Neutron Diffusion Theory Research Articles

Development of 3D neutron noise simulator based on GFEM with unstructured tetrahedron elements

In the present study, the neutron noise, i.e. the stationary fluctuation of the neutron flux around its mean value is calculated based on the 2G, 3D neutron diffusion theory. To this end, the static neutron calculation is performed at the first stage. The spatial discretization of the neutron diffusion equation is performed based on linear approximation of Galerkin Finite Element Method (GFEM) using unstructured tetrahedron elements. Using power iteration method, neutron flux and corresponding eigen-value are obtained. The results are then benchmarked against the valid results for VVER-1000 (3D) benchmark problem. In the second stage, the neutron noise equation is solved using GFEM and Green’s function method for the absorber of variable strength noise source. Two procedures are used to validate the performed neutron noise calculation. The calculated neutron noise distributions are displayed in the different axial layers in the reactor core and its variation in axial direction is investigated. The main novelty of the present paper is the solution of the neutron noise equations in the three dimensional geometries using unstructured tetrahedron elements.

Status on development and verification of reactivity initiated accident analysis code for PWR (NODAL3)

A coupled neutronics thermal-hydraulics code NODAL3 has been developed based on the nodal few-group neutron diffusion theory in 3-dimensional Cartesian geometry for a typical pressurized water reactor (PWR) static and transient analyses, especially for reactivity initiated accidents (RIA).The spatial variables are treated by using a polynomial nodal method (PNM) while for the neutron dynamic solver the adiabatic and improved quasi-static methods are adopted. A simple single channel thermal-hydraulics module and its steam table is implemented into the code. Verification works on static and transient benchmarks are being conducted to assess the accuracy of the code. For the static benchmark verification, the IAEA-2D, IAEA-3D, BIBLIS and KOEBERG light water reactor (LWR) benchmark problems were selected, while for the transient benchmark verification, the OECD NEACRP 3-D LWR Core Transient Benchmark and NEA-NSC 3-D/1-D PWR Core Transient Benchmark (Uncontrolled Withdrawal of Control Rods at Zero Power). Excellent agreement of the NODAL3 results with the reference solutions and other validated nodal codes was confirmed

Applicability of adiabatic approximation on neutron noise analysis in molten salt reactor

In this paper, the applicability of the adiabatic approximation on the neutron noise analysis in the molten salt reactor (MSR) is investigated. The neutronic model considering the fuel recirculation is established based on one-group neutron diffusion theory. In linear perturbation theory, the fluctuation equations are derived due to the smallness of the perturbation. Following the standard procedure, the kinetic equations for the fluctuations of amplitude factors, which are dependent on the unknown fluctuations of shape functions, are obtained. If the fluctuations of shape functions are neglected, the solutions of both the total neutron noise and its point kinetic term are inaccurate. In order to estimate the effects of the fluctuations of shape functions and the interference between the point kinetic and space-dependent terms, the adiabatic approximation is applied. The solutions are compared to the exact ones and those obtained in the point kinetic approximation to figure out the applicability of the adiabatic approximation. The results show that the adiabatic approximation is accurate enough for low velocities. With higher velocities, the solutions obtained in both the point kinetic and adiabatic approximations gradually deviate from the exact ones, whereas the adiabatic approximation can still provide good estimation of the frequency- and space-dependent behavior of neutron noise in the plateau region. In particular, the spatial oscillation of the amplitude of neutron noise can be well reconstructed. Moreover, the characteristic peak, which is related to the inverse of fuel recirculation, is absent in the point kinetic approximation but can be observed in the adiabatic approximation. Therefore, the adiabatic approximation can be regarded as an effective tool for reconstructing and interpreting the neutron noise in MSR.

Numerical solution of the neutron diffusion equations as eigenvalue problem in the nuclear reactors using GDQ method

The most widely used mathematical description of the neutron distribution in nuclear reactors is provided by neutron diffusion theory. If fission source does not balance the leakage and absorption terms, to use steady state diffusion equation we multiply the source term by a constant 1/k, where k is multiplication factor. The equation may be rewritten as an eigenvalue equation. In this paper, Generalised Differential Quadrature (GDQ) method is presented to solve the multi-groups neutron diffusion equations for nuclear reactors as an eigenvalue problem. The main idea of the GDQ is that the derivative of a function at a sample point can be approximated as a weighted linear summation of the value of the function at all of the points in the domain. The comparison between GDQ and Finite Difference (FD) methods shows a significant improvement in the convergence rate for GDQ method with a smaller number of grid points.

