Time-dependent Neutron Diffusion Equation Research Articles

A new strategy for validating transient neutronic solver using the measurement data during the dynamic rod worth measurement process

A new strategy was proposed in this study to validate the transient neutronic solver. This new strategy features in employing the measurement data of the ex-core detector signals and reactivity worth from the dynamic rod worth measurement (DRWM). Meanwhile, the control-rod movement during the DRWM is simulated explicitly by the transient neutronic solver based on the time-dependent neutron diffusion equation and the response function. By comparing the simulated results with measurement data, this strategy can strongly guarantee the accuracy and reliability of the transient neutronic solver in engineering applications. Additionally, different from the limited transient benchmarks and transient facilities, this new strategy has sufficient measurement data for transient validation benefiting from the widely applied DRWM technique in PWRs across the world. The application of such a new strategy has been demonstrated based on the CNP1000 reactor operated in China. The ex-core detector signals and reactivity worth with respect to each individual control rod were studied and provided. Good agreements have been achieved between measurement and simulation. This newly proposed strategy can be a feasible and reliable approach to validate the transient neutronic solver.

A finite volume method based multi-group neutron diffusion model for space-time nuclear reactor kinetics problems in RZ geometry

The transient analysis of Fast Breeder Reactor (FBR) requires the development of high fidelity space-time nuclear reactor kinetics codes for the realistic prediction of spatial and temporal behavior of neutron flux and fission power along with the feedback effects. So, a space-time nuclear reactor kinetics model has been developed for solving the coupled time dependent multi-group neutron diffusion equations and the delayed neutron precursor equations in axi-symmetric cylindrical (r-z) geometry. The numerical model employs an orthogonal mesh finite volume method for spatial discretization and implicit method for time discretization. A computer code named MANDIR-KRZ (Multi-group Analysis of Neutron DIffusion in Reactor for Kinetics problems in RZ geometry) has been developed and employed to reconstruct the keff, transient neutron flux and power results of different benchmark problems. Mathematical formulation of the numerical model and results of the code for Dodds and FBR transient benchmark problems are discussed in this paper.

Development and benchmarking of SIMULATE5-K-VVER

SIMULATE5-K-VVER (S5K-VVER) is the newest addition to the CMS5-VVER modeling suite developed by Studsvik Scandpower, Inc. (SSP). In S5K-VVER, the coupled time-dependent multigroup neutron diffusion equation, delayed neutron precursor concentrations, and thermal hydraulic models are solved in triangular nodal geometry to model transients in Water-Water Energetic Reactors (VVERs).The neutron kinetics equations in S5K-VVER are solved in a two-step process: temporal integration and spatial integration. In S5K-VVER there are two options for temporal integration: the Frequency Transform Method (FTM) and a Backward-Euler Method (BEM).S5K-VVER is then used to solve a series of benchmark problems including the AER-DYN-001 and AER-DYN-002 numerical transient benchmarks. Finally, the Kalinin-3 benchmark is modeled which describes the shutting-off of a single Main Coolant Pump (MCP) in the Kalinin-3 VVER-1000/320 nuclear power plant. Compared to plant data, the S5K-VVER solution agrees within 1.5% of core total power during this transient.

Steady-state and transient simulations of the MTR research reactor using the high-fidelity N-TH coupling method

The tremendous advancement in computer technology makes it feasible to perform high-fidelity and multi-physics numerical simulations of reactors, allowing for a more accurate and comprehensive description of the phenomena that occur within nuclear reactors. The application of the high-fidelity coupled neutronic and thermal-hydraulic codes to execute the safety analysis and assessment of the fresh highly enriched uranium core of the IAEA-10 MW MTR research reactor is presented in this research. A three-dimensional coupled neutronic and thermal-hydraulic code is realized using the open-source C++ library OpenFOAM. The coupled code used in this study utilizes an internal coupling approach, where the identical program and grid system are utilized to solve both the neutron physical and thermal-hydraulic fields. The neutron physical field in the reactor is solved by the three-dimensional multi-group time-dependent neutron diffusion equations, and the CFD methodology is applied in solving the thermal-hydraulic governing equations. The global and local steady-state calculation results of MTR are obtained, which are in accordance with the results issued from the IAEA TECDOC and other previous relevant studies. The unprotected loss of flow accident transient condition is considered to conduct the simulation and analysis of the MTR research reactor. The evolutions of temporal and spatial distributions of important parameters of the MTR research reactor are acquired under the ULOFA transient condition, revealing the responses of the reactor core during the transient accident without SCRAM. The results obtained show that the high-fidelity N-TH coupling code can be utilized to analyze the safety of the MTR research reactor.

