Lesson 4: Computer method overview

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 !

ErSErErErErErJ fa ,,,,,,

K-effective eigenvalue Changes , the number of neutrons per fission

ErErErErErJ fa ,,,,,

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 Quadrature integration: We only find the flux in

a few particular directions, then find the scalar fluxes (needed to get F, A, and L) by weighted averaging of these

We call this the “Discrete ordinates” technique

• Bottom line for us: More angles=more accuracy but more computer

resources

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Energy treatment: Multigroup

E20E21E22E23E24E25E26E27

Energy

Energy group structure

Divide the energy range 0 MeV to ~ 20 MeV into “groups” First group is the HIGHEST energy range (to follow the

neutrons) Instead of having cross sections and fluxes as a continuous

function of energy, we assume that all neutrons in a give group act “alike” and can be represented with a single total cross sections, a single absorption cross section, etc.

Scattering becomes “group-to-group” instead of “energy to energy”

This results in “within-group scattering”—where a neutron loses energy in a scattering event, but not enough to lower its energy to the next lower group

Energy treatment: Multigroup (cont’d) To get the group cross sections right, you have to know

the energy shape (spectrum) of the flux within each of these groups

We don’t know this shape, so we guess. Usual best guess:

NOTICE: These shapes do NOT depend on what material you are talking about Precalculated into libraries

( ) ~fast E E

int1( ) ~EE

( ) ~EkT

thermal E Ee

6-11

This works pretty well EXCEPT for resonance materials, where the 1/E flux shape is “disturbed” by huge drops at resonance energies

If we look at the flux inside narrow and separated resonances, we find that the flux looks more like:

This shape will be different for every material mixture in the problem

Resonance treatments

( ) ~ s

t

EE E

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The result of this is that the cross section libraries can be PARTIALLY precalculated:

All nonresonance materials Even resonance material for energy groups that do not

have resonances in them Approximations of the cross sections in resonance groups

of resonance materials SCALE calls these “MASTER” libraries

BUT Accurate cross sections for resonance groups requires a

special treatment that needs to know the actual materials in the problem—so has to wait until you run YOUR case

First step of analysis:”MASTER””WORKING” library

Cross section libraries

Space treatment: Cell centered finite difference

Mathematical basis: Subdivide the space into homogeneous cells, integrate transport equation over each cell to get something like:

Then we “sweep” over the spatial grid from known to unknown cells

2/1,, jin

jin ,2/1, jin ,2/1,

2/1,, jin

, nij nijS

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

Particle track of two 10 MeV fission neutrons

Fissile

Absorbed

Fission

Monte Carlo

We will cover the mathematical details in 3 steps

1. General overview of MC approach2. Example walkthrough3. 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:1. Where the particle is born2. Initial particle energy3. Initial particle direction4. Distance to next collision5. Type of collision6. Outcome of scattering collision (E, direction)

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

Advantages Disadvantages

DiscreteOrdinates(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 requirements 1/N1/2 error convergence