9. Deterministic Transport

Introduction by S. M. Bowman

SCALE deterministic transport capabilities enable criticality safety, depletion, sensitivity, and uncertainty analysis, as well as hybrid approaches to Monte Carlo analysis. SCALE provides a one-dimensional (1D) transport solver for eigenvalue neutronics and fixed source neutron-gamma analysis with XSDRN, two-dimensional (2D) eigenvalue neutronics with NEWT, and a three-dimensional (3D) transport solver for hybrid acceleration of Monte Carlo fixed source and eigenvalue calculations with Denovo. Generally, the use of these transport solvers in SCALE is best accessed through the capability specific sequences: CSAS for criticality safety, TRITON for 1D and 2D depletion, TSUNAMI-1D for sensitivity and uncertainty analysis, and MAVRIC for 3D fixed source hybrid Monte Carlo analysis.

XSDRN

XSDRN is a multigroup discrete-ordinates code that solves the 1D Boltzmann equation in slab, cylindrical, or spherical coordinates. Alternatively, the user can select P1 diffusion theory, infinite medium theory, or Bn theory. A variety of calculational types is available, including fixed source, eigenvalue, or search calculations. In SCALE, XSDRN is used for several purposes: eigenvalue (k_eff) determination; cross section collapsing; and computation of fundamental-mode or generalized adjoint functions for sensitivity analysis.

NEWT

NEWT (New ESC-based Weighting Transport code) is a multigroup discrete-ordinates radiation transport computer code with flexible meshing capabilities that allow 2D neutron transport calculations using complex geometric models. The differencing scheme employed by NEWT-the Extended Step Characteristic approach-allows a computational mesh based on arbitrary polygons. Such a mesh can be used to closely approximate curved or irregular surfaces to provide the capability to model problems that were formerly difficult or impractical to model directly with discrete-ordinates methods. Automated grid generation capabilities provide a simplified user input specification in which elementary bodies can be defined and placed within a problem domain. NEWT can be used for eigenvalue, critical-buckling correction, and source calculations, and it can be used to prepare collapsed weighted cross sections in AMPX working library format.

Like other SCALE modules, NEWT can be run as a standalone module or as part of a SCALE sequence. NEWT has been incorporated into SCALE TRITON control module sequences. TRITON can be used simply to prepare cross sections for a NEWT transport calculation and then automatically execute NEWT. TRITON also provides the capability to perform 2D depletion calculations in which the transport capabilities of NEWT are combined with multiple ORIGEN depletion calculations to perform 2D depletion of complex geometries. In the TRITON depletion sequence, NEWT can also be used to generate lattice-physics parameters and cross sections for use in subsequent nodal core simulator calculations.

DENOVO

Denovo [DTESSC10], is a parallel 3D discrete-ordinates code available in SCALE as part of two control module sequences for different applications, as described below. Because Denovo can only be run in SCALE via the Monaco with Automated Variance Reduction using Importance Calculations (MAVRIC) or Denovo Eigenvalue Calculation (DEVC), it is not documented separately in the section entitled “Deterministic Transport” in this manual.

The MAVRIC hybrid Monte Carlo radiation shielding sequence employs the Consistent Adjoint Driven Importance Sampling (CADIS) and Forward-Weighted CADIS (FW-CADIS) methodologies. Denovo is used to generate adjoint (and, for FW-CADIS, forward) scalar fluxes for the CADIS methods in MAVRIC. This adjoint flux information is then used by MAVRIC to construct a space- and energy-dependent importance map (i.e., weight windows) to be used for biasing during Monte Carlo particle transport and as a mesh-based biased source distribution. For use in MAVRIC/CADIS, it is highly desirable that the S_N code be fast, positive, and robust. The phase-space shape of the forward and adjoint fluxes, as opposed to a highly accurate solution, is the most important quality for Monte Carlo weight-window generation. Accordingly, Denovo provides a step-characteristics spatial differencing option that produces positive scalar fluxes as long as the source (volume plus in-scatter) is positive. Denovo uses an orthogonal, nonuniform mesh that is ideal for CADIS applications because of the speed and robustness of calculations on this mesh type. Denovo can be run stand-alone in MAVRIC to perform fixed source calculations using the PARM=forward (for forward Denovo) or PARM=adjoint (for adjoint Denovo). See the MAVRIC chapter for details.

The other sequence that uses Denovo is the DEVC sequence. DEVC generates a reasonably accurate starting source through a Denovo eigenvalue calculation. Denovo can be run stand-alone in DEVC for calculating criticality eigenvalue problems. This sequence reads an input file very similar to a CSAS6 input file that contains an extra block of input for describing the Denovo mesh grid and calculational parameters.

