Lesson 5 Objectives

1
Lesson 5 Objectives

• Review of multigroup equations
• Matrix formulation
• Definition of outer (energy) iteration
• General eigenvalues
• Solution of k-effective eigenvalue equation

2
Energy treatment

• We now turn to the ENERGY numeric approximation
  solutions .
• The idea is to convert the continuous dimensions
  to discretized form.
• Calculus gt Algebra
• Steps we will follow
• Derivation of multigroup form
• Reduction of group coupling to outer iteration in
  matrix form
• Solution strategies that avoid matrix
  manipulation

3
Definition of Multigroup

• All of the deterministic methods (and many Monte
  Carlo) represent energy variable using multigroup
  formalism
• Basic idea is that the energy variable is divided
  into contiguous regions (called groups)
• Note that it is traditional to number groups from
  high energy to low.

4
Derivation of multigroup equations

• Divide and conquer technique common to many
  numerical methods Discretization
• Differs from other numerical methods by having an
  assumed UNDERLYING shape
• The solution to the multigroup equations gives a
  shape to a guide function that corrects the
  assumed underlying shape
• Must keep this in mind when plotting the spectrum

5
Derivation of MG equations (2)

• From original energy balance equation
• Assume an underlying shapean assumption of the
  RELATIVE spectrum shape,
• Superimpose a guide function made up of linear
  combinations of basis functions

6
Derivation of MG equations (3)

• Substitute the guide function
• into the balance condition and operate on it
  with to get the Generalized group
  g
  
  equation

7
Derivation of MG equations (4)

• Most common basis functions Multigroup
  membership functions
• Using this, we have the simplification

8
Derivation of MG equations (5)

• With proper definition of group cross sections
  (see Eqn. 4.39 in text) we can put this equation
  into the standard multigroup form
• Strong balance condition (at each E) has become a
  weak balance (only total balance within a group)

9
MG cross section definitions
10
Important points to make

• The assumed shapes f(E) take the mathematical
  role of weight functions in formation of group
  cross sections
• Because the group flux definition does NOT
  involve the division by the group width it is NOT
  a density in energy.
• The numerical value goes up and down with group
  SIZE.
• Therefore, if you GRAPH group fluxes, you should
  divide by the group width before comparing to
  continuous spectra

11
Important points to make (2)

• An even better comparison to continue spectra
  would be fg,calc f(E) / fg
• We do not have to predict a spectral shape f(E)
  that is good for ALL energies, but just accurate
  over the limited range of each group.
• Therefore, as groups get smaller, the selection
  of an accurate f(E) gets less important
• Results in a histogram guide function

12
Finding the group spectra

• There are two common ways to find the f(E) for
  neutrons
• Assuming a shape Use general physical
  understanding to deduce the expected spectral
  shapes fission, 1/E, Maxwellian
• Calculating a shape Use a simplified problem
  that can be approximately solved to get a shape
  resonance processing techniques, finegroup to
  multigroup

13
MG cross section definitions (2)

• Be sure to be able to perform these integrals if
  I provide you with energy-dependent cross
  sections and fluxes
• Hardest Scattering (elastic) because of the
  limited energy range of neutron after scatter
• The final integral involves the intersection of
  the destination group range (Eg,Eg-1) and the
  range that can physically be reached (aE,E)

14
Finegroup to multigroup

• Bootstrap technique whereby
• Assumed spectrum shapes are used to form
  finegroup cross sections (Ggt200)
• Simplified-geometry calculations are done with
  these large datasets.
• The resulting finegroup spectra are used to
  collapse fine-group XSs to multigroup

Multi-group structure (Group 3)
E2
E3
Energy
E20
E21
E22
E23
E24
E25
E26
E27
Fine-group structure
15
Finegroup to multigroup (2)

• Energy collapsing equation
• Using the calculated finegroup fluxes, we
  conserve reaction rates to get new cross
  sections
• Assumes multigroup flux will be

16
Preparation of MG cross-sections

• Basic information is in ENDF/B files system
  containing
• Tabulated pointwise cross-sections for
  non-resonance cross sections
• Resolved resonance parameters
• Unresolved resonance statistical parameters
• Scattering transfer functions pointwise in
  energy and pointwise OR Legendre in angle

17
Preparation of MG XSs (2)

• As described in the text, there are a number of
  computer code collections that will prepare cross
  sections from ENDF/B files
• NJOY (LANL)
• AMPX (ORNL)
• The job is in 2 parts
• Non-resonance cross sections using assumed
  spectrum shapes (non-problem-dependent)
• Resonance processing (problem dependent)

18
Energy solution strategies Fixed source

• The resulting energy group equations are coupled
  through scattering and fission
• We will first deal with non-fission situation,
  where the group equation is

19
Matrix formulation

• Write this in matrix operator form by defining
  operators
• Combine them into a single matrix operator

