1. Nodal Methods for Core Neutron Diffusion Calculations
Reactor Numerical Analysis and Design 1st Semester of 2008
Lecture Note 10
Nodal Methods for Core Neutron Diffusion Calculations
May 8, 2008
Prof. Joo Han-gyu
Department of Nuclear Engineering

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. 3-D Steady-State Multigroup Neutron Diffusion Equation
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 obtaining 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

10. 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

11. 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

12. One Node Formulation Given Conditions Aim
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

13. 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

14. 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

15. 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

16. Determination of Buckling Eigenvalues
Characteristic Equation
For Nontrivial Solution
Eigen-Buckling (Roots of Characteristic Equation)
Fundamental Mode
Second Harmonics Mode

17. 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

18. 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

19. Flux Components

20. 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

21. 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

22. Comparison of Accuracy for 3 Nodal Methods
NEACRP L336 C5G7 MOX Benchmark
Error of Various Nodal Schemes
Thermal Flux
UOX FA
MOX FA
* Reference=ANM 4x4 Calculation
Fission Source

23. 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 facilitates 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