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Delayed Neutron and Nuclear Reactor Kinetics

  • Yoshiaki Oka
Chapter
Part of the An Advanced Course in Nuclear Engineering book series (ACNE)

Abstract

Introduction Reactor power changes when the temperature and position of the control rods of a nuclear reactor are changed. This change is unique to each reactor, and its characteristics are called “nuclear reactor kinetics.”

The control rods are made of strong neutron-absorbing materials, and when they are inserted into the reactor, the reaction rate of neutron absorption increases. The reactor becomes subcritical and its power decreases. Conversely, the reaction rate of neutron absorption decreases when the control rods are withdrawn; the reactor becomes supercritical and its power increases. The reaction rate of neutron absorption changes when the reactor temperature is changed and, therefore the reactor power changes.

The reactor power is proportional to the number of fission reactions per second in the nuclear reactor. As fission reactions are caused by neutrons, the number of their reactions is proportional to the total number of neutrons in the reactor. However, the number of neutrons varies depending on the neutron production rate due to the fission reactions, the rate of neutron absorption by the nuclear fuel and reactor structure materials, and the rate of neutron leakage from the reactor.

Keywords

Fission Reaction Neutron Absorption Prompt Neutron Delayed Neutron Neutron Leakage 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction Reactor power changes when the temperature and position of the control rods of a nuclear reactor are changed. This change is unique to each reactor, and its characteristics are called “nuclear reactor kinetics.”

The control rods are made of strong neutron-absorbing materials, and when they are inserted into the reactor, the reaction rate of neutron absorption increases. The reactor becomes subcritical and its power decreases. Conversely, the reaction rate of neutron absorption decreases when the control rods are withdrawn; the reactor becomes supercritical and its power increases. The reaction rate of neutron absorption changes when the reactor temperature is changed and, therefore the reactor power changes.

The reactor power is proportional to the number of fission reactions per second in the nuclear reactor. As fission reactions are caused by neutrons, the number of their reactions is proportional to the total number of neutrons in the reactor. However, the number of neutrons varies depending on the neutron production rate due to the fission reactions, the rate of neutron absorption by the nuclear fuel and reactor structure materials, and the rate of neutron leakage from the reactor.

1.1 Fission Chain Reactions

When a neutron collides with a heavy nucleus such as 235U or 239Pu, it splits the nucleus into two nuclei which are called fission fragments. In this process, multiple neutrons, several gamma rays, and neutrinos are also released. Multiple neutrons are produced by a single fission reaction. If one of these neutrons causes another fission event, it results in another emission of neutrons, followed by more fission events, and so on. This is called a fission chain reaction. Figure 1.1 illustrates the concept of the fission chain reaction.
Fig. 1.1

Conceptual diagram of fission chain reaction

Because neutrons play a key role in maintaining the fission chain reaction, it is important to understand their behavior with regard to designing a reactor and analyzing its characteristics. A multiplication factor is used to analyze the fission chain reaction quantitatively. The effective neutron multiplication factor, k, is defined as the ratio between neutron production and loss (absorption plus neutron leakage from the reactor). In an infinitely sized reactor, the neutron loss consists of only neutron absorption, and k is expressed as the infinite neutron multiplication factor, k∞. However, the actual reactor size is limited and, in addition to the neutron absorption, neutron leakage from the reactor needs to be considered. In this case, the multiplication factor is expressed as the effective neutron multiplication factor, k eff.
$$ {k_{\mathrm{ eff}}}=\frac{{\mathrm{ Neutron}\mathrm{ production}\mathrm{ rate}}}{{\mathrm{ Neutron}\mathrm{ loss}\mathrm{ rate}(\mathrm{ the}\mathrm{ sum}\mathrm{ of}\mathrm{ absorption}\mathrm{ rate}\mathrm{ and}\mathrm{ leakage}\mathrm{ rate})}} $$
(1.1)

If k eff = 1, the number of neutrons is constant in the reactor. In other words, the fission rate is constant and the constant energy release continues. At this state, the reactor is critical. If k eff < 1, the number of neutrons decreases gradually in the reactor with progression of the fission chain reaction. In this state, the reactor is subcritical. However, if “k eff > 1,” the chain reaction rate increases and the reactor is supercritical.