Numerical solution of the neutron diffusion equations as eigenvalue problem in the nuclear reactors using GDQ method

The most widely used mathematical description of the neutron distribution in nuclear reactors is provided by neutron diffusion theory. If fission source does not balance the leakage and absorption terms, to use steady state diffusion equation we multiply the source term by a constant 1/k, where k is multiplication factor. The equation may be rewritten as an eigenvalue equation. In this paper, Generalised Differential Quadrature (GDQ) method is presented to solve the multi-groups neutron diffusion equations for nuclear reactors as an eigenvalue problem. The main idea of the GDQ is that the derivative of a function at a sample point can be approximated as a weighted linear summation of the value of the function at all of the points in the domain. The comparison between GDQ and Finite Difference (FD) methods shows a significant improvement in the convergence rate for GDQ method with a smaller number of grid points.

Impact of new evaluated nuclear data libraries on core characteristics of innovative reactor designs

The impact of new evaluated nuclear data libraries (ENDF/B, JENDL and JEFF) on core characteristics of thermal and fast spectra innovative reactor designs was investigated. The innovative reactor design with thermal neutron spectrum is represented by small-sized, long-life CANDLE (Constant Axial shape of Neutron flux, nuclide densities and power shape During Life of Energy producing reactors) HTGRs (High Temperature Gas-cooled Reactors) with uranium and thorium fuel cycles, while the one with the fast neutron spectrum is represented by compact, sodium-cooled B&BRs (Breed-and-Burn Reactors). The CANDLE HTGR core characteristics (i.e. kinf, keff, discharge burnup, burning region moving velocity, core life time and axial power peaking factor) were evaluated by a dedicated deterministic neutronics and depletion code based on few-group neutron diffusion theory in 2-D RZ geometry, while the required microscopic cross sections were prepared by using SRAC2006 with JENDL-4.0, ENDF/B-VI and JEFF-3.1 based new SRAC2006 libraries. The B&BR core characteristics were evaluated using a continuous energy Monte Carlo neutron transport code, SERPENT, with JENDL-4.0, ENDF/B-VII.0, ENDF/B-VII.1 and JEFF-3.1.1 nuclear data based new libraries. The impact of new evaluated nuclear data libraries for innovative CANDLE HTGR core characteristics was found significantly larger for thorium fuel cycle than the one for uranium fuel cycle, especially in the discharge burnup, burning region moving velocity which affected the core life time significantly. The findings indicated the needs of more accurate nuclear data libraries for nuclides involved in thorium fuel cycle. The impact of new evaluated nuclear data libraries for innovative B&BR core characteristics was found in a slightly large variation for keff evolution during burnup which in turn affected the estimated core life time. However, a good agreement within the statistical error on the integral kinetic parameters, Doppler reactivity coefficients, coolant density reactivity coefficients, axial and radial expansion reactivity coefficients, power density distributions can be observed among the libraries. In addition, the buildup of fission products and minor actinides were also similar.

Characters of neutron noise in full-size molten salt reactor

In the present paper, the frequency-dependent and space-dependent behavior of the neutron noise in a full-size Molten Salt Reactor (MSR) is investigated. The theoretical models considering the fuel circulation are established based on one-group neutron diffusion theory. Green’s function of the neutron noise induced by a propagating perturbation is introduced with linear noise theory, due to the small perturbation. The equations are numerically calculated by developing a code, in which the eigenfunction expansion method is adopted. The static results show that the effective delayed neutron fraction changes non-monotonically with the increasing fuel velocity. In the dynamic case, the results are compared to those obtained in 1D MSR and a traditional reactor, in order to figure out the effects of both the fuel circulation and the system size. It is found that there is no difference in 1D and 3D MSR systems from the view of fuel circulation, i.e., the fuel circulation enhances the spatial neutronic coupling and leads to the stronger point kinetic effect. The more prominent space-dependent effect founded in 3D traditional reactors is also observed in the MSR, due to the looser neutronic coupling and the unique singularity of Green’s function in the location of the perturbation. Another interesting finding is that Green’s function for low frequencies changes non-monotonically with increasing velocity. The largest magnitude of Green’s function is observed at the velocity where the effective delayed neutron fraction reaches its minimum. Finally, the neutron noise induced by a specific propagating perturbation is calculated and analyzed.

