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

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

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 2 One-speed neutron diffusion in a finite medium Steady State Diffusion Equation A B 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!! x

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 3 One-speed neutron diffusion in a multiplying medium More realistic 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: (1)absorption (1)leakage

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

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

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

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 7 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 More on One-Speed Diffusion

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 8 Spherical Bare Reactor (one-speed diffusion) Minimum leakage  minimum fuel to achieve criticality. x r r0r0  HW 23  Continue! Reactor More on One-Speed Diffusion

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 9 HW 24 Infinite planer source in an infinite medium. x a a/2 a 0 /2 Source HW 25   More on One-Speed Diffusion Infinite planer source in a finite medium.

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 10 More on One-Speed Diffusion Infinite planer source in a multi-region medium. FiniteInfinite 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 are lost 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 Removal cross section = fission + capture + scattering to group 2 Depends on thermal flux. Fast diffusion coefficient or

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 15 Two-Group Neutron Diffusion Thermal diffusion coefficient Thermal absorption cross section = fission + capture. 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: Geometrical Review Cramer’s rule!

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