Homogenization Methods for Full Core Solution of the Pn Transport Equations with 3-D Cross Sections Andrew Hall October 16, 2015
Outline Resource-Renewable Boiling Water Reactor (RBWR) Current Neutron Diffusion Methods for RBWR Analysis Advanced Methods: Quasi-diffusion Methods Numerical Results Summary and Conclusions 2
LWR-RBWR Comparison 3 4 1 2 1: BWR Fuel Assembly 2: ABWR Core 3: RBWR Core Layout 4: RBWR Assembly 3
Resource-Renewable Boiling Water Reactor (RBWR) The RBWR is a reactor design originally proposed by Hitachi which is capable of achieving a conversion ratio of 1.0 Design features include: Short, parfait style core Tight pitch fuel lattice Smaller coolant mass flow-rate Large exit void fraction Less negative core void reactivity coefficient Y-shaped control blades 4
normalized neutron spectra per unit lethargy RBWR Characteristics Hard neutron spectrum compared to typical Light Water Reactors (LWRs) Average core void fraction of 53% compared to 36% for the ABWR 0.3 0.25 0.2 Thermal Reactor RBWR ARR PWR Fast Reactor Double peaked axial power distribution provides large axial heterogeneity compared to radial The use of 2-D cross sections has difficulty capturing these axial heterogeneities 0.15 0.1 0.05 0 10-2 10 0 10 2 10 4 10 6 Neutron Energy (ev) 5
Coupled Code System for RBWR Simulation Lattice Code: Serpent GENPMAXS Neutron Flux Solver: PARCS T/H: PATHS Cross Section Library (PMAX) Equilibrium Search 6
Monte Carlo XSEC Homogenization The traditional homogenization performed for the RBWR involves modeling a 3-D assembly and generating cross sections for various axial levels For the RBWR, a 12-group energy structure was used to collapse the cross sections Energy groups and regions with low neutron populations require additional neutron histories (slows down the simulation) The thermal energy groups have a larger uncertainty compared to the fast energy groups Group Number Upper Energy (ev) 1 1.0000E+7 2 3.6788E+6 3 2.2313E+6 4 1.3534E+6 5 4.9787E+5 6 1.8316E+5 7 4.0868E+4 8 5.5308E+3 9 1.3007E+2 10 3.9279E+0 11 1.4450E+0 12 6.2500E-1 Minimum 8.2500E-5 One of the main quantities of interest is the diffusion coefficient D which has no tabulated continuous-energy data It is determined using the Monte Carlo estimate for the transport cross section This value is used for all directions Σ tr,g = K k=1 φ k,g Σ t,k,g Σ s1,k,g K k=1 φ k,g D g = 1 3Σ tr,g 7
Axial Discontinuity Factors (ZDFs) In addition to the group constants provided from the 3-D Serpent calculation, axial discontinuity factors (ZDFs) are determined for the interfaces between axial segments Het Hom For axially heterogeneous cores, the material interface can lead to steep flux gradients which diffusion theory has difficulty capturing In this situation an axial discontinuity factor (ZDF) is desired based on a boundary problem and not because of homogenization (radial discontinuity factors) ZDFs are based on the same definition as traditional discontinuity factors, where the homogeneous flux is made discontinuous to conserve continuity of current Φ + i f + i = Φ i+1 f i+1 This application of ZDFs was discussed at previous Serpent meetings f i + = Φ i + Φ i +, f i+1 = Φ i+1 Φ i+1 8
PARCS/PATHS Core Simulator All of this group constant information is used within the nodal code PARCS for assembly and full core problems For hexagonal lattices, PARCS solves a coupled 2-D radial diffusion equation and 1-D axial diffusion equation Due to the axial heterogeneity of the RBWR, diffusion has difficulty solving this axial diffusion problem even with the use of ZDFs This is because during the generation of ZDFs, the values can become negative or very large which can cause instabilities within the simulation for large core systems with feedback or burnup 9
Full Core Results with Bounded ZDFs Instead, a bounding approximation was applied to the ZDFs that improved stability but introduced error into the simulation This allowed us to produce full-core equilibrium results using PARCS/PATHS Axial Power distribution Radial Power distribution Though this method provided a stable solution, error was introduced due to the ZDF bounding Investigated methods that reduce or eliminate this source of error
Higher-Order Axial Solutions To improve the accuracy of the RBWR simulation, we wanted to improve upon the use of diffusion for the axial solver Looked at implementing spherical harmonics equations (P1, P2, P3) as well as Quasi-diffusion The focus of the work presented here will be based on the Quasi-diffusion method that was used All of the results are based on 1-D solutions for the axial direction of an RBWR-type assembly 11
Quasi-diffusion Equations The Quasi-diffusion equation is based on the use of Eddington factors defined as: d Ω Ω 4π u Ω v ψ E uv = d Ω ψ These expressions are based on the diffusion equation without estimating the angular flux as a linear function of angle 4π The Eddington factors are calculated from the Monte Carlo simulation by determining the angular weighted fluxes φ 2,uv = d 4π Ω Ω u Ω v ψ φ 2,uv = φ 2,xx φ 2,xy φ 2,xz φ 2,yx φ 2,yy φ 2,yz E uv = φ 2,zx φ 2,zy φ 2,zz If the Eddington factor is used as an approximation within the 1-D transport equation, this produces the 1-D Quasi-diffusion equation: d d dx 1 Σ tr (r, E) dx E j r, E φ 0 (r, E) + Σ t (r, E)φ 0 (r, E) = de Σ s r, E E φ 0 r, E 0 φ 2,xx φ 0 φ 2,yx φ 0 φ 2,zx φ 0 φ 2,xy φ 0 φ 2,yy φ 0 φ 2,zy φ 0 + λχ(r, E) deνσ f r, E φ 0 (r, E ) 0 φ 2,xz φ 0 φ 2,yz φ 0 φ 2,zz φ 0 12
Quasi-diffusion Equations (Cont.) In 1-D there are two equivalent ways to solve the Quasi-diffusion equation The first method involves multiplying and dividing by the Eddington factor and solving for the product E j x, E φ 0 (x, E) (Only valid in 1-D) d dx 1 Σ tr x, E d dx E j x, E φ 0 x, E + Σ t x, E E j x, E E j x, E φ 0 x, E = de Σ s x, E E x, E 0 E j E j x, E φ 0 x, E + λχ x, E de νσ f x, E 0 E j x, E E j x, E φ 0 (x, E ) The second method involves using the Eddington factor as a discontinuity factor d dx D(x, E) d dx 3E j x, E φ 0 x, E + Σ t x, E φ 0 x, E = de Σ s x, E E φ 0 x, E + λχ x, E deνσ f x, E φ 0 (x, E ) 0 For this study, the first method is used 0 13
Eddington Factors from Serpent As shown in the previous slides, Eddington factors were calculated from Serpent by adding additional angular weighted tallies when scoring the flux This produces a 3x3 Matrix for each group constant universe and energy group Some useful properties of Eddington factors: For typical LWRs with non-axially varying fuel, the diagonal elements of this 3x3 Matrix are close to 1/3 Off-diagonal elements are close to 0 The values for Eddington factors can only vary between 0 and 1 If the diagonal elements of this Matrix are 1/3 and the off-diagonal elements are 0, then the Quasi-diffusion equation reduces to the Diffusion equation First investigated a pin cell problem for an RBWR-type geometry to evaluate the magnitude of the Eddington factors 14
RBWR Pin Cell: Fuel vs. Non-Fuel Eddington Factors UB LF Coolant IB UF LB Wanted to develop a deeper understanding of the physics happening within the fuel and non-fuel regions Developed a pin-cell based on the RBWR assembly Tallied the particles traveling in each separate region
Fuel Eddington Factor Non-Fuel Eddington Factor Fast Group: Fuel vs. Non-Fuel Eddington Factors 0.5 0.45 Fuel Group 1 - Fast Most Group zz xx yy 0.5 0.45 Non-Fuel Group 1 - Fast Most Group zz xx yy 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0 50 100 150 Axial Height (cm) 0.25 0 50 100 150 Axial Height (cm) There is little difference between the fuel and non-fuel Eddington factors for the fast group Looked next at the thermal most group
Fuel Eddington Factor Non-Fuel Eddington Factor Thermal Group: Fuel vs. Non-Fuel Eddington Factors 0.5 0.45 Fuel Group 12 - Thermal Most Group zz xx yy 0.5 0.45 Non-Fuel Group 12 - Thermal Most Group zz xx yy 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0 50 100 150 Axial Height (cm) 0.25 0 50 100 150 Axial Height (cm) There is a notable difference between the fuel and non-fuel Eddington factors for the thermal group in the fissile regions Next slide compares just the axial (zz) Eddington Factors
Eddington Factor (Ezz) Eddington Factor (Ezz) Axial Eddington Factors (zz): Fuel vs. Non-Fuel Eddington Factors 0.5 Group 1 - Axial (zz) Fuel vs. Non-Fuel Eddington Factors Fuel Non-Fuel 0.5 Group 12 - Axial (zz) Fuel vs. Non-Fuel Eddington Factors Fuel Non-Fuel 0.45 0.45 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0 50 100 150 Axial Height (cm) 0.25 0 50 100 150 Axial Height (cm)
Pin Cell Discussion From the pin cell analysis, it is clear that the Eddington factors can deviate significantly from 1/3 The goal is that these values will act as a correction to improve the axial solution and reduce the need for ZDFs Created a full 3-D assembly in Serpent and generated cross sections and group constants Calculated assembly results using P1, P2, P3 and Quasidiffusion to compare the accuracy 19
Assembly Solution Created an RBWR problem with no axial reflectors and reflective boundary conditions on all sides Serpent 3-D cross sections and values 34 axial regions and 12 energy groups Improvement noticed in eigenvalue from P1 -> P3 and Quasi-diffusion Compared the group 1 and 12 flux distributions Also compared assembly Eddington factors for E xx, E yy and E zz Solver Eigenvalue Difference from Serpent (pcm) Serpent 1.04192 - P1 1.03429-763 P2 1.04477 285 P3 1.04209 17 Quasi-diffusion 1.04238 46 20
Assembly Flux Comparison 21
Eddington Factors from Serpent Group 1 (fast) and Group 12 (thermal) are shown for illustration If E zz =1/3, neutrons travel isotropically If E zz >1/3, neutrons favor traveling axially If E zz <1/3, neutrons favor traveling radially Mean Free Path: Fast: ~5-6cm Thermal: ~1-2cm
Solution with Discontinuity Factors Each of these methods can also be used with ZDFs to reproduce the reference solution Solver Reference (Serpent) Use of Discontinuity Eigenvalue Difference from Factors Serpent (pcm) - 1.04192 - P1 No DFs 1.03429-763 P1 With DFs 1.04192 0 P2 No DFs 1.04477 285 P2 With DFs 1.04192 0 P3 No DFs 1.04209 17 P3 With DFs 1.04192 0 QD No DFs 1.04238 46 QD With DFs 1.04192 0 Further details can be found in my thesis 23
Assembly Analysis Discussion Quasi-diffusion improved the axial solution for an RBWRtype assembly compared to typical diffusion The Quasi-diffusion method provided similar results for the eigenvalue and flux as the P3 approximation ZDFs can be used for higher-order methods to reproduce the exact transport solution from Serpent Further improvements can be made by refining the axial meshing within Serpent but increases the computational burden to achieve statistical accuracy within each GCU 24
Summary and Conclusions For axially heterogeneous cores such as the RBWR, diffusion theory is unable to capture the axial streaming effect. Higher-order transport corrections are required such as P3 or Quasi-diffusion Implemented tallies within the Serpent Monte Carlo code to calculate Eddington factors for the Quasi-diffusion method The generation of Eddington factors is computationally inexpensive using Monte Carlo methods Currently working on implementing the Quasi-diffusion method into the nodal code PARCS 25
Questions?