Neutron Diffusion Theory

We use a one speed or one group approximation where

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Unformatted text preview: d be valid if the second derivative of the flux does not vary appreciably. 6. Rapidly time-varying flux It was assumed that the flux is independent of time. It is possible to relax this requirement if the change in ϕ is small during the time for a neutron to travel a few mean free paths. If the slow neutrons in a reactor have velocity of v ≈ 10 5 cm/sec, and their travel over three scattering mean free paths, then we can state that the time variation of the flux must satisfy the condition: φ dφ >> dt 3λs 3λs ≈ 5 [sec] v 10 which states that the time needed for the neutron to travel three mean free paths is smaller than the characteristic time for the neutron flux variation. 5. DERIVATION OF THE NEUTRON DIFFUSION EQUATION: To derive the neutron diffusion equation we adopt the following assumptions: 1. We use a one-speed or one-group approximation where the neutrons can be characterized by a single average kinetic energy. 2. We characterize the neutron distribution in the reactor by the particle density n(r,t) which is the number of neutrons per unit volume at a position r at time t. Its relationship to the flux is: φ ( r ,t ) = v n ( r , t ) We consider an arbitrary volume V and write the balance equation: ⎡Time rate of change ⎤ ⎢of the number of ⎥ = ⎡ Pr oduction rate ⎤ − ⎡ Absorptions ⎤ − ⎡ Net leakage from ⎤ ⎥ ⎢in V ⎥ ⎢ the surface of V ⎥ ⎢ ⎥ ⎢in V ⎣ ⎦⎣ ⎦⎣ ⎦ ⎢ neutrons in V ⎥ ⎣ ⎦ The first term is expressed mathematically as: ⎤ d⎡ 1 ⎤ 1⎡ ∂ ⎤ d⎡ ⎢ ∫ n( r, t ) dV ⎥ = ⎢ ∫ φ ( r, t ) dV ⎥ = ⎢ ∫ φ ( r, t ) dV ⎥ dt ⎣V ⎦ dt ⎣V v ⎦ v ⎣V ∂t ⎦ The production rate can be written as: ∫ S (r , t ) dV V The absorp...
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This note was uploaded on 02/28/2013 for the course NUC 150 taught by Professor N/a during the Spring '13 term at Berkeley.

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