Most of the methods required for multiphysics Monte Carlo simulation have been individually developed and well studied. A new multiphysics Monte Carlo code named MCS has been developed for the analysis of large reactors.
Introduction
Background
At the same time, there have been attempts to apply the Monte Carlo method to multiphysics simulation. Thus, we have decided to develop a new Monte Carlo code called MCS aimed at large-scale power reactor simulation since 2013 [14].
Objective of Thesis
The MCS Monte Carlo Code
Transport Equation
Program Flow
The distance to the next collision is sampled as shown in Figure II-2 by the macro total cross section Σt with the following equation:. where ξ is a pseudorandom number sampled from a uniform distribution on [0,1]. If dcol < dsur, as shown in Figure II-4, move the particle position to the collision site and continue the collision simulation.
Pseudo-Random Number Generation
Geometry
Nuclear Physics
Tallies
Convergence Check
Statistics
Numerical Tests
- ICSBEP
- INDC Benchmark
- The Doppler-Defect Benchmark
- BEAVRS Cycle 1 HZP
The standard problem is the artificial pin problem consisting of fuel and water regions as shown in Figure II-10. The RMS error of the counted result and the measured data is 5% and the distribution is shown in Figure II-18.
Parallel Algorithm
Hash-Indexing
The scoring performances of MCS are tested against the Monte Carlo benchmark [36], a core containing 63,624 fuel cells. The effective neutron multiplication factor and number of particle histories treated per second are shown in Table II-8.
Inter-Cycle Correlation
Introduction
Inter-Cycle Correlation of Tally
- Underestimation of Variance in Tally
- Numerical Test
The cumulative quantity in cycle i is the average of M histories, taking into account that there is no correlation between particles in the same cycle. The numerical test is performed using a one-dimensional homogeneous plate geometry using one group of sections as shown in Table III-1. The thickness of the panel is 10 cm, and reflective borders are placed on both sides.
The nu-cleavage reaction at the left half region was counted and studied in this section. The intercycle covariance was estimated using 100 inactive cycles, 100,000 active cycles, and 100,000 histories per cycle. Because of this poor convergence of covariance, it is almost impossible to count in practical simulation.
As shown in the figure, the true standard deviation is estimated using Eq. III.7) agrees very well for all cycles, and the true standard deviation is estimated using Eq.
Subcycle Method
- Underestimation of Variance with Subcycle
- Numerical Tests
It should be noted that the real variance and the sum of Ck, A, are independent of the number of subcycles and the history per subcycle. As discussed, the real-to-apparent ratio depends on the number of subcycles L and can be estimated by Eqs. The solid line represents the measured ratio and the dashed line the estimated value by Eq.
As shown in the figure, the estimated ratio agrees well with the measured value after about 100 cycles.
Study on BEAVRS
The real to apparent standard deviation increases as the mesh size increases (mesh → pin → assembly). The real-to-apparent ratio depending on the number of subcycles can be estimated using measured variance and covariance with a subcycle as Figure III-7. The standard deviation of mesh power in Figure III-18 and Figure III-19 shows a similar trend as with one subcycle, but the average ratio is moved to 1.0 representing the smaller inter-cycle correlation.
The ratio between real and apparent ratios also shows a smoother distribution, as expected, as shown in Figure III-21. This may explain the real and apparent assembly power ratio shown in Figure III.23. The root mean (RMS) of the assembly power standard deviation is compared to the ideal convergence slope in Figure III-24.
To see the convergence slope of each assembly, the slopes are shown in Figure III-25 for 6 selected assemblies.
Feedbacks for Power Reactor Simulation
Introduction
Depletion
- Numerical Tests
- Equilibrium Xenon Feedback
- Memory Requirement
The VERA-1C problem consists of a single Westinghouse type 17×17 fuel cell in a BOC as shown in Figure IV-2. The VERA-2C is an assembly problem consisting of fuel pins 1C and a guide tube as shown in Figure IV-3. The equilibrium xenon feedback function is implemented in the MCS, as the target purpose of this function is to prevent steady-state xenon oscillation.
When using equilibrium xenon feedback, MCS first prepares the cumulative fission yield of 135I and. The density calculated by the equilibrium xenon function and the one from burnout solver must be the same. This proves that the MCS equilibrium xenon function updates the xenon number density, the same as the depletion solution implemented in MCS.
The dependence of Mburnup in Eq. IV.4) on the number of burn-up cells and the number of processors is illustrated in Figure IV-8.
On-The-Fly Doppler Broadening
- Multipole Representation
- Interpolation
In the resolved energy range, MCS uses a sqrt-linear scheme as follows: where σ is the intercept. Usually there are 20 cross sections on one energy point and the cross section will be chosen by a random number. The cross-section scale in ptable can vary with temperature, and makxsf chooses the scale at.
Since MCS takes the cross section from both scales and interpolates it as shown in Figure IV-12. It should be noted that the same random number should be used for the cross-section sample. The accuracy of in-flight interpolation for thermal distribution data is tested with the VERA 1C benchmark problem.