As the concept of the fission chain reaction of Fig. 1.1 shows, neutrons are generated by nuclear fission and are lost when captured by nuclei or when leaked to the outside of reactor. When the generated neutrons are absorbed by a fissile nuclide (235U, for example) or by a fissionable nuclide (238U, for example) a certain rate of neutron generation triggers a fission. This produces new neutrons. If generation of a neutron is thought of as its “birth” and its loss is a “death,” the fission chain reaction is a process whereby “child” neutrons are born of “parent” neutrons. The effective multiplication factor can also be defined as a ratio of the number of neutrons between two consecutive generations.
$$ {k_{\mathrm{ eff}}}=\frac{{\mathrm{ Number}\mathrm{ of}\mathrm{ neutrons}\mathrm{ of}\mathrm{ a}\mathrm{ generation}}}{{\mathrm{ Number}\mathrm{ of}\mathrm{ neutrons}\mathrm{ of}\mathrm{ the}\mathrm{ preceeding}\mathrm{ generation}}} $$
(1.2)

Actually, however, it is difficult to determine the length of a generation of neutrons. This is because some neutrons trigger fission reactions immediately after they are reproduced but other neutrons trigger after their moderation to thermal neutrons. Neutron capture occurs randomly, and so does leakage of neutrons to the outside of the reactor. The following discussion uses Eq. (1.1) to define the effective multiplication factor.

When a reactor operates with a constant power, its effective multiplication factor is equal to 1. The reactor is “critical”. When the reactor is shut down, it is “subcritical”. During startup, the reactor is controlled so that it becomes “supercritical”; the neutron production rate is increased above the neutron loss rate, and the number of neutrons in the reactor is gradually increased. When the reactor reaches the required power rating, it is returned to the critical state and is operated with a constant output. If the reactor is required to change from low power to high power, it is controlled to reach the supercritical state. If the reactor is required to return to low power, it is controlled to reach the subcritical state; when it reaches the required power, the reactor is returned to critical.

The nuclear reactor kinetics usually explains an increase or decrease in the number of neutrons in the entire core. In other words, the spatial distribution of neutrons is not considered in the core. Here, a point-wise reactor approximation is used, where the core is represented by one point and no space variable is considered. Reactor kinetics that considers the spatial distribution is called space–time kinetics.

1.2 Change in Multiplication Factor and Nuclear Reactor Kinetics

In order to operate a nuclear reactor at constant power, the effective multiplication factor (k eff) must be equal to 1; that is, the neutron production by fission must be accurately balanced with the neutron loss due to the neutron absorption and leakage. Although the k eff value is changed by various factors, they can be grouped by the times needed for them to occur which are as follows:
  • Short period (from seconds to minutes): startup, shutdown, and disturbances during operation (including changes of temperature, pressure, and moderator density)

  • Intermediate period (from hours to days): generation and decay of fission products (xenon and samarium) having strong neutron absorption

  • Long period (from months to years): burnup (consumption) of nuclear fuel and accumulation of fission products

It is important to estimate of a change in the number of neutrons and a change in the power with time that occur when k eff changes. This is covered by the reactor kinetics and nuclear plant dynamics. The nuclear reactor kinetics covers the change in the number of neutrons and the change in power due to a short-period change in multiplication factor. Over a long period, the change in multiplication factor is compensated for by control rods, chemical shim, and burnable poison. Details for this are presented in Part II  Chaps. 3 6 of this book. In an intermediate period, the change in multiplication factor is estimated using the generation and decay model of xenon and samarium. It is different from the nuclear reactor kinetics model.

1.3 Prompt Neutron and Delayed Neutron

Most neutrons (99.35 % for 235U fission by thermal neutrons) are emitted immediately by a nuclear fission event. These are called “prompt neutrons.” A few neutrons are emitted a little after nuclear fission occurs and they are called “delayed neutrons.”

The delayed neutrons are primarily produced from the decay of the fission products emitting neutrons. The fission products that emit delayed neutrons are called delayed neutron precursors. There are many delayed neutron precursors such as 87Br, 88Br, 137I, 138I, and 139I and they have different half-lives. The delayed neutron precursors are treated in six groups with different half-lives for analysis of nuclear reactor kinetics. Table 1.1 shows data of delayed neutrons that are generated by thermal fission of 235U.
Table 1.1

Data of a delayed neutron generated by thermal fission of uranium-235

Group

Half-life (s)

Decay constant, λ i (s–1)

Delayed neutron fraction, β i

1

55.72

0.012 4

0.000 215

2

22.72

0.030 5

0.001 424

3

6.22

0.111

0.001 274

4

2.30

0.301

0.002 568

5

0.610

1.14

0.000 748

6

0.230

3.01

0.000 273

Total delayed neutron fraction β: 0.006 5

The delayed neutron precursor of the longest half-life is 87Br. Delayed neutron data differ from those of the fission nuclides and are between the data of thermal fission and fast fission neutrons. The data need to be used correctly depending on the type of the fuel and neutron spectrum of the reactor. Table 1.2 lists the delayed neutron fraction for each fission nuclide.
Table 1.2

