The transient behavior of the neutron flux in a reactor core is represented by the diffusion equation (5):

dnjdt = exp(— Д2т)Лтс(1 — /3)2а<£ + exp(— B2rD)p £ ).{С{ + DVty — Елф

1 (1.3)

which expresses the fact that the rate of increase of neutrons is equal to the prompt and delayed neutron production reduced by leakage and ab­sorption (5).

The delayed neutrons’ concentrations are represented by;

dCJdt = (к^^фір) -Xfii, і = 1, 2,…, N (1.4)

which shows the radioactive decay of the precursors of the delayed neutrons.

Equation (1.3) is space, time, and energy dependent [и(г, t, £)]. We could more accurately start with a set of equations each of which applies to a different energy. Here are diffusion equations for three discrete energy bands as an example:

Fast group (> 1 MeV). Here fast neutrons are born, there is no resonance capture, but leakage occurs and the neutrons scatter down in energy to the next group

dnF/dt = [£„,(! —- Р)£афт/р] + Df — Е$ф$ (1.5)

In the fast reactor 2F is small.

Epithermal group (1 keV-0.5 MeV). Here delayed neutrons are born, and further neutrons arrive from the fast group after energy degradation

N

dnFjdt = pEF exp(—B2r — Td])^ j + P ^ + DF V2<f>^ (1.6)

t-i

In the fast reactor 27E is even smaller than 27F.

Thermal group (0.025 eV). Here all neutrons arrive after further energy degradation in a thermal reactor, although very few neutrons would reach this range in the fast reactor core

dn-i/dt EF exp(—52Tj))^E И — p (1.7)

These three equations may be summed to give Eq. (1.3) by defining a new diffusion coefficient D as:

D = DT + [p exp(—BH)DF Р2ф?/Р2фт] + [exp(—B2rT>)DF Р2фъ/Р2фт]

(1.8)

n = nFp exp(—B2t) + nF exp(— BHj)) + nT

Thus the original single diffusion equation can be used to represent the single average group of neutrons. So long as D is chosen correctly the equation is more accurate than is, at first sight, apparent. Note that in a steady state, Eq. (1.3) reduces to the critical condition keg = 1.0.

This reduction to a single group equation would also apply in the case of a fast reactor; however, one should note that it is an approximation that may be used in kinetic calculations while the accuracy would certainly not be adequate for steady-state physics calculations.