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

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-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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flux Components

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

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

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

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