Lesson 4: Computer method overview Neutron transport overview Comparison of deterministic vs. Monte Carlo User-level knowledge of Discrete ordinates User-level knowledge of Monte Carlo Advantages and disadvantages of each

Neutron transport overview

Neutron balance equation Scalar flux/current balance Used in shielding, reactor theory, crit. safety, kinetics Problem: No source=No solution !

K-effective eigenvalue Changes n, the number of neutrons per fission Advantages: Everybody uses it Guaranteed real solution Good measure of distance from criticality for reactors Disadvantages: Weak physical basis nu does not really change to make a system critical Not a good measure of distance from criticality for CS

Neutron transport Scalar vs. angular flux Boltzmann transport equation Neutron accounting balance General terms of a balance in energy, angle, space Deterministic Subdivide energy, angle, space and solve equation Get k-effective and flux Monte Carlo Numerical simulation of transport Get particular flux-related answers, not flux everywhere

Discrete ordinates overview

Deterministic grid solution Subdivide everything: Energy : Multigroup Space: Chop up space into “mesh cells” Angle: Only allow particles to travel in particular directions Use balance condition to figure out (and save) the flux as a function of space, energy, direction Spatial and angular flux for Group 1 (10 MeV-20 MeV) Source

Directional treatment Mathematical basis: Quadrature integration Bottom line for us: More angles=more accuracy but more computer resources

Energy treatment: Multigroup Mathematical basis: Reduce continuous energy by “grouping” neutrons in energy ranges Define: We want to conserve reaction rates: So the proper group cross sections are rigorously found to be: Do you see a problem here?

Energy treatment, cont’d “Weight” cross sections with a “guess” of flux energy (“spectrum”) shape Cross sections are only as good as the assumed shapes Common assumption: (Fission, 1/E, Maxwellian) for smooth cross sections + spectrum from resonance treatments for resonance cross sections Bottom Line for us: More groups are (theoretically) better but takes longer In practice it depends on the group structure and the assumed spectra within the groups We choose the group structure by choosing from the available SCALE libraries: 27 group, 44 group, 238 group, etc.

Space treatment: Cell centered finite difference Mathematical basis: Subdivide the space into homogeneous cells, integrate transport equation over each cell to get something like:

Space treatment, cont’d Bottom Line for us: More cells are better but takes longer Cell size should be <1 mean free path SCALE picks the spatial discretization automatically, but you can control it (Something of a kludge, as you will see later in the course)

Monte Carlo overview

Statistical solution (“simulation”) Continuous in Energy, Space, Angle Sample (poll) by following “typical” neutrons Drop a million neutrons into the system and see how many new neutrons are created: ratio is k-effective Absorbed Fissile Fission Particle track of two 10 MeV fission neutrons

Monte Carlo We will cover the mathematical details in 3 steps General overview of MC approach Example walkthrough Special considerations for criticality calculations Goal: Give you just enough details for you to be an intelligent user

General Overview of MC Monte Carlo: Stochastic approach Statistical simulation of individual particle histories Keep score of quantities you care about (for us MOSTLY k-effective, but we also want to know where fission is occurring) Gives results PLUS standard deviation = statistical measure of how reliable the answer is

Simple Walkthrough Six types of decisions to be made: Where the particle is born Initial particle energy Initial particle direction Distance to next collision Type of collision Outcome of scattering collision (E, direction) How are these decisions made?

Decision 1: Where particle is born 3 choices: Set of fixed points (first “generation” only) Uniformly distributed (first generation only) KENO picks point in geometry Rejected if not in fuel After first generation, previous generation’s sites used to start new fissions

Decision 2: Initial particle energy From a fission neutron energy spectrum Complicated algorithms based on advanced mathematical treatments from spectrum measurements

Decision 3: Initial particle direction Easy one for us because all fission is isotropic Must choose the “longitude” and “latitude” that the particle would cross a unit sphere centered on original location Mathematical results: m=cosine of angle from polar axis F=azimuthal angle (longitude)

Decision 4: Distance to next collision Let s=distance traveled in medium with St Prob. of colliding in ds at distance s =(Prob. of surviving to s)x(Prob. of colliding in ds|survived to s) =(e- St s)(Stds) So, the PDF is Ste- St s This results in:

Decision 5: Type of collision Most straight-forward of all because straight from cross sections: Ss / St =probability of scatter Sa / St =probability of absorption Analog: If x< Ss / St , it is a scatter Otherwise particle is absorbed (lost)

Decision 6: Outcome of scattering collision In simplest case (isotropic scattering), a combination of Decisions 3 and 5: “Decision 5-like” choice of new energy group Decision 3 gets new direction In more complicated (normal) case, must deal with the fact that the outcoming neutron direction distribution depends on how much energy is lost

Special input variables user must provide Must deal w/ “generations”=outer iteration Fix a fission source spatial shape Find new fission source shape and eigenvalue User must specify # of generations AND # of histories per generation AND # of generation to “skip” Skipped generations allow the original lousy spatial fission distribution to improve before we really start keeping “score” KENO defaults: 203 generations of 1000 histories per generation, skipping first 3

Advantages and disadvantages of each Discrete Ordinates (deterministic) Fast (1D, 2D) Accurate for simple geometries Delivers answer everywhere: = Complete spatial, energy, angular map of the flux 1/N (or better) error convergence Slow (3D) Multigroup energy required Geometry must be approximated Large computer memory requirements User must determine accuracy by repeated calculations Monte Carlo (stochastic) “Exact” geometry Continuous energy possible Estimate of accuracy given Slow (1,2,3D) Large computer time 1/N1/2 error convergence