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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). 1 Steady State Diffusion Equation One-speed neutron diffusion in a finite medium • At the interface A B A B d A dB J A J B DA DB dx dx x • 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!! Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 2 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: neutronproductionrate(S) k (1) absorption neutronabsorptionrate( A) (1) leakage keff neutronproductionrate( S ) neutronabsorptionrate( A) neutronleakage rate( LE ) Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 3 More realistic multiplying medium keff A Pnon leak k A LE LE SA surface area S V Volum e non- leakageprobability For a critical reactor: Keff = 1 K > 1 LE SA a 2 1 3 S V a a Steady state homogeneous reactor 2 0 a k (r ) a (r ) D (r ) k 1 2 2 2 (r ) B (r ) 0 B 2 L Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). Material buckling 4 More on One-Speed Diffusion HW 21 Show that for a critical homogeneous reactor Pnon leak a a 1 2 2 2 2 B L 1 a D a B D Infinite Slab Reactor (one-speed diffusion) z • Vacuum beyond. • Return current = 0. Reactor x d = linear extrapolation distance a/2 = 0.71 tr (for plane surfaces) a = 2.13 D. a0/2 d d Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 5 More on One-Speed Diffusion HW 22 d 2 For the infinite slab 2 B 2 0 . Show that the dx general solution ( x) A cos Bx C sin Bx With BC’s a0 )0 2 d ( x) 0 dx x 0 ( Flux is symmetric about the origin. ( x) A cos Bx A 0 a0 a0 a0 3 5 ( ) A cos B( ) 0 B( ) , , ,... 2 2 2 2 2 2 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 6 More on One-Speed Diffusion HW 22 (continued) a0 3 5 B( ) , , ,... 2 2 2 2 3 5 a0 , , ,... B B B Fundamental mode, the only mode significant in critical reactors. ( x) 0 cos a0 x B a0 Geometrical Buckling For a critical reactor, the geometrical buckling is equal to the material buckling. 2 k 1 To achieve criticality Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). a L2 0 7 More on One-Speed Diffusion Spherical Bare Reactor (one-speed diffusion) 6a 4a 4 3 3 a 3 a 2 2 Minimum leakage minimum fuel to achieve criticality. 2 d 2 d HW 23 2 B 0 2 dr r dr A C cos Br sin Br r r Reactor r C r sin , r0 r r0 B Continue! Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). x r0 8 More on One-Speed Diffusion HW 24 Infinite planer source in an infinite medium. SL x / L d 2 ( x) 1 S ( x) e 2 ( x) 2 dx L D 2D HW 25 Infinite planer source in a finite medium. SL sinha0 2 x / 2L 2D cosh(a0 / 2L) Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). x a/2 a a0/2 Source 9 More on One-Speed Diffusion Infinite planer source in a multi-region medium. 1 ( a / 2) 2 ( a / 2) d1 d2 D1 D2 Infinite Finite Infinite dx dx x a / 2 m ore x a / 2 BC Project 2 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 10 Back to Multiplication Factor k k = fp, P keff fPnon leak eff k non leak 1 • Fast from thermal, a • Fast from fast, . (i) f (i) i • Thermal from fast, p. • Thermal available for fission afuel f fuel mod erator poison a clad a a a Thinking QUIZ • For each thermal neutron absorbed, how many fast neutrons are produced? Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 11 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). 12 Two-Group Neutron Diffusion 1 (r ) 10 MeV ( E, r )dE Fast 1eV 1eV 2 (r ) ( E , r )dE Therm al 0 1 f 1 1 2 f 2 2 keff 2 2 D1 1 D2 2 a1 1 a 2 2 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 13 Two-Group Neutron Diffusion 2 0 S1 (r ) a1 1 (r ) D1 1 (r ) Depends on thermal flux. Fast diffusion Removal cross section coefficient = fission + capture + scattering to group 2 2 0 f 1 1 (r ) f 2 2 (r ) a1 1 (r ) D1 1 (r ) or 0 k 2 a 2 2 (r ) a1 1 (r ) D1 1 (r ) Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 14 Two-Group Neutron Diffusion 2 0 S2 (r ) a 2 2 (r ) D2 2 (r ) Depends on fast flux. Thermal absorption cross section = fission + capture. Thermal diffusion coefficient 2 0 s12 1 (r ) a 2 2 (r ) D2 2 (r ) or 2 0 a1 1 (r ) a 2 2 (r ) D2 2 (r ) Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 15 Two-Group Neutron Diffusion 0 k 2 a 2 2 (r ) a1 1 (r ) D1 1 (r ) 2 0 a1 1 (r ) a 2 2 (r ) D2 2 (r ) • A coupled system of equations; both depend on both fluxes. • For a critical, steady state system: 2 1 (r ) B 1 (r ) 0 2 2 2 ( r ) B 2 ( r ) 0 2 Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). Review Cramer’s rule! Geometrical 16