Two-Group Theory of a Ring of N Cylindrical Rods or Zones in a Reflected Reactor

The critical condition is obtained for a system consisting of a ring of N equally spaced identical cylindrical rods in a reflected cylindrical reactor. The fluxes in each region are expressed in terms of a Fourier Series expansion of the angular dependence of the flux about each rod. The imposition of the boundary conditions gives a set of linear homogeneous equations, from which the critical determinant is deduced. Matrix theory is used throughout, which facilitates the treatment of the problem, and which in the case of a bare reactor provides a method of elimination of constants alternative to that given by Avery. The derivation is also valid for a system containing a ring of N multiplying or nonmultiplying zones. A little modification of this theory leads, without difficulty, to the solution of the problem of a ring of N control rods, which are “black” to thermal neutrons.