## A POINT MODEL OF REACTOR KINETICS

Figure 2.4 depicts a conceptual model of nuclear reactor kinetics whose variables are defined by

N—total number of free neutrons in the reactor at any time t l—life expectancy of a free neutron in the reactor N + dN—total number of free neutrons in the reactor at t + l G—expected number of neutrons born after a fission

8 Not temporarily withheld in precursors.

Figure 2.4 Conceptual Model of Neutron Kinetics |

J

1 — b—conditional probability of prompt fission with b = Yh bj

j=1

bj—conditional probability of creating a neutron precursor of the jth group

tj—time constant for the radioactive decay of the jth-precursor group

Cj—total number of neutrons temporarily withheld in the jth — precursor group

CA—neutron production rate from the artificial start-up source 1 — F—probability that a neutron is parasitically absorbed or escapes from the core

J—number of identifiably different precursor-groups

For clarity, just one group of delayed neutron precursors is shown in the diagram, though in practice a full description would generally involve no more than six.[32] As shown in Table 2.2, the above parameters depend on the type of reactor (fast or thermal), and also on its current geometry and temperature distribution. A 1-D diffusion model for tracking these parameters with changes of internal temperature distribution and control rod position is discussed in Ref. [117]. Because the expected lifetime l of a free neutron is so much shorter than the time constants of the radioactive decay of precursors, and because these decay processes are Poissonian, then the number of precursor-atoms in the jth-group which release their

neutrons over the period 1 is Cjl/tj. The number of these precursor — atoms created in the same interval is seen from Figure 2.4 to be FfijGN, so that to a first approximation

-d — = FbjGN/1 — Cj/ tj for 1 < j < J

Neutrons in a reactor core originate from prompt fission, the radioactive decay of precursors and the artificial start-up source. Reference again to Figure 2.4 enables the neutron population after the time internal 1 to be derived as

j

N + SN = F( 1 — b)GN + Cjl/tj + CaI (2.23)

j=1

Because a target nucleus absorbs a neutron prior to its fission, the actual increase in the neutron population over this time interval is SN, so that to a first approximation10

Defining

K = FG

10 Implicit in the notation of equation (2.22), whose left-hand side could otherwise be just SN.

reduces equations (2.22) and (2.24) to

dC;

l I = KbjN — Cj1/tj for 1 < j < J

dN’ J

l Ht =[K(1 — b) — 1]N + Cj l/tj + ICa

j=1

Some pertinent parameters for typical thermal and fast reactors are specified in Table 2.2, and the leakage and absorption factor 1 — F is derived from diffusion equation simulations as about 0.2. The much shorter life expectancy l of a neutron in a fast reactor results from its much greater fuel enrichment of around 20% and the absence of a moderator.

It is seen from Figure 2.4 that

FGN = Number of prompt and delayed neutrons created during the lifetime of their free parents

so from equation (2.25)

In that equation (2.25) allows for both the escape (leakage) and parasitic capture of neutrons, the above equation evidently justifies the description of K as the effective multiplication factor. If the free neutron — parents produce their equal number of prompt and delayed offspring so that the effective multiplication factor is unity, then the population appears stable. Indeed neglecting the relatively small contribution of the start-up source, equation (2.26) confirms that

K = 1 (2.28)

is a necessary and sufficient condition for the neutron population in a reactor to remain numerically constant.

Because the combined mass of fragments from a fission is less than that of their fissile parent atom [58], the deficit 8m appears in the form of their kinetic energy 8m c2. Collisions with surrounding materials rapidly

degrade this into heat, which for either U-235 or Pu-239 amounts to about 32 pJ per fission. It follows therefore from Figure 2.4 that

FN

Reactor power = 32—— x 10—12 W (2.29)

Nuclear power reactors are seen, therefore, to have enormous neutron populations, and for simulation purposes equations (2.26) are conveniently scaled, say by trillions of neutrons.