Three dimensional neutronic/thermal-hydraulic coupled simulation of MSR in transient state condition

MSR (molten salt reactor) use liquid molten salt as coolant and fuel solvent, which was the only one liquid reactor of six Generation IV reactor types. As a liquid reactor the physical property of reactor was significantly influenced by fuel salt flow and the conventional analysis methods applied in solid fuel reactors are not applicable for this type of reactors. The present work developed a three dimensional neutronic/thermal-hydraulic coupled code investigated the neutronics and thermo-hydraulics characteristics of the core in transient condition based on neutron diffusion theory and numerical heat transfer. The code consists of two group neutron diffusion equations for fast and thermal neutron fluxes and six group balance equations for delayed neutron precursors. The code was separately validated by neutron benchmark and flow and heat transfer benchmark. Three different transient conditions was analyzed with inlet temperature drop, reactivity jump and pump coastdown. The results provide some valuable information in design and research this kind of reactor.

Fissioncore: A desktop-computer simulation of a fission-bomb core

A computer program, fissioncore, has been developed to deterministically simulate the growth of the number of neutrons within an exploding fission-bomb core. The program allows users to explore the dependence of criticality conditions on parameters such as nuclear cross-sections, core radius, number of secondary neutrons liberated per fission, and the distance between nuclei. Simulations clearly illustrate the existence of a critical radius given a particular set of parameter values, as well as how the exponential growth of the neutron population (the condition that characterizes criticality) depends on these parameters. No understanding of neutron diffusion theory is necessary to appreciate the logic of the program or the results. The code is freely available in FORTRAN, C, and Java and is configured so that modifications to accommodate more refined physical conditions are possible.

Three dimensional neutronic/thermal-hydraulic coupled simulation of MSR in steady state condition

MSR (molten salt reactor) uses liquid molten salt as the coolant and fuel solvent, making it the only liquid reactor among the six generation IV reactor types. As a liquid reactor the physical properties of the reactor are significantly influenced by the fuel salt flow therefore conventional analysis methods applied in solid fuel reactors are not applicable for this type of reactors. The present work developed a three dimensional neutronic/thermal-hydraulic coupled code and applied it to investigate the thermal-hydraulic characteristics of the core in steady state condition based on neutron diffusion theory and numerical heat transfer. The code consists of two group neutron diffusion equations for fast and thermal neutron fluxes and six group balance equations for delayed neutron precursors. The temperature distribution, neutron fluxes and delayed neutron precursors distribution of the core in steady state conditions was studied, and the result analyzed when inlet temperature and velocity were changed. From simulation it was found that the inlet temperature has little influence to neutron distribution however inlet velocity affects the delayed neutron distribution in steady state condition. The results provide some valuable information in design and research of this kind of reactor.

NEMSQR: A 3-D multi group diffusion theory code based on nodal expansion method for square geometry

A three dimensional, multigroup, neutron diffusion theory based computer code NEMSQR (Nodal Expansion Method for SQuare geometry) is developed for square geometry in order to perform reactor core calculation. The code is based on Nodal Expansion Method (NEM). In this method, an inhomogeneous matrix equation, which involves spatial moments of nodal flux distribution and surface averaged partial currents across the nodal surfaces, is derived using fourth order polynomial approximation to spatial dependence of nodal flux. Discontinuity factor is incorporated into the code to reduce homogenization error. This code is used for calculation of effective multiplication factor, neutron flux (both direct and adjoint flux), integral kinetics parameters and subcritical count in presence of external neutron source. The numerical studies reported here for several benchmark problems related to light water reactor as well as fast breeder reactor demonstrate the accuracy of the code.

Numerical analysis for a molten salt reactor in the presence of localized perturbations

Molten salt reactors (MSRs) have seen a marked resurgence of interest over the past few decades, highlighted by their inclusion as one of the six Generation IV reactor types. The MSRs are characterized by using the fluid-fuel, so that their technologies are fundamentally different from those used in the conventional solid-fuel reactors. In this paper, the attention is focused on the behaviors of an MSR in the presence of localized perturbations caused by fissile precipitates and gas bubbles. A neutron kinetic model considering the fuel salt flow is established based on the neutron diffusion theory, which consists of two-group neutron diffusion equations for the fast and thermal neutron fluxes and six-group balance equations for delayed neutron precursors, and the group constants dependent on the temperature are calculated by the code DRAGON. In addition, the k-epsilon turbulent model is adopted to establish the flow and heat transfer. The thermo-hydraulic and neutronic models are coupled through the temperature, heat source and velocity. The effects of the localized perturbation on the distributions of power, temperature, neutron fluxes and delayed neutron precursors are obtained and discussed in detail. The results provide some valuable information for the research and design of this new generation reactor.

Assessment of the traditional neutron-diffusion core-analysis method for the analysis of the Super Critical Water Reactor

The key design quantities of the pressure-tube-based (PT-based) Super Critical Water Reactor (SCWR) core design are expected to be computed with the traditional core-analysis code which solves the two-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the lattice code. Two issues