20
Matrix formulation (2)

• The resulting matrix relationship is

21
Matrix solution strategies (3)

• Although we generally do NOT form a matrix in our
  computer codes (too large and sparse), our
  numerical multigroup treatments can be formally
  considered in matrix terms.
• If we subdivided the H matrix into lower,
  diagonal, and upper parts, i.e.,

22
Matrix solution strategies (4)

• The same two basic iterative approaches we saw in
  the DT solution are used here as well Jacobi and
  Gauss-Seidel
• Jacobi (simultaneous update) uses
• Gauss-Seidel (successive update) uses
• where is the iteration counter

23
Matrix solution strategies (5)

• In practice, the groups are solved one at a time
  (group 1, then group 2, etc.) A single sweep
  through all groups is called an OUTER ITERATION.
  (The solution of the spatial flux for each group
  is the INNER iteration we studied before.)
• The G-S approach takes advantage of the fact
  that, for groups of LOWER number, the current
  iteration has already been done.
• For many problems (esp. photon and fast group
  structures) one outer iteration is all that is
  required (no up-scatter)
• Bottom Line We can worry about space and
  direction for ONE group at a time

24
Eigenvalue Calculations

• Returning to the multigroup balance equation
• Without external sources, we get the homogeneous
  equation
• Two characteristics of the solution
• Any constant times a solution is a solution.
• There probably isnt a meaningful solution

25
Eigenvalue solution normalization

• For the first point, we generally either
  normalize to 1 fission neutron
• or to a desired power level
• where k is a conversion constant (e.g., 200
  MeV/fission)

26
Eigenvalue approach

• For the second problem (i.e., no meaningful
  solution), we deal with it by adding a term with
  a constant that we can adjust to achieve balance
  in the equation.
• We will discuss four different eigenvalue
  formulations
• Lambda (k-effective) eigenvalue
• Alpha (time-absorption) eigenvalue
• B2 (buckling) eigenvalue
• Material search eigenvalue

27
Lambda (k-effective) eigenvalue

• The first (and most common) eigenvalue form
  involves dividing n, the number of neutrons
  emitted per fission
• Keep largest of multiple eigenvalues

28
Lambda eigenvalue (2)

• The criticality state is given by
• Advantages
• Everybody uses it
• Guaranteed real solution
• Fairly intuitive
• Good measure of distance from criticality for
  reactors
• Disadvantages
• No physical basis
• Not a good measure of distance from criticality
  for CS

29
Alpha (time-absorption) eigenvalue

• The second eigenvalue form involves adding a term
  to the removal term
• Keep largest of multiple eigenvalues

30
Alpha eigenvalue (3)

• The criticality state is given by
• Advantages
• Physical basis
• Intuitive for kinetics work
• Disadvantages
• No guaranteed real solution
• Not intuitive for reactor design or CS work

31
B2 (buckling) eigenvalue

• The third eigenvalue form also involves adding a
  term to the removal term
• Physical basis is the diffusion theory
  approximation of leakage by

32
B2 eigenvalue (3)

• The criticality state is given by
• Advantages
• Physical basis
• Good measure of distance from criticality
• Disadvantages
• No guaranteed real solution
• Not intuitive for kinetics or CS work

33
Material search eigenvalue

• The last eigenvalue is a number density of a
  key nuclide, Nj

• Advantages
• Physical basis
• Great for design
• Guaranteed real solution (fuel isotopes)
• Disadvantages
• No guaranteed real solution (non-fuel isotopes)

34
K-effective solutions

• For k-effective calculations, the cross sections
  are the same, but the matrix changes subtly
• which looks more complicated but is actually the
  same thing

35
K-effective solutions (2)

• Recognizing that the term in parenthesis is just
  a renormalization constant, we set it to 1 and
  break it out to give us a fixed source matrix
• with the eigenvalue becoming

36
K-effective solutions (2)

• This comes down to
• Solve for the flux produced by the fission
  neutron energy distribution
• See what k-effective supports a unit source
  with the resulting group fluxes
• (If you want something OTHER than
  k-effectivee.g., Boron content or B2--change
  that something by trial and error until
  k-effective1)

37
HW5

• For group spanning 1 keV-4 keV, find the group
  total and scattering to the next lower group
  (which goes down to 0.5 keV), if
• Assume flux is 1/E and use A6

38
HW5

• Using the 3 group assembly macroscopic cross
  sections in PUBLIC area file HW5.DAT
• Find k-infinity for the exposures provided
  (MWD/assembly)
• For the 0 exposure cross sections, find the
  buckling eigenvalue and translate into a critical
  reactor size (with H/D1)
• Find the 0 exposure B-10 concentration for
  criticality, assuming group 3 absorption cross
  section of 2000 b
• For the k-infinity problem for 0 exposure,
  collapse the cross sections to 1 group