1. 1 POWER-KAERI Development of a Hexagonal Solution Module for the PARCS Code May, 2000 Progress Review

2. 2 POWER-KAERI Project Overview oObjective  Implement an Efficient Hexagonal Neutronic Solver in the PARCS code oWork Scope  Develop Hexagonal Solution Methods for Spatial Kinetics Calculation Satisfying : -Fastness for Coupled 3D Kinetics/System T-H Calculations -Accuracy for Solution Fidelity -Versatility for Wide Range of Applications (MultiGroup, MultiRegion within a Hexagon)  Implement a Hexagonal Solution Module in the PARCS Code -Keep both Rectangular and Hexagonal Solvers in one Code (DMM Essential) -Enable Coupled Calculation with System T-H Codes  Verify Performance for -Hexagonal Eigenvalue Benchmark Problems -Transient Benchmark Problems Involving VVER Reactors oSchedule  Nov. 99 – April 00: Investigate Various Hexagonal Solvers and Select two (Based on EVP Solution Performance)  May 00: Implement the two Solvers into PARCS  June 00 : Perform VVER1000 Rod Ejection Transient Benchmarks with RELAP/PARCS  July 00 – Aug. 00: Refine the Solvers and Prepare Documentation

3. 3 POWER-KAERI Investigated Hexagonal Solvers oConformal Mapping  Employs Prebuilt Mapping Functions to Transform a Hexagon to a Renangle  Accurate for Practical Applications  Vulnerable to Large Errors under Strong Flux Gradient Conditions oAnalytic Function Expansion Nodal (AFEN) Method  Two-Dimensional Expansion using 12 Trigonometric and Exponential Functions  Most Accurate  No Transverse Integration, Analytic Solution Basis  Hard to Expand to Multigroup oLocal Fine-Mesh Finite Difference Method (LFMFDM)  Nodal Coupling Resolved by Fine Mesh FDM Solution to Two-Node Problems in the framework of CMFD  Fast and Accurate (Accuracy Adjustable)  Evolved to One-Node Formulation oHigher Order Polynomial Expansion Nodal (HOPEN) Method  Expansion using 2D Polynomials on a Triangle Basis  Sufficiently Accurate with 6 Triangles per Hexagon, Further Mesh Refinement Possible  No Limitations on Energy Groups and Allows Multiple regions within a Hexagon  Evolved to Triangular-Z Polynomial Expansion Method (TPEN)

4. 4 POWER-KAERI PARCS Hexagonal Solver Overview oCMFD Formulation  Keep the Same Solution Methods as the Rectangular Solver -Eigenvalue Calculation by Wielandt Shift Method -Transient FSP Formulated by Theta Method and Analytic Precursor Integration  Linear System Solver -Currently, Krylov Solver for Hexagonal Geometry -SOR or CCSI solver might replace the Krylov Solver for Flexibility in Symmetry Handling oDual Nodal Solvers  Fine-Mesh FDM Solver -Transverse-Integrated  1D in Character -2nd Order Transverse Current Approximation along the Surfaces of the Hexagon -Surface Current Source Method Employed at the External Boundaries -Currently, Two-Node FDM -One-Node FDM will Replace the Two-Node Solver for Speed  TPEN Solver -Separate Radial and Axial Directions -No Transverse Integration in the Radial Solution  Direct 2D Solution -Axial Direction Solution Resolved by NEM

5. 5 POWER-KAERI Two-Node FDM Solver oNeutron Balance Equation for a Trapezoid oConstraints on Node-Average Fluxes oResulting Linear System (LHS only)  x y Two Node Geometry

6. 6 POWER-KAERI Transverse Current Approximation oQuadratic Representation of Transverse Currents oThree Vector Addition Scheme at Corner  Use only at the interior surfaces of the hexagon

7. 7 POWER-KAERI Surface Current Source Method oTo Determine the Current Profile at the External Surface oUtilizes Precalculated Response of Corner Current to the Unit Current Source Placed a Segment of the other Surfaces oUse Fine Mesh FDM to Obtain the Response for the Boundary Composition - Needed only Once for a Core

