01TWMND

A.A. 2021/22

2021/22

Nuclear fission reactor physics and transport theory

The course provides the specific physical and mathematical bases to be used in the study of the physics of fission and fusion reactors. The course is made of two parts. In the first, the kinetic theory is treated. The classic Boltzmann equation is deduced and the relationship between kinetic theory and fluid models is studied. In the second part the neutron transport equation and the physical principles of nuclear fission reactors are illustrated. Some simplified models and approximations for the description of the neutronics of multiplying systems are then presented, together with the fundamentals of criticality theory, reactor dynamics and perturbation theory.

Nuclear fission reactor physics and transport theory

The course nuclear fission reactor physics and transport theory aims at providing competences regarding the modelling of neutron interaction with matter, with a specific focus on the phenomena occurring in a fission nuclear reactors, complemented by a more general approach to Boltzmann kinetic theory for gases and fluids. The lectures on fission reactor physics, amounting to 75% of the course content, illustrates the physico-mathematical model adopted for the description of neutron behavior in nuclear reactors. The physical significance of the linear transport model and the various approaches to its numerical, deterministic solution are presented. The resulting set of information provided allow the energy and nuclear engineering master student to acquire the necessary competences to predict and interpret the physical behaviour of a nuclear system, in both steady-state and transient conditions. The course also include a set of lectures focused on Boltzmann kinetic theory (25% of the course content). Boltzmann kinetic theory provides a solid, common background for the understanding of the neutronic phenomena occurring in a nuclear reactors, as well as the particle interactions in plasma physics and fluid-dynamics phenomena, which are then treated in additional details in other courses of the master program in energy and nuclear engineering .

Nuclear fission reactor physics and transport theory

The part of the course on transport theory should enable the students to understand the fundamentals of kinetic theory and the link to neutronics and fluid-dynamics. The student should also acquire the basic knowledge on some of the mathematical physical models for the design of the cores of nuclear reactors. He should be able to carry out calculations and simulations of the neutronics of multiplying systems, criticality calculations and evaluations on the dynamics of nuclear reactors, and he should acquire the capability to physically interpret the results.

Nuclear fission reactor physics and transport theory

At the end of the course the student should: ELO 1 - know the linear transport equation for neutrons in its different mathematical forms (integro-differential and integral); ELO 2 - know the physico-mathematical models and approximation adopted for the design of the cores of nuclear reactors; ELO 3 - know the concept of criticality in a multiplying assembly and its mathematical representation; ELO 4 - understand the physical phenomena at the basis of the steady-state and dynamic behaviour of a nuclear reactor and how they can be described through physico-mathematical models; ELO 5 - be able to write down the balance equation for neutrons in steady-state and time-dependent conditions under different approximation (diffusion, spherical harmonics expansion, discrete ordinates, ...); ELO 6 - know the hypotheses, limitations and field of application of perturbation theory; ELO 7 - know the hypotheses and limitations of point kinetic models and, more in general, of approximate models for the transient analysis of nuclear reactors; ELO 8 - perform the solution to simplified models for the description of neutron behavior in a multiplying system and physically interpret the results; ELO 9 - know the fundamentals of kinetic theory; ELO10 - understand the physical significance of kinetic entropy and its link to thermodynamics; ELO 11 - understand the connection of kinetic theory to neutronics and fluid-dynamics.

Nuclear fission reactor physics and transport theory

The student should have the basic knowledge of mathematics and physics, as can be acquired in the courses of a bachelor?s program in industrial engineering.

Nuclear fission reactor physics and transport theory

The student should have the basic knowledge of mathematics and physics, as can be acquired in the courses of a bachelor?s program in industrial engineering. A previous knowledge on basic aspects of nuclear reactor physics as the ones given in the frame of the bachelor program in energy engineering is surely helpful but not compulsory for the understanding of the contents of the course.

