Time-dependent Neutron Transport Research Articles

A re-activation model for 169Yb intensity modulated brachytherapy sources accounting for spatiotemporal isotopic composition.

Intensity modulated brachytherapy based on partially shielded intracavitary and interstitial applicators is possible with a cost-effective 169Yb production method. 169Yb is a traditionally expensive isotope suitable for this purpose, with an average γ-ray energy of 93keV. Re-activating a single 169Yb source multiple times in a nuclear reactor between clinical uses was shown to theoretically reduce cost by approximately 75% relative to conventional single-activation sources. With re-activation, substantial spatiotemporal variation in isotopic source composition is expected between activations via 168Yb burnup and 169Yb decay, resulting in time dependent neutron transmission, precursor usage, and reactor time needed per re-activation. To introduce a generalized model of radioactive source production that accounts for spatiotemporal variation in isotopic source composition to improve the efficiency estimate of the 169Yb production process, with and without re-activation. A time-dependent thermal neutron transport, isotope transmutation, and decay model was developed. Thermal neutron flux within partitioned sub-volumes of a cylindrical active source was calculated by raytracing through the spatiotemporal dependent isotopic composition throughout the source, accounting for thermal neutron attenuation along each ray. The model was benchmarked, generalized, and applied to a variety of active source dimensions with radii ranging from 0.4 to 1.0mm, lengths from 2.5 to 10.5mm, and volumes from 0.31 to 7.85 mm3, at thermal neutron fluxes from 1×1014to1×1015 n cm-2 s-1. The 168Yb-Yb2O3 density was 8.5g cm-3 with 82% 168Yb-enrichment. As an example, a reference re-activatable 169Yb active source (RRS) constructed of 82%-enriched 168Yb-Yb2O3 precursor was modeled, with 0.6mm diameter, 10.5mm length, 3 mm3 volume, 8.5g cm-3 density, and a thermal neutron activation flux of 4 × 1014 neutronscm-2s-1. The average clinical 169Yb activity for a 0.99 versus 0.31 mm3 source dropped from 20.1 to 7.5Ci for a 4×1014ncm-2s-1 activation flux and from 20.9 to 8.7Ci for a 1×1015ncm-2s-1 activation flux. For thermal neutron fluxes ≥2×1014ncm-2s-1, total precursor and reactor time per clinic-year were maximized at a source volume of 0.99 mm3 and reached a near minimum at 3mm3. When the spatiotemporal isotopic composition effect was accounted for, average thermal neutron transmission increased over RRS lifetime from 23.6% to 55.9%. A 28% reduction (42.5 days to 30.6 days) in the reactor time needed per clinic-year for the RRS is predicted relative to a model that does not account for spatiotemporal isotopic composition effects. Accounting for spatiotemporal isotopic composition effects within the RRS results in a 28% reduction in the reactor time per clinic-year relative to the case in which such changes are not accounted for. Smaller volume sources had a disadvantage in that average clinical 169Yb activity decreased substantially below 20Ci for source volumes under 1mm3. Increasing source volume above 3 mm3 adds little value in precursor and reactor time savings and has a geometric disadvantage.

Time-dependent neutron transport method based on three-dimensional method of characteristics

The 3D Method of Characteristics (MOC) has high numerical accuracy, geometrical flexibility, making it suitable for the simulation of advanced reactors with complex geometry, while the development of transient solvers based on it is still relatively limited. In this work, transient method was developed based on 3D MOC solvers of OpenMOC and predictor–corrector quasi-static method and the C5G7-TD benchmark problems were used to verify the implementation. In 2D cases and 3D moderator density change cases, the results of OpenMOC were in good agreement with the results of other transient codes. In 3D control rods movement cases, there are oscillation errors between the existing results of different transient codes due to the control rod cusping effect. This work is based on the refined axial modeling of 3D MOC, which avoids the cusping effect and provides new results of C5G7-TD benchmark.

