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# ex10_022808 - EP 471 Engineering Problem Solving II...

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3/7/09 EP 471 -- Engineering Problem Solving II Exercise 10 Eigenvalue Problems: Calculation of Critical, One-Speed Flux Shapes One of the applications of the power iteration method comes in finding critical flux shapes in nuclear reactor cores. Here we have an eigenvalue problem where our interest is confined to the fundamental mode; we can’t have any negative neutron populations, and the first (fundamental) mode shape is the solution that’s positive everywhere. This is also a problem that doesn’t lend itself to the bvp4c utility because the composition of the reactor changes discontinuously between the core and the reflector. A qualitative flux shape is shown below: ϕ x R C The core region (C) contains the nuclear fuel. The reflector region (R) may simply be water; its purpose is to increase the neutron population at the edge of the core. The governing equation in the core region is: C fC C aC C C k dx d D ϕ ν ϕ ϕ Σ = Σ + 1 2 2 and in the reflector region: 0 2 2 = Σ + R aR R R dx d D ϕ ϕ The coefficients D C , Σ aC , νΣ fC , D R and Σ aR are constant coefficients within the core and reflector regions respectively, but they change discontinuously at the boundary between core and reflector. The eigenvalue is k , the neutron multiplication factor. The reactor is critical when k = 1 (as many neutrons exist in the next generation as in the previous generation, or the power is constant). These equations follow from something analogous to a continuity equation in fluid mechanics: they are “conservation of neutron” equations instead of “conservation of mass” equations.

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ex10_022808 - EP 471 Engineering Problem Solving II...

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