Arn 02-0-reactor theory


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Arn 02-0-reactor theory

  1. 1. Nuclear Reactor Theory - Nuclear Reactor Analysis -
  2. 2. FRM-II
  3. 3. The Neutron Flux <ul><li>The Neutron Flux is the Main Variable in Nuclear Reactor Theory </li></ul><ul><ul><li>To design and analyze a nuclear reactor it is necessary to predict: </li></ul></ul><ul><ul><ul><li>How the neutrons are spatially distributed. </li></ul></ul></ul><ul><ul><ul><li>How the neutron population evolves with time. </li></ul></ul></ul><ul><ul><li>An exact calculation would need to track the neutrons as they move in the system. </li></ul></ul><ul><ul><li>This is not possible with current computer capabilities </li></ul></ul><ul><ul><li>We need to use Approximations: </li></ul></ul><ul><ul><ul><li>Monte Carlo Methods. </li></ul></ul></ul><ul><ul><ul><li>Analytic Neutron Transport Methods. </li></ul></ul></ul><ul><ul><ul><li>Neutron Diffusion Approximation. </li></ul></ul></ul>
  4. 4. The Neutron Flux <ul><li>In a reactor neutrons have many different energies: </li></ul><ul><li>The neutrons move with velocities </li></ul><ul><li>The neutrons interact with a probability per unit length </li></ul><ul><li>The reaction rate can be written as </li></ul><ul><li>The neutron FLUX is </li></ul>
  5. 5. The Neutron Flux <ul><li>Typical Neutron Energy Spectrum </li></ul>Ref.:
  6. 6. FRM-II
  7. 7. The Diffusion Approximation: Fick´s Law <ul><li>Neutrons DIFFUSE in the medium as Chemical Species do in solution. </li></ul><ul><ul><li>A net flow of neutrons exists from HIGH Flux to LOW Flux regions. </li></ul></ul><ul><ul><li>For a one-dimemsional, one-energy system: </li></ul></ul><ul><ul><li>J x is the NET number of neutrons that pass per unit time through an area perpendicular to the direction x. </li></ul></ul><ul><ul><li>D is the Diffusion Coefficent (cm) </li></ul></ul>Neutron Current Density Vector x  (x) J x Gradient of Flux
  8. 8. The Equation of Continuity <ul><li>It is the conservation equation of neutrons in a medium </li></ul><ul><li>The Total number of neutrons in a Volume V is </li></ul><ul><li>The Rate of Change is: </li></ul>V r
  9. 9. The Equation of Continuity <ul><li>The Rate of Production in the Volume V is </li></ul><ul><li>The Rate of Absorption in the Volume V is </li></ul><ul><li>Rate of Leakage through the surface A </li></ul>Surface J x J z J y
  10. 10. The Diffusion Equation <ul><li>The Continuity Equation </li></ul><ul><li>The Diffusion Eq . </li></ul>Fick´s Law Diffusion Length The integrands must satisfy The integration is over the same volume V
  11. 11. The Diffusion Equation <ul><li>Boundary Conditions </li></ul><ul><ul><li>The neutron flux can be found by solving the diffusion equation. </li></ul></ul><ul><ul><li>It requires the specification of BOUNDARY conditions for the FLUX. </li></ul></ul><ul><ul><ul><li>Physical: </li></ul></ul></ul><ul><ul><ul><ul><li>The flux must be always POSITIVE. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>The flux must be FINITE. </li></ul></ul></ul></ul><ul><ul><ul><li>Geometry: </li></ul></ul></ul><ul><ul><ul><ul><li>Vacuum Boundary Condition: Unreflected Core </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Interface Boundary Conditions for two adjacent regions </li></ul></ul></ul></ul>A B n d  x)  d)=0
