Nodal Methods for Core Neutron Diffusion Calculations

1
Nodal Methods for Core Neutron Diffusion
Calculations
Reactor Numerical Analysis and Design

• October 2006
• Han Gyu Joo
• Seoul National University
• T. Downar
• Purdue University

2
Contents

• Transverse Integration and Resulting
  One-Dimensional Neutron Diffusion Equation
• Treatment of Transverse leakage
• Nodal Expansion Method with One-Node Formulation
• Polynomial Intra-nodal Flux Expansion
• Response Matrix Formulation
• Iterative Solution Sequence
• Analytic Nodal Method with Two-Node Formulation
• Two-Node Problem
• Analytic Solution of Two-Group, One-D Neutron
  Diffusion Eqn.
• Implementation with the CMFD Framework
• Semi-Analytic Nodal Method
• Polynomial Intra-nodal Source Expansion
• Analytic Solution for One Node

3
Introduction

• 3-D Steady-State Multigroup Neutron Diffusion
  Equation
• Fick's Law of Diffusion for Current out of Flux
• Computational Node in 3-D Space
• Property assumed constant within each homogenized
  node
• FDM accurate only if the node size is
  sufficiently small (1cm)
• Nodal methods to achieve high accuracy with large
  nodes (20 cm)

4
Nodal Balance Equation (NBE)

• Volume Averaging of Diffusion Equation for a Node
• Integrate over the node volume then divide by
  volume
• Volume Average Flux
• Integration of the Divergence Term using Gauss
  Theorem
• Surface Average Current
• Nodal Balance Equation for Average Quantities of
  Interest (Nodal Power)

5
Need for Transverse Integration

• NBE Solution Consideration
• Information on 6 surface average currents only
  required for obtain the node average flux which
  will determine the nodal power
• Surface Average Currents
• Average of Flux Derivative on a Surface
• Equals to Derivative of Average Flux at the
  Surface
• Better to work with the neutron diffusion
  equation for average flux rather than one for the
  point wise flux (3-D)
• Transverse Integration
• Set a direction of interest (e.g. x)
• Perform integration within node over
• 2-D plane normal to the direction, then
• divide by plane area

6
Normalization of Variables

• Normalized Independent Variables
• Transformation of Integration and Derivative
  Operator
• Simplified Averaging
• Normalized 3-D Diffusion Equation

7
Transverse Integrated Quantities

• Transverse Integration of Leakage Term
• Plane Average One-Dimensional Flux
• Line Average Surface Current at Arbitrary
  Position x

8
Transverse Integrated One-Dimensional Neutron
Diffusion Equation

• Transverse Integration of 3-D Neutron Diffusion
  Equation
• Define Transverse Leakage to Move to RHS
• Transverse Integrated One-Dimensional Neutron
  Diffusion Equation (Final Form)
• Diffusion Equivalent Group Constant

9
Transverse Integrated One-dimensional Neutron
Diffusion Equations

• Set of 3 Directional 1-D Neutron Diffusion
  Equations
• 3-D Partial Differential Equation
• ? Three 1-D
  Ordinary Differential Equations
• Coupled through average transverse leakage term
• Exact if the proper transverse leakages are used
• Approximation on Transverse Leakage
• Quadratic Shape (2nd order polynomial)
• based on observation that change of flux
  distribution is not sensitive to change of
  transverse leakage
• Iteratively update transverse leakage

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Transverse Leakage Approximation

• Quadratic Approximation in Each Node
• Average TL Conservation Scheme to Determine l1
  and l2
• Use three node average transverse leakages
• Values of own node and two adjacent nodes
• Impose constraint of conserving the averages of
  two adjacent nodes

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Alternative Schemes for Determining T.L. Coef.

• Apply linear shape to determine surface values
• use continuity of gradient of transverse leakage
  to determine surface value of TL
• Determine l1 and l2 by making the polynomial pass
  through the two boundary points as well as
  satisfying the average

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Nodal Expansion Method

• Intranodal Flux Expansion of 1-D Flux
• Approximate 1-D Flux by 4th Order Polynomial
• Basis Functions
• Not Orthogonal Function
• Integration in Range 0,1 results 0.
• 2nd Order Transverse Leakage

13
One Node Formulation

• Given Conditions
• Incoming Partial Currents at Both Boundaries
• Quartic Intranodal Variation of Source
• Aim
• Solve for flux expansion
• Then update the outgoing partial current and
  source polynomial

14
NEM Solution Sequence

• Determination of 5 coefficients of solution in
  terms of incoming partial currents and node
  average flux
• 0th order Node Average Flux
• 1st and 2nd order- Surface Average Flux
• Treatment of a1 and a2
• They are unknown since that they contains the
  outgoing partial currents yet to be determined
• In traditional NEM, however, it is assumed that
  they are known from the surface average fluxes
  obtained at previous iteration step while node
  average flux will be determined from 3-D neutron
  diffusion equation

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Weighted Residual Method

• Three Physical Constraints
• 2 Incoming Current Boundary Conditions
• 1 Nodal Balance
• Two-Additional Conditions Required to Determine 5
  Coeff.
• Weighted Residual Method for 1-D Neutron Diff.
  Eqn.
• 1st Moment of Neutron Diffusion Equation
• 
  
• contains a1 which is unknown in principle
• 2nd Moment of Neutron Diffusion Equation
• contains a2 which is unknown in principle

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Outgoing Current Relations

• Response of Outgoing Partial Current to Node
  Average Flux and Incoming Partial Currents.
• Net Current at Boundary
• Substitution of net current given in terms of
  expansion coefficients yields
• where is relative diffusivity.

