1. 7/21/2018
Analysis and quantification of modelling errors introduced in the deterministic calculational path applied to a mini-core problem
SAIP 2015 conference 01 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)
University of Johannesburg 2014 Postgraduate Symposium
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2. Outline Introduction Problem statement Neutronics modelling
7/21/2018
Outline
Introduction
Problem statement
Neutronics modelling
Methodology
Results and discussions
Conclusion 2
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3. 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 3
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4. 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 4
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5. (Duderstadt and Hamilton 1976)
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.
(Duderstadt and Hamilton 1976) 5
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6. 7/21/2018
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.
Remove and add 2-step process to the previois slide 6
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7. 7/21/2018
Neutronics modelling...
Here is a two-step deterministic calculational path.
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.
Use nodal cross sections in the diffusion calculation to simulate full core.
Add the reaction rate equation 7
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8. Neutronics modelling... Homogenization and energy condensation
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Neutronics modelling...
Homogenization and energy condensation
The 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.
Take out the 100, condensation (cont. to groups) and collapsing (fine to few) 8
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9. Reaction rates preserved
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Neutronics modelling...
Reaction rates preserved
Add the reaction rate equation
(Smith 1980) 9
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10. (Duderstadt and Hamilton 1976)
Neutronics modelling…
Diffusion approximation
The diffusion equation is derived from transport equation
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 10
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11. Neutronics modelling…
7/21/2018
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).
Emphasis on the first 3 errors. Add sub-bullet stating that we preserves rxn rte and leakages 11
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12. Methodology
We want to numerically quantify the errors made in a mini-core problem.
With reflective boundary conditions
Fuel-Water model 12
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13. Methodology: Codes Code systems used: SCALE6.1 (NEWT) OSCAR-4 (MGRAC)
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 13
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14. Methodology: Calculations
7/21/2018
Methodology: Calculations
The scheme proposed here is to analyse the 1st three errors.
Explain one in detail. Add text (1) and (2) ---1 To 5 1 2 1 3 4 5
Spectral error
Homogenization error
Diffusion error 14
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15. The scheme proposed here is to analyse environmental error.
7/21/2018
Methodology: Calculations
The scheme proposed here is to analyse environmental error.
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.
Correct the right and wrong 15
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16. Results and discussion
7/21/2018
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 =
Integral measure of the difference betw two systems
An absolute difference of greater than 1000 is considered significant 16
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17. Table 1: Errors from the first 3 simplifications
7/21/2018
Results and discussion
Integral measure of the difference betw two systems
An absolute difference of greater than 1000 is considered significant
2-groups
4-groups
6-groups
Combined error (pcm)
-3472
-4406
-4113
Table 1: Errors from the first 3 simplifications 17
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18. Table 2: Environmental dependency in 6 energy groups
Results and discussion
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. 18
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19. Results and discussion
7/21/2018
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.
Integral measure of the difference betw two systems
An absolute difference of greater than 1000 is considered significant 19
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20. Conclusion
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. 20
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21. THANK YOU 21
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