Nuclear fission reactor physics and transport theory

1 Kinetic theory of gases 1.1 Rarefied gases 1.2 Binary collisions 1.3 Boltzmann equation 1.4 H theorem 1.5 Equations of fluid-dynamics 2 Design calculation of multiplying structures 2.1 Generation of nuclear data; 2.2 Criticality calculation; multiplication eigenvalue and physical meaning; 2.3 Integration of the neutronic calculation in the design of a nuclear reactor. 3 Neutronic models 3.1 Multigroup diffusion theory; 3.2 Homogeneous and heterogeneous reactors; reflected reactors; 3.3 Neutron transport theory. 3.4 Slowing-down of neutrons in matter 4 Perturbation theory 4.1 Perturbation methods for eigenvalues; 4.2 Generalized perturbation techniques. 5 Nuclear reactor dynamics 5.1 Factorization methods for the solution of the neutronic equations; point kinetics and quasi-static method; 5.2 Non-linear feed-back effects; 5.3 Transmutation phenomena. 6 Mathematical and computational techniques 6.1 Spherical harmonics method 6.2 Discrete ordinate method 6.3 Analytical solution of the linear transport equation

Nuclear fission reactor physics and transport theory

1. Kinetic theory (20h) 1.1 Ergodic hypothesis 1.2 Canonical distribution. Examples 1.3 Boltzmann theory for the gases 1.4 Theorem of entropy 1.5 Deduction of the equations of the fluid dynamics 2 Design calculation of multiplying structures (15h) 2.1 Generation of nuclear data; 2.2 Criticality calculation; multiplication eigenvalue and physical meaning; 2.3 Integration of the neutronic calculation in the design of a nuclear reactor. 3 Neutronic models (12h) 3.1 Neutron transport theory 3.2 Multigroup models 3.3 Diffusion theory; 3.4 Homogeneous and heterogeneous reactors; reflected reactors; 4 Mathematical and computational techniques (16h) 4.1 Spherical harmonics method 4.2 Discrete ordinate method 4.3 Analytical solution of the linear transport equation 5 Perturbation theory (8h) 5.1 Perturbation methods for eigenvalues; 5.2 Generalized perturbation techniques. 6 Nuclear reactor dynamics (9h) 6.1 Factorization methods for the solution of the neutronic equations; point kinetics and quasi-static method; 6.2 Non-linear feed-back effects;

Nuclear fission reactor physics and transport theory

Nuclear fission reactor physics and transport theory

Nuclear fission reactor physics and transport theory

In the sessions applications of the theory presented in the lectures are proposed, such as: - Simulation of the collisional process in a gas - analytical solution of a 0D kinetic model - Analitycal solution of the neutron transport equation in the infinite medium

Nuclear fission reactor physics and transport theory

The course is composed by lectures and exercise sessions, where the concepts illustrated are complemented by numerical applications on simplified cases to improve understanding. The examples of solution presented can be treated both on paper and using computational tools such as MATLAB. Examples of simplified problems illustrated during the exercise sessions: - Simulation of the collisional process in a gas - Analytical solution of a 0D kinetic model - Analytical solution of the neutron transport equation in the infinite medium: calculation of singularities on comparison to diffusion - Criticality problems in different configurations (reflected systems, multigroup model, ...): numerical solution of the eigenvalue problem - Application of perturbation theory in two-group diffusion: simulation of control rod worth - Solution of neutron dynamic models: point kinetics and quasi-statics

Nuclear fission reactor physics and transport theory

- Bell, G. I., and Glasstone, S. Nuclear Reactor Theory, Van Nostrand Reinhold Inc.,U.S., 1970 - A. F. Henry, Nuclear-reactor Analysis, MIT Press, 1975 - K. M. Case, P. F. Zweifel, Linear transport theory, Addison-Wesley, 1967 - R. V. Meghreblian, D. K. Holmes, Reactor Analysis, McGraw-Hill, 1960

