### RERTR Publications:

Analysis Methods for Thermal Research and Test Reactors

#### ANL/RERTR/TM-29

COMPUTING CONTROL ROD WORTHS

IN THERMAL RESEARCH REACTORS

##### 5.0 SUMMARY AND CONCLUSIONS

The most reliable method for calculating control rod worths makes use
of a Monte Carlo code, such as MCNP^{14}, 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 P_{5}
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 TiB_{2}-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.