apply knowledge to determine: </li></ul><ul><li>Neutron reaction rates (R) </li></ul><ul><li>Power distribution (P) </li></ul><ul><li>Multiplication Factor (K) </li></ul><ul><li>Reactivity Coefficients ( ρ ) </li></ul>
    33. 33. Boltzmann Transport Equation <ul><li>Describes the distribution of neutrons in a host medium as a function of their position ( r ), energy (E), direction of motion ( Ω ) and time (t) </li></ul><ul><li>Derived by following the principal of neutron conservation in an infinitesimal region of space, time and direction. </li></ul><ul><li>The density of neutrons is very high hence we need to calculate only their ensemble average behaviour to solve for the local fission rate </li></ul>Balance term Leakage/Streaming Term Removal mechanisms (absorption and scatter out) Double differential scatter cross-section Flux term for neutron behaviour Source term
    34. 34. Balance term <ul><li>Describes the time rate of change of neutron flux with respect to distance ( r ), energy (E), direction ( Ω ) and time (t) </li></ul>
    35. 35. Leakage term <ul><li>Also known as the streaming term </li></ul><ul><li>Measures the net rate at which neutrons are entering or leaving the volume </li></ul>
    36. 36. Removal Mechanisms <ul><li>Σ t = Σ a + Σ s </li></ul><ul><li>Σ a - neutrons “lost” to the system by absorption </li></ul><ul><li>Σ s - neutrons “lost” to the system via scatter </li></ul>
    37. 37. Double Differential Scatter Cross-Section <ul><li>Gain mechanism </li></ul><ul><li>Describes the gain due to neutrons scattering into dE about E and d Ω about Ω from other energies E` and directions Ω ` </li></ul><ul><li>Also known as the “in-scattering” term </li></ul>
    38. 38. Angular flux <ul><li>Gain mechanism </li></ul>
    39. 39. Source neutron density <ul><li>Can be either Loss (-ve) or Gain (+ve) mechanism </li></ul><ul><li>External to the system </li></ul><ul><li>Describes neutrons streaming into the volume V through the surface s </li></ul>
    40. 40. Strategies for solving <ul><li>Monte Carlo </li></ul><ul><li>Angular variation simplification </li></ul><ul><li>Energy variation simplification </li></ul><ul><li>Space and Time simplification </li></ul>
    41. 41. Monte Carlo (1) <ul><li>Statistical method based on inventing particles and following their histories </li></ul><ul><li>Mimics microscopic physics of the problem by using: </li></ul><ul><ul><li>Total cross-sections ( Σ t ) as a sum of absorption ( Σ a ) , fission ( Σ f ) and scatter ( Σ s ) cross-sections </li></ul></ul><ul><ul><li>Number densities </li></ul></ul><ul><ul><li>Scatter dynamics </li></ul></ul><ul><li>The premise is to follow many histories in order to approximate the real world </li></ul>
    42. 42. Monte Carlo (2) <ul><li>Optimisation techniques: </li></ul><ul><ul><li>Histories not allowed to be stopped by absorption </li></ul></ul><ul><ul><li>Treat all interactions as scatters but adjust weight such that continuing particle probability = scatter probability </li></ul></ul><ul><ul><li>Splitting to keep number of particles significant in important regions: rouletting </li></ul></ul>
    43. 43. Angular Variation Simplification <ul><li>Break up angular variation into N different portions called Discrete Ordinate or S N theory, assuming no variation within a portion </li></ul><ul><li>Represent the angular variation of φ by a functional form usually with Legendre polynomials. </li></ul><ul><ul><li>Known as P N theory using N Legendre polynomials to describe angular variation. </li></ul></ul>
    44. 44. Energy Variation Simplification <ul><li>Similar process to S N theory where the energy range is broken up into discrete energy groups assuming no variation. </li></ul>
