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Nuclear Reactor Theory Intermezzo

Nuclear Reactor Theory Intermezzo. William D’haeseleer. Nuclear Reactor Theory. Intermezzo: One-Speed Diffusion Theory of a Nuclear Reactor See Duderstadt & Hamilton: § 5.III A D § 5. IV [to a large extent to be studied independently].

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Nuclear Reactor Theory Intermezzo

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  1. Nuclear Reactor TheoryIntermezzo William D’haeseleer William D’haeseleer BNEN – NRT 2010-2011

  2. Nuclear Reactor Theory Intermezzo: One-Speed Diffusion Theory of a Nuclear Reactor See Duderstadt & Hamilton: § 5.III AD § 5. IV [to a large extent to be studied independently] William D’haeseleer BNEN – NRT 2010-2011

  3. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation Consider mono-energetic neutrons only → time-dependent neutron diffusion equation Consider s as the neutron source due to fission William D’haeseleer BNEN – NRT 2010-2011

  4. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation Evolution of flux dependent upon • geometry (diffusion; leakage) • material composition (absorption; fission) → For arbitrary geometry & composition Steady state only arises when reactor is critical; i.e., when k = 1 William D’haeseleer BNEN – NRT 2010-2011

  5. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation In general, k ≠ 1; therefore apply a “trick” to find k = f (dimensions, material composition) then set k = 1 relationship between dimensions & material composition for criticality ! William D’haeseleer BNEN – NRT 2010-2011

  6. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation ≡ B² called“Material Buckling” Bm2 -- for k = 1 -- At this stage, k still unknown, since B² is not known William D’haeseleer BNEN – NRT 2010-2011

  7. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation Note (1): • suppose one writes → one could introduce f = fuel utilization factor William D’haeseleer BNEN – NRT 2010-2011

  8. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation Note (2):Consider an infinite reactor in such a reactor, Φ independent of position (uniform throughout) reduces to William D’haeseleer BNEN – NRT 2010-2011

  9. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation Note (3):Consider now general case (finitereactor) • Take now as a definition: Then, source term → the time-dependent diffusion equation William D’haeseleer BNEN – NRT 2010-2011

  10. Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation Note (4):still finite reactor • Time independent case: Or for a critical reactor, k = 1 William D’haeseleer BNEN – NRT 2010-2011

  11. Nuclear Reactor TheoryMono-energetic critical equation Reactor is critical if Material Buckling Geometric Buckling; B12=Bg2 “critical equation” William D’haeseleer BNEN – NRT 2010-2011

  12. Nuclear Reactor TheoryMono-energetic critical equation “critical equation” determines requirement to have a critical reactor: - material composition - dimensions etc. determines requirement to have a critical reactor: - material composition - dimensions etc. Alternative expression of critical equation: William D’haeseleer BNEN – NRT 2010-2011

  13. Nuclear Reactor TheoryMono-energetic critical equation Physical meaning of critical equation Consider a bare reactor of arbitrary geometry. # of neutrons leaking out of the system # of neutrons absorbed William D’haeseleer BNEN – NRT 2010-2011

  14. Nuclear Reactor TheoryMono-energetic critical equation Physical meaning of critical equation Non-leakage probability*: From: William D’haeseleer BNEN – NRT 2010-2011 *Note: symbols PL and PNL are used interchangeably and mean the same!

  15. Nuclear Reactor TheoryMono-energetic critical equation Physical meaning of critical equation Interpretation of Physical meaning of critical equation Interpretation of # of neutrons absorbed Gives rise to a release of fission neutrons: William D’haeseleer BNEN – NRT 2010-2011

  16. Nuclear Reactor TheoryMono-energetic critical equation Physical meaning of critical equation Physical meaning of critical equation Of these neutrons, only a fraction PL does not leak out, and gives rise to absorption in the next generation: Hence:  For a critical reactor: William D’haeseleer BNEN – NRT 2010-2011

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