Perturbation Theory for Method of Characteristics

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Perturbation Theory for Method of Characteristics

• Igor R. Suslov, Oleg G. Komlev, Ivan V. Tormyshev
• Institute of Physics and Power Engineering,
• Obninsk, Russia

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INTRODUCTION MCCG3D code development for core
calculation without homogenization

• ISTC project 1836
• visualization
• Use of parameterized cross-sections library
• verification
• validation

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Structure of ISTC project 1836

• Block 1. Generalized Model for Reactor
  Calculations
• Block 2. Libraries of nuclear data, group
  cross-sections
• Block 3. Neutron transport and perturbation
  theory
• Block 4. Fuel burn-up
• Block 5. Matrices of sensitivities and
  covariance's
• Block 6. Analysis of errors and their impact on
  reactor functionals
• Block 7. Analysis of several accident scenarios,
  determination of safety boundaries

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Main tendency

• High accuracy and reliability of neutronics
• Very detailed description both geometry and
  neutrons (and gammas) distributuions
• Numerical expenses are not decisive factor

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GUI for MCCG3D
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C5G7MOX z0.
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MOX SA
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C5G7MOX y6.875
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Fast Neutrons, ?5G7MOX
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Fast Neutrons distribution near boundary UO2-MOX
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MCCG3D verification

• C5G7MOX OECD Benchmark
• Benchmarks within project ISTC 1836

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Percentage of fuel pins within confidence
intervals
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Applications of Perturbation Theory Calculations

• Assessment of reactor property variations caused
  by variations of cross-sections, nuclide
  densities, geometry and dimensions.
• Some specific applications
• Evaluation of critical experiment
• Adjustment of neutron cross-sections on the basis
  of experiments
• Optimization studies
• Determination of requirements for permissible
  neutron cross-section inaccuracies and
  technological tolerances
• Neutron cross-section and technological
  uncertainty constituent of uncertainty in reactor
  physics properties.

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Perturbation Theory For Transport Equation
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Perturbation Theory In Terms Of Long
Characteristics Method (MCCG3D Code)

• 1. Angular quadrature formula

2. Space quadrature formula

• Numerical volume of n-th cell in m-th direction.

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Perturbation Theory In Terms Of Long
Characteristics Method (MCCG3D Code)(2)

• 3. Total cross-section term in PT formula

- quadrature weight of i-th trajectory in m-th
direction for n-th FD volume.
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Perturbation Theory In Terms Of Long
Characteristics Method (MCCG3D Code)(3)

• 4. Calculation of sensitivity coefficients for
  nuclide concentrations.
• Nuclide concentration derivatives of
  cross-sections are necessary.
• It is based on calculation of sensitivity
  coefficients for cross-sections (composite
  function derivative)

- sensitivity coefficients for nuclide
concentrations.
- nuclide concentration derivatives of
cross-sections.
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KEFSFGG Code Sensitivity Coefficient Calculations

• 1. Nature of problem solved
• KEFSFGG computes multiplication factor
  sensitivity coefficients for cross-sections and
  nuclide concentrations.
• Method of solution
• First order perturbation theory on neutron
  transport equations in irregular geometry
  (without necessity of reactor core
  homogenization).
• Input data
• Regular and adjoint angular fluxes , FD volumes,
  quadrature weights and directions (all of them
  computed by MCCG3D code).
• Necessary cross-sections and theirs derivatives.

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KEFSFGG Verification

• 1. The main objectives of numerical experiments.
• Development of benchmark problem collection (both
  numerical and analytical problems).
• Confirmation of direct-adjoint solution
  consistency for homogeneous and inhomogeneous
  transport equations.
• Comparison of Keff variations obtained by
  perturbation theory and direct calculations.

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KEFSFGG Verification. consistency of direct and
adjoint solutions.

• 1. Consistency criterion for homogeneous
  transport equation

2. Consistency criterion for inhomogeneous
transport equations (in operator form)
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KEFSFGG Verification. Numerical benchmark
descriptions.

• 1. Benchmark problem 1.

• 1D cylindrical geometry (diameter is equal to 40
  cm)
• two group cross-sections.

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KEFSFGG Verification. Numerical studies of the
benchmark problem 1.

• Direct-adjoint solution consistency criterion is
  met regular and adjoint multiplication factors
  are equal for all finite-difference approximation
  parameters.
• Comparison of Keff variations obtained by
  perturbation theory and direct calculations

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KEFSFGG Verification. Numerical studies of the
benchmark problem 1.
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The model of VENUS-2 without homogenization for
the MCCG3D code
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KEFSFGG Verification. Numerical studies of
VENUS-2 model (1).

• Regular-adjoint solution consistency criterion is
  met

• Comparison of Keff variations obtained by
  perturbation theory and direct calculations

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KEFSFGG Verification. Numerical studies of
VENUS-2 model(2).

• Perturbation theory calculations of Keff
  variations tends to direct ones.

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KEFSFGG Verification. Numerical benchmark
descriptions (3).

• 3. Benchmark problem 3.

• 2D model of P39G8 fuel assembly (VVER-1000 MOX
  fuel cycle)
• Four group cell averaged cross-sections obtained
  by TRIFON calculations.

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KEFSFGG Verification. Numerical studies of VVER
SA.
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KEFSFGG Verification. Numerical studies of VVER
SA(2).

• Comparison of Keff obtained by perturbation
  theory and direct calculations.

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KEFSFGG Verification. Analytical benchmark
description.

• 1. Benchmark problem 4.

• 1D cylindrical geometry, two layers (diameters
  are equal to 1 and 2 cm)
• two group cross-sections
• discrete ordinates method
• external neutron source
• vacuum external boundary conditions
• isotropy condition at r0

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Thank you !!!