1. Lesson 6: Computer method overview  Neutron transport overviews  Comparison of deterministic vs. Monte Carlo  User-level knowledge of Monte Carlo  Example calc: Approach to critical

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, the number of neutrons per fission  Advantages:  Everybody uses it  Guaranteed real solution  Fairly intuitive (if you don’t take it too seriously)  Good measure of distance from criticality for reactors  Disadvantages:  No physical basis  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. Deterministic grid solution  Subdivide everything: Energy, Space, Angle  “Eulerian” grid: Fixed in space  Use balance condition to figure out each piece Spatial and angular flux for Group 1 (10 MeV-20 MeV) Source

7. Stochastic solution  Continuous in Energy, Space, Angle  “Lagrangian” grid: Follows the particle  Sample (poll) by following “typical” neutrons Particle track of two 10 MeV fission neutrons Fissile Absorbed Fission

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

9. Monte Carlo overview

10. Monte Carlo  We will cover the mathematical details in 3 steps 1. General overview of MC approach 2. Example walkthrough 3. Special considerations for criticality calculations  Goal: Give you just enough details for you to be an intelligent user

11. General Overview of MC  Monte Carlo: Stochastic approach  Statistical simulation of individual particle histories  Keep score of quantities you care about  Most of mathematically interesting features come from “variance reduction methods”  Gives results PLUS standard deviation = statistical measure of how reliable the answer is

12. Mathematical basis  Statistical simulation driven by random number generator: 0<  <1, with a uniform distribution: p(  d  prob. of picking  in ( ,  +d  )=d   Score keeping driven by statistical formula:

13. Simple Walkthrough  Six types of decisions to be made: 1. Where the particle is born 2. Initial particle energy 3. Initial particle direction 4. Distance to next collision 5. Type of collision 6. Outcome of scattering collision (E, direction)  How are these decisions made?

14. Decision 1: Where particle is born  3 choices: 1. Set of fixed points (first “generation” only) 2. 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

15. 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

16. 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:   =cosine of angle from polar axis   =azimuthal angle (longitude)

17. Decision 4: Distance to next collision  Let s=distance traveled in medium with  t  Prob. of colliding in ds at distance s =(Prob. of surviving to s)x(Prob. of colliding in ds|survived to s) =(e -  t s) x(  t s)  Using methods from NE582, this results in:

18. Decision 5: Type of collision  Most straight-forward of all because straight from cross sections:   s /  t =probability of scatter   a /  t =probability of absorption  Therefore, if  <  s /  t, 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

19. 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

20. Special considerations for criticality safety  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

21. Approach to critical

22.  This is a brief overview of the basic techniques of running a critical experiment  The very same techniques are used to bring a reactor from shutdown state to an initial zero- power critical condition  The basic idea is to use the equations of subcritical multiplication to sneak up on the critical value of some parameter (i.e., one of the nine MAGICMERV parameters)

23. Approach to critical (cont’d)  We begin with the basic definition of k- effective (which we will call “k” in our equations)  The ratio of fission neutrons produced in any “generation” to those in the previous generation  Actually, since the flux increases linearly with fission rate (if you are careful) any flux- dependent parameter can be used in place of the fission neutron production We will use the response of an external detector We will use the response of an external detector  In both practice and theory, there must be an external neutron source to give the system something to multiply

24. Approach to critical (cont’d)  If we assume an external source rate of S neutrons/generation, we can see that: 1. In the first generation, S neutrons will be released. Since no fission has taken place yet, these will be the only neutrons in the system. 2. In the second generation, S new neutrons will be released and the previous generation’s S neutrons will cause kS(<S) new neutrons to give us a population of:

25. Approach to critical (cont’d) 3. In the next generation, the same thing happens: S new neutrons will be released and the previous population is multiplied by k, giving us: 4. Following this through to an infinite series:

26. Approach to critical (cont’d) 5. The 1/(1-k) term is a MULTIPLIER of the source. It appears from first glance that this equation can be used to find k, but it really can’t, because we never really know what the original k is. 6. But it CAN tell you is how much (relatively) closer your most recent change got you to k=1.  For example, if you make a change and your detector reading goes from 100 counts/sec to 200 counts/sec, you DO know that you went HALFWAY to k=1. But you can’t tell whether you went from k=.8 to.9 or (maybe) 0.96 to 0.98

27. Approach to critical (cont’d)  This is useful because it tells you that ANOTHER change of (about) the same magnitude would get you to critical. 7. Since criticality corresponds to the M=1/(1-k) term (the MULTIPLIER of the source) going infinite, we have found it is more useful to instead watch its inverse.  Criticality corresponds to 1/M going to 0. (Inclass example as time allows)