A bare cubic reactor core of edge length is a single homogeneous lattice with…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Nuclear Physics · Partial differential equations and boundary-value problems

Energy from Physics, Ch The critical reactor: neutron diffusion, buckling and the size of a core

Problem

A bare cubic reactor core of edge length

a

is a single homogeneous lattice with infinite multiplication factor

k_\infty

and migration area

M^2

, the two lumped one-group parameters. Neutrons are lost only by leaking through the six faces of the cube, where the extrapolated flux is taken to vanish. Place the origin at one corner, so the core occupies

0\le x,y,z\le a

. ((a)) Starting from the one-group diffusion equation

D\nabla^2\phi-\Sigma_a\phi+\nu\Sigma_f\phi=0

, write it in the form

\nabla^2\phi+B_m^2\phi=0

and give

B_m^2

in terms of

k_\infty

and

M^2

. ((b)) Separate variables,

\phi(x,y,z)=X(x)Y(y)Z(z)

, apply

\phi=0

on all six faces, and keep only the solution that is positive everywhere inside the cube. Give this fundamental flux shape and its geometric buckling

B_g^2

. ((c)) State the critical condition and use it to find the edge length

a

of a bare cubic core with

k_\infty=1.07

and

M^2=720\,\mathrm{cm^2}

Related theory

Nuclear Physics

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