An approximate method of homogenizing a heterogeneous reactor

Abstract

It is pointed out that the mathematical procedure for homogenizing a heterogeneous reactor can be prescribed in a variety of ways, not necessarily all leading to the same parameters. Thus the constants of a homogenized reactor cannot be uniquely defined. The general principles of homogenization are considered, using k_eff as a functional, and it is demonstrated that the use of effective cross-sections for capture and fission derived by averaging over the neutron spectrum in an infinite lattice leads to an error in k_eff proportional to the Laplacian. In particular it is impossible to determine a unique diffusion coefficient, even for a strictly defined direction. Any attempt to increase the accuracy results in the effective parameters ceasing to have universal applicability, and they become characteristic of the form and dimensions of the reactor. Formulae are presented for plane, cylindrical and spherical geometries which specify the diffusion coefficient in finite reactors as a function of direction.