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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 1 HW 14 More on Moderators Calculate the moderating power and ratio for pure D 2 O as well as for D 2 O contaminated with a) 0.25% and b) 1% H 2 O. Comment on the results. In CANDU systems there is a need for heavy water upgradors.

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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 2 More on Moderators

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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 3 More on Moderators Continuous slowing down model or Fermi model. The scattering of neutrons is isotropic in the CM system, thus is independent on neutron energy. also represents the average increase in lethargy per collision, i.e. after n collisions the neutron lethargy will be increased by n units. Materials of low mass number is large Fermi model is inapplicable.

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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 4 More on Moderators Moderator-to-fuel ratio Moderator-to-fuel ratio N m /N u. Ratio leakage a of the moderator f. Ratio slowing down time p leakage. Water moderated reactors, for example, should be under moderated. T ratio (why).

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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 5 One-Speed Interactions Particular general. Recall: Neutrons dont have a chance to interact with each other (review test!) Simultaneous beams, different intensities, same energy: F t = t (I A + I B + I C + …) = t (n A + n B + n C + …)v In a reactor, if neutrons are moving in all directions n = n A + n B + n C + … R t = t nv = t

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 6 Neutrons per cm 3 at r whose velocity vector lies within d about. Same argument as before One-Speed Interactions where

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 7 Multiple Energy Interactions Neutrons per cm 3 at r with energy interval ( E, E+dE ) whose velocity vector lies within d about. Generalize to include energy Thus knowing the material properties t and the neutron flux as a function of space and energy, we can calculate the interaction rate throughout the reactor. Scalar

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 8 Neutron Current Similarly and so on … Redefine as Scalar Neutron current density From larger flux to smaller flux! Neutrons are not pushed! More scattering in one direction than in the other.

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 9 Net flow of neutrons per second per unit area normal to the x direction: In general: Equation of Continuity Rate of change in neutron density Production rate Absorption rate Leakage in/out rate Volume Source distribution function Surface area bounding Normal to A Equation of Continuity

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 10 Using Gauss Divergence Theorem Equation of Continuity

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 11 For steady state operation For non-spacial dependence Delayed sources? Equation of Continuity

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 12 Ficks Law Assumptions: 1.The medium is infinite. 2.The medium is uniform 3.There are no neutron sources in the medium. 4.Scattering is isotropic in the lab. coordinate system. 5.The neutron flux is a slowly varying function of position. 6.The neutron flux is not a function of time. Restrictive! Applicability??

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 13 Ficks Law Diffusion: random walk of an ensemble of particles from region of high concentration to region of small concentration. Flow is proportional to the negative gradient of the concentration.

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 14 x y z r dA z Ficks Law scattered r Number of neutrons scattered per second from d at r and going through dA z Slowly varying Isotropic Removed (assuming no buildup)

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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 15 Ficks Law

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Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 16 Ficks Law HW 15 and show that and generalize Diffusion coefficient Ficks law The current density is proportional to the negative of the gradient of the neutron flux.

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Also known as Gauss’ Theorem

Also known as Gauss’ Theorem

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