Foundations in Applied Nuclear Engineering Analysis

The following sections are included:

• Dedication
• Preface to Second Edition
• About the Author
• Contents

Nuclear engineering analysis is a very broad subject title. Nuclear engineering and medical physics are disciplines that are ultimately very applied; while the “pure sciences” are often afforded the luxury of only considering hypothetical problems, nuclear engineers must respond to delivering real engineering systems to benefit humanity, e.g. providing nuclear power, advanced imaging devices, curing cancer, and the like. Therefore, nuclear engineers must design systems and devices that are based on nuclear interactions using subatomic particles that are not detectable by the ordinary human senses…

The study of probability began in the 17_th Century as a result of a desire to determine the outcome of games of chance. By definition, a probability is a real number bounded between [0, 1]. Often, probabilities are expressed as percentages, or positive outcomes in a finite number of trials, e.g. your odds of winning are “1 in 3”. More formally…

Most problems of practical interest in nuclear engineering and medical physics require the use of numerical algorithms to yield a solution. In fact, many of the quantities of interest can only be solved numerically. No matter what is being computed numerically, it is a fundamental truth that there are inherent assumptions, approximations, and limitations, and also errors associated with numerical computations. Therefore, it is important for one to have an appreciation for the issues involved in accomplishing numerical calculations…

In order to understand several concepts used in the mathematics of solving problems in nuclear engineering, it is important to have experience with complex numbers. Complex numbers play a role in the solution of differential equations, as well as in spherical harmonics functions, and other applications involving nuclear engineering. Therefore, this chapter covers a very brief look at complex numbers.

A differential equation, according to the Dictionary of Mathematics, is: An Ordinary Differential Equation (ODE) is an equation that defines a relationship between an independent variable x and a dependent variable y, and one or more derivatives of y with respect to x, the solution of which leads to a mathematical identity when the solution is substituted back into the equation. Also, a Partial Differential Equation (PDE) is a differential equation that involves more than one independent variable, with dependent variable partial derivatives of the independent variables in an equation…

Power series are useful tools that can be used to expand other functions, solve equations, provide for assessment of intervals of convergence, used as trial functions, and are applied in all areas of engineering. Included in this discussion are Taylor's Series, which are extremely important in numerical approximations. We present here a brief discussion of Power Series with several applications.

We already discussed constant coefficients and methods of solutions. Now we discuss more general solution methods and applications. First we'll review analytic functions and points of interest.

This chapter is concerns fundamental concepts in linear algebra, focusing on vectors, matrices, and their applications in linear systems.

This chapter concerns fundamental concepts series expansions of functions as a preamble to the solution of partial differential equations. Mastery of these topics are important in order to be well prepared to solve partial differential equations.

Up to this point, we have covered a number of topics that form a mosaic of tools and methods useful in solving applied differential equations problems. There remain some critical applied engineering concepts that are essential to completing the suite of tools desirable for an engineer to become highly effective at solving differential equations, both ordinary and partial. This chapter, titled “Applied Solution Methods—Part I” covers those remaining topics. Following this chapter is “Applied Solution Methods—Part 2 PDEs and Heat Transfer” which focuses on problem solving methods based on the heat equation and the neutron diffusion equation.

In this chapter, we consider both ODEs and partial differential equations, or PDEs, solved using common methods encountered in nuclear engineering applications and related problems of interest. Fundamentally, PDEs are typically solved by applying methods similar to those used for ordinary differential equations, but with the added complexity of separating multivariate conditions, and applying orthogonality to yield a linearly independent solution basis. For this reason, we focus on solving the heat equation. Then, after covering essential concepts here, we will consider problems involving neutron transport and diffusion theory.

We now consider problems involving neutron transport and neutron diffusion theory; because the emphasis of this presentation is on methods, we do not present strict end-to-end theory to support/not support the use of diffusion theory in the applied problems considered, as this is best covered in a dedicated course on reactor physics. We begin with the transport equation for completeness, and then merge into the diffusion equation…

Due to their complexity, which is evident for seemingly simple problems, partial differential equations are often best solved numerically. This chapter presents the fundamental knowledge needed in order to derive the systems of equations needed for numerical finite difference solutions of PDEs. These systems of equations can then be directly solved via linear algebra methods discussed previously.

The following sections are included:

• Appendix: Selected Problems; Sets I–V
• Bibliography
• Index