5.0 SUMMARY AND CONCLUSIONS

The most reliable method for calculating control rod worths makes use of a Monte Carlo code, such as MCNP¹⁴, which can model the reactor, the fuel assemblies, and the shim-safety rods in considerable detail. For diffusion-theory calculations, however, special methods are needed because of steep flux gradients near the surface of strong neutron absorbers. These special methods fall into two categories. In the first category, pairs of group- and mesh-dependent effective diffusion parameters are found for the absorber. In the second category the absorber is isolated from the diffusion calculation by specifying group-dependent internal boundary conditions (current-to-flux ratios) on the surface of the absorber material. High-order transport calculations are required to determine the effective diffusion parameters or the internal boundary conditions. In general, these special methods are needed in multigroup diffusion calculations only for the low-energy groups. For those intermediate and fast groups for which S_a/S_s << 1 for the absorber the unmodified diffusion parameters may be used. Potentially, the effective diffusion parameter method is somewhat more accurate than the method using internal boundary conditions. This is because the first method uses two adjusted parameters (D_eff and S_a-eff) for each group whereas the second uses only one (J/f). For symmetric situations where fluxes and currents are the same on each side of the absorber, the two methods give essentially identical results.

This paper describes methods for calculating effective diffusion parameters and internal boundary conditions. For slab-like absorbers (thickness much less than transverse dimensions), the effective diffusion parameters are expressed in terms of the a and b blackness coefficients and the mesh spacing in the absorber. For best results, spectrum-weighted blackness coefficients evaluated in the P₅ approximation are used. For those low-energy groups for which S_a/S_s >> 1, the modified zero-scatter approximation may be used for the blackness coefficients, namely a_0m and b_0m. For non-slab-like absorbers effective diffusion parameters are found by adjusting the absorber cross sections until the reaction rate ratio for absorption in the rod to fission in a neighboring fuel region matches that of a corresponding Monte Carlo or discrete ordinates transport calculation. Fine-mesh transport calculations are used to determine internal boundary conditions from fluxes and currents at the surface of the absorber. However, values must not be used which exceed radius-dependent current-to-flux ratios for perfectly black absorbers. Methods for calculating these "black" limits are discussed.

All these methods are illustrated by calculating control rod worths for a number of absorber materials (Cd, Ag-In-Cd, Hf, borated stainless steel, and TiB₂-Al6351) in several research reactors. In general, diffusion-theory worth calculations using these methods are found to be in reasonable agreement with detailed Monte Carlo results and with experimental measurements.