The controlled and uncontrolled chain reaction of the fission of 235U is used in nuclear power plants and weapons, respectively. The neutron balance is quantita­tively determined using the effective neutron multiplication factor (k), which is the average number of neutrons produced from one fission that cause an additional fission:

k = —

Up

where Up and ns are the number of primary and secondary neutrons, respectively.

One of the most important properties of the fission is that two to three neutrons per fission are released (see Eq. (6.21)), which can initiate new fission steps. The

condition of the sustainable chain reaction is that at least one of the released neu­trons should initiate an additional fission. If the average number of the neutrons ini­tiating new fission (k) is 1, the released energy becomes constant. In a stationary state, this is the case in nuclear reactors.

When the number of neutrons initiating additional fission is more than 1, the released energy exponentially increases. This is the case, for example, at the startup of nuclear reactors, or when the power produced by nuclear reactors is to be increased. Nuclear weapons are designed to operate in this way.

In fission reactions, the energy of the released neutrons is usually high (1—2MeV); the additional fission, however, can be initiated only by slow or ther­mal neutrons (<0.1 eV). For this reason, the velocity or energy of the neutrons has to decrease, which significantly influences the neutron multiplication factor. When the fissile material is assumed to be in an infinite quantity, the multiplication factor (kN) is given by the so-called four-factor formula as follows:

k® = єpfП (7-2)

In this equation, є is the fast fission factor, which takes into consideration that the fast neutron can initiate another fission to a small degree (by 1—3%); P is the resonance escape probability, the fraction of neutrons escaping capture while slowing down. The value of p usually ranges from 0.6 to 0.9 and is increased by all factors assisting the slowing down of the neutrons (e. g., by the improvement of the moderators) by decreasing the size of the fuel and by increasing its distance from the fuel rods. The thermal utilization factor, f, is the ratio of the thermal neutrons initiating additional fission to the number of thermal neutrons captured by another reaction (e. g., by nuclides other than fissile ones). And n is the thermal neutron yield, that is, the number released in the fission process.

If the size of the fuel is finite, the effective multiplication factor (keff in Eq. (7.1)) is used; keff<k®. At keff< 1, the chain reaction stops because of the continuous decrease of the neutrons. The reactor is subcritical. When keff = 1, the rate of the chain reaction is constant, and the reactor is critical. When keff > 1, the number of the neutrons, and, as a consequence, the number of fission reactions, increases and the reactor is supercritical.

A characteristic property of the reactor is the reactivity (p):

keff ~ 1
keff

The value of p can be negative, zero, or positive, depending on whether the reactor is subcritical, critical, or supercritical, respectively. Since the fissile material is continuously used up by fission, the fission products can also capture neutrons, and a certain excess of reactivity is required for the critical operation.

7.1.1 The Main Parts of Nuclear Reactors

The very simple scheme of a nuclear reactor and the connecting energetic units are shown in Figure 7.2. The arrangement of the fuel and control rods in the reactor vessel is shown in Figure 7.3.

Figure 7.3 The arrangement of the fuel and control rods in the reactor vessel.

Coolant to heat
exchanger

Fuel rod Control rod

The most important parts of the nuclear reactors are the fuel elements, modera­tor, reflector, cooling system, control rods, and shielding. In the following sections, the material, properties, and operation of these parts will be discussed.