Comparison of Sensitivity Analysis Techniques in Monte

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Comparison of Sensitivity Analysis Techniques in
Monte Carlo Codes for Multi-Region Criticality
Calculations

• Bradley T. Rearden
• Douglas E. Peplow
• Oak Ridge National Laboratory

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Sensitivity and Uncertainty Analysis Techniques

• New techniques recently developed at ORNL to
  determine applicability of critical experiments
  for criticality code validation.
• Use energy-dependent sensitivity coefficients and
  cross section covariance data to assess
  similarity of systems.

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Sensitivity Methods in Monte Carlo Codes

• Differential Sampling
• Calculates the derivatives of the response with
  respect to some parameter at the same time the
  response is calculated.
• First, second, third, etc. and cross derivatives
  can be computed.
• Available in MCNP and can be used for fixed
  source or criticality problems.
• MCNP manual cautions The track length estimate
  of keff in KCODE criticality calculations assumes
  the fundamental eigenvector (fission
  distribution) is unchanged in the perturbed
  configuration.
• This means that the source dependence on the
  parameter of interest is not taken into account
  by the differential sampling, which could lead to
  incorrect values of the derivatives if that
  source dependence is large.

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Sensitivity Methods in Monte Carlo Codes

• Perturbation Theory
• Independent forward and adjoint calculations
• Group-wise fluxes are computed and folded with
  cross-section data to produce sensitivity
  coefficients.
• Only valid for small perturbations with linear
  response.
• Employed inSEN3 analysis sequence using KENO V.a
  in SCALE.

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GODIVA Sphere

• 8.7414 cm bare HEU metal
• 18.74 g/cm3
• 93.71 235U, and 5.27 238U, 1.02 234U
• ENDF/B-V data were used in both codes to ensure
  consistency

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Sensitivity Calculations

• Sensitivity of keff to fuel density was
  investigated.
• Over entire system
• Only in outer 2-cm thick shell
• Code generated sensitivity coefficients
• Direct Recalculation
• /- 10 in MCNP
• /- 5 in KENO V.a

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Sensitivity Results

• Reasonable results for both codes for whole
  system.
• MCNP over-predicts outer shell by 40.

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Explanation

• Differential Sampling
• When density of whole system is perturbed,
  uniform change in source distribution.
• Only outer 2-cm thick shell is perturbed,
  non-uniform change in source distribution.
• Perturbation Theory
• Forward fluxes are weighted with adjoint flux or
  importance function

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Possible Solution for Differential Operator

• Fixed source problem where source is function of
  material density, accumulator stores sum of the
  relative derivatives of interaction
  probabilities, Pij
• To include effect of source that is proportional
  to density (Scr), start accumulator with
  derivative of source with respect to density

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Application to Criticality Problems

• Must be extended from generation to generation.
• Effect of each energy-dependent cross-section of
  interest would have to be taken into account.
• Would provide powerful tool for analysts with
  ability to predict non-linear response to
  perturbations.
• For now, perturbation theory provides adequate
  results for criticality applications.

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