DIFFUSION OF NEUTRONS

1. DIFFUSION OF NEUTRONS OVERVIEW Basic Physical Assumptions Generic Transport Equation Diffusion Equation Fermi’s Age Equation Solutions to Reactor Equation

2. Basic Physical Assumptions • Neutrons are dimensionless points • Neutron – neutron interactions are neglected • Neutrons travel in straight lines • Collisions are instantaneous • Background material properties are isotropic • Properties of background material are known and time-independent T10: Diffusion of Neutrons

3. (a) (b) Physical Model T10: Diffusion of Neutrons

4. v χm θ v´ rc rm b rc Collision Model T10: Diffusion of Neutrons

5. Initial Definitions ey z v W q ey ex j r y x T10: Diffusion of Neutrons

6. Neutron Density T10: Diffusion of Neutrons

7. Angular Flux and Current Density J dS T10: Diffusion of Neutrons

8. Generic Transport Equation Arbitrary volume V We ignore macroscopic forces T10: Diffusion of Neutrons

9. Generic Transport Equation Gauss Theorem: T10: Diffusion of Neutrons

10. Substantial Derivative z r y x T10: Diffusion of Neutrons

11. Transport (Boltzmann) Equation T10: Diffusion of Neutrons

12. Collision Term z r y x T10: Diffusion of Neutrons

13. Neutron Transport Equation T10: Diffusion of Neutrons

14. z y x Boundary Condition Outgoing direction Outward normal Ω Incoming direction ns r Volume V Ω Rs Surface S V T10: Diffusion of Neutrons

15. Difficulties • Mathematical structure is too involved • Mixed type equation (integro-differential), no way to reduce it to a differential equation • Boundary conditions are given only for a halve of the values • Too many variables (7 in general) • Angular variable T10: Diffusion of Neutrons

16. Angular Measures 180 Solar disks T10: Diffusion of Neutrons

17. Plane Angles R φ T10: Diffusion of Neutrons

18. Solid Angles T10: Diffusion of Neutrons

19. One-Group Diffusion Model • Infinite homogeneous and isotropic medium • Neutron scattering is isotropic in Lab-system • Weak absorptionΣa << Σs • All neutrons have the same velosity v. (One-Speed Approximation) • The neutron flux is slowly varying function of position T10: Diffusion of Neutrons

20. Derivation Isotropic scattering z y x r = 0 is most important T10: Diffusion of Neutrons

21. Derivation II Taylor’s series at the origin: T10: Diffusion of Neutrons

22. Derivation III T10: Diffusion of Neutrons

23. Fick’s Law CM-System → Lab-System: T10: Diffusion of Neutrons

24. Transport Mean Free Path Information about the original direction is lost Y Transport correction = A number of anisotropic collisions is replaced by one isotropic Y Y ls lscosY lscosY2 ltr T10: Diffusion of Neutrons

25. Diffusion Equation T10: Diffusion of Neutrons

26. Leakage Rate z Jz dz dx (x,y,z) dy y x T10: Diffusion of Neutrons

27. Diffusion Equation Time-dependent: Time-independent: Time-independent from a steady source T10: Diffusion of Neutrons

28. Laplace’s Operator Cartesian geometry Cylindrical geometry Spherical geometry T10: Diffusion of Neutrons

29. Symmetries Slab geometry Spherical geometry Cylindrical geometry z y x z n = 0 for slab n = 1 for cylindrical n = 2 for spherical T10: Diffusion of Neutrons

30. General Properties • Flux is finite and non-negative • Flux preserves the symmetry • No return from a free surface • Flux and current are continues • Diffusion equation describes the balance of neutrons T10: Diffusion of Neutrons

31. for +z - direction: for -z - direction: Interface Conditions A B z T10: Diffusion of Neutrons

32. Boundary Condition Transport equation Free surface Diffusion eq. Straight line extrapolation from x = 0 towards vacuum: T10: Diffusion of Neutrons

33. PlaneInfinite Source in Infinite Medium Transport equation Q0 x = 0 T10: Diffusion of Neutrons

34. Point Source in Infinite Medium r T10: Diffusion of Neutrons

35. PlaneInfinite Source in Slab Medium Infinite: Q0 Slab: x = -a/2 x = 0 x = a/2 T10: Diffusion of Neutrons

36. PlaneInfinite Source with Reflector Q0 2 1 1 2 Reflector Reflector a Bare slab T10: Diffusion of Neutrons

37. Age of Neutrons Energy • q(E) - number of neutrons, which per cubic-centimeter and second pass energy E. • q(E) = [n×cm-3 s-1] • X-sections depend on E: D(E),Σs(E),... Q E0 q(E) E Slowing down medium: Mean Total Slowing down distance Can be shown T10: Diffusion of Neutrons

38. Fermi’s Age Equation E+dE q(E+dE) E q(E) T10: Diffusion of Neutrons

39. Fermi’s Age Equation II τ~ time T10: Diffusion of Neutrons

40. Solutions to the Age Equation No absorption x = 0 r No absorption T10: Diffusion of Neutrons

41. Slowing Down Density for Different Fermi’s Ages T10: Diffusion of Neutrons

42. Migration Area (Length) Thermal neutron absorbed Fast neutron borne r rs rth Fast neutron thermalized T10: Diffusion of Neutrons

43. Diffusion and Slowing Down Parameters for Various Moderators T10: Diffusion of Neutrons

44. Neutrons in Multiplying Medium Assumption: T10: Diffusion of Neutrons

45. Principles of a Nuclear Reactor Leakage E N2 2 MeV N1 Fast fission n n/fission Energy Slowing down Resonance abs. ν ≈ 2.5 Non-fissile abs. Non-fuel abs. 1 eV Fission 200 MeV/fission Leakage T10: Diffusion of Neutrons

46. T10: Diffusion of Neutrons

47. T10: Diffusion of Neutrons

48. Buckling as Curvature Large core Small core T10: Diffusion of Neutrons

49. Criticality Condition T10: Diffusion of Neutrons

50. Matrix Differential operator Transport operator Eigenvalues T10: Diffusion of Neutrons