Evaluation of time dependent neutron transport algorithms using 2D modular ray tracing MOC

Three algorithms for solving the multigroup time dependent neutron transport in two dimensions utilizing the method of characteristics (MOC) are developed in this study. These algorithms are distinguished base on the average angular flux calculation along the characteristic line (track) and the applied approximation to the angular flux time derivative. Since the calculation of the scalar flux in the characteristics method requires the average angular flux along the track, two approaches for implementation of track averaged angular flux are presented. In the first approach, the effect of the angular flux in the previous time step as an extra term added to the source is considered. Second, the average angular flux is applied similar to the common method used in the steady state form. Moreover, two states for the angular flux time derivative in implementing the transient neutron transport solution are considered. First, the angular dependency of the angular flux in time derivative is preserved. Second, isotropic scalar flux approximation is applied to the time derivative.Considering the vital role of ray tracing and improving the non-physical disturbances effect caused by ray tracing methods, the latest modular ray tracing (MRT) method is utilized for the transient calculation. For verification, the well-known TWIGL benchmark is modeled and the results are compared with MPACT and DeCART codes. Additionally, the heterogeneous benchmark problems MOXK_2G and MOXK_7G are simulated and the results are compared with DeCART and VARIANT_K codes.

High-Fidelity Transient Analysis with STREAM Based on the Predictor-Corrector Quasi-Static Method

The predictor-corrector quasi-static method (PCQM) is used to solve the transient problem in the STREAM code, a steady-state and transient reactor analysis code with the method of characteristics. In PCQM, the angular neutron flux undergoes a factorized split to form the product of shape and amplitude functions. The time-dependent neutron transport equation is solved to obtain the shape function whereas the amplitude function is obtained by resolving the exact point kinetics equations (EPKEs). A two-level coarse mesh finite difference technique is implemented to reduce the transient running time of the transport solution. Moreover, high-order polynomial interpolation is applied to the kinetics parameters utilized in EPKEs to reduce the error when the reactivity insertion is nonlinear. Several numerical benchmarks are solved to justify the application of the procedure, proving that the method maintains solution accuracy.

A Novel Hybrid Deterministic and Monte Carlo Neutron Transport Formulation and Algorithm (tRAPID) for Accurate and Fast 3-D Reactor Kinetics

The available three-dimensional (3-D), time-dependent neutron transport algorithms and codes (deterministic or Monte Carlo) are very computationally intensive and are impractical for the simulation of real-world reactors. Henceforth, commonly approximate forms of the transport equation (e.g., diffusion or SPn) are used with expected loss of accuracy. We have developed a hybrid deterministic and Monte Carlo algorithm that not only preserve a Monte Carlo–level accuracy but can achieve a solution in seconds or minutes. This algorithm has been incorporated into the RAPID code system and tested for a number of benchmark problems. This novel time-dependent algorithm, referred to as tRAPID, utilizes a transient fission matrix methodology and allows for fast and accurate simulation of 3-D time-dependent neutron transport problems. The tRAPID algorithm is used to calculate neutron kinetics parameters (such as l eff and Rossi- α ) and 3-D time-dependent prompt and delayed fission source distributions for two reference models: the Flattop-Pu critical assembly and the Jožef Stefan Institute TRIGA Mark-II benchmark core. Results are compared to experiments reported in the International Criticality Safety Benchmark Evaluation Project Handbook as well as to a reference Serpent Monte Carlo calculation. The tRAPID results are in excellent agreement with both the experimental data and Serpent predictions, while requiring minimal computing resources.

An overview of power reactor kinetics and control in load-following operation modes

Previous work done on reactor kinetics and control in load-following operation modes available in open literature is reviewed. The analysis is focused on, however not limited to pressurized water reactors. Different approximations of the time-dependent neutron transport problem as well as different control algorithms are described in detail and compared. Due to lack of published information the majority of the comparisons was done on qualitative level. In order to facilitate future testing and intercomparisons of models and algorithms, two so-called reference scenarios with time-dependent power demand are defined: a scenario to test the limitations of the load-following capabilities of the nuclear facilities and a second, quasi-realistic scenario.

Analysis of Population Control Techniques for Time-Dependent and Eigenvalue Monte Carlo Neutron Transport Calculations

An extensive study of population control techniques (PCTs) for time-dependent and eigenvalue Monte Carlo (MC) neutron transport calculations is presented. We define PCT as a technique that takes a censused population and returns a controlled, unbiased population. A new perspective based on an abstraction of particle census and population control is explored, paving the way to improved understanding and application of the concepts. Five distinct PCTs identified from the literature are reviewed: simple sampling, splitting-roulette (SR), combing (CO), modified combing, and duplicate-discard (DD). A theoretical analysis of how much uncertainty is introduced to a population by each PCT is presented. Parallel algorithms for the PCTs, applicable for both time-dependent and eigenvalue MC simulations, are proposed. The relative performance of the PCTs based on run time and tally mean error or standard deviation is assessed by solving time-dependent and eigenvalue test problems. It is found that SR and CO are equally the most performing techniques, closely followed by DD.