• 9.1. XSDRNPM: A One-Dimensional Discrete-Ordinates Code for Transport Analysis
  • 9.1.1. Introduction
    • 9.1.1.1. Functions performed
      • 9.1.1.1.1. Background on XSDRNPM
    • 9.1.1.2. Applications in SCALE
      • 9.1.1.2.1. Notes on the use of various spectral calculational options
    • 9.1.1.3. Selection of output cross-section library formats
  • 9.1.2. Theory and Procedures
    • 9.1.2.1. One-dimensional discrete-ordinates theory
    • 9.1.2.2. Multigroup one-dimensional Boltzmann equation
    • 9.1.2.3. Scattering source term
      • 9.1.2.3.1. Slab and Spherical Geometries
      • 9.1.2.3.2. Cylindrical Geometry
    • 9.1.2.4. Discrete-ordinates difference equations
      • 9.1.2.4.1. Discrete-ordinates equation for a slab
      • 9.1.2.4.2. Discrete-ordinates equations for sphere and cylinder
      • 9.1.2.4.3. \(S_n\) quadratures for slabs
      • 9.1.2.4.4. \(S_n\) quadratures for spheres
      • 9.1.2.4.5. \(S_n\) quadratures for cylinders
    • 9.1.2.5. Weighted-difference formulation for discrete-ordinates equations
    • 9.1.2.6. Boundary conditions
    • 9.1.2.7. Fixed sources
    • 9.1.2.8. Dimension search calculations
      • 9.1.2.8.1. Zone-width search (IEVT = 4)
      • 9.1.2.8.2. Outer radius search (IEVT = 5)
      • 9.1.2.8.3. Buckling search (IEVT = 6)
      • 9.1.2.8.4. Search calculation strategy
    • 9.1.2.9. Alpa Search
      • 9.1.2.9.1. Alpha search
      • 9.1.2.9.2. Direct-buckling search
    • 9.1.2.10. Iteration and convergence tests
    • 9.1.2.11. Group banding (scaling rebalance)
    • 9.1.2.12. Buckling correction
    • 9.1.2.13. Void streaming correction
    • 9.1.2.14. Cross-section weighting
      • 9.1.2.14.1. “Cell Weighting”
      • 9.1.2.14.2. “Zone” weighting
      • 9.1.2.14.3. “Region” weighting
      • 9.1.2.14.4. “Inner-cell” weighting
      • 9.1.2.14.5. Multigroup weighting equations
      • 9.1.2.14.6. Transfer matrices
      • 9.1.2.14.7. Weighting of \(\bar{v}\)
      • 9.1.2.14.8. Transport cross sections
        • 9.1.2.14.8.1. Outscatter approximation (inconsistent method)
        • 9.1.2.14.8.2. Inscatter approximation (consistent method)
        • 9.1.2.14.8.3. Weighting function for transport cross section
    • 9.1.2.15. Adjoint calculations
      • 9.1.2.15.1. Generalized adjoint calculations
    • 9.1.2.16. Coupled neutron-photon calculations
    • 9.1.2.17. Diffusion theory option
    • 9.1.2.18. Infinite-medium theory option
    • 9.1.2.19. B_N theory option
  • 9.1.3. XSDRNPM Input Data
    • 9.1.3.1. Abbreviated XSDRNPM input description
  • 9.1.4. XSDRNPM Input/Output Assignments
  • 9.1.5. XSDRN Sample Problem
  • 9.1.6. Output Cross Sections
  • 9.1.7. Error messages
  • 9.1.8. Appendices
    • 9.1.8.1. Special XSDRNPM Files
      • 9.1.8.1.1. Activity file
      • 9.1.8.1.2. Balance table file
      • 9.1.8.1.3. Input and derived data file
    • 9.1.8.2. XSDRNPM Mixed Cross Sections
• 9.2. NEWT: A New Transport Algorithm for Two-Dimensional Discrete-Ordinates Analysis in Non-Orthogonal Geometries
  • 9.2.1. Introduction
    • 9.2.1.1. How to use this manual
    • 9.2.1.2. Background
    • 9.2.1.3. Discrete-ordinates solution on an arbitrary grid
    • 9.2.1.4. Functions performed
  • 9.2.2. Theory and Procedures
    • 9.2.2.1. Boltzmann transport equation
    • 9.2.2.2. The step characteristic approximation
    • 9.2.2.3. The Extended Step Characteristic approach
      • 9.2.2.3.1. Cell properties and geometries
      • 9.2.2.3.2. Relationships between cells
      • 9.2.2.3.3. The set of characteristic directions
      • 9.2.2.3.4. Angular flux at a cell boundary
      • 9.2.2.3.5. Mapping a characteristic vector into the two-dimensional problem domain
      • 9.2.2.3.6. Neutron balance within a computational cell
    • 9.2.2.4. Coarse-mesh finite-difference acceleration
    • 9.2.2.5. Assembly discontinuity factors
  • 9.2.3. Input Formats
    • 9.2.3.1. Overview of newt data blocks
    • 9.2.3.2. Parameter block
      • 9.2.3.2.1. Convergence and acceleration parameters
      • 9.2.3.2.2. Output editing
      • 9.2.3.2.3. Angular quadrature
      • 9.2.3.2.4. Control options
        • 9.2.3.2.4.1. Geometry processing options
