Course Outline:

1. Definition and Characteristics of Nodal Methods

2. Derivation of Global Nodal Equations Having the Finite-Difference Form

3. Standard Nodal methods – Features, Deficiencies, and Range of Application

4. Modern Nodal Methods:

   1. Transverse-Integrated Methods:
      1. Transverse Integrated Procedure
      2. 1D Transverse Integrated Flux Representation:
         1. Polynomial Methods
         2. Analytical Methods
         3. Semi-Analytical Methods
      3. Nodal Interface Formulation:
         1. Partial Current Formulation
         2. Net Current Formulation
         3. Surface Flux Formulation
      4. Transverse Leakage Approximation
   2. Other Types

5. Approaches for Treating Geometry Peculiarities:

   1. Direct Derivations in Cartesian, Hexagonal –Z and Cylindrical Geometry’s
   2. Conformal Mapping

6. Convergence and Iterative Solution of Nodal Equations

   1. Convergence Criteria and Analysis
   2. Iterative methods and Acceleration Techniques
   3. Non-linear Iterative Strategy and Multi-Level Optimization

7. Dehomogenization and Flux Reconstruction Schemes:

   1. Non-Separable Within-Node Flux Distribution
   2. Interpolation for Evaluation of Corner Values
   3. Calculation of Pin Powers

8. Nodal Depletion Models and High Order Treatment of Nodal Heterogeneity

9. Transient Nodal Methods

10. Application to Multidimensional Neutron Transport Problems

11. Verification & Validation of Nodal Methods:

    1. General Remarks on Benchmarking
    2. Types of LWR Benchmarks: 2D and 3D Test Problems; Mathematical Steady-State Problems, Depletion and Fuel management Problems, and Transient Benchmark Problems Without and With Thermal-Hydraulic Feedback
    3. Qualification Procedures

The course is focused on learning advanced nodal computational methods for analyzing Light-Water Reactors. From a mathematical viewpoint, nodal methods of reactor physics are special finite volume methods. The course discusses approaches to exploit the inherent advantages of the cell-centered mixed type finite volume methods for solution of the neutron diffusion equations. Accurate and efficient techniques for the calculation of flux gradients on the surface of the finite volumes are introduced. Further effective iterative methods to solve resulting large sparse linear equation systems are presented. The course combines learning the nodal methods theory with computer projects, using codes based on nodal methods, which gives student an insight of the accuracy and efficiency of used techniques.

Instructor: Dr. Kostadin N. Ivanov, 865-0046, 230 Reber Building

E-mail: kni1@psu.edu

Lectures: T., Th., 2:30-3:45pm
Computer Labs: Th., 2:30-3:35pm
Office Hours: Mon., Fri. 10:00-11:00am

Text and References:

• Lecture Notes and Handouts
• Computer Code Input Manuals
• Benchmark Problem Specifications
• Methods of Steady-State Reactor Physics in Nuclear Design, R. J. J. Stammler and M. J. Abate, (1983) (Academic Press), Chapter XI, “Nodal Equations for 3D Reactor Calculations”, p.393
• “Progress in Nodal Methods for the Solution of the Neutron Diffusion and Transport Equations”, R. D. Lawrence, Progress in Nuclear Energy, Vol. 17, No. 3, p. 271-301, (1986)
• “Diffusion Theory Methods for Spatial Kinetics Calculations”, T. M. Sutton and B. N. Aviles, Progress in Nuclear Energy, Vol. 30, No. 2, pp. 119-182, (1996)
• Other Selected Papers

   

  Pragmatic and Computer Oriented Approach:

  • Aiming at improving the understanding of theory, computerization and practice of nodal reactor analysis methods
  • Combination of academics exercises (homework) with practical applications (computer assignments)
  • Special emphasis on the numerical aspects of nodal reactor analysis
  • Lessons from the extensive industrial experience with the development and application of computational tools utilizing nodal techniques

   

  The course will consist of:

  • Five homework problems on specific topics
  • Four projects (with different time spent on them)
  • Two exams

   

  The grading of the course is as follows:

  • Homework 30%
  • Projects 50%
  • TExams 20%

   

  Assumed Knowledge:

  • Group diffusion theory
  • General aspects of reactor core design
  • Standard numerical methods
  • Basic principles of computer programming

   

   

   

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