VITAS: A multi-purpose simulation code for the solution of neutron transport problems based on variational nodal methods

Existing neutronic whole-core simulation codes are problem-oriented. With the advancement of computer technology and the development of complicated micro-nuclear reactors, there is a pressing need to create solution methods with broad applicability and the ability to accommodate more complex, high-fidelity, unstructured geometries. This necessity motivates the development of a neutron transport code that can adapt to different types of nuclear reactors. This work describes the code design and theoretical model of VITAS, a multi-purpose simulation code for handling steady-state and time-dependent neutron transport problems. This code incorporates algorithms and models inside a unified variational nodal framework based on the second-order neutron transport equation. The most recent version of VITAS employs the isotropic scattering approximation and permits arbitrary spatial and angular refinement. VITAS is compatible with Cartesian geometry, hexagonal geometry, triangular geometry, and Cartesian geometry with finite element subdivisions in the radial direction (Cartesian-FE). The three-dimensional modeling can be performed by axially extruding the two-dimensional mesh. Users can choose the mesh type and expansion order on-demand for optimal computational efficiency.In this paper, VITAS is validated using benchmark problems, i.e., the TAKEDA3 benchmark and the transient LMW benchmark with Cartesian homogenized geometry, the C5G7 and C5G7-TD exercises with Cartesian-FE geometry, the TAKEDA4 benchmark and the Buckner and Stewart benchmark with hexagonal-z homogenized geometry, and the Dodds benchmark with cylindrical-z geometry described by triangular-z unstructured meshes. The space-angle truncation is extensively examined to demonstrate the asymptotic convergence behavior with p- and h-refinement. The results indicate that VITAS has the potential to become a versatile toolset for solving a variety of nuclear reactor problems.

Application of stiffness confinement method within variational nodal method for solving time-dependent neutron transport equation

The stiffness confinement method (SCM) is implemented in the variational nodal method (VNM) framework to solve time-dependent multi-group neutron transport equations. With frequency-transform, the time-dependent neutron equation is decoupled from the precursor equation and transformed into a dynamic eigenvalue problem which can be solved with the in-house VNM code VITAS. An improved time-source iteration method within the VNM-SCM framework is proposed to accelerate constructing response matrices and reduces memory usage. Based on the VNM-SCM scheme, the kinetics module of VITAS is developed for the Cartesian and hexagonal-z geometry and verified by several benchmark problems.Numerical results demonstrate that VNM-SCM can perform accurate and efficient transient calculations with large time-step sizes. For the two-dimensional TWIGL benchmark, the VNM-SCM produces a maximum core power error of -0.57% with the time-step size of 20 ms. For the three-dimensional LMW benchmark, the VNM-SCM results in a maximum core power error of 0.95% with the time-step size of 2 s. The VNM-SCM also yields an accurate neutron flux prediction for the hexagonal-z geometry problem. In addition, the effect of the angular expansion is investigated systematically. Examinations with the three-dimensional LMW benchmark display an error reduction in the core power error from 0.131% to 0.011% from P1 to P3 interfacial angular expansion, with a substantial computing time increase of 6.14 hours. The study indicates that the high-order angular expansion significantly increases the computing time but contributes little to the accuracy of the core power. Besides, it was found that the initial steady-state calculation dominates the shape error of the flux in the transient calculation.

Pin-by-Pin Coupled Transient Monte Carlo Analysis Using the iMC Code

In this article, we present a coupled multi-physics Monte Carlo reactor transient analysis framework implemented in the KAIST Monte Carlo iMC code. In the multi-physics framework, the time-dependent neutron transport calculation and the transient heat transfer analysis are done based on the predictor–corrector quasi-static Monte Carlo method and the three-dimensional finite element method, respectively. Using this high-fidelity analysis framework, we demonstrated the negative temperature feedback effect in two pressurized water reactor (PWR) transient scenarios. First, a 3-D burnable absorber-loaded fuel assembly was considered with all reflective boundary conditions. In this simple problem, a positive reactivity-induced transient was analyzed to characterize the reactor responses in view of the pin-wise power and temperature distribution. Second, the iMC multi-physics analysis is applied to a control rod withdrawal transient in the TMI-1 mini core problem, and detailed time-dependent results were provided and compared with the Serpent/SUBCHANFLOW analysis. In both cases, independent MC runs were performed to quantify the uncertainty of the multi-physics MC transient analysis.

A Stable Method for Solving the Simplified Time-Dependent Neutron Transport Equation with Background Motion

The neutron transport equation is usually solved against a stationary background medium. When the background material is moving, the transport equation will need to be modified. Solving the transport equation with moving material is quite complicated, especially for the curved coordinate system because of the double angular redistributions. In this study, the discretization method of the simplified transport equation considering the moving-material effect is implemented in three-dimensional cylindrical geometry. Directly solving this modified transport equation with the standard solution technique is problematic since the advection term introduced by moving material may render the transport solver numerically unstable. The speed ratio λ is defined for stability analysis. A forced-stable method is proposed in this study to achieve good numerical stability for any material speeds and time-step sizes. The accuracy of this new method is verified using manufactured solutions. Steady numerical results demonstrate that the effects introduced by background motion cannot be neglected as the material speed starts to approach one-tenth of the neutron speed. Moreover, transient analysis indicates that the moving background has a considerable impact on the criticality of a system.

ANALYSIS OF TIME-EIGENVALUE AND EIGENFUNCTIONS IN THE CROCUS BENCHMARK

Time-dependent neutron transport in non-critical state can be expressed by the natural mode equation. In order to estimate the dominant eigenvalue and eigenfunction of the natural mode, CEA had extended the α-k method and developed the generalized iterated fission probability method (G-IFP) in the TRIPOLI-4® code. CRIEPI has chosen to compute those quantities by a time-dependent neutron transport calculation, and has thus developed a time-dependent neutron transport technique based on k-power iteration (TDPI) in MCNP-5. In this work, we compare the two approaches by computing the dominant eigenvalue and the direct and adjoint eigenfunctions for the CROCUS benchmark. The model has previously been qualified for keffs and kinetic parameters by TRIPOLI-4 and MCNP-5. The eigenvalues of the natural mode equations by α-k and TDPI are in good agreement with each other, and closely follow those predicted by the inhour equation. Neutron spectra and spatial distributions (flux and fission neutron emission) obtained by the two methods are also in good agreement. Similar results are also obtained for the adjoint fundamental eigenfunctions. These findings substantiate the coherence of both calculation strategies for natural mode.

Adjoint flux calculation of natural mode equation by time dependent neutron transport

ABSTRACT The physical implication of the eigenfunction of the adjoint of the natural mode equation is studied. The eigenfunction at a phase space position is theoretically derived to be proportional to the amplitude of the neutron flux sufficiently long time after a source is placed at the position. Meanwhile, the time change rate and distribution of the neutron flux originated in the source must converge to those of the fundamental mode of the natural mode equation. Based on the proportional relation, the adjoint flux is calculated for a MOX- and UO2-fueled lattice by a continuous-energy Monte Carlo time-dependent neutron transport calculation. The calculated distributions of the adjoint flux agree well with those of the iterated fission probability both in the prompt and delayed critical states.

A new Monte Carlo approach for solution of the time dependent neutron transport equation based on nodal discretization to simulate the xenon oscillation with feedback

In this paper a probabilistic methodology based on core nodalization is proposed to estimate the core power in the presence of xenon oscillation. A time-dependent Monte Carlo neutron transport code named MCSP-NOD is developed for dynamic analysis in arbitrary 3D geometries to simulate xenon oscillations as well as sub-critical condition with feedbacks. The new code is based on the approach adopted in MCNP-NOD which was previously introduced as a tool for core transient analysis using the MCNPX platform. As before, the core is divided into nodes of arbitrary dimensions, and all terms of the transport equation e.g. interaction rates, leakage ratio are estimated using the MC techniques. However, as a new option the concentration of iodine and xenon are also estimated which enables us to predict the oscillatory behavior of reactor power following poison oscillation. These quantities are then employed within the time-dependent neutron transport equation for each node independently to compute the neutron population. Simulations prove the robustness of the method.

Time-dependent integral transport in one-dimensional infinite media using dimensionless variables and the reduced collision formulation

A new method for calculating time-dependent integral neutron transport single collision fluxes has been developed for one-dimensional infinite media problems using dimensionless variables and the reduced collision formulation. By transforming the integral scalar flux equation into dimensionless variables, one may immediately integrate over the time domain and reduce the problem complexity to integrations strictly over the spatial domain, significantly reducing the computational work required to obtain a solution. The results have been compared to published benchmark solutions for validation. Further, reformulating the scalar flux as a Green’s function has allowed for the determination of an analytical expression for the relative error of any collision level scalar flux to the summed scalar flux. This provides the absolute number of collisions required for convergence at any given mean free time, using previous work in one-dimensional Cartesian transport and derived work herein for the one-dimensional spherical transport.

