Propagation noise calculations in VVER-type reactor core

Neutron noise induced by propagating disturbances in VVER-type reactor core is addressed in this paper. The spatial discretization of the governing equations is based on the box-scheme finite difference method for triangular-z geometry. Using the derived equations, a 3-D 2-group neutron noise simulator (called TRIDYN-3) is developed for hexagonal-structured reactor core, by which the discrete form of both the forward and adjoint reactor dynamic transfer functions (in the frequency domain) can be calculated. In addition, both types of noise sources, namely point-like and traveling perturbations, can be modeled by TRIDYN-3. The results are then benchmarked in different cases. Considering the noise source as propagating perturbations of the macroscopic absorption cross sections, the induced neutron noise is calculated throughout the reactor core. For the first time, adjoint approach is applied and examined for modeling moving noise sources. Moreover, the space- and frequency-dependence of the propagation noise are investigated in this paper.     

Identification and localization of absorbers of variable strength in nuclear reactors

This paper investigates the possibility of localising a noise source of the type “absorber of variable strength” (or reactor oscillator) from as few as five neutron detectors evenly distributed throughout the core of a commercial nuclear reactor. The novelty of this investigation lies with the fact that the calculations are performed for a realistic 2-D heterogeneous reactor in the 2-group diffusion approximation, via the prior determination of the corresponding reactor transfer function. It is first demonstrated that the response of such a reactor to a localized perturbation deviates significantly from point-kinetics. The space-dependence of the induced neutron noise thus carries enough information about the location of the noise source, which makes it possible to determine its position from a few detector readings. The identification of the type of noise source is easily performed from the in-phase behaviour of the induced neutron noise. Different unfolding techniques are finally tested. All these techniques rely on the use of the reactor transfer function. One of these techniques is based on the comparison between the actual measured neutron noise and the neutron noise calculated for every possible location of the noise source. This technique is very reliable and almost insensitive to the contamination of the detector signals by background noise, but also extremely CPU consuming. Another technique, based on the piece-wise inversion of the reactor transfer function and requiring little CPU effort, was developed. Although this technique is much less reliable when background noise is present, this technique is useful to indicate a region of the reactor where a noise source is likely to be located.     

Numerical tools applied to power reactor noise analysis

In order to be able to calculate the space- and frequency-dependent neutron noise in real inhomogeneous systems in two-group theory, a code was developed for the calculation of the Green's function (dynamic transfer function) of such systems. This paper reports on the development as well as the test and application of the numerical tools employed. The code that was developed yields the space-dependence of the fluctuations of the neutron flux induced by fluctuating properties of the medium in the two-group diffusion approximation and in a two-dimensional representation of heterogeneous systems, for both critical systems and non-critical systems with an external source. Some applications of these tools to power reactor noise analysis are then described, including the unfolding of the parameters of the noise source from the induced neutron noise, measured at a few discrete locations throughout the core. Other concrete applications concern the study of the space-dependence of the Decay Ratio in Boiling Water Reactors, the noise-based estimation of the Moderator Temperature Coefficient of reactivity in Pressurized Water Reactors, the modeling of the beam- and shell-mode core-barrel vibrations in Pressurized Water Reactors, and the investigation of the validity of the point-kinetic approximation in subcritical systems driven by an external source. In most of these applications, calculations performed using the code are compared with at-power plant measurements. Power reactor noise analysis applications of the above type, i.e. core monitoring without disturbing plant operation, is of particular interest in the framework of the extensive program of power uprates worldwide.     

ASYNT — An adjoint synthesis method for neutron irradiation estimation on the vessels of VVER/PWR reactors

The new ASYNT method is developed and proposed for neutron fluence calculations. This method uses the solution of adjoint neutron transport equation for flux/responses evaluation. The evaluation of flux/responses is reduced to the space and energy integration of the product of 3D adjoint solution and the neutron source distribution, determined by realized loading patterns and operational regimes. The adjoint solution does not depend on the neutron source distribution and is obtained only once for every surveillance site, response and type of reactor. The application of this method results in separability of azimuthal and axial dependence in the 3D adjoint solution. That is why the 3D adjoint solution could be synthesised from the 2D and 1D adjoint solutions. The circular cylinder reactor core presentation of the solution axial dependence is the only approximation used in the ASYNT method.     