Delayed neutron fraction of nuclides

Nuclide

Total delayed neutron fraction, β

232Th

0.020 3a

233U

0.002 6

235U

0.006 5

238U

0.014 8a

239Pu

0.002 0a

aGenerated by fast fission

The delayed neutrons have approximately 0.4-MeV average energy, which is lower than the approximate 2-MeV average energy of the prompt neutrons. Therefore, the fraction of delayed neutrons that is leaked outside the reactor and lost disappear is slightly smaller than that of the prompt neutrons. The fraction of delayed neutrons that contributes to the fission chain reactions is slightly larger than that of the prompt neutrons. This effect is considered in the analysis of nuclear reactor kinetics. A slightly larger delayed neutron fraction is used than the absolute value “β” depending on the reactor and the effect is shown as “β eff.” If the reactor has a large core volume, neutron leakage is very small during moderation and there is almost no difference between them. The “β eff” value depends on the reactor size and neutron spectrum. Although the delayed neutron fraction is low, it slows down the transient change of the reactor and, therefore, it plays a very important role in the reactor control.

The distribution of prompt neutron energies can be expressed by the following function:
$$ \chi (E)=0.453{{\mathrm{ e}}^{-1.012E }}\sinh\sqrt{2.19E } $$
(1.3)
where, E is the neutron energy in MeV.

1.4 Kinetic Parameters

This section explains the parameters commonly used for description of reactor kinetics. The definition of effective multiplication factor, k eff, has been given by Eq. (1.1) or (1.2).

The “reactivity” indicates a degree of deviation from the critical state, and it can be defined as follows:
$$ \rho =\frac{{{k_{\mathrm{ eff}}}-1}}{{{k_{\mathrm{ eff}}}}} $$
(1.4)

If the reactor is supercritical, k eff > 1 and the value of ρ is positive. If the reactor is subcritical, k eff < 1 and the value of ρ is negative. ρ takes a value within the range of −∞ < ρ < 1.

The reactivity is expressed as a numerical value or a percentage. It is shown in the French unit of “pcm” (10−5), in the English unit of “milli-k” (10−3), or in the American unit of “dollars” ($) and “cents” (¢). A dollar is equal to the value ρ divided by generation rate β of delayed neutrons, and 1 $ is equal to 100 ¢.

If k eff = 1, that is, if ρ = 0, it is strictly said to be the “delayed critical” state. The generation of neutrons in the reactor (including the generation of delayed neutrons caused by the decay of delayed neutron precursors) is equal to the loss of neutrons. When the reactivity is 1 $, the generation of prompt neutrons is equal to its loss. This is called the “prompt critical” state. The state above the prompt critical is called the “prompt supercritical.”

The prompt neutron lifetime can be defined by the following equation:
$$ l\equiv \frac{{\mathrm{ Total}\mathrm{ number}\mathrm{ of}\mathrm{ neutrons}\mathrm{ in}\mathrm{ reactor}}}{{\mathrm{ Extinction}\mathrm{ rate}\mathrm{ of}\mathrm{ neutrons}}} $$
(1.5)
The prompt neutron lifetime can be expressed as an average time from generation of a prompt neutron to its absorption. Because the neutron slowing-down time is much shorter than the time when the neutron is diffused and absorbed, the prompt neutron lifetime of the thermal reactor is almost the same as the diffusion time of a thermal neutron. The diffusion time and slowing-down time for various moderators are shown in Table 1.3.
Table 1.3

Diffusion time and slowing-down time of various moderators

Moderator

Diffusion time (ms)

Slowing-down time (μs)

Light water (H2O)

0.205

1.0

Heavy water (D2O)

100a

8.1

Beryllium (Be)

3.46

9.3

Graphite

13.0

23

aDepends on the purity of heavy water

Because the neutrons are not moderated to become thermal neutrons in the fast reactor, the prompt neutron lifetime is an order of 10−5–10−7 s.

The prompt neutron generation time can be defined by the following equation:
$$ \varLambda \equiv \frac{{\mathrm{ Total}\mathrm{ number}\mathrm{ of}\mathrm{ neutrons}\mathrm{ in}\mathrm{ reactor}}}{{\mathrm{ Neutron}\mathrm{ generation}\mathrm{ rate}}} $$
(1.6)

The prompt neutron generation time is equal to the prompt neutron lifetime divided by the effective multiplication factor.

Among the kinetics parameters described here, the denominator is the neutron loss rate obtained in Eqs. (1.1) and (1.5) for effective multiplication factor k eff and prompt neutron lifetime and the denominator is the neutron generation rate for equation of reactivity ρ and prompt neutron generation time Λ. If k eff = 1, values and Λ become equal to each other. In the kinetics equations described below, a pair of k eff and values or a pair of ρ and Λ values should be used.

Copyright information

© Springer Japan 2013

Authors and Affiliations

  • Yoshiaki Oka
    • 1
  1. 1.Tokyo UniversityTokyoJapan

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