Neutronic Analysis For Nuclear Reactor Systems

This chapter introduces fundamental properties of the neutron. It covers reactions induced by neutrons, nuclear fission, slowing down of neutrons in infinite media, diffusion theory, the few-group approximation, point kinetics, and fission product poisoning. It emphasizes the nuclear physics bases of reactor design and its relationship to reactor engineering problems.

It is essential to know the spatial and energy distributions of the neutrons in a field in a nuclear fission reactor, D–T (or D–D) fusion reactor, or other nuclear reactors populated with large numbers of neutrons. It is obvious why the spatial distribution should be known, and because neutron reactions vary widely with energy, the energy distribution is also a critical parameter. The neutron energy distribution is often called the neutron spectrum. The neutron distribution satisfies transport equation. It is usually difficult to solve this equation, and often approximated equation so-called diffusion equation is solved instead. In this chapter only overview of transport equation and diffusion equation of neutrons is presented, and methods for solving these equations are presented in the following sections.

The fundamental aspect of keeping a reactor critical was discussed in Chap. 2, and we found out that the most principle evaluation quantity of the nuclear design calculation is the effective multiplication factor (\( \overline{\sigma}\left(v,T\right)=\frac{1}{v}{\displaystyle \int {d}^3V\left|v-V\right|\sigma \left(\left|v-V\right|\right)}M\left(V,T\right) \)) and neutron flux distribution. We also so far have noticed that the excess reactivity, control rod worth, reactivity coefficient, power distribution, etc., are undoubtedly inseparable from the nuclear design calculation. Some quantities among them can be derived by secondary calculations from the effective multiplication factor or neutron flux distribution that was also discussed in Sect. 2.15 of Chap. 2. In this chapter we treat the theory and mechanism to be able to analysis and calculate the effective multiplication factor and neutron flux distribution and possibly show numerical analysis and computer codes involved with solving the diffusion equation in a one-dimensional and one-group models. The goal of this chapter is also for readers to understand simple reactor systems, the notion of criticality, what it means both physically and mathematically, how to analytically solve steady-state flux for simple geometries, and finally how to numerically solve the steady state for more arbitrarily complex geometries.

In this chapter we derive the multigroup diffusion equation (MGDE), and we illustrate how do we solve them in a way that allows us to calculate an accurate eigenvalue and accurate reaction rates. Since the cross sections vary wildly by multiple orders of magnitude over the energy range in a typical nuclear reactor, the major problem is determining the accurate multigroup cross sections for the design problem under consideration.

The constructive techniques of functional analysis, using a computer code, allow us to build up directly, in their original domain of definition, solutions to linear transport equation. FEMP code is a computer code written in FORTRAN 77, to approximate the Boltzmann transport equation in one-dimensional form using a spherical harmonic for the angular variable and a linear finite element for the spatial variable.

In neutronic analysis for nuclear reactor systems, we look at three types of reactors depending upon the average energy of neutrons, which cause the bulk of the fission in the system. Thermal reactors, in which most fission are, are induced by neutrons, and these neutrons more or less are in thermal equilibrium with the atoms in the reactor and maintain energy below approximately 0.3 eV. Intermediate reactor or resonance reactors, in which neutrons have energy above thermal up to 10 keV and most of these neutrons are responsible for fission, lie in the resonance region of the heavy elements. This type of reactor is also largely responsible for producing fissions. Finally, the third one is the fast reactor, in which fissions are induced primarily by neutrons with energy of the order of 100 keV and above. Finally, yet importantly, one should be aware of the fact that in a thermal reactor, the fission neutrons are slowed down by the use of a moderator, and that is a mass of material, such as carbon, beryllium, or water, which is distributed throughout the fueled region or core of the reactor [1].

In this chapter, we will study the Doppler broadening of resonances. The Doppler effect improves reactor stability. Broadened resonance or heating of a fuel results in a higher probability of absorption, thus causes negative reactivity insertion or reduction of reactor power. One of the most important virtues of the optical model is that it takes into account the existence of giant or broad resonances in the total cross section as part of neutronic analysis for nuclear reactor systems. For resonances of energy levels, which are spaced widely apart, we can describe the energy dependency of the absorption cross section via the Breit–Wigner single-level resonance formula.

When deterministic neutron transport methods are applied to lattice or whole-core problems, the multigroup approximation is usually applied to the cross-sectional treatment for the energy domain. Due to the complicated energy behavior of resonance cross sections, the weighting spectrum for collapsing multigroup cross sections is very dependent on energy and space, which becomes a crucial challenge when analyzing a lattice or full-core configuration.

Although accurate determination of the thermal spectrum also requires advanced computational methods, average oversimplified spectra often serve as a reasonable first approximation in performing rudimentary reactor calculations. The main aspect of nuclear reactor analysis, as we have learned so far, is multigroup diffusion theory. In previous chapters, we developed the general form of multigroup diffusion equations and recommended a strategy for their solution. However, these set of equations contained various parameters known as group constants formally defined as the average over the energy-dependent intergroup flux which must be determined before these equations play formal important roles. In this chapter, we introduce the calculation of neutron energy spectrum characterizing fast neutrons, and as a result, the calculation of fast neutron spectra as well as generation of fast group constants will be of concern. At the conclusion, we will deal with the development of the theory of neutron slowing down and resonance absorption.

