If a control rod is inserted a distance x into the core, as shown in Figure 1.22, the change in reactivity (neglecting any effects other than neutron capture) is proportional to the integral of фф* over the length of the rod times the difference between the capture cross-section of

the material making up the rod and that of the material (presumably coolant) it displaces — i. e.

Ap a J ^<pg<pg( ^ ^ dx. (1.46)

The result is an S-shaped curve of reactivity against position as shown in Figure 1.22. The reactivity change on inserting a control rod fully into the core is usually called the “worth” of the rod.

This first-order perturbation theory estimate is not an accurate indication of the worth because the distributions of ф and ф* are altered by the presence of the rod. Figure 1.23 shows the axial variation of the total flux ф with control rods withdrawn and with all the control rods inserted a third of the way into the core. The presence of the rods in the top of the core pushes the flux towards the bottom.

Figure 1.24 gives an impression of the flux distribution around a single partially-inserted rod and indicates how the flux is depressed in its vicinity. The result is that the reactivity worth of one control rod

Control

Figure 1.24 Flux depression in the vicinity of a control rod.

depends on the position of all the others, and there are many control rods and shut-off rods in a large reactor. The worth of a rod is lower if the neighbouring rods are inserted and higher if they are not.

It is normal, however, to move the control rods together to keep flux distortion across the core to a minimum. If this were not done the power density might be higher on one side of the core than the other, causing greater non-uniformity of coolant outlet temperature and therefore more entropy gain due to mixing. There would also be variations of the fuel burnup rate.