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Response Matrix

• Three-Directional Outgoing Currents Described
  Altogether
• a3 and a4 are treated as known by using a1 and a2
  which are approximated by previously known
  surface fluxes
• otherwise, to solve rigorously, need to solve for
  13 unknowns including a3 and a4 for each
  direction simultaneously.

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Solution of 3-D Nodal Balance Equation

• 3-D Neutron Diffusion Equation for Node Average
  Flux
• Substitution of outgoing partial current to 3-D
  neutron diffusion equation yields node average
  flux.
• Then use the response matrix to update outgoing
  currents

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One-Node NEM Iterative Solution Sequence

• For a given group
• Determine sequentially
• Source expansion coeff.
• a1 and a2 from previous surface fluxes
• a3 and a4 using source moments and a1 and a2
• node average flux
• outgoing current
• Move to next group
• Move to next node once all groups are done
• Group sweep and node sweep can be reversed (node
  sweep then group sweep)
• Update eigenvalue

20
Analytic Nodal Method for 2-G Problem

• 1D, Two-Group Diffusion Equation
• All source terms except transverse leakage now on
  LHS
• Analytic Solution Homogeneous Particular Sol.
• Trial Homogeneous Solution

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Determination of Buckling Eigenvalues

• Characteristic Equation
• For Nontrivial Solution
• Eigen-Buckling (Roots of Characteristic Equation)
• Fundamental Mode
• Second Harmonics Mode

22
Homogeneous Solutions

• Each Group Homogenous Solution
• Fundamental Mode
• Second-Harmonics Mode
• Combined Homogenous Solution
• Linearly Dependent Group 1 and Group 2 Equations
• Fast-to-Thermal Flux Ratio

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Particular Solution

• Particular Solution for Quadratic Transverse
  Leakage
• Determined Solely by Transverse Leakage!
• General Solution in a Node
• 4 Coefficients to determine for the 2 group
  problem

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Two-Node ANM Solution

• Boundary Condition and Given Parameters
• Quadratic Transverse Leakage for Two Nodes, keff
• Node-Average Fluxes for Two Nodes
• 8 Unknown Coefficients
• 4 per node x 2 nodes
• 8 Constraints ? Unique Solution
• 4 Node Average Fluxes (2 Groups x 2 Nodes)
• 2 Flux Continuity at Interface (2 Groups)
• 2 Current Continuity at Interface (2 Groups)
• Solution Sequence
• Assume Node-Average Flux
• Solve for Net Currents for each Direction from
  2-Node
• Update Node-Average Flux from Nodal Balance
• Repeat

25
Semi-analytic Nodal Method

• Transverse Integrated One-Dimensional Neutron
  Diffusion Equation for a Node and for a Group
• Approximation of Source with 4-th Order Legendre
  Polynomial
• Analytic Solution of Second Order Differential
  Equation
• Exponential Homogeneous and Polynomial Particular
  Solutions

26
One-Node, One-Group SANM Formulation

• Incoming Current Boundary Condition
• Problem Statement (2D for easier illustration)
• Solve 2 coupled second order differential
  equations (one for x and the other for y, given 2
  BCs, respectively)

• Coupled through average transverse leakage term
• Higher order terms in transverse leakage assumed
  to be fixed (only constant term representing the
  average would vary by the simultaneous solution
  of the two)
• ?All source terms except the average transverse
  leakage are given (Q tilde)

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Solution Sequence

• Normalization of Variables
• Balance Equation to Solve for x-dir
• Homogeneous and Particular Solution
• General Solution

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Solution Sequence

• Determine particular solution coefficients for
  non-constant terms (c1.. c4)
• Express the two homogeneous solution coefficients
  in terms of incoming boundary conditions
• Coefficient B contains unknown node average flux
• Outgoing current is given in terms of average
  flux as well

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Solution Sequence

• Three Dimensional Nodal Balance Equation for
  Simultaneous Solution of Node Average Flux and
  All Outgoing Currents
• Solve for average flux first
• Coeff. B determined ? All the coeff. known
• Then use average flux to determine the outgoing
  current which is to update the incoming current
• Determine source expansion coefficient
• Fourth order Legendre expansion of sinh and cosh
  functions
• Move to next group by updating the scattering
  source

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Accuracy of SANM in Two-Group Application

• NEACRP L336 C5G7 MOX Benchmark

Thermal Flux
Error of Various Nodal Schemes
MOX FA
UOX FA
ReferenceANM 4x4 Calculation
Fission Source
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Performance of Various Nodal Methods in NEACRP
A1 Transient Problem

• Ejected Rod Worth

Transient Core Power Behavior
? SANM performs almost as accurately as ANM
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Summary and Conclusions

• Transverse integrated method is an innovative way
  of solving 3-D neutron diffusion equation which
  is to convert the 3-D partial differential
  equation into 3 ordinary differential equations
  based on the observation that the impact of
  transverse leakage onto the a directional current
  is weak. Transverse leakage is thus approximated
  by a second order polynomial and iteratively
  updated.
• NEM is simple and efficient as long as the
  fission source iteration scheme is applied. It
  thus facilitate multigroup calculations. It loses
  accuracy for highly varying flux problems.
• One-node formulation is easier to implement, but
  slower in convergence than the two-node
  formulation
• ANM has the best accuracy, but it is not amenable
  for multigroup problems
• SANM would be the best choice in practical
  applications for its simplicity, multigroup
  applicability, and comparable accuracy to ANM