Strategies of Preconditioner Updates for Sequences of Linear Systems Associated with the Neutron Diffusion

The time-dependent neutron diffusion equation approximates the neutronic power evolution inside a nuclear reactor core. Applying a Galerkin finite element method for the spatial discretization of these equations leads to a stiff semi-discrete system of ordinary differential equations. For time discretization, an implicit scheme is used, which implies solving a large and sparse linear system of equations for each time step. The GMRES method is used to solve these systems because of its fast convergence when a suitable preconditioner is provided. This work explores several matrix-free strategies based on different updated preconditioners, which are constructed by low-rank updates of a given initial preconditioner. They are two tuned preconditioners based on the bad and good Broyden’s methods, initially developed for nonlinear equations and optimization problems, and spectral preconditioners. The efficiency of the resulting preconditioners under study is closely related to the selection of the subspace used to construct the update. Our numerical results show the effectiveness of these methodologies in terms of CPU time and storage for different nuclear benchmark transients, even if the initial preconditioner is not good enough.

A hybrid finite-difference/nodal method for benchmarking the HEXNEM3 coarse-mesh nodal method for time-dependent neutron diffusion applications

The solution accuracy of the recently developed modal analytical coarse-mesh finite-difference (ACMFD) formulation of the HEXNEM3 nodal flux expansion method for solving the time-dependent two-group neutron diffusion equation in hexagonal-z geometry is verified through a series of benchmark problems in both WWER-440 and WWER-1000 geometries. Reference solutions to these mathematical test problems are obtained by a hybrid finite-difference/nodal method developed specifically for the purpose of this study. The accuracy of the hybrid scheme for solving the neutron diffusion equation is also benchmarked against a set of test problems with published reference solutions. The coded implementations of the newly formulated HEXNEM3 method and the simpler but computationally expensive hybrid method demonstrated a very good mutual agreement in all test cases presented in this article.

Semi wavelet-based improved quasi static method for the analysis of PHWR transients

A new computational method, by incorporating wavelets in the improved quasi static (IQS) scheme, is developed and presented here for the analysis of CANDU PHWR and TWIGL reactor transients. According to this method the time dependent neutron diffusion equation is split into amplitude function and shape function. The amplitude function is solved using the Haar wavelet and its solution is used in the estimation of shape function. The advantage of this computational method is that the concept of micro time step and macro time step, adopted in the IQS scheme, does not arise here at all and the transient can be estimated with only one time step, i.e. single time step. In this method, the transient estimation methodology is different from the improved quasi static scheme. This computational method is applied to estimate the transients in CANDU 3D-PHWR benchmark, CANDU-2D and TWIGL reactors. In TWIGL reactor, a Doppler temperature feedback in introduced and the transient is estimated with feedback. The estimated transients are compared with the reference. From the comparison of results, it is observed that this method is capable of estimating the reactivity initiated transients in heterogeneous reactors to a good accuracy, by adopting only one time step. This computational method is first of its kind in the estimation of reactor transients. It is also shown that as the order of resolution of wavelet is increased, the error in the estimation of transient is reduced.

