Geometrical buckling and solutions of various shapes of reactors
Geometrical buckling is a measure of neutron leakage, while material buckling is a measure of neutron production minus absorption. With this terminology the criticality condition may also be stated as the material and geometric buckling being equal:
Bm = Bg
The quantity Bg2 is called the geometrical buckling of the reactor and depends only on the geometry. This term is derived from the notion that the neutron flux distribution is somehow ‘‘buckled’’ in a homogeneous finite reactor. It can be derived the geometrical buckling is the negative relative curvature of the neutron flux (Bg2 = ∇2Ф(x) / Ф(x)). In a small reactor the neutron flux have more concave downward or ‘‘buckled’’ curvature (higher Bg2) than in a large one.
The value of geometrical buckling for infinite slab reactor can be derived, when the vacuum boundary condition is applied on the solution of diffusion equation. The physically acceptable solution for infinite slab reactor is:
Φ(x) = C.cos(Bg x)
The vacuum boundary condition requires the relative neutron flux near the boundary to have a slope of -1/d, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary. This zero flux boundary condition is more straightforward and is can be written mathematically as:
Therefore, the solution must be Φ(ae/2) = C.cos(Bg .ae/2) = 0 and the values of geometrical buckling, Bg, are limited to Bg = nπ/a_e, where n is any odd integer. The only one physically acceptable odd integer is n=1, because higher values of n would give cosine functions which would become negative for some values of x. The solution of the diffusion equation is: