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 SAIP 2015 Annual Conference 1

Outline Introduction Problem statement Neutronics modelling 7/21/2018 Outline Introduction Problem statement Neutronics modelling Methodology Results and discussions Conclusion 2 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 2

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 SAIP 2015 Annual Conference

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 SAIP 2015 Annual Conference

(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 SAIP 2015 Annual Conference

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 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 6

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 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 7

Neutronics modelling... Homogenization and energy condensation 7/21/2018 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 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 8

Reaction rates preserved 7/21/2018 Neutronics modelling... Reaction rates preserved Add the reaction rate equation (Smith 1980) 9 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 9

(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 SAIP 2015 Annual Conference

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 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 11

Methodology We want to numerically quantify the errors made in a mini-core problem. With reflective boundary conditions Fuel-Water model 12 SAIP 2015 Annual Conference

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 SAIP 2015 Annual Conference

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 SAIP 2015 Annual Conference University of Johannesburg 2014 Postgraduate Symposium 14

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 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 15

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 = 1.17073 Integral measure of the difference betw two systems An absolute difference of greater than 1000 is considered significant 16 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 16

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 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 17

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 SAIP 2015 Annual Conference

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 University of Johannesburg 2014 Postgraduate Symposium SAIP 2015 Annual Conference 19

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 SAIP 2015 Annual Conference

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