Estimation of time-dependent neutron transport from point source based on Monte Carlo power iteration

ABSTRACT This study proposes a method for calculating time-dependent neutron transport from a point source with a continuous-energy Monte Carlo code. To deal with neutron multiplication and attenuation in orders of magnitude, the power iteration method conventionally used to estimate the effective multiplication factor keff was utilized. The time of a neutron flying in a cycle from emission of its ancestor at the point source was estimated. In the estimation, the decay time of the delayed neutron precursor was considered. The neutron flux was tallied in time bins in each cycle. The source strength in the cycle was considered as the product of keff estimators from the first to the previous cycle. By summing up the tallied flux multiplied by the strength, the neutron flux variation with time was obtained. This method was verified against a UO2 fuel lattice moderated and reflected by light water.

A new approach for solution of time dependent neutron transport equation based on nodal discretization using MCNPX code with feedback

This paper proposes a new method for solving the time-dependent neutron transport equation based on nodal discretization using the MCNPX code. Most valid nodal codes are based on the diffusion theory with differences in approximating the leakage term until now. However, the Monte Carlo (MC) method is able to estimate transport parameters without approximations usual in diffusion method. Therefore, improving the nodal approach via the MC techniques can substantially reduce the errors caused by diffusion approximations. In the proposed method, the reactor core is divided into nodes of arbitrary dimensions, and all terms of the transport equation e.g. interaction rates and leakage ratio are estimated using MCNPX. They are then employed within the time-dependent neutron transport equation for each node independently to compute the neutron population. Based on this approach, a time-dependent code namely MCNP-NOD (MCNPX code based on a NODal discretization) was developed for solving time-dependent transport equation in an arbitrary geometry considering feed backs. The MCNP-NOD is able to simulate multi-group processes using appropriate libraries. Several test problems are examined to evaluate the method.

Cross-verification of SUHAM-TD and nTracer reactivity insertion transient solutions without materials homogenization approximation using OECD/NEA C5G7-TD benchmark

Recent advances in the development of methods and tools for enhanced accuracy reactor simulations have allowed moving towards time-dependent neutron transport solution without spatial cross-sections homogenization approximation. Although the full core reactor dynamic problems are still too much challenging for the modern codes, reduced order problems can be solved already today. Examples of the codes allowing such advanced transient calculations are, among several others, the SUHAM-TD and nTracer codes, which are discussed in this work. In order to perform an intermediate verification of the SUHAM-TD code, which is under development at the Kurchatov Institute, and to perform an independent verification of the nTracer code of the Seoul National University by its user team at the Paul Scherrer Institute, performances of these two codes have been assessed using the ongoing OECD/NEA C5G7-TD Phase-I benchmark exercises. In particular, two cases of positive reactivity insertion were selected for detailed comparison, being the most relevant from the point of view of future practical applications.

Singular Eigenfunction Expansion Solution for Time-Dependent Neutron Transport with Delayed Neutrons

We develop an analytic solution for time-dependent neutron transport with delayed neutrons using the singular eigenfunction expansion method. Our approach is based on a technique for solving time-dependent neutron-transport problems without delayed neutrons (Case and Zweifel, Linear Transport Theory, Addison-Wesley, 1967), which we effectively generalize to include the presence of delayed-neutron precursors. In particular, we obtain eigenfunctions composed of two parts: one corresponding to the neutron angular flux and one corresponding to the delayed-neutron precursor concentration. We further demonstrate that these eigenfunctions are complete. We also provide numerical results for an example problem.

Reactivity determination using the hybrid transport point kinetics and the area method

This paper presents an application of the hybrid transport point kinetic (HTPK) technique to the reactivity determination in subcritical reactor configurations. The mathematical model of the HTPK, initially proposed by Picca et al. (2011) to simulate the time-dependent neutron transport, is here extended to incorporate delayed emissions. The classical area method (Sjӧstrand, 1956), developed to invert the point kinetic (PK) model, is then adapted to accommodate the peculiarities of the HTPK approach, to allow its analytical inversion. This novel inverse neutron kinetic methodology is tested on a three-region reactor configuration, showing the interesting performance of the approach based on the HTPK model as compared to the standard area method and highlighting its potential to overcome some of the limitations of the PK-based inversion.