8. 8 POWER-KAERI TPEN Solver Development Overview oOne-Node Hexagon Formulation  To use TPEN within the Framework of CMFD  Partial Incoming Currents and Hexagon Corner Point Fluxes are Used as Constraint for the One-Node TPEN Solver oCMFD vs. CMR Comparison  CMFD turned out to be more efficient in terms of the number of nodal updates oCMFD Solver  Point and Line-SOR  Convenience in Handling Various Symmetries  Wielandt Shift Method for Accelerating Eigenvalue Convergence oGlobal Iteration Logic Refinement  Symmetric Gauss-Seidel Sweep (both ways) in the One-Node Nodal Calculation  Use of Node Average Flux Ratios (Post-CMFD Flux/Post-Nodal Flux) to Update the BC for the One-Node Nodal Calculation -J_in, Phi_corner, Flux Moments

9. 9 POWER-KAERI Triangular PEN Formulation oUnknowns Selected for a Triangle (9 in total per Group) oFlux Expansion for a Triangle x p u  Nodal Volume Average Flux,  Moments  Fluxes at three Corners,  Surface average fluxes at three surfaces

10. 10 POWER-KAERI Constraints Used in TPEN oNodal Balance for the Triangle oTwo 1-st Order Weighted Residual Balance (x and y directions) oSurface Average Current Continuity oCorner Point Balance (CPB)

11. 11 POWER-KAERI Hexagonal TPEN Formulation oBoundary Conditions Given only at the Hexagon Boundary oUnknowns in the Interior (31 in total)

12. 12 POWER-KAERI Hexagonal TPEN Formulation oConstraints to Determine the 31 Unknowns  6 Nodal Balance Equations for 6 Node Average Flux  12 WRM Equations for 12 Moments  6 Net Current Continuity Conditions for 6 Inner Surface Flux  6 Incoming Current Conditions for 6 Outgoing Currents  1 Net Leakage Balance Equation for 1 Center Flux

13. 13 POWER-KAERI Hexagon TPEN Linear System oThe linear system was solved analytically by using Mathematica.

14. 14 POWER-KAERI Eigenvalue Benchmark Problems Examined

15. 15 POWER-KAERI TPEN Calculation Flow Inner Iteration(SOR) F.S. Calculation IF2 ? IF1 ? n=1 m=1 n=n+1 m=m+1 l=1 l=l+1 Calculation of Multiplier, f TPEN Solution NEM Axial Solution CPB Solution Update Triangular Flux, Moments, Currents From f Update Backward Sweep ? yes no yes no IF3 ? no yes End

16. 16 POWER-KAERI Comparison of Solution Accuracy

17. 17 POWER-KAERI Comparison of Calculation Performance Computation Time Method/Code CPU time(sec)* CMR(AFEN-NEM)/MASTER18.2 1-Node CMFD(TPEN) 9.8 2-Node CMFD(FDM) 8.2 Method/CodeCMR 1-Node CMFD(TPEN) CPU time(sec) 10.49.8 No. of Outer Iterations/ No. of Nodal Updates 156/3251/15 T nodal /T total (%) 64.066.0 Comparison of Iteration Characteristics of CMFD and CMR for Accelerating TPEN * Pentinum III 500 MHz (VVER440 3D Problem Only)

18. 18 POWER-KAERI oTo Reduce the Computational Burden of the Two-Node FDM Problems oIncoming Partial Currents are Chosen as BC Instead of Node Avg. Flux oSolves Three Directions Simultaneously  FDM Formulation for a System of three second order 1-D Diffusion Equations (Coupled through the transverse leakage terms appearing on the RHS)  Balance Equation at each Mesh  Unknowns (3*N+4) -3*N Fine Mesh Flux -3 Adjusted Transverse Leakage Source ( ) -Node (Hexagon) Averaged Flux  Equations -3*N Mesh Balance Equation -3 Node Average Flux Constraints -1 Nodal Balance Equation oLinear System can be Solved by Gaussian Elimination very Effectively One-Node FDM x y