Study of the MTC estimation by noise analysis in 2-D heterogeneous systems

The effect of a heterogeneous distribution of the temperature noise on the MTC estimation by noise analysis is investigated. This investigation relies on 2-group diffusion theory, and all the calculations are performed in a 2-D realistic heterogeneous core. It is shown, similarly to the 1-D case, that the main reason of the MTC underestimation by noise analysis compared to its design-predicted value lies with the fact that the temperature noise might not be homogeneous in the core, and therefore using the local temperature noise in the MTC noise estimation gives erroneous results. A new MTC estimator, which was previously proposed for 1-D 1-group homogeneous cases and which is able to take this heterogeneity into account, was extended to 2-D 2-group heterogeneous cases. It was proven that this new estimator is always able to give a correct MTC estimation with an accuracy of 3%. This small discrepancy comes from the fact that the reactor does not behave in a point-kinetic way, contrary to the assumptions used in the noise estimators. This discrepancy is however quite small.     

Comparison between triangular and hexagonal modeling of a hexagonal-structured reactor core using box method

A hexagonal-structured reactor core (e.g. VVER-type) is mostly modeled by structured triangular and hexagonal mesh zones. Although both the triangular and hexagonal models give good approximations over the neutronic calculation of the core, there are some differences between them that seem necessary to be clarified. For this purpose, the neutronic calculations of a hexagonal-structured reactor core have to be performed using the structured triangular and hexagonal meshes based on box method of discretisation and then the results of two models should be benchmarked in different cases.In this paper, the box method of discretisation is derived for triangular and hexagonal meshes. Then, two 2-D 2-group static simulators for triangular and hexagonal geometries (called TRIDIF-2 and HEXDIF-2, respectively) are developed using the box method. The results are benchmarked against the well-known CITATION computer code in case of a VVER-1000 reactor core. Furthermore, the relative powers calculated by the TRIDIF-2 and HEXDIF-2 along with the ones obtained by the CITATION code are compared with the verified results which have been presented in the Final Safety Analysis Report (FSAR) of the aforementioned reactor.Different benchmark cases revealed the reliability of the box method in contrast with the CITATION code. Furthermore, it is shown that the triangular modeling of the core is more acceptable compared with the hexagonal one.     

Improvement of Neutron Source Introduction Method for Absolute Measurements of Low Reactor Power

The neutron source introduction method was applied to absolute measurements of low reactor power at the Static Experiment Critical Facility STACY. To obtain the effective neutron source intensity more accurately, which is a key parameter for the source introduction method, the neutron source is newly defined as fission neutrons from the first fission reaction caused by neutrons emitted from the external neutron source. To obtain the newly defined effective neutron source intensity, the probability that a neutron from the external neutron source causes a fission reaction is calculated using the Monte Carlo code MCNP. This calculation took into consideration the three-dimensional complicated core structures. Furthermore, the fission reaction distribution, fundamental mode forward and adjoint flux distribution in a critical state were calculated using the three-dimensional transport code THREEDANT. Following the principle of the neutron source introduction method, an external neutron source was inserted near the STACY core tank and the reactor power was measured. The reactor powers by the neutron source introduction method were in good agreement with the ones from the analyses of the FP activity generated by high power operation.     

An analytically solvable,axially non-homogeneous reactor model—I. General solutions

In conceptual and model studies of neutron noise and diagnostics, and even in most practical applications, simple homogeneous reactor models have been used so far, in which closed form analytical solutions are possible. In this paper a less trivial, axially non-homogeneous reactor model is used. In this model both the static flux and the noise equations can still be solved by analytical methods. The model consists of a 2-D homogeneous rectangular core in which a δ-function (Feinberg-Galanin) control rod is inserted partially. Solution of both the static flux and the dynamic equations (the latter corresponding to a rod manoeuvring experiment in which the rod is moved up and down periodically) can be given by two different analytical methodologies. Both are calculated and discussed in the paper. This model will be used later in concrete diagnostic applications as well as in studies of reactor kinetic approximations.     

