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Steady State Diffusion Equation

Steady State Diffusion Equation. Scalar flux, vector current. HW 20. Study example 5.3 and solve problem 5.8 in Lamarsh. Steady State Diffusion Equation. One-speed neutron diffusion in a finite medium. At the interface What if A or B is a vacuum? Linear extrapolation distance.

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Steady State Diffusion Equation

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  1. Steady State Diffusion Equation Scalar flux, vector current. HW 20 Study example 5.3 and solve problem 5.8 in Lamarsh. Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  2. Steady State Diffusion Equation One-speed neutron diffusion in a finite medium • At the interface • What if A or B is a vacuum? • Linear extrapolation distance. • Bare slab with central infinite planar source (Lamarsh). • Same but with medium surrounding the slab. • Maybe we will be back to this after you try it!! A B x Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  3. More realistic multiplying medium One-speed neutron diffusion in a multiplying medium • The reactor core is a finite multiplying medium. • Neutron flux? • Reaction rates? • Power distribution in the reactor core? • Recall: • Critical (or steady-state): • Number of neutrons produced by fission = number of neutrons lost by: • absorption • leakage Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  4. More realistic multiplying medium For a critical reactor: Keff = 1 K > 1 Steady state homogeneous reactor Material buckling Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  5. More on One-Speed Diffusion HW 21 Show that for a critical homogeneous reactor Infinite Slab Reactor (one-speed diffusion) z  • Vacuum beyond. • Return current = 0. • d = linear extrapolation distance • = 0.71 tr (for plane surfaces) • = 2.13 D. Reactor x a/2 a a0/2 d d Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  6. More on One-Speed Diffusion HW 22 For the infinite slab . Show that the general solution With BC’s Flux is symmetric about the origin.  Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  7. More on One-Speed Diffusion HW 22 (continued) Fundamental mode, the only mode significant in critical reactors. For a critical reactor, the geometrical buckling is equal to the material buckling. To achieve criticality Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  8. More on One-Speed Diffusion Spherical Bare Reactor (one-speed diffusion) Minimum leakage  minimum fuel to achieve criticality. HW 23  Reactor x  r Continue! r0 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  9. More on One-Speed Diffusion HW 24 Infinite planer source in an infinite medium.   HW 25 ? Infinite planer source in a finite medium. x a/2 a a0/2 Source Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  10. More on One-Speed Diffusion Infinite planer source in a multi-region medium. Infinite Finite Infinite  Project 2 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  11. Back to Multiplication Factor k = fp,  • Fast from thermal, • Fast from fast, . • Thermal from fast, p. • Thermal available for fission • Thinking QUIZ • For each thermal neutron absorbed, how many fast neutrons are produced? Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  12. Two-Group Neutron Diffusion • Introductory to multi-group. • All neutrons are either in a fast or in a thermal energy group. • Boundary between two groups is set to 1 eV. • Thermal neutrons diffuse in a medium and cause fission (or are captured) or leak out from the system. • Source for thermal neutrons is provided by the slowing down of fast neutrons (born in fission). • Fast neutrons arelost by slowing down due to elastic scattering in the medium or leak out from the system (or fission or capture). • Source for fast neutrons is thermal neutron fission. Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  13. Two-Group Neutron Diffusion Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  14. Two-Group Neutron Diffusion Fast Fast diffusion coefficient Depends on thermal flux. Removal cross section = fission + capture + scattering to group 2 or Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  15. Two-Group Neutron Diffusion Thermal Thermal absorption cross section = fission + capture. Thermal diffusion coefficient Depends on fast flux. or Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

  16. Two-Group Neutron Diffusion • A coupled system of equations; both depend on both fluxes. • For a critical, steady state system: Review Cramer’s rule! Geometrical Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

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