  12. 12. The Diffusion Equation <ul><li>The Diffusion Variables </li></ul><ul><li>Validity of the Fick´s Law Approximation: </li></ul><ul><ul><li>Fick´s Law IS NOT an exact relation, but an approximation. </li></ul></ul><ul><ul><li>It is not valid: </li></ul></ul><ul><ul><ul><li>In a medium that strongly absorbs neutrons (e.g. near control rods). </li></ul></ul></ul><ul><ul><ul><li>Within about three mean free paths of either a neutron source or the surface of a medium. </li></ul></ul></ul><ul><ul><ul><li>For strongly anisotropic neutron scattering. </li></ul></ul></ul>Low A High A Reflectors have LOW A
  13. 13. One-group Reactor Equation <ul><li>The design of a reactor requires </li></ul><ul><ul><li>The calculations of the conditions necessary for criticality. </li></ul></ul><ul><ul><li>The calculation of the distribution of neutrons to determine the power distribution in the system: </li></ul></ul><ul><ul><ul><li>Establish the thermal conditions. </li></ul></ul></ul><ul><ul><ul><li>Determine the needs for heat removal during operation and abnormal conditions. </li></ul></ul></ul><ul><li>The simplest equation is for a “bare” Fast Reactor </li></ul><ul><ul><li>One-group Flux and Neutronic Parameters </li></ul></ul><ul><ul><li>One-group Reactor Equation: </li></ul></ul>
  14. 14. One-group Reactor Equation <ul><li>The source of neutrons </li></ul><ul><ul><li>In a steady state reactor the source of neutrons is mainly the fissions in the fuel. </li></ul></ul><ul><ul><li>The average number of neutrons per fission is </li></ul></ul><ul><ul><li>The source can be expressed in terms of the rate of absorptions as </li></ul></ul><ul><ul><li>And in Terms of k ∞ </li></ul></ul>
  15. 15. One-group Reactor Equation <ul><ul><li>In Steady-state, if the fission source does not balance the Leakage and the absorption, the equation is not satisfied. </li></ul></ul><ul><ul><li>The source term is multiplied by 1/k eff . </li></ul></ul><ul><ul><li>The Buckling is defined as </li></ul></ul><ul><ul><li>And the k eff is </li></ul></ul>Material Buckling Geometric Buckling
  16. 16. Criticality of a Bare Reactor <ul><li>The necessary condition for the reactor to be critical is </li></ul><ul><li>In terms of the Buckling for a critical reactor: </li></ul><ul><ul><li>The equation determines the conditions under which a bare reactor is critical. </li></ul></ul><ul><ul><ul><li>For a given geometry, which determines the buckling, the composition can be calculated. </li></ul></ul></ul><ul><ul><ul><li>For a given composition, the “ CRITICAL ” buckling can be computed and the geometric dimensions obtained. </li></ul></ul></ul>k eff accounts also for the leakage Material Properties
  17. 17. One-group Critical Reactor Equation <ul><li>Examples of Solutions </li></ul>Power Energy per fission The Solution for the Flux is: Buckling Spherical Critical Reactor The flux is a function only of the radius r Homogeneous Reactor The power is given by the integral There are many possible values of B that will satisfy the boundary conditions , but the geometrical buckling is the FIRST eigenvalue B 1 r R
  18. 18. One-group Critical Reactor Equation <ul><li>Examples of Solutions </li></ul>Finite Cylindrical Critical Reactor The flux is a function of the radius r and z Two Functions: The solution: Homogeneous Reactor z r H/2 H/2 R
  19. 19. One-group Critical Reactor Equation <ul><li>Maximum-to-Average Flux and Power </li></ul><ul><ul><li>The maximum value of the flux  max in a uniform bare reactor is always found at the center. </li></ul></ul><ul><ul><li>The power density is also highest at the center. </li></ul></ul><ul><ul><li>The maximun-to-average flux ratio  is a measure of the overall variation of the flux in the system. </li></ul></ul><ul><li>For a spherical bare reactor </li></ul>Too large for a real reactor. Real reactors have FLATTER Flux distributions by using reflectors and distributing the fuel.
