Pi is Exactly Three!!!

The title of this article certainly caught your attention, didn't it? This is a statement made by Professor Frink in the sixteenth episode of the twelfth season of the Simpsons series called "Bye Bye Nerdie". Since it worked for him to get the attention of the scientists attending his talk, I thought I should try it here too and I certainly hope it worked.

Like it or not, mathematics, or most people simply call it math, has been and will always be an integral part of our lives. For lots of people, the most tangible application of math will usually be related to the financial aspect of one's life. But for others, like engineers, scientists, financial analysts, and of course math teachers or professors, math somehow manages to show up more often throughout their daily activities. For the nuclear industry, math is indeed one of its supporting pillars. With that in mind, I would like to take us on a short journey through a few applications of math in the field of nuclear engineering.

Linear Algebra and Matrices

The heart of linear algebra is solving linear equations. For example, it could be as simple as a child trying to figure out the price of an ice cream cone, knowing that it costs five dollars to get four cones. As the number of equations involved getting larger, one would use matrices and perform matrix computation on the system of equations. In the field of nuclear engineering, as in other engineering fields, large matrix computation is typically related to solving discretized differential equations, using for example the finite difference or finite element methods. Sometimes, non-linear problems (for example, neutron diffusion equations with feedback mechanisms) are linearized to get first order approximation within the overall iterative method. The resulting system of equations is then treated like any other linear system of equations using the knowledge of linear algebra and matrix computation.

Calculus and Differential Equations

The knowledge of calculus and differential equations (both ordinary and partial differential equations) is essential for a successful career in the nuclear engineering field. The knowledge is demanded from the very beginning - the concept of radioactive decay. For reactor physics application, the knowledge is required even when solving the neutron diffusion equation in the simplest form - in other word, when solving one-group, one-region, Cartesian geometry problem. Needless to say, the calculations become more demanding when cylindrical or spherical coordinate systems are used. Finally, calculus of variation or variational method is another type of analyses typically used in this field that requires lots of calculus.

Complex Number

A complex number is defined as a subset of the number system which has real and imaginary components. The first practical use of complex number in the nuclear engineering field is related to reactor dynamics and control. One important component of the stability analyses within reactor dynamic is the understanding of the Laplace Transfer operation which not only requires the knowledge of complex number but also of calculus. Another application is conformal mapping which is useful in modeling and analyzing reactor cores with hexagonal lattice.

Mathematical Logic

The main application of mathematical logic in nuclear reactor operation is related to the reactor trip system. For example, in Canada Deuterium Uranium (CANDU) reactors, the Regional/Neutron Overpower Protection system consists of two shutdown systems. In each shutdown system, there are three safety channels, each of which consists of up to around 20 detectors. Each safety channel trips when one of the detector trips. The overall shutdown system will trip following certain trip logics. The knowledge of Mathematical Logic would be essential in designing such system.

Game Theory

Game Theory can be described as a branch of mathematics which studies mathematical models of strategic interactions among some participants who are rational and are assumed to perform optimal action based on available information. From this description, it should be obvious that the application would not be strictly analytical. Typical applications of game theory in nuclear engineering field are related to nuclear non-proliferation treaty, nuclear stability, and nuclear safeguard.

Optimization

An optimization process can be simply described as a process of selecting certain combination of inputs such that the best outcome is attained. The definition of the "best" outcome depends on the problem being solved. It could sometimes be related to finding a minimum value (e.g., minimizing the number of detectors installed in the core of a reactor or minimizing the dose received by a patient during a radiotherapy procedure), while for another problem is related to finding a maximum value (e.g., maximizing the fuel utilization or maximizing the safety margin). Optimizations are usually invoked during the design process of a new system or when trying to improve certain process or existing system. The following are some nuclear engineering fields where optimization techniques have been successfully utilized: nuclear fuel design, in-core nuclear fuel management, safety analyses, management of irradiated nuclear fuels as well as radioactive wastes, and energy production.

Needless to say, there are many other branches of math which have applications in nuclear engineering such as probability and statistics, topology, numerical analysis, and cryptography, to name a few. In closing, through this short article on the roles of math in the nuclear engineering field I hope nuclear engineering students who were wandering why they had to take so many math courses will wander no more. As the field of nuclear engineering advances, there will be additional branches of math utilized within the analyses. Finally, I hope we will be applying and taking advantage of math more often in our lives. I hope I can count on you.