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About this book

This expanded new edition develops the theory of nuclear reactors from the fundamentals of fission to the operating characteristics of modern reactors. The first half of the book emphasizes reactor criticality analysis and all of the fundamentals that go into modern calculations. Simplified one group diffusion theory models are presented and extended into sophisticated multi-group transport theory models. The second half of the book deals with the two main topics of interest related to operating reactors – reactor kinetics/dynamics, and in-core fuel management. Additional chapters have been added to expand and bring the material up-to-date and include the utilization of more computer codes. Code models and detailed data sets are provided along with example problems making this a useful text for students and researchers wishing to develop an understanding of nuclear power and its implementation in today’s modern energy spectrum.

Covers the fundamentals of neutronic analysis for nuclear reactor systems to help understand nuclear reactor theory;

Describes the benefits, uses, safety features, and challenges related to implementation of Small Modular Reactors;

Provides examples, data sets, and code to assist the reader in obtaining mastery over the subjects.

Table of Contents


Chapter 1. Neutron Physics Background

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.
Bahman Zohuri

Chapter 2. Modeling Neutron Transport and Interactions

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.
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Chapter 3. Spatial Effects in Modeling Neutron Diffusion: One-Group Models

The fundamental aspect of keeping a reactor critical was discussed in Chap. 2, and we found out that the most principal evaluation quantity of the nuclear design calculation is the effective multiplication factor \( \left(\overline{\sigma}\left(\mathrm{v},T\right)=\frac{1}{\mathrm{v}}\int {d}^3V\left|\mathrm{v}-V\right|\sigma \left(\left|\mathrm{v}-V\right|\right)M\Big(V,T\Big)\right) \) 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 so far. In this chapter we treat the theory and mechanism to be able to analyze 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 one-dimensional and one-group models. The goal of this chapter is also for the reader to understand simple reactor systems, the notion of criticality, what it means both physically and mathematically, how to analytically solve for the steady-state flux for simple geometries, and finally how to numerically solve the steady state for more arbitrarily complex geometries.
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Chapter 4. Energy Effects in Modeling Neutron Diffusion: Two-Group Models

In this chapter, we derive the multi-group 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 becomes one of determining accurate multi-group cross sections for the design problem under consideration.
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Chapter 5. Numerical Methods in Modeling Neutron Diffusion

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.
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Chapter 6. Slowing Down Theory

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:
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Chapter 7. Resonance Processing

In this chapter, we will study the Doppler broadening of resonances. Doppler effect improves reactor stability. Broadened resonance or heating of a fuel results in a higher probability of absorption, thus causing 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 Breit-Wigner single-level resonance formula.
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Chapter 8. Heterogeneous Reactors and Wigner-Seitz Cells

When deterministic neutron transport methods are applied to lattice or whole-core problems, the multi-group 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 multi-group cross sections is very dependent on energy and space, which becomes a crucial challenge when analyzing a lattice or full-core configuration.
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Chapter 9. Thermal Spectra and Thermal Cross Sections

Although accurate determination of the thermal spectrum also requires advanced computational methods, averages over simplified 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 a multi-group diffusion theory. In previous chapters, we developed the general form of multi-group diffusion equations and recommended a strategy for their solution. However, these set of equations contained various parameters known as group constants formally defined as average over the energy-dependent intergroup fluxed which must be determined before these equations play a formal important roles. In this chapter, we introduce the calculation of neutron energy spectrum characterizing fast neutrons, and as 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.
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Chapter 10. Perturbation Theory for Reactor Neutronics

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, for a uniform occurrence of perturbation throughout the entire reactor or a region of it, 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 non-uniform, and there are numerous examples of such non-uniform perturbations.
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Chapter 11. Reactor Kinetics and Point Kinetics

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 harmonic 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, keff, period, neutron density, and power level.
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Chapter 12. Reactor Dynamics

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 time dependence of neutron population and therefore the power level of the reactor. Such situation may take place, for a number of 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 require attention and adjustment to day-to-day operation of 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 predict the time behavior of the neutron population in a reactor core driven by changes in reactor multiplication, which are not circumstances that are totally controlled by the operator of power plant and reactor core. Furthermore, variables such as indirect accessibility to control such as the fuel temperature or coolant density distribution throughout reactor do have impact to the situation. However, these variables depend on the reactor power level and hence the neutron fluxes itself. Additionally, study of the time dependence of the related process, which is involved in determining the core multiplication as a function of power level of the reactor multiplication, is subject of our study in this chapter, and it is called nuclear reactor dynamics. This usually involves detailed modeling of the entire nuclear steam supply system, which is part of feedback system as well.
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Chapter 13. Reactor Stability

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 affect 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 practiced for the 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 and safety management systems are the underlying causes of most accidents.
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Chapter 14. Numerical Modeling for Time-Dependent Problems

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.
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Chapter 15. Fission Product Buildup and Decay

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 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.
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Chapter 16. Fuel Burnup and Fuel Management

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 onsite for cooling and storage. Some utilities are moving their spent fuel after several years in the pool into 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.
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Chapter 17. Why Nuclear Power Plant Energy

The major growth in the electricity production industry in the last 30 years has centered on the expansion of natural gas power plants based on gas turbine cycles. The most popular extension of the simple Brayton gas turbine has been the combined cycle power plant with the Air Brayton cycle serving as the topping cycle and the steam Rankine cycle serving as the bottoming cycle for new generation of nuclear power plants that are known as GEN-IV. The Air Brayton cycle is an open-air cycle and the steam Rankine cycle is a closed cycle. The Air Brayton cycle for a natural gas-driven power plant must be an open cycle, where the air is drawn in from the environment and exhausted with the products of combustion to the environment. This technique is suggested as an innovative approach to GEN-IV nuclear power plants in form and type of Small Modular Reactors (SMRs). The hot exhaust from the Air Brayton cycle passes through a Heat Recovery Steam Generator (HSRG) prior to exhausting to the environment in a combined cycle. The HRSG serves the same purpose as a boiler for the conventional steam Rankine cycle [1, 2].
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Chapter 18. Small Modular Reactors and Innovative Efficient Enhancement Design

The smaller-sized nuclear reactors were becoming instrumental during the pioneering days of commercial nuclear power to facilitate the development and demonstration of early reactor technologies and to establish operational experience for the fledgling nuclear power industry starting with the first powered US Navy nuclear submarine. As part of innovative approach in design of small modular reactors (SMRs) in the new era of nuclear power energy to meet the demand for electricity due to growth of population, researchers are in quest of a more efficient electrical outpower by these types of reactors known as advanced small modular reactors (AdvSMRs) suggested as the air Brayton cycle is an open-air cycle and the steam Rankine cycle is a closed cycle. The air Brayton cycle for a natural gas-driven power plant must be an open cycle, where the air is drawn in from the environment and exhausted with the products of combustion to the environment.
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Chapter 19. Design and Analysis of Core Design for Small Modular Reactors

The pronuclear energy advocates are lobbying that the sustainable development of the world’s energy sector cannot be achieved without extensive use of nuclear energy and the advantages of nuclear-related technologies, including upcoming new generation of the small modular reactors in the near future horizon. The dawn of these SMRs requires new design and analysis no matter if they are falling into light water reactor (LWR), pressurized water reactor (PWR), or even multi-application small light water reactor (MASLWR) categories, depending on the vendor involved with these new technologies and consequently safety standards and their nonproliferation requirements as well. This chapter visits these standards for core design and generally elaborated on them with understanding that readers need to refer just beyond this book and this chapter for more details.
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