  20. 20. Multi-group Reactor Equation <ul><li>For thermal reactors and for accurate solutions it is necessary to solve the diffusion equation the energy dependency to obtain </li></ul><ul><ul><li>More precise description of Cross Section energy dependency. </li></ul></ul><ul><ul><li>More accurate reaction rates (fission, absorption, scattering, etc.) </li></ul></ul><ul><ul><li>The process of moderation and resonance absorption. </li></ul></ul><ul><ul><li>The thermal and fast fission rates. </li></ul></ul><ul><li>The energy spectrum is divided into “ENERGY GROUPS”: g 1 , g 2 ,…,g N </li></ul>g 1 g 2 g N …… .. Discretization of the Neutron Energy for Multi-group Calculations
  21. 21. Multi-group Reactor Equation <ul><li>The “transfer” of neutrons between groups is accounted for by: </li></ul><ul><ul><li>Scattering Cross-sections (Transfer X_sections) </li></ul></ul><ul><ul><li>Fission Spectrum </li></ul></ul><ul><li>The Multi-group diffusion equations are: </li></ul><ul><li>For Fluxes and X-sections defined as: </li></ul>Group 1 ( g 1 ) Group 2 ( g 1 ) Group n-1 ( g n-1 ) Group N ( g N ) … . Energy Increasing Energy Groups for a N-Group Diffusion Calculation Transfer out of g Transfer into g
  22. 22. Multigroup Diffusion Core Analysis Codes <ul><li>Modern Core Analysis Codes </li></ul><ul><ul><li>They use the Multi-group Diffusion Equations in two or several groups </li></ul></ul><ul><ul><ul><li>Two to six Groups: Fast and Thermal + additional resonance region groups. </li></ul></ul></ul><ul><ul><ul><li>Cross sections are obtained form Advanced Transport based Lattice-Codes (e.g. CASMO-4, WIMS, NEWT): </li></ul></ul></ul><ul><ul><ul><ul><li>Energy Averages maintaining Reaction Rates. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Spatial-Material Averages: Heterogeneous Cores. </li></ul></ul></ul></ul><ul><ul><ul><li>Corrections to the Diffusion Approximation: </li></ul></ul></ul><ul><ul><ul><ul><li>Neutronic Information from Lattice-Codes with Neutron Transport Corrections. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Advanced formulations of NET NEUTRON CURRENTS across interfaces and in highly absorbing regions. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Algorithms to “reconstruct” the local flux at the fuel rod level. </li></ul></ul></ul></ul><ul><ul><li>Examples: SIMULATE, DIF3D, PARCS. </li></ul></ul>
  23. 23. State-of-the-Art Nodal Methods Ref.: ief/ief-6/2/htr2-flu.html Node i,j,k Ref.: The Multi-group equations are solved for each node i,j,k in which the reactor is divided. The nodes “homogenize” the heterogeneous reactor.
  24. 24. Neutron Transport <ul><li>Transport theory is based on the Boltzmann Equation developed for the kinetic theory of gases. </li></ul><ul><ul><li>The development of nuclear reactors in the 1940 applied the equation to the transport of neutrons in </li></ul></ul><ul><ul><ul><li>Reactor design and </li></ul></ul></ul><ul><ul><ul><li>Radiation shielding. </li></ul></ul></ul><ul><ul><li>Analytical solutions are very difficult for real 3D-configurations. </li></ul></ul><ul><ul><li>Today, the Transport Equation is solved numerically by discretizing the </li></ul></ul><ul><ul><ul><li>Angular, </li></ul></ul></ul><ul><ul><ul><li>Energy, and </li></ul></ul></ul><ul><ul><ul><li>Time Dependence of the neutron flux and the cross sections. </li></ul></ul></ul><ul><li>It is a more accurate description of the neutron </li></ul><ul><li>field than the diffusion equation. </li></ul>
  25. 25. Neutron Transport <ul><li>Neutron Transport Methods account for the angular direction </li></ul>Fission Source Scattering External Neutron Source Time variation and removal of neutrons Angular Flux and Neutron Density z x y dV dA