    45. 45. Space and time <ul><li>Assume separation of spatial and temporal effects </li></ul><ul><li>Use discrete ordinate approach to calculate values (e.g. φ ) at different spatial mesh points. </li></ul><ul><li>Often different mesh sizes are used in different regions depending on accuracy required for solution: </li></ul><ul><ul><li>Fine mesh used over detailed geometry </li></ul></ul><ul><ul><li>Coarse mesh employed on rest of model </li></ul></ul><ul><li>Large number of calculations at each mesh point (i.e. 10 directional values, 5 energy groups) can quickly lead to large numbers and huge computational cost </li></ul>
    46. 46. <ul><li>Point Kinetics Equations </li></ul><ul><li>Delayed Neutron Fraction </li></ul><ul><li>Importance of Delayed Neutrons </li></ul><ul><li>Prompt Criticality </li></ul>Learning Outcomes:
    47. 47. Requisite Knowledge (1) Where: k= number of neutrons in current generation number of neutrons in previous generation And: ρ = k - 1 k State Multiplication Factor (k) Reactivity (ρ) Sub Critical k < 1 ρ< 0 Critical k = 1 ρ = 0 Super Critical k > 1 ρ > 0
    48. 48. Requisite Knowledge (2)
    49. 49. Delayed Neutron Fraction (1) <ul><li>If β is the fraction of delayed fission fragments then (1- β ) represents the fraction of prompt neutrons </li></ul><ul><li>Of the total number of fast neutrons produced for thermal neutron absorption (1- β ) η are emitted instantaneously </li></ul><ul><ul><ul><ul><li>η – average no of fast fission neutrons emitted as the result of the capture of one thermal neutron in the fuel material </li></ul></ul></ul></ul><ul><li>Βη therefore represents the fraction of delayed neutrons released over time </li></ul><ul><li>The Multiplication factor can therefore be said to exist of two parts: </li></ul><ul><ul><li>k (1- β ) - prompt neutron multiplication factor </li></ul></ul><ul><ul><li>k β – delayed neutron multiplication factor </li></ul></ul>
    50. 50. Delayed Neutron Fraction (2) <ul><li>If in the reactor operation the prompt neutron multiplication factor k (1- β ) is adjusted to be just less than (or equal to) unity </li></ul><ul><li>then </li></ul><ul><li>The rate of increase of neutrons form one generation to the next will be determined by the delayed neutrons </li></ul><ul><li>Since β is approx. 0.0075 for thermal fission this can be realised by having an effective multiplication factor (k eff ) between 1 and 1.0075 </li></ul><ul><li>consequently </li></ul><ul><li>The neutron flux and power level will increase relatively slowly and adequate control is possible. </li></ul>
    51. 51. Importance of Delayed Neutrons <ul><li>Delayed neutrons are responsible for the ability to control the rate at which power can rise in a reactor.   </li></ul><ul><li>If a nuclear reactor happened to be prompt critical - even very slightly - the number of neutrons would increase exponentially and very quickly the reactor would become uncontrollable </li></ul><ul><li>By widening the margins of non-operation and supercriticality and allowing more time to regulate the reactor, delayed neutrons are essential to inherent reactor safety </li></ul>
    52. 52. Prompt Criticality <ul><li>When the effective multiplication factor (k eff ) is equal to 1.0075, the reactor is described as prompt critical </li></ul><ul><li>The nuclear fission chain can be maintained by the prompt neutrons alone. </li></ul><ul><li>If k exceeds this value, multiplication will occur due to prompt neutrons alone irrespective of delayed neutron population resulting in a rapid exponential increase in flux and power – neutron prompt super critical </li></ul><ul><li>This is to be avoided at all costs c.f Chernobyl </li></ul>
    53. 53. <ul><li>Reactor Systems </li></ul><ul><li>Gas Cooled Reactors – Magnox & AGR </li></ul><ul><li>Light Water Reactors – PWR & BWR </li></ul><ul><li>Pressure Tube Reactors – CANDU & RBMK-1000 </li></ul><ul><li>Fast Reactors </li></ul>Learning Outcomes:
    54. 54. Gas Cooled Reactors – Magnox Fuel Uranium Tetrafluoride and Magnesium Control Safety Rods: Boron Steel Bulk Rods: Boron Steel Fine Control: Mild Steel Coolant CO 2 Cladding Magnox Can Moderator Graphite
    55. 55. Gas Cooled Reactors – AGR Fuel UO 2 fuel pellets Cladding Stainless Steel Control Coarse rods: Cr-Mo-B Alloy Fine rods: Cr-Mo Moderator Graphite Coolant CO 2