Bifurcation Analysis of Spatial Xenon Oscillations in Large Pressurized Heavy Water Reactors Using Multipoint Reactor Kinetics with Thermal-Hydraulic Feedback

Conventionally, the stability of xenon oscillations is estimated by solution of the time-dependent neutron diffusion equation, coupled with iodine and xenon equations, by finding out the damping ratios in each case. This is performed for different initial perturbations and core burnup conditions and is a very time-consuming and tedious process. Some earlier studies include linear stability estimation, which is valid for small perturbations, but not much work has been done in nonlinear stability analysis for spatial xenon oscillations in particular. In this paper, an approach for carrying out bifurcation analysis of xenon oscillations in large pressurized heavy water reactors (PHWRs) is demonstrated using reduced-order models. The reduced-order model for studying spatial xenon oscillations consists of multipoint kinetic equations coupled with xenon and iodine equations along with explicit fuel and coolant temperature feedback. Both subcritical Hopf bifurcation and supercritical Hopf bifurcation in different parameter planes exist, which leads to unstable limit cycles in the linearly stable region (subcritical Hopf bifurcation) and stable limit cycles in the linearly unstable region (supercritical Hopf bifurcation). The stability map provides a total picture of the stability of the out-of-phase oscillations in a PHWR. Depending on the value of the fuel temperature coefficient of reactivity and coolant temperature coefficient of reactivity, one can determine the operating power level above which the out-of-phase xenon oscillations start to grow. This model can be used to analyze nonlinear stability characteristics without spatial power control, which is helpful in identification of stable/unstable regimes in different parameter spaces and is likely to aid in reactor design.

ADAPTIVE TIME-STEP CONTROL FOR THE MODAL METHOD TO INTEGRATE THE MULTIGROUP NEUTRON DIFFUSION EQUATION

The distribution of the power inside a reactor core can be described by the time dependent multigroup neutron diffusion equation. One of the approaches to integrate this time-dependent equation is the modal method, that assumes that the solution can be described by the sum of amplitude function multiplied by shape functions of modes. These shape functions can be computed by solving a _-modes problems. The modal method has a great interest when the distribution of the power cannot be well approximated by only one shape function, mainly, when local perturbations are applied during the transient. Usually, the shape functions of the modal methods are updated for the time-dependent equations with a constant time-step size to obtain accurate results. In this work, we propose a modal methodology with an adaptive control time-step to update the eigenfunctions associated with the modes. This algorithm improves efficiency because of time is not spent solving the systems to a level of accuracy beyond relevance and reduces the step size if they detect a numerical instability. Step size controllers require an error estimation. Different error estimations are considered and analyzed in a benchmark problem with a out of phase local perturbation.

Reconstruction of neutron flux distribution by nodal synthesis method using online in-core neutron detector readings

The safety and optimal performance of nuclear reactors require online monitoring in the core. The present paper describes a method that avoids the solution of the time-dependent neutron diffusion equation, and it uses online readings of the fixed in-core neutron detectors to reconstruct the three-dimensional (3D) neutron flux distribution. The essential idea of the nodal synthesis method is the separation of time and space-dependence of the neutron flux distribution. The time-dependent section of the flux distribution is determined by in-core neutron detector readings, and the space-dependent section is obtained from pre-computed harmonics of the neutron diffusion equation. In online computation, time-dependent coefficients are only computed, and flux distribution is reconstructed by pre-computed harmonics. To calculate the diffusion equation harmonics, the nodal expansion method and the Gram-Schmidt process are used. Online reconstruction of neutron flux distribution in the Bushehr Nuclear Power Plant with VVER-1000/446 reactor type is simulated, and the results are benchmarked by valid results. The maximum relative percent error between reconstructed results and the valid results is about 5%, and the computing time is about 0.5 s.

Adaptive time-step control for modal methods to integrate the neutron diffusion equation

The solution of the time-dependent neutron diffusion equation can be approximated using quasi-static methods that factorise the neutronic flux as the product of a time dependent function times a shape function that depends both on space and time. A generalization of this technique is the updated modal method. This strategy assumes that the neutron flux can be decomposed into a sum of amplitudes multiplied by some shape functions. These functions, known as modes, come from the solution of the eigenvalue problems associated with the static neutron diffusion equation that are being updated along the transient. In previous works, the time step used to update the modes is set to a fixed value and this implies the need of using small time-steps to obtain accurate results and, consequently, a high computational cost. In this work, we propose the use of an adaptive control time-step that reduces automatically the time-step when the algorithm detects large errors and increases this value when it is not necessary to use small steps. Several strategies to compute the modes updating time step are proposed and their performance is tested for different transients in benchmark reactors with rectangular and hexagonal geometry.