An object-oriented approach to simulation of IRIS dynamic response

In this paper the development of an adequate modelling and simulation tool for Dynamics and Control tasks is presented. The key features of the developed simulator are: “Modularity” - the system model is built by connecting the models of its components, which are written independently of their boundary conditions; “Openness” - the code of each component model is clearly readable and close to the original equations and easily customised by the experienced user; “Efficiency” - the simulation code is fast; “Tool support” - the simulation tool is based on reliable, tested and well-documented software.To achieve these objectives, the Modelica language was used as a basis for the development of the simulator. The Modelica language is the result of recent advances in the field of object-oriented, multi-physics, dynamic system modelling. The language definition is open-source and it has already been successfully adopted in several industrial fields.The test bed for the application of the object-oriented approach has been the new generation, integral type, IRIS nuclear reactor. IRIS (International Reactor Innovative and Secure) is a pressurized light water cooled, small/medium power (335 MWe) reactor reactor, under development by an international consortium of nineteen organizations from ten countries. The preliminary design has been completed and the pre-application licensing process with the US-Nuclear Regulatory Commission (NRC) is underway.To provide the required capabilities for the analysis, specific models for the nuclear reactor components have been developed, to be applied for the dynamic simulation of the IRIS integral reactor, albeit keeping general validity for PWR plants. The following Modelica models have been written to satisfy the IRIS modelling requirements and are presented in this paper: point reactor kinetic, fuel heat transfer, control rods model, and a once-through type steam generator, thus obtaining a specific library of nuclear models and components. As far as other classical power generation plant components are concerned, the Thermo Power open library, developed at Politecnico di Milano as well, has been adopted and is briefly presented in the paper. Originally conceived for conventional, fossil-fired plants, the highly modular approach allowed to effectively reuse the models of the balance of plant systems, which have been connected to the models of the nuclear power generation process, to obtain a system simulator for the IRIS reactor.Finally, preliminary results of the code validation process and the reactor dynamics are presented.     

Galerkin and Generalized Least Squares finite element: A comparative study for multi-group diffusion solvers

In this paper, the solution of multi-group neutron/adjoint equation using Finite Element Method (FEM) for hexagonal and rectangular reactor cores is reported. The spatial discretization of the neutron diffusion equation is performed based on two different Finite Element Methods (FEMs) using unstructured triangular elements generated by Gambit software. Calculations are performed using Galerkin and Generalized Least Squares FEMs; based on which results are compared. Using the power iteration method for the neutron and adjoint calculations, the neutron and adjoint flux distributions with the corresponding eigenvalues are obtained. The results are then validated against the valid results for the IAEA-2D andBIBLIS-2D benchmark problems. The results of GFEM-2D (developed based on Galerkin FEM) and GELES-2D (developed based on Generalized Least Squares FEM) computer codes are also compared with results obtained from DONJON4 computer code. To investigate the validation of developed computer codes for the calculation with more than two energy groups, the calculations are performed for a benchmark problem with seven energy groups. To investigate the dependency of the results to the number of elements, a sensitivity analysis of the calculations to the number of elements is performed.     

Multi-group theory of neutron noise induced by vibrating boundaries

The paper extends the one-group analysis of the neutron noise induced by fluctuating boundaries [Ann. Nucl. Energy 27(2000)1385] to the general multi-group non-homogeneous model. The full solution is given through the Green's function of the static problem, the static flux, and a quantity describing the boundary movements. A multi-group absorber model is proposed to represent the perturbation, which turns out to be very useful, for instance, to derive the point reactor and adiabatic approximations of the neutron noise arising from the oscillating boundaries. Finally, an equivalent solution is given in terms of the adjoint function.     

CORE SIM: A multi-purpose neutronic tool for research and education

This paper deals with the development, validation, and demonstration of an innovative neutronic tool. The novelty of the tool resides in its versatility, since many different systems can be investigated and different kinds of calculations can be performed. More precisely, both critical systems and subcritical systems with an external neutron source can be studied, and static and dynamic cases in the frequency domain (i.e. for stationary fluctuations) can be considered. In addition, the tool has the ability to determine the different eigenfunctions of any nuclear core. For each situation, the static neutron flux, the different eigenmodes and eigenvalues, the first-order neutron noise, and their adjoint functions are estimated, as well as the effective multiplication factor of the system. The main advantages of the tool, which is entirely MatLab based, lie with the robustness of the implemented numerical algorithms, its high portability between different computer platforms and operative systems, and finally its ease of use since no input deck writing is required. The present version of the tool, which is based on two-group diffusion theory, is mostly suited to investigate thermal systems. The definition of both the static and dynamic core configurations directly from the static macroscopic cross-sections and their fluctuations, respectively, makes the tool particularly well suited for research and education. Some of the many benchmark cases used to validate the tool are briefly reported. The static and dynamic capabilities of the tool are also demonstrated for the following configurations: a vibrating control rod, a perturbation traveling upwards with the core flow, and a high frequency localized perturbation. The tool is freely available on direct request to the author of the present paper.     

Development of two-energy group,two-dimensional,frequency dependent detector adjoint function based on the nodal method

A nodalization technique has been demonstrated to calculate the response of a detector to a vibrating absorber in a reactor core using a concept of local/global components, based on the frequency dependent detector adjoint function. The technique was developed for two-energy group one-dimensional or one-energy group two-dimensional reactor core geometry. The purpose of this research was to expand the applicability of a nodalization model technique to calculate the real and the imaginary parts of the detector adjoint function for two-energy group two-dimensional reactor geometry. The frequency dependent detector adjoint functions presented by complex equations were expanded into real and imaginary parts. In the nodalization technique, the flux or detector adjoint function is expanded into polynomials about the center point of each node. A computer code was developed to calculate static flux for two-energy group, two-dimensional reactor geometry. The eigen value (k_eff) and static flux were calculated for the Iowa State University UTR-10 reactor and the results were compared against the values calculated using the computer code exterminator. The eigen values were within less than 0.1% agreement. The phase angle and the detector adjoint function for the frequency of 10 rad/s were calculated for a detector located in the center of a 60×60 cm reactor. The phase angle calculated by the nodalization model technique varied from 0.2° near the source to 0.4° away from the source. These values are well within the range of the phase angle value of 0.2° calculated using the zero power transfer function. The thermal detector adjoint function peaked in the center as expected. The discontinuity in the current of the real thermal detector adjoint function at the detector position was observed as expected. The average current based on the polynomials on the left node of the interface and the right node of the interface matched within 1% of the average value at the interface. The current of the imaginary fast and thermal detector adjoint function on both sides of the interface varied ±2% from the average value at the interface. No discontinuity in the current was observed in the case of the fast real and imaginary and thermal imaginary components of the detector adjoint function at the detector location.     

Estimation of the fast neutron fluence at control rod tips using a 3-D diffusion/2-D transport multi-step calculation scheme

This paper presents the development and application of a methodology to estimate, using conventional deterministic lattice/core analysis methods, the fast neutron fluence at the tips of control rods inserted during operation in PWR reactors. The developed methodology is based on a 3-D Nodal Diffusion/2-D Lattice Transport multi-step calculation scheme, in which the operating history of the nuclear environment around the tip is tracked in 3-D core follow analyses and used thereafter to provide boundary conditions for 2-D transport calculations to compute the fast neutron flux. In subsequent steps, radial and axial correction factors, both based on fast flux results from the 3-D core simulator, are applied to the calculated 2-D transport flux in order to take into account radial leakage effects as well as axial flux gradients around the tip. The fluence is finally estimated through a time-integration of the corrected 2-D transport fast flux. The developed methodology has been applied to estimate the fluence for a total of 15 control rods, over 21 operating cycles of a Swiss nuclear power plant.     