Nuclear fission reactor physics and transport theory

- Bell, G. I., and Glasstone, S. Nuclear Reactor Theory, Van Nostrand Reinhold Inc.,U.S., 1970 - A. F. Henry, Nuclear-reactor Analysis, MIT Press, 1975 - K. M. Case, P. F. Zweifel, Linear transport theory, Addison-Wesley, 1967 - R. V. Meghreblian, D. K. Holmes, Reactor Analysis, McGraw-Hill, 1960 - K. Huang, Statistical Mechanics, Wiley, 2008 - D. Tong, Lectures on Statistical Physics, University of Cambridge,2012

Nuclear fission reactor physics and transport theory

**Modalita di esame:** Prova orale obbligatoria;

Nuclear fission reactor physics and transport theory

The evaluation is carried out by an oral examination, to verify that the student knows the basic principles of particle transport, the fundamentals of nuclear reactor physics and the methods to solve neutron transport problems in steady state and in transient regimes. In the first part of the exam the student can present an argument of his choice. In the second part the student is asked to answer questions on both general transport theory and nuclear reactor physics.

Nuclear fission reactor physics and transport theory

**Exam:** Compulsory oral exam;

Nuclear fission reactor physics and transport theory

The evaluation is carried out by an oral examination, to verify that the student knows the basic principles of particle transport, the fundamentals of nuclear reactor physics and the methods to solve neutron transport problems in steady state and in transient regimes. In the first part of the exam the student may be asked to present a subject of their choice, and the resulting discussion will in principle cover the whole program of the course. Due to the characteristics of the two, complementary parts of the course, students are allowed to be interviewed on the two course parts in separate sessions. The final grade is the average of the two grades obtained, weighted on the fraction of course hours devoted to each part.

Nuclear fission reactor physics and transport theory

**Modalita di esame:** Prova orale obbligatoria;

Nuclear fission reactor physics and transport theory

The evaluation is carried out by an oral examination, to verify that the student knows the basic principles of particle transport, the fundamentals of nuclear reactor physics and the methods to solve neutron transport problems in steady state and in transient regimes. In the first part of the exam the student can present an argument of his choice. In the second part the student is asked to answer questions on both general transport theory and nuclear reactor physics.

Nuclear fission reactor physics and transport theory

**Exam:** Compulsory oral exam;

Nuclear fission reactor physics and transport theory

The evaluation is carried out by an oral examination, to verify that the student knows the basic principles of particle transport, the fundamentals of nuclear reactor physics and the methods to solve neutron transport problems in steady state and in transient regimes. In the first part of the exam the student may be asked to present a subject of their choice, and the resulting discussion will in principle cover the whole program of the course. Due to the characteristics of the two, complementary parts of the course, students are allowed to be interviewed on the two course parts in separate sessions. The final grade is the average of the two grades obtained, weighted on the fraction of course hours devoted to each part.

Nuclear fission reactor physics and transport theory

**Modalita di esame:** Prova orale obbligatoria;

Nuclear fission reactor physics and transport theory

The evaluation is carried out by an oral examination, to verify that the student knows the basic principles of particle transport, the fundamentals of nuclear reactor physics and the methods to solve neutron transport problems in steady state and in transient regimes. In the first part of the exam the student can present an argument of his choice. In the second part the student is asked to answer questions on both general transport theory and nuclear reactor physics.

Nuclear fission reactor physics and transport theory

**Exam:** Compulsory oral exam;

Nuclear fission reactor physics and transport theory

The evaluation is carried out by an oral examination, to verify that the student knows the basic principles of particle transport, the fundamentals of nuclear reactor physics and the methods to solve neutron transport problems in steady state and in transient regimes. In the first part of the exam the student may be asked to present a subject of their choice, and the resulting discussion will in principle cover the whole program of the course. Due to the characteristics of the two, complementary parts of the course, students are allowed to be interviewed on the two course parts in separate sessions. The final grade is the average of the two grades obtained, weighted on the fraction of course hours devoted to each part.

© Politecnico di Torino

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY