INSTRUCTORS: J. K. Shultis, Office WD-125, (913) 532-5626; email jks@ksu.edu; R.E. Faw, Office WD-127, (913) 532-5963; email faw@ksu.edu

PREREQUISITES: Knowledge of (1) neutron-particle interactions, (2) basic nuclear science, (3) neutron diffusion and transport equation (4) calculus, complex analysis, and transform methods, (6) a programming language.

COURSE GOALS: (1) Understand the rigorous development of particle transport and its application to a wide variety of disciplines. (2) Learn techniques used to obtain exact solutions of the transport equation, and (3) gain insight into how diffusion and hydrodynamic approximations can be obtained from the Boltzmann equation.

TOPICS:

1. Introduction:

Transport theory: history and range of application. Generic transport equation (TE). Evaluation of the collision term for neutrons. The streaming term in different geometries. Examples of the TE used in neutron, gas dynamics, and plasma problems

2. Collision Models:

2.1 Neutron-nucleus interactions: general properties for different types of interactions; scattering of thermal neutrons; cross section models. Simplest form of the neutron TE (1-D, 1-speed, isotropic scattering). Integral form of the TE. Energy multigroup approximation.

2.2 Radiative Transfer: Types of photon transport, history. Derivation of the radiative transfer equation (RTE). Absorption and emission coefficients, and their calculation in gases. Local thermodynamic equilibrium (LTE) and the Planck distribution. RTE with LTE and analogy to neutron TE. Averaging coefficients. Global radiation equilibrium. Other approximation to the RTE: grey medium, picket-fence model, non-LTE problems.

2.3 Transport Equation for High Energy Photons: Types of interactions, and use of Compton wavelength.

2.4 Transport Equations for High Energy Electrons: Continuous slowing down approximation. Fokker-Planck Approximation. Recent papers on electron transport calculational methods.

2.5 Kinetic Theory of Gases: Assumptions and derivation of the Boltzmann binary collision term and the form of the Boltzmann transport equation (BTE). The Maxwellian equilibrium distribution as a solution of the BTE. The H-theorem. Collision invariants. Linearized Boltzmann equation. Cross section models for BTE.

3. Solving the One-Speed Transport Equation:

3.1 Simplest Form: One-Speed neutron TE with isotropic scattering in various geometries (integro-differential and integral forms).

3.2 Formal solution of integral TE by orders-of-scattering method.

3.3 Summary of important results in complex analysis

3.4 Review of integral transport and separation of variable techniques as applied to the diffusion equation.

3.5 Integral transform applied to TE for plane source in infinite medium.

3.6 Singular eigenfunction method applied to infinite medium Green's function, the half-space albedo problem, and the Milne problem.

4. Continuum Descriptions:

4.1 General approach for obtaining diffusion-like equations.

4.2 The one-speed diffusion equation approximation--TE moments approach.

4.3 The diffusion equation derived using operator projection methods.

4.4 The hydrodynamics equations. Definition of hydrodynamic quantities. Maxwell's transfer (conservation) equations. Formal form of the general hydrodynamics equations (HDE). Equilibrium and Euler's equations for invicid fluids. Chapman-Enskog expansions and the derivation of Navier-Stokes and Fourier's heat conduction equations. Higher order approximations (Burnett equations).

5. Moments Method:

5.1 TE for high-energy photons. Theorems (existence, uniqueness, scaling plane-density variations, uniform flux)

5.2 Method of moments (collided and uncollided moments; spatial and angular moments, moments for doses, point and plane sources).

5.3 Reconstruction of flux from the moments: polynomial expansion and function fitting methods.

5.4 Treatment of electron transport.

EVALUATIONS: Biweekly problem assignments will be given. Each assignment requires an individual formal writeup. An assignment grade will be based not only on the correctness of the results, but also on the completeness of the writeup and the discussion of the results. We expect to see originality in your work with your writeups including results and analyses of issues not specifically requested. Finally, you are expected to treat your assignments with a professional attitude by meeting the time deadlines assigned. There will be no examinations in this class.