Time-dependent neutron diffusion analysis using finite element method for a block-type VHTR core design

The very high-temperature gas-cooled reactor (VHTR) has attracted interest owing to its high passive safety. A reactor analysis tool is required to analyze the safety characteristics of a VHTR in detail, and such a tool should be capable of a transient calculation for a reactivity insertion accident such as control rod ejection or withdrawal. This work presents a transient analysis of a block-type VHTR core. Time-dependent neutron diffusion equation is solved by the finite element method. A simplified thermal fluid analysis tool is also implemented to consider thermal feedback. In addition, a new method is introduced to resolve the control rod cusping effect without additional computation or mesh reconstruction. The above methods are applied to a reactor physics code CAPP, developed at Korea Atomic Energy Research Institute (KAERI). The methods are tested on three-dimensional VHTR problems that include control rods and show encouraging results.

Three-dimensional time-dependent neutron diffusion simulation using average current nodal expansion method

In this work, the average current nodal expansion method was developed for the time-dependent neutronic simulation of transients in a nuclear reactor's core. For this purpose, an adopted iterative algorithm was proposed for solving the 3-D time-dependent neutron diffusion equation. In the average current nodal expansion method, the domain of the reactor core can be modeled by coarse meshes for neutronic calculation associated with reasonable precision of results. The discretization of time differential terms in the time-dependent equations was fulfilled, according to the implicit scheme. The proposed strategy was implemented in some kinetic problems including an infinite slab reactor, TWIGL 2-D seed-blanket reactor, and 3-D LMW LWR. At first, the steady-state solution was carried out for each test case, and then, the dynamic neutronic calculation was performed during the time for a specified transient scenario. Obtained results of static and dynamic solutions were verified in comparison with well-known references. Results can indicate the ability of the developed calculator to simulate transients in a nuclear reactor's core.

Three-Dimensional Pin-Resolved Transient Diffusion Analysis of PWR Core by the Hybrid Coarse-Mesh Finite Difference Algorithm

This paper presents the methodology and performance of the Hybrid Coarse-Mesh Finite Difference (HCMFD) algorithm for transient pinwise analyses of three-dimensional (3-D) pressurized water reactor (PWR) problems. The time-dependent neutron diffusion equations and their applications in two steps of the HCMFD algorithm, i.e., local and global iterations, are introduced in detail. Taking into account the characteristics of the local-global nonlinear HCMFD iterations, an optimization strategy to minimize the computing time of the transient HCMFD calculation is established by focusing on the balance between the number of local and global calculations. Based on the optimization strategy, the actual computational performance of the transient HCMFD algorithm, in view of both computing time and accuracy, is evaluated for the core of a big-sized conventional PWR in this work. To demonstrate the effectiveness of the optimized iteration strategy, various slow and fast transients including a rod ejection transient are simulated by the transient HCMFD algorithm. It is clearly shown that a 3-D pin-resolved whole-core transient solution for a big PWR can be obtained in a reasonably short computing time by the transient 3-D HCMFD algorithm.

Modal methods for the neutron diffusion equation using different spatial modes

The behaviour of the neutrons inside a nuclear reactor core can be modelled by using the time dependent neutron diffusion equation. Different time schemes have been used to integrate this equation. One possibility is to use a modal method, which is based on the expansion of the neutron flux in terms of spatial modes that are the eigenfunctions associated with a given configuration of the reactor core. Several spatial modes can be defined for the neutron diffusion equation such as the λ, α and γ-modes. In this work, the λ, the α and the γ-modes have been used to develop different modal kinetics equations, using a high order finite element method for the spatial discretization of the neutron diffusion equation. The performance of the different modal kinetic equations has been tested and compared using two 3D transient benchmark problems.

