Neutron Diffusion Theory

Normally boundary conditions are both and n specified

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Unformatted text preview: ∂⎞ 1 ∂⎛ 2 ∂⎞ 1 ⎜ sin(θ ) ⎟+ 2 ⎜r ⎟+ 2 ∂θ ⎠ r sin 2 (θ ) ∂ϕ 2 r 2 ∂r ⎝ ∂r ⎠ r sin(θ ) ∂θ ⎝ (31) In case of symmetry around the angular variables θ and ϕ , these equations simplify in the one dimensional case into: ∇2 ≡ Where: 1 d⎛ α d⎞ ⎜r ⎟ r α dr ⎝ dr ⎠ α = 0, for the cartesian coordinates, α = 1, for the cylindrical coordinates, α = 2, for the spherical coordinates, and the partial derivative has been replaced by the total derivative. When the flux is not a function of time, we use the steady state diffusion equation, or the scalar Helmholtz equation; D∇2φ − Σa φ + S = 0 (32) Which is a partial differential equation of the elliptic type. The Helmholtz equation can be written as: ∇2φ ( r ) − Where L2 = 1 S (r) φ (r) = − 2 L D D , L is the diffusion length. Σa (33) If on the other hand, the neutron density and flux are independent of position, we can write from Eqs. 28, considering div J = 0 : 1 d φ (t ) = S (t ) − Σ aφ (t ) v dt (34) which is a time dependent equation in the flux. 7. BOUNDARY CONDITIONS FOR THE STEADY-STATE DIFFUSION EQUATION Mathematically, for the Helmholtz equation the following boundary conditions are needed: ∂φ , or a linear combination of the two must be specified. either φ or the normal derivative ∂n ∂φ cannot be specified independently. Normally, boundary conditions are Both φ and ∂n specified based on physical arguments that do not violate this condition. 1. Vacuum Boundary Conditions: The mean free path of neutron in air is much larger than in the reactor, so that it is possible to treat it as a vacuum in reactor calculations. If we consider no neutrons reflected from the vacuum back to the reactor core (Fig. 3), then we can write the equivalent to Eq. 17: J − ( x) = 1 Σs 1 Σ s ∂φ φ0 + =0 4 Σt 6 Σ t 2 ∂x 0 (35) Recalling that from the Fick’s law derivation: D= Σs 3Σ t We can write: J − ( x) = From which: 2 1 Σs D ∂φ φ0 + =0 4 Σt 2 ∂x 0 1 ∂φ 1 Σs =− φ0 ∂x 0 2 D Σt (36) (37) If at the boundary the material is mostly a scatterer (Σ a << Σ t ) , then substitute for D = λtr 3 Σs Σt ≈ 1 , and we can : 1 ∂φ 1 31 ≈− ≈− φ0 ∂x 0 2D 2 λtr (38) However from the geometry of Fig. 3, if the diffusion theory is linearly extrapolated, we can write: φ0 d = tan(θ ) = − ∂φ ∂x (39) 0 where d is the “extrapolated length”. Comparing Eqs. 38 and 39, we get: 1 ∂φ 31 1 ≈− =− φ0 ...
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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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