# Preface

The design of the reactor core is based on the description of the production, transport and absorption of neutrons. As neutrons move within a medium, viz. gas, liquid or solid, they collide with the nuclei of the atoms in the medium. During such collisions, neutrons may be absorbed by the nuclei, which are either elastic or inelastic. Absorption of neutrons may result in a loss or an increase in the number of neutrons by fission. Fission neutrons usually have different energies and move in different directions than incident neutrons. As a result, there may be scattering of neutrons, which changes the position, energy and direction of the motion of the neutrons. Furthermore, the scattering of neutron collision inside a reactor depends upon the geometry of the reactor, diffusion coefficient and absorption coefficient. It may also be noted that the parameters responsible for the diffusion (i.e. the scattering of neutron) may not always be crisp, rather they may be uncertain.

In general, these uncertainties occur due to the vague, imprecise and incomplete information about the variables and parameters, as a result of errors in measurement, observation, experiment or applying different operating conditions or due to maintenance-induced errors, which are uncertain in nature. So, to overcome these uncertainties, one may use the fuzzy/interval or stochastic environmental parameters and variables in place of crisp (fixed) parameters. With these uncertainties, the governing differential equations turn fuzzy/ interval or stochastic. Practically, it is sometimes difficult to obtain the solution of fuzzy equations due to the complexity in the fuzzy arithmetic; for example, addition and multiplication are not the inverse operations of subtraction and division, respectively. On the other hand, we may model the problem as stochastic only when sufficient data are available. Moreover, the combination (hybridization) of fuzzy and stochastic is also a new vista. These were found to be important and challenging areas of study in recent years. As such, one may need to understand the nuclear diffusion principles/theories corresponding to reliable and efficient techniques for the solution of such uncertain problems.

Accordingly, the objective of this book is to provide first the basic concepts of reactor physics as well as neutron diffusion theory. The main aim of the book, however, is about handling uncertainty in neutron diffusion problems. Hence, uncertainties (i.e. fuzzy, interval, stochastic) and their applications in nuclear diffusion problems have been included here in a systematic manner, along with the recent developments. This book may be an essential reference for students, scholars, practitioners, researchers and academicians in the assorted fields of engineering and science, particularly nuclear engineering.

Chapter 1 describes the preliminaries of basic reactor principles. Here, a few important and related terminologies for the nuclear reactor are briefly explained. In Chapter 2, neutron diffusion theory and the formulation of the neutron diffusion equation have been presented, which give first-hand scientific insights to study various nuclear design problems. The transportation of the scattered neutrons is modelled and formulated mathematically. In Chapter 3, the fundamentals of uncertainties are discussed. In this chapter, uncertainties are addressed with respect to three categories, viz. interval, fuzzy and probabilistic. The operations of these uncertainties are demonstrated through various examples. Furthermore, Chapter 4 elaborates the uncertain modelling (considering interval/fuzzy parameters as uncertain) of neutron diffusion. The factors involved in the reactor system, which are responsible for uncertainness, are modelled in terms of interval/fuzzy.

One-group models with respect to crisp parameters are explained in Chapter 5. This chapter includes both the analytical and numerical approaches to investigate one- group models, whereas in Chapter 6, uncertain (considering interval/fuzzy parameters as uncertain) one-group models are discussed, and example problems are investigated. In this chapter, the uncertain neutron diffusion equation for a bare square homogeneous reactor is discussed, which has been modelled by the fuzzy finite element method. The multigroup neutron diffusion equation has been generalized in Chapter 7, and again the finite element method has been used to solve the same. In Chapter 8, the uncertain multigroup neutron diffusion model is investigated. Accordingly, a benchmark problem is solved under an uncertain environment, and the sensitivity of the uncertain parameters is also analyzed.

Chapter 9 includes the theory of point kinetic diffusion. Here, different terminologies related to point kinetic diffusion of the one-group bare reactor are discussed, and the point kinetic diffusion equation for crisp parameters is formulated. Next, the stochastic point kinetic diffusion equation is modelled in Chapter 10. The basic concept of the birth-death stochastic process and stochastic point kinetic diffusion are discussed here. Finally, in Chapter 11, hybridized uncertainty (i.e. both probabilistic and fuzzy) is considered, and uncertain point kinetic diffusion is modelled. Furthermore, the hybridized uncertainty for this model is demonstrated through an example problem.

This book aims to provide a systematic understanding of nuclear diffusion theory along with uncertainty, viz. fuzzy/interval/stochastic and hybrid methods. The book will certainly prove to be a benchmark for graduate and postgraduate students, teachers, engineers and researchers in this field. It provides comprehensive results and an up-to-date and self-contained review of the topic along with application-oriented treatment of the use of newly developed methods of different fuzzy, stochastic and hybrid computations in various domains of nuclear engineering and sciences. It is worth mentioning that the presented methods may very well be used in/extended to various other engineering disciplines, viz. electronics, marine, chemical and mining, and sciences such as physics, chemistry and biotechnology, where one wants to model physical problems with respect to non-probabilistic (interval/fuzzy) and hybrid uncertainties for understanding the real scenario. ^{[1]} ^{[2]}