A modal ACMFD formulation of the HEXNEM3 method for solving the time-dependent neutron diffusion equation

An analytical coarse-mesh finite-difference (ACMFD) formulation of the HEXNEM3 nodal flux expansion method for solving the two-group neutron diffusion equation in hexagonal-z geometry is developed. In the time-dependent case, where an implicit differencing scheme in time is required for stability, nodal flux expansion necessitates iterating on the group sources. In order to avoid convergence issues, modal decomposition through matrix diagonalisation in the energy domain is first performed, and then the HEXNEM3 nodal expansion model is applied to the modes instead of fluxes, with boundary and continuity conditions on the scalar flux and the net current. The ACMFD formulation of HEXNEM3 has the advantage to produce an explicit non-homogeneous linear algebraic system for either the node-averaged fluxes or modes in all groups and nodes of a three-dimensional problem, with a free choice of any appropriate solution method.

Results of two-dimensional hexagonal dynamic benchmark problems solved using diffusion kinetic code

A two-dimensional nodal kinetics code was developed to solve for the time-dependent neutron diffusion equation in the hexagonal geometry. Direct space time method was used for solving the time-dependent multi-group diffusion equation. The space part was solved using higher order nodal expansion method, wherein one-dimensional neutron flux in each node and energy group was approximated using higher order polynomial, while the first-order time derivative was approximated by finite difference implicit scheme. The code was used to solve two problems based on (1) VVER-440 and (2) HWR.

Development of a GPU-accelerated 3D neutron dynamics code for PB-FHR

Pebble bed fluoride-salt cooled high temperature reactor (PB-FHR) is a kind of novel nuclear energy systems, combining advanced techniques such as coated particle fuel and molten-salt coolant to achieve better safety and economy performance. The compact core of a PB-FHR is geometrically complicated due to the randomly-packed pebble bed with fuel motion in a complex core chamber, which yields a highly detailed core-modeling for accurate neutron dynamics analysis. Three-dimensional fine mesh finite volume method is expected to achieve better accuracy because of its strong geometry adaptability in detailed core-modeling, but is much more time-consuming compared to other methods such as coarse mesh nodal method. A GPU-accelerated 3D fine mesh neutron dynamics code (GAND) is developed, using conjugate gradient method (CG) to solve the time-dependent multi-group neutron diffusion equations in r-z-θ coordinates. The GAND code is verified by a cylindrical reactor benchmark with good agreement, and is preliminarily applied to a PB-FHR core in both static and transient analysis. Speed-up ratio and other performance of GAND code using different equation preconditioners is studied, a best speed-up ratio of 21.65 has been achieved using Neumann polynomials-preconditioned CG.

PARCS code multi-group neutron diffusion constants generation using Monte Carlo method

PARCS code is a three-dimensional (3D) reactor core simulator which solves the steady-state and time-dependent multi-group neutron diffusion equations if the multi-group diffusion constants (MGDCs) are provided. The MGDCs are mostly prepared for reactor physics problems using deterministic lattice codes. Beside approximation in the geometry, a lattice code inherently applies estimates to the neutron transport model. On the other hand, the geometric flexibility and use of continuous energy cross sections data library associated with the Monte Carlo (MC) method makes it a good candidate for the generation of highly accurate multi-group cross sections. In this study, a new MC based methodology is applied to generate the MGDCs which can be utilized in the PARCS code input file directly or as PMAXS files for a reactor core simulation. To achieve this, a new tool in MATLAB software is developed to compute the MGDCs from the MCNPX 2.7 MC code outputs. Verification of the proposed method for two-group constants generation is carried out using Tehran research reactor (TRR) core simulation in different steady state conditions. The calculated values of axial and radial power distributions and multiplication factor using the PARCS code are verified against the MCNPX 2.7 code results. The results illustrate that the proposed method has high accuracy in MGDCs generation.

Moving meshes to solve the time-dependent neutron diffusion equation in hexagonal geometry

To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. Here the spatial discretization of this equation is done using a finite element method that permits h–p refinements for different geometries. This means that the accuracy of the solution can be improved refining the spatial mesh (h-refinement) and also increasing the degree of the polynomial expansions used in the finite element method (p-refinement). Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying one-dimensional and three-dimensional benchmark problems.