Central Rod in One-velocity Model

The present treatment of the problem of computing the effectiveness of a control rod is carried out on the basis of two analytical models. In the application of the first, the one-velocity model, attention is directed toward displaying some of the elementary physical ideas involved in the general problem. The model is used here primarily for purposes of illustration it is not intended that it be considered an accurate tool for computing control-rod effectiveness. The second method utilizes the two-group model, and it is expected that the results obtained from this formulation will be useful for the solution of practical problems. 

In the various control-rod problems considered here it is always assumed that the rods have circular cross sections and that they are inserted with axes parallel to the axis of the reactor. Calculations of rod effectiveness are carried out only for the case of the fully inserted 

Finally, it is pointed out that, unless otherwise stated, control-rod effectiveness is measured in the present work by the reactivity of the 

As a first elementary example of a control-rod calculation consider the case of a bare cylindrical thermal reactor operating at steady state with a fully inserted central gray control rod. By gray rod we mean a rod composed of a material which, although a reasonably good neutron absorber, is not so strong an absorber that it cannot support some neutron flux. Also, let us assume that diffusion theory is applicable within the interior of this rod. It should be noted that these assumptions are not unlike those imposed for fuel lumps in the calculation of the thermal utilization for heterogeneous lattices. 

The general geometric configuration of this reactor with control rod is shown in Fig. 11.1. The radius of the reactor is R and its height 2k. The radius of the rod is 6. At steady state the appropriate one-velocity neutron-flux equations are 

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