However, on-the-fly interpolation must agree with maxxsf since they use the same interpolation scheme.
Thermal hydraulics Coupling
- Mapping
- Feedback Strategy
- VERA HFP Assembly
If we look at global index as shown in Figure IV-18, the index from 1 to 6 is for first pin, and next 6 indexes are for second pin. Much research has been done on the coupling of neutronics code and thermal hydraulic code system. In the framework of Picard iteration, neutronics code and thermal hydraulics solver are independent, but they exchange the variables that are input parameter for each code.
If the temperature does not converge, the thermal hydraulic state including fuel temperature, coolant temperature and coolant density is updated. When coupling Monte Carlo code and thermal hydraulic code, the statistical uncertainty must be taken into account when deciding the convergence criteria. The Problem #6 assembly is composed of 264 fuel pins of 3.1% enriched uranium fuel and 25 guide tubes as shown in Figure IV-20.
The axial mean radial distribution of fission reaction rate and fuel temperature are compared in Figure IV-23 and Figure IV-24.
Critical Boron Concentration
Power Reactor Simulation
Introduction
Sensitivity Study on BEAVRS Cycle 1
- Burnup Step Sensitivity
- History Sensitivity
- Mesh Structure Sensitivity
- Error Propagation
Thus, it is important to estimate the effect of the uncertainty of the thermal hydraulic state on the Monte Carlo multiphysics solution. It should be mentioned that the uncertainty of the thermal hydraulics would be proportional to the uncertainty of the power, which is inversely proportional to the √N. These figures show that the uncertainty of the state of the thermal hydraulics does not have much influence on the neutronic solution.
It should be noted that the standard deviation used to plot these figures is not the estimated standard deviation of all cycles, but of one cycle. Because Case 2 uses 10 times more histories, the standard deviation of Case 2 is smaller than Case 1. The distribution of axial splitting in a core with different axial grids is compared in Figure V-9.
The important observation in this figure is that the linear power results of 8 simulations are in good agreement despite large standard deviation.
Variance Overestimation in Tally
Because of the higher correlation, the number of series needed to estimate the unbiased standard deviation increases to 60, as shown in Figure V-15.
BEAVRS Cycle 1
- Simulation Result
- Statistical Uncertainty
BEAVRS provides the measured detector signal at 24 points, highlighted in a blue dot, along with the inlet coolant temperature, rod position and power. To produce a detector signal under certain conditions, MCS simulations were performed with restart capability by utilizing the number density produced by previous simulation with 100% full rod off condition. Benchmark provides a tilt-corrected detector signal along with the original detector signal, since the original signal has a lot of tilt.
The detector signal was calculated by inserting an input instrument tube around 235 U and calculating the split response rate. The maximum, minimum and RMS difference of the detector signal for 22 burn points are compared and summarized in Table II. The detector signal agrees very well with 1.5-3% RMS error while the standard deviation is about 1-2%.
The calculation time to meet the 95/95 criteria is estimated using the true standard deviation by assuming that the standard deviation will decrease by 1/sqrt(N).
BEAVRS Cycle 2
- Refueling
- Cycle 2 Simulation
The BOC HZP is performed prior to the Cycle 2 simulation to check whether the modeling has been performed correctly. The critical boron concentration of four data agree very well within 25 ppm, except for the MCS and ST/RK at the beginning of the cycle. The critical boron concentration of MCS at BOC is therefore much higher than that of ST/RK.
The operating history of cycle 2 is much simpler than that of cycle 1, as shown in figure V-32, and there are 14 data points providing detector signal marked in blue. Unlikely cycle 1, the data points of cycle 2 are almost 100% power and all bar out of state. In general, MCS result of Cycle 2 shows good agreement and the local quantity count results seem reasonable, which proves the ability of MCS multicycle simulation.
The local quantity survey results look reasonable, proving the capability of the MCS cycle simulation.
Conclusion
Viitanen, et al., "The Serpent Monte Carlo Code: Status, Development and Applications in 2013," Annals of Nuclear Energy, vol. Zhang, et al., "Preliminary Simulation Results of the BEAVRS Three-Dimensional Cycle 1 Wholecore Depletion of the UNIST Monte Carlo Code MCS," proc. Sun, et al., "Improved Methods for Handling Massive Numbers in Reactor Monte Carlo Code RMC," Proc.
Nakagawa, “Error estimates and their biases in Monte Carlo eigenvalue calculations,” Nuclear Science and Engineering. Hyunsuk Lee, Sooyoung Choi, and Deokjung Lee*, “A hybrid Monte Carlo method/method of characteristics for efficient analysis of neutron transport,” Nucl. Jiankai Yu, Hyunsuk Lee, Hanjoo Kim, Peng Zhang, Deokjung Lee*, “Preliminary Coupling of Thermal/Hydraulic Solvers in Monte Carlo MCS Code for Practical LWR Analysis”, Ann.
Hyunsuk Lee, Chidong Kong and Deokjung Lee*, “Status of Monte Carlo Code Development at UNIST”, PHYSOR2014, Kyoto, Japan, 28 September 3 October (2014) [oral presentation].