    56. 56. Light Water Reactors – PWR Fuel UO 2 fuel pellets (3-4% U 235 ) Cladding Zircaloy4 Burnable Poison Boric acid in primary circuit water Control Boron carbide alloy Moderator/Coolant Water (H 2 O)
    57. 57. Light Water Reactors – BWR Fuel Sintered UO 2 fuel pellets (2-3% U 235 ) Cladding Zircaloy Burnable Poison Gadolinium Oxide Control Boron alloy Moderator/Coolant H 2 O
    58. 58. Pressure Tube Reactors – CANDU Fuel UO 2 fuel pellets Cladding Zircaloy Control Short term: Gd 2 O 3 Long term: D 2 0 Moderator Deuterium (D 2 O) Coolant Deuterium (D 2 O) in separate circuit to moderator
    59. 59. Pressure Tube Reactors – RBMK-1000
    60. 60. Fast Reactors <ul><li>The fast breeder reactor (FBR) uses a plutonium fuel rather than uranium. </li></ul><ul><li>The Pu is surrounded by rods of U-238 which absorb neutrons and are transmuted into Pu-239. </li></ul><ul><li>As the plutonium in the core becomes depleted it creates or breeds more plutonium from the uranium around it. </li></ul><ul><li>Because of the extreme temperatures surrounding the reactor a special coolant of liquid sodium (Na) is used to transfer heat to the steam generator. </li></ul><ul><li>The role of the steam generator is to generate steam which can then drive a turbine. </li></ul>
    61. 61. Reactor Summary Reactor Type Plant Design Fuel Clad Burnable Poison Control Moderator/Coolant Problems Graphite Moderated Reactors Magnox Uranium Tetrafluoride + Magnesium Magnox Can None Safety Rods: Boron Steel Bulk Rods: Boron Steel Fine Control: Mild Steel Graphite Moderator/ CO 2 Coolant (i) Creep – irradiation and temperature (ii) Ratchetting of fuel elements (iii) CO 2 Oxidation of fuel (iv) Fuel element swelling (v) Fin waving Advanced Gas Cooled Reactor (AGR) UO 2 Fuel Pellets Stainless Steel Gadolinium Coarse Rods: Cr-Mo-B Alloy Fine Rods: Cr-Mo Graphite Moderator/ CO 2 Coolant <ul><li>Water vapour </li></ul><ul><li>C deposition on fuel </li></ul><ul><li>Pellet clad interaction – clad fracture </li></ul><ul><li>End cap failures </li></ul><ul><li>Spalled oxides </li></ul><ul><li>Boiler cracking </li></ul>Light Water Reactors (LWR) Pressurised Water Reactor (PWR) UO 2 Fuel Pellets Slight Enrichment (3-4% U 235 ) Zircaloy4 Boric Acid (in primary circuit water) Ag-In-Cd Alloy or Boron Carbide Alloy Water (H 2 O) <ul><li>Coolant radioactivity during operation </li></ul><ul><li>Radiolysis of H 2 0 – recombination potentially explosive </li></ul><ul><li>Corrosion products in primary circuit </li></ul><ul><li>LOCA issues </li></ul>Boiling Water Reactor (BWR) Sintered UO 2 Fuel Pellets(2-3% U 235 ) Zircaloy Gadolineum Oxide (Gd 2 O 3 ) Cruciform – Probably a Boron alloy Water (H 2 O) <ul><li>Cannot inhibit radiolytic breakdown of H 2 O </li></ul><ul><li>Oxygenated water in core leads to SCC in pipework </li></ul>Pressure Tube Reactors CANDU Natural UO 2 Fuel Pellets Zircaloy None Short term: Gadolinium oxide control rods Long term: poisoning by moderator Deuterium (D 2 O) Moderator and coolant in separate circuits (i) Failure of pressure tubes (ii) Boiler problems (iii) Pressure tube bowing under irradiation RBMK-1000 Enriched UO 2 Fuel Pellets (2% U 235 ) Zircaloy None Boron control rods (211) Moderator: Graphite Coolant: H 2 O <ul><li>Positive void coefficient </li></ul><ul><li>Graphite tipped control rods </li></ul>
    62. 62. References <ul><li>Content </li></ul><ul><ul><li>Physics and Technology of Nuclear Reactors Course, Course Notes 2007-2008, School of Physics and Astronomy, University of Birmingham </li></ul></ul><ul><ul><li>Nuclear Reactor Theory, Glasstone and Edlund, Second Printing, Macmillan and Co Limited </li></ul></ul><ul><ul><li>Atomic Archive - </li></ul></ul><ul><ul><li>Encyclopedia Britannica - </li></ul></ul><ul><ul><li>European Nuclear Society - </li></ul></ul><ul><li>Assorted reactor images courtesy of: </li></ul><ul><ul><li> </li></ul></ul><ul><ul><li> </li></ul></ul>