Lesson 3 Objectives

1
Lesson 3 Objectives

• Numerical approximation to DE
• Matrix form of numerical DE
• Matrix solution strategies
• Inner iteration
• Outer (power) iteration
• Integral transport methods
• Theory
• Application to 1D slab
• Solution in 1D slab

2
Diffusion Theory Derivation

• Diffusion theory Simplified treatment of
  direction components of neutron flux
• Same spatial (finite difference) and energy
  (multigroup) treatments as transport theory
• One speed Multiple groups coupled through
  scattering and fission
• In a homogeneous 2D region

3
Numerical solution

• Neutron balance (integrate over cell/cell size)

4
Numerical solution (2)

• where

5
Numerical solution (3)

• Dealing with derivatives
• Solve for edge flux

6
Numerical solution (4)

• Substitute back into equation
• For boundary cells
• Reflection Vacuum

7
Matrix form

• The cell balance is now
• In matrix form, this is

8
Matrix form (2)

• Note that A tri-diagonal and diagonally dominant

9
Solution strategy (inner iteration)

• Direct Gauss elimination
• Iterative
• Jacobi (Simultaneous replacement) iterative
  scheme
• Gauss-Siedel (Successive replacement scheme)

10
Jacobi scheme

• Set all
• For each cell i,i1,,N
• Repeat to convergence
• Variation Even/odd sweeps

11
Gauss-Siedel scheme

• Set all
• For each cell i,i1,,N
• Repeat to convergence
• Variation Alternating direction

12
2D versions of Jacobi and Gauss-Siedel

• Include leakage in y-direction
• Leakage across a boundary EXACTLY like a 1D
  version of the line between them

13
Solution strategies (outer itern)

• Set
• Update source from fission
• Solve inner iteration for
• Update eigenvalue
• Normalize flux
• Repeat to 2 until convergence

14
Integral transport methods

• Used for pin cell and assembly cross sections
• Differs 2 ways from Discrete Ordinates
• 1. Uses the integral form of the equation No
  derivative terms
• 2. The angular variable is integrated out, so the
  basic unknown is NOT the angular flux,
  but its integral, the scalar flux

15
I.T. methods (2)

• The Q term includes fission, scatter, and
  external source
• The integrals in the exponentials are line
  integrals of the total cross section along the
  direction of travel.
• We refer to these as the OPTICAL DISTANCE between
  the two points and
• This corresponds to the number of mean free paths
  between the two points (and commutes).

16
I.T. methods (3)

• If we let R go to infinity, the second term is
  not needed
• Since the source is isotropic, we only need the
  scalar flux to feed it (e.g., fission
  scattering sources), so it makes sense for us to
  integrate this equation over angle to get
• Notice that the COMBINATION of integrating over
  all DIRECTIONS and all DISTANCES away from any
  point Integration over all space

17
I.T. methods (4)

• Note that
• Using a spherical coordinate system, we have
• These two can be substituted into the previous
  equation to give us
• This is the general form of the Integral
  Transport Equation

18
Application to 2D lattice

• We begin with a calculational grid
• where the variables A and L are AREA and LENGTH,
  respectively.

19
Application to 2D lattice (2)

• The average flux in each mesh area is
• And the average outgoing partial currents for the
  boundary segments are

20
Application to 2D lattice (3)

• We use these and define

21
Application to 2D lattice (4)

• We note that the number of transfer coefficients
  to calculate can be reduced using reciprocity
  relationships

22
Application to 2D lattice (5)

• These four sets of transfer coefficients can be
  written as four matrices PAA, PAL,PLA and PLL,
  with elements
• If we define f as the collision rate and
  similarly take spatial densities out of the
  partial currents and source (s is fission and
  external source)

23
Application to 2D lattice (6)

• We can write our equations in matrix form
• If we again separate the within-cell scatter
• where

24
Application to 2D lattice (7)

• we can write the matrix relations as
• Of course, the incoming partial currents are the
  outgoing values from some other edge
• where j is the side opposite j (in some sense).

25
Homework 3

• Solve the following problem for the average
  fluxes in the three regions, using diffusion
  theory methods. (Let average m0.)
• Solve it numerically without subdividing the
  three regions using
• a. Direct inversion of matrix (Gaussian
  elimination)
• b. Jacobi iteration
• c. Gauss-Siedel iteration