The effectiveness of control rods decreases with time as a result of the burn-up of the absorbers. The burn-up of a control rod can be easily calculated once the flux distribution inside the rod is known from a transport calculation. The flux distribution obtained from a transport calculation (S„ or collision probability method) must first be normalized to the value corresponding to full reactor power. In the transport calcula­tion the rod is subdivided in s concentrical shells (e. g. the regions of a collision probability method) and for each shell j and energy group і an unnormalized flux щ is obtained. The diffusion calculation performed over the complete core gives for each group і the diffusion theory flux ф, (asymptotic flux) at the outer boundary of the rod, normalized to full reactor power. The net current entering the rod for each group і is given by

J. =§<*.. (13.1)

This expression follows from Fick’s law and from the definition of the extrapolation length d.

The total number of neutrons absorbed in group і per unit length of control rod is

Пі = Jilrrr (13.2)

where r is the control-rod radius.

The normalization factor / for the fluxes <pti can be obtained considering that

Пі = І) X. iiVicpaf (13.3)

j-i

where V) = volume of shell j per unit control rod length,

2ац = macroscopic absorption cross-section of shell j, group i.

With the normalized fluxes ftpa the reaction rates and burn-up of each shell can be calculated.

Practically only the outermost sheet of absorber burns because the flux decreases very rapidly inside the rod, so that in black rods the effect of burn-up is practically equivalent to a reduction of the absorber diameter. As burn-up proceeds the transport calculation has to be repeated in order to obtain the flux distribution for the new burn-up time steps.

Another important problem is posed by the heat production in the control rods, which must be known in order to provide the necessary cooling. This heat production results from absorption of y-rays and of neutrons.

Usually the main absorbing material of a control rod is l0B. The absorption of a neutron in l0B takes place according to the following reactions:

,0B + n —— — 7Li + a + y(2.79 MeV)

—7Li* + a + 7(2.31 MeV) (13.4)

?Li*——————————— — 7Li + 7(0.48 MeV)

As these ys do not have a very high energy, one can assume that they are absorbed by the metal components of the rod, so that each neutron absorbed in Boron produces

2.79 MeV = 4.47 x КГ13 W sec.

Another important control-rod material is iron.

The ys produced by Fe(n, y) reactions are usually only a small fraction of the y-flux of the core. This y-flux is usually calculated with computer codes used for shielding purposes. Once this flux is known the heat production can be calculated using the у-absorption coefficients for each control-rod component. The fine structure of the y-flux in the control rod is often neglected in these calculations.