二维输运燃耗耦合程序开发与验证

为准确计算反应堆内燃耗问题,建立了基于二维离散纵标法及BATEMAN燃耗方法的输运燃耗耦合计算方法,并开发相应的计算程序。基于ENDF/B-Ⅶ评价库开发了175群中子和42群光子截面数据库MUSE-F1.0,采用OECD/NEA发布的MOX燃料快堆基准题对耦合计算方法及程序系统进行验证计算。结果表明,耦合计算程序结果与基准题吻合良好,误差在8%以内,初步验证了耦合计算程序在快堆嬗变工程应用中的可行性。     

Comparative study of 2-group and diffusion theories for the calculation of the neutron noise in 1D 2-region systems

In this paper, the neutron noise, i.e. the stationary fluctuations of the neutron flux around its mean value, is calculated in 2-group P₁ and diffusion theories for a 2-region slab reactor using Green’s function technique. The applicability of diffusion theory for different types and locations of the perturbation, as well as different frequencies, is assessed. Material data, i.e. nuclear cross-sections and kinetic parameters, representative of a Light Water Reactor (LWR) and of a Heavy Water Reactor (HWR), respectively, are used in this work. It is demonstrated that for practical situations in LWRs and HWRs, there is no significant advantage to use P₁ theory since diffusion theory gives acceptable results. The largest deviations between the two formalisms are observed in regions of large gradients of the static neutron flux, such as close to the reflector interface and close to the perturbation. Such observations are in agreement with theoretical expectations. This study also indicates that neglecting the effect of cross-section perturbation on the diffusion coefficient gives a rather small impact on the solution. This allows drastically simplifying the determination of the neutron noise. When using numerical techniques for such a determination the memory requirements and computational effort can be significantly reduced.     

Transport model based on three-dimensional cross-section generation for TRIGA core analysis

Three-dimensional (3-D) transport model for the Pennsylvania State University Breazeale Reactor (PSBR) core analysis has been developed based on the discrete ordinates (S_n) method. The effective fine- and broad-group structures for the TRIGA cross-section libraries were selected based on CPXSD (Contributon and Point-wise Cross-Section Driven) methodology. The study shows results of the following effective broad-group energy structures – a 12-group structure in 2-D geometry vs. a 26-group structure in 3-D geometry. Different 3-D pin/core configurations were used to verify and validate the selected effective group structures. The results agree with continuous energy cross-section Monte Carlo calculations for eigenvalues and normalized pin-power distributions, which are used as a reference in this research.     

Comparison of single and multicell heat transfer analyses near nuclear reactor assembly walls

A “multicell” approach to the problem of heat transfer near the wall of a nuclear reactor fuel assembly is compared to the “single cell” approach. Steady state, fully developed heat transfer is considered in assemblies without grid or wire spacers. Results of the multicell approach indicate significant discrepancies in temperature predictions occur when calculations are based on eight fuel elements as compared to the results based on a single fuel element (cell). The multicell analysis includes the effect of mass flux distribution across the subassembly, and the resulting heat transfer trends are not consistent with the single cell approach. These trends are discussed and the utility of the multicell approach is demonstrated.     

Influence of the external neutron sources in the criticality prediction using 1/M curve

The influence of external neutron sources in the process to obtain the criticality condition is estimated. To reach this objective, the three-dimensional neutron diffusion equation in two groups of energy is solved, for a subcritical PWR reactor core with external neutron sources. The results are compared with the solution of the corresponding problem without external neutron sources, that is an eigenvalue problem. The method developed for this purposes it makes use of both the nodal method (for calculation of the neutron flux) and the finite differences method (for calculation of the adjoint flux). A coarse mesh finite difference method was developed for the adjoint flux calculation, which uses the output of the nodal expansion method. The results regarding the influence of the external neutron source presence for attaining criticality have shown that far from criticality it is necessary to calculate the reactivity values of the system.