      • 9.2.3.2.5. Critical spectrum options
      • 9.2.3.2.6. File unit options
    • 9.2.3.3. Material Block
    • 9.2.3.4. Source block
    • 9.2.3.5. Collapse block
    • 9.2.3.6. Geometry block
      • 9.2.3.6.1. Bodies
        • 9.2.3.6.1.1. Shapes
        • 9.2.3.6.1.2. Cylinder
        • 9.2.3.6.1.3. Cuboid
        • 9.2.3.6.1.4. Hexprism and rhexprism
        • 9.2.3.6.1.5. Wedge
        • 9.2.3.6.1.6. Polygon
        • 9.2.3.6.1.7. Example of shape specifications
        • 9.2.3.6.1.8. Shape modifier commands
        • 9.2.3.6.1.9. ORIGIN
        • 9.2.3.6.1.10. Rotate
        • 9.2.3.6.1.11. Chord
        • 9.2.3.6.1.12. Com
        • 9.2.3.6.1.13. Sides
        • 9.2.3.6.1.14. Holes
        • 9.2.3.6.1.15. Array placement
      • 9.2.3.6.2. Media specifications
        • 9.2.3.6.2.1. Unit boundary
      • 9.2.3.6.3. Geometry block examples
        • 9.2.3.6.3.1. Simple pin cell
        • 9.2.3.6.3.2. Hexagonal assembly
        • 9.2.3.6.3.3. Advanced CANDU reactor ACR-700 assembly
      • 9.2.3.6.4. Summary of geometry specifications
        • 9.2.3.6.4.1. Unit definition statements
        • 9.2.3.6.4.2. Geometry modifiers
    • 9.2.3.7. Boundary conditions
      • 9.2.3.7.1. Boundary conditions descriptions
        • 9.2.3.7.1.1. Reflective boundary condition
        • 9.2.3.7.1.2. White boundary condition
        • 9.2.3.7.1.3. Vacuum boundary condition
        • 9.2.3.7.1.4. Periodic boundary condition
      • 9.2.3.7.2. Boundary condition specification
    • 9.2.3.8. General cross section weighting
      • 9.2.3.8.1. Scattering cross section transfer matrix weighting
      • 9.2.3.8.2. Weighting of the collapsed fission spectrum, \(\chi\)
      • 9.2.3.8.3. Weighting of the number of neutrons per fission
      • 9.2.3.8.4. Weighting of (n,2n), (n,3n), and (n,4n) cross sections
      • 9.2.3.8.5. Calculation and weighting of transport cross sections
        • 9.2.3.8.5.1. Outscatter approximation (inconsistent method)
        • 9.2.3.8.5.2. Inscatter approximation (consistent method)
        • 9.2.3.8.5.3. Weighting function for transport cross section
    • 9.2.3.9. Array definition
      • 9.2.3.9.1. Array types
      • 9.2.3.9.2. Pin-power edits
      • 9.2.3.9.3. Fill operations
      • 9.2.3.9.4. Examples of array definitions
    • 9.2.3.10. Homogenization block
    • 9.2.3.11. Assembly discontinuity factors
    • 9.2.3.12. Flux planes
    • 9.2.3.13. Mixing table block
  • 9.2.4. Examples of Inputs
    • 9.2.4.1. Sample 1
    • 9.2.4.2. Sample 2
      • 9.2.4.2.1. Sample 3
    • 9.2.4.3. Sample 4
    • 9.2.4.4. Sample 5
  • 9.2.5. Description of Output
    • 9.2.5.1. NEWT banner
    • 9.2.5.2. Input summary
      • 9.2.5.2.1. Control options
      • 9.2.5.2.2. Output options
      • 9.2.5.2.3. Input/output unit assignments
      • 9.2.5.2.4. Convergence control parameters
      • 9.2.5.2.5. Pin-power edit requests
      • 9.2.5.2.6. Geometry specifications
      • 9.2.5.2.7. Homogenization region specifications
      • 9.2.5.2.8. Material specifications
      • 9.2.5.2.9. Derived parameters
      • 9.2.5.2.10. Energy group structure listing
      • 9.2.5.2.11. Quadrature parameters
      • 9.2.5.2.12. Mixture volumes listing
      • 9.2.5.2.13. Mixing table listing
      • 9.2.5.2.14. Nuclide cross sections
      • 9.2.5.2.15. Mixture cross sections
    • 9.2.5.3. Iteration history
    • 9.2.5.4. Four-factor formula
    • 9.2.5.5. Fine-group balance tables
    • 9.2.5.6. Planar fluxes and currents
    • 9.2.5.7. Pin-power edits
    • 9.2.5.8. Broad-group collapse
      • 9.2.5.8.1. Broad-group summary data
      • 9.2.5.8.2. Broad-group cross section data
    • 9.2.5.9. Critical spectrum edit
    • 9.2.5.10. Assembly discontinuity factors
    • 9.2.5.11. Groupwise form factors
      • 9.2.5.11.1. Homogenized cross sections
    • 9.2.5.12. End-of-calculation banner
      • 9.2.5.12.1. Postscript graphics files
    • 9.2.5.13. Media zone edits

References

DTESSC10

Thomas M. Evans, Alissa S. Stafford, Rachel N. Slaybaugh, and Kevin T. Clarno. Denovo: A new three-dimensional parallel discrete ordinates code in SCALE. Nuclear technology, 171(2):171–200, 2010.