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. 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

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

9. 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?

10. 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.

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

12. 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)

13. Monte Carlo overview

14. 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

15. 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

16. 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

17. 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?

18. 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

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

20. 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)

21. 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:

22. 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)

23. 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

24. 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

25. 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