Perturbation theory in neutronic analysis for nuclear reactor system is often necessary, when we analyze and compute the effect of small changes on the behavior of a reactor. On the other hand, if there is a uniform occurrence of perturbation throughout the entire reactor or a region, then, we use methods that we have so far discussed and presented in previous chapters, although we never encounter uniform perturbation in practice of reactor operations. However, most of the changes, which occur in the operation mode of a reactor, are nonuniform, and there are numerous examples of such nonuniform perturbations.

The point kinetics model can be obtained directly from the space- and time-dependent transport equations. However, these equations are too complicated to be of any practical application. The diffusion approximation, obtained by keeping only the PI terms of the spherical harmonics expansion in the angular variable of the directional flux, is frequently used in neutronic analysis. This chapter discusses reactor characteristics that change because of changing reactivity. A basic approach using a minimum of mathematics has been followed. Emphasis has been placed on distinguishing between prompt and delayed neutrons and showing relationships among reactor variables, k _eff, period, neutron density, and power level.

In order for nuclear fission power to operate at a constant power level, the rate of neutron production via fission reactions must be exactly balanced by neutron loss via absorption and leakage. If we deviate from this simple balancing role, it would cause in a time dependence of neutron population and therefore the power level of the reactor. Such situation may take place, for a number reasons, such as reactor operator may have a requirement to change the reactor power level by temporarily altering the control fuel rod so it will change the core or source multiplication, or there may be long-term changes in core multiplication due to fuel depletion and isotopic buildup. Other examples may also be encountered that requires attention and adjustment to the day-to-day operation of the reactor, such as unforeseen accident or failure of primary coolant pump system, etc. The topic of nuclear kinetic reactor as we have learned in the previous chapter is handling this situation by allowing us to be able to predict the time behavior of the neutron population in a reactor core driven by changes in reactor multiplication, which is not a circumstance that is totally controlled by the operator of power plant and reactor core. Furthermore, variables such as indirect accessibility to control such fuel temperature or coolant density distribution throughout the reactor do have impact to the situation. However, these variables depend on the reactor power level and hence the neutron fluxes itself. Additionally, the study of the time dependence of the related process, which is involved with determining the core multiplication as a function of power level of the reactor multiplication, is the subject of our study in this chapter, and it is called nuclear reactor dynamics. This usually involves with detailed modeling of the entire nuclear steam supply system, which is part of feedback system as well.

Understanding time-dependent behaviors of nuclear reactors and the methods of their control is essential to the operation and safety of nuclear power plants. This chapter provides researchers and engineers in nuclear engineering very general yet comprehensive information on the fundamental theory of nuclear reactor kinetics and control and the state-of-the-art practice in actual plants, as well as the idea of how to bridge the two. The dynamics and stability of engineering equipment that affects their economical and operation from safety and reliable operation point of view. In this chapter, we will talk about the existing knowledge that is today’s practice for design of reactor power plants and their stabilities as well as available techniques to designers. Although, stable power processes are never guaranteed. An assortment of unstable behaviors wrecks power apparatus, including mechanical vibration, malfunctioning control apparatus, unstable fluid flow, unstable boiling of liquids, or combinations thereof. Failures and weaknesses of safety management systems are the underlying causes of most accidents.

The possibility of a plutonium-fueled nuclear-powered reactor, such as a fast breeder reactor that could produce more fuel than it consumed, was first raised during World War II, in the United States by scientists involved in Manhattan Project and the US Atomic Bomb Program. In the past two decades, the Soviet Union, the United Kingdom, France, Germany, Japan, and India followed the United States in developing a nationalized plutonium breeder reactor programs, while Belgium, Italy, and the Netherlands collaborated with the French and German programs. In all of these programs, the main driver of this effort was the hope of solving the long-term energy supply problem using the large-scale deployment of fissional nuclear energy for electric power. Breeder reactors, such as plutonium-fueled breeder reactors, appeared to offer a way to avoid a potential shortage of the low-cost uranium required to support such an ambitious vision using other kinds of reactors, including today’s new generation of power reactors known as GEN IV.

Nuclear fission products are the atomic fragments left after a large atomic nucleus undergoes nuclear fission. Typically, a nucleus that has a large atomic mass like uranium could fission by splitting into two smaller nuclei, along with a few neutrons. This process results in the release of heat energy, such as kinetic energy of the nuclei, and gamma rays. The fission products themselves are often unstable and radioactive, due to being relatively neutron rich for their high atomic number, and many of them quickly undergo beta decay. This releases additional energy in the form of beta particles, antineutrinos, and gamma rays. Thus, fission events normally result in beta radiation and antineutrinos, even though these particles are not produced directly by the fission event itself.

Nuclear fuel is removed from a reactor every few years when it can no longer economically keep a chain reaction going. This “spent” fuel remains radioactive and must be managed. At first, it goes into a pool on-site for cooling and storage. Some utilities are moving their spent fuel after several years in the pool into the US Nuclear Regulatory Commission (NRC) certified dry storage casks. These casks are specially designed to contain the radioactivity and allow hot spent fuel to cool further. In contrast to fossil fuel, the fuel in nuclear reactors cannot be converted since the fuel undergoes changes during its use in the reactor, which require the fuel elements to be exchanged.