TO: C. K. Beck and Reactor Committee
FROM: R. L. Murray and A. C. Menius, Jr.
SUBJECT: Control Rod-Power Characteristic

For purposes of instrument and control-rod design
it is necessary to know the effect of changes in control-
rod position in the reactor power level. Such data are
normally obtained empirically; but lacking a precise graph
from Los Alamos, a rough theoretical calculation was made.

The power is shown to be approximately proportional
to the excess reactivity; the latter in turn is given by an
S-shaped curve which is similar to a displaced sine curve.
The variation with position of rod of the power is shown in
the attached figure, adjusted to fit a total value of
20,000 microres.

The arguments loading to this graph are given
in the appendix of this note.

The effect of a cadmium or boron control rod is to depress the neutron flux,
usually to zero at the boundary. If such a rod were inserted axially in a
circular cylinder of length much greater than the diameter, the neutron flux
would be represented, in a bare reactor, by the function

where Jo is the Bessel function and the dimensions are given in the sketch, Fig. 1.

(This is to be contrasted with the undisturbed flux

The critical condition for such a system would be written

where M² is an effective migration length for neutrons.

(Undisturbed, the k is given by the same relation without the a)

In a finite cylindrical bare pile, there is an axial sinusoidal variation
in flux density, , "modulating" the flux formulas given above; see
Fig. 2.

The approximation is now made that the effective k of a finite cylinder with a rod partially inserted is the average of the undisturbed value kv and the disturbed value kd, with weighting factors pro- portional to the axial flux that is affected, namely the respective areas under the flux curve

It is easy to show that the effective k by this criterion is

If r1 is the excess reactivity value of the
rod and the "rod-in" position corresponds to
r = 0, then the excess reactivity r at any

distance the rod is pulled up, Z, is given by

In order to translate this into a power graph it is necessary only to note
that if the pile is just critical at zero power with the rod in and the
solution at the temperature of the cooling water, for any other steady state
at power P the excess reactivity due to the rod must just balance the drop due
to the solution temperature rise, to again reach k = 1.

Calculations on heat removal and experimental data from Los Alamos indicate
an essentially linear rise in allowed power with temperature. Thus
P/P1 = T/T1
where (P1,T1) is the maximum operating point, and (P,T) is any other.

If the temperature effect on reactivity is written

where [alpha] is the temperature coefficient, then the rod-power correlation is

This is plotted for the following assumed values of the constants

Pl = 10 kw [alpha] = 240 [mu]re/°C rl = 1.7 x 104 [mu]re T1 = 70°C

Thus