Analysis and quantification of modelling errors introduced in the deterministic calculational path applied to a mini-core problem

SAIP 2015 conference01 July 2015

Speaker: Mr. T.P. Gina(1), (2)

Supervisors: Prof. S.H. Connell(1)

Mrs. S.A. Groenewald(2)

Dr. W.R. Joubert (2)

Affiliation: UJ(1), Necsa(2)

Outline

1. Introduction2. Problem statement3. Neutronics modelling4. Methodology5. Results and discussions6. Conclusion

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Introduction

• What is reactor modelling?– Set of algorithms & computer codes to perform reactor

calculations– Understand and predict core behaviour

• Why is reactor modelling error analysis important?

• Part of a bigger study focused on improving the errors made in modelling MTRs.– Modelling error analysis is done on a mini-core problem– The approach defined will be applied to a full-core MTR model

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Problem statement

• To analyze and quantify errors introduced by simplifications made in the deterministic calculational path for a mini-core problem– Energy group condensation– Spatial homogenization– Diffusion approximation– Environmental dependency

• Investigate individual and combined effect on the calculational path

• This study will contribute to a broader understanding of the current calculational path and its limitations

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Neutronics modelling

• Determines neutron flux distribution

• It describes the motion and interaction of neutrons with nuclei in the reactor core.

– 7 independent variables [x, y, z, θ, ϕ, E, t]– Flux is the dependent variable.

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(Duderstadt and Hamilton 1976)

Neutronics modelling...

• For day-to-day reactor calculations, the deterministic approach is used to solve transport equation because of its calculational efficiency

• This approach involves discretizing variables of the transport equation to set of equations.

• The transport equation is solved numerically• The deterministic method is applied to reactor

analysis calculations via a two-step process.

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Neutronics modelling...

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Here is a two-step deterministic calculational path.1.Perform a detailed 2D transport calculation on each assembly type.

– Use solution to simplify geometry and energy representation

– Produce spatially homogenized, energy condensed assembly cross section for each reactor component.

2.Use nodal cross sections in the diffusion calculation to simulate full core.

Neutronics modelling...

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Homogenization and energy condensationThe simplifications made in geometry and energy representation in the node involve performing a fine energy group heterogeneous transport calculation.

Heterogeneous flux is used as weighting function to homogenise and collapse cross-sections to fewer (10s) energy groups

Each node has a constant set of few-groups homogenized nodal parameters that preserve the transport solution in an average sense.

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

Cro

ss s

ectio

n (b

arn)

Energy (eV)

Total cross section for U235in 238 and 6 energy groups

U235 total in 238 groups

U235 total in 6 groups

Neutronics modelling...

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Reaction rates preserved

(Smith 1980)

Neutronics modelling…

• Diffusion approximation– The diffusion equation is derived from transport

equation

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Fick’s Law

(Duderstadt and Hamilton 1976)

Diffusion approx. theory is valid:•Slowly varying current density in time

•Isotropic scattering

•Angular flux distribution is linearly anisotropic

Diffusion approx. theory is invalid:•In strongly absorbing media

•Near the boundary where material properties change dramatically over mfp type distances

•Near localized sources

Neutronics modelling…

• Environmental dependency– Due to the transport solution’s approximate boundary conditions– Cross-sections are generated in an environment that is not exact

to the environment where they’ll be used in the core calculation.– Using cross sections from an infinite environment for the fuel

elements in a different core environment an environmental error is introduced in the model.

• The first 3 errors are typically addressed by using the equivalence theory (ET).

• ET reproduces node-integrated parameters of the known heterogeneous solution (Smith K.S).

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Methodology

• We want to numerically quantify the errors made in a mini-core problem.

• With reflective boundary conditions

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Fuel-Water model

Methodology: Codes

• Code systems used:– SCALE6.1 (NEWT)

• 2D transport solver• Uses Sn (Discrete Ordinate Method)

– OSCAR-4 (MGRAC)• 3D diffusion solver• Uses the Multi-group Analytic Nodal Diffusion Method

– Serpent• Uses the Monte Carlo stochastic method• 3D and continuous energy

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Methodology: Calculations

1 2 1 3 4 5

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Spectral error Homogenization error Diffusion error

The scheme proposed here is to analyse the 1st three errors.

Methodology: Calculations

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•The Serpent code was used to generate correct and approximated fuel cross sections.•Functionality exists to generate nodal equivalence parameters from Serpent calculations.•Two MGRAC calculations were set up

– One with no environmental error and one with environmental error

•Compare k-effs.

The scheme proposed here is to analyse environmental error.

Results and discussion

• The k-eff is measure of criticality• Error in k-eff is measured in pcm as:

• Error in k-eff > 500pcm is considered large.

• Reference k-eff = 1.17073

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Results and discussion

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2-groups 4-groups 6-groups

Combined error (pcm) -3472 -4406 -4113

Table 1: Errors from the first 3 simplifications

Results and discussion

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Table 2: Environmental dependency in 6 energy groups

• The Serpent calculation and the 6-groups homogenized diffusion calculation are equivalent (within some statistical margin) because of ET.

• After ET resolved the first 3 errors an environmental error of 733 pcm remains.

Results and discussion

• Spectral error and reactivity increase with the decrease in number of groups

• Homogenization error is small

• Diffusion error is up to 6000 pcm larger in 2-groups

• All 3 simplifications reduce calculational time.

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Conclusion

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• All simplification to the deterministic calculational path were investigated for a mini-core problem.

• Spectral, diffusion and environmental error were significant for a mini-core problem in 6-groups.

• ET was successfully used to resolve all errors except environmental.

• Future work– Environmental error will be investigated further.– More models will be investigated.– Results will be used in an on-going research project

to improve current calculational path.

THANK YOU

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