Lesson 4: Computer method overview

1. 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

2. Neutron transport overview

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

4. 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

5. 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

6. Discrete ordinates overview

7. Source Spatial and angular flux for Group 1 (10 MeV-20 MeV) 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

8. Discrete ordinates • We will cover the mathematical details in 3 steps • Numerical methods to determine the neutron flux field from source • Numerical treatment of source • Putting it together into a solution strategy • Goal: Give you just enough details for you to be an intelligent user • Cocktail party knowledge

9. I. Numerical treatment • Subset of NE581 • We will use the easiest form: 1D slab: • Discretization in all 3 dimensions: space, energy, direction • Space: Relatively smooth, but equation has spatial derivatives—Finite difference method • Energy: Extremely non-smooth (resonances), but no derivatives—Multigroup treatment (complexity reduction) • Direction: Smooth, no derivatives—Quadrature integration • Named for the directional treatment

10. Directional treatment • Mathematical basis: Quadrature integration • Lots of math research done in defining optimum (m,w) sets (“quadratures”) • Most common—Gaussian quadratures • mn are roots of Pn(m) • wn are chosen to perfectly integrate the first n moments

11. Application to our problem • Solve for the flux only in particular directions: • Scalar flux found with quadratureintegration • Bottom line for us: • More angles=more accuracy but more computer resources • SCALE defaults to 8 directions (S8) • Available=4, 8, 16, 32,…

12. 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?

13. Energy treatment, cont’d • “Weight” cross sections with an assumed flux 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.

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

15. Space treatment, cont’d • One equation, five unknown y values • Incoming boundary fluxes are known as the outgoing of neighbor • Eliminates two unknown • Still left with 1 equation, 3 unknowns, so we must assume another relationship among the three unknowns. • Step: • Guaranteed positive flux, but not very accurate • Diamond Difference: • Surprisingly accurate, but requires a “negative flux fixup”

16. 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)

17. Monte Carlo overview

18. 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

19. 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

20. 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

21. Mathematical basis • Statistical simulation driven by random number generator: 0<x<1, with a uniform distribution: p(x)dx=prob. of picking x in (x, x+dx)=dx • Score keeping driven by statistical formula:

22. 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?

23. 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

24. Decision 2: Initial particle energy • From a fission neutron energy spectrum • Complicated algorithms based on advanced mathematical treatments • Rejection method based on “bounding box” idea

25. 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=azimuthalangle (longitude)

26. 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:

27. 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) • “Weighting in lieu of absorption”=fancy term for following the non-absorbing fraction of the particle

28. 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 energy/direction couplings from elastic and inelastic scattering physics

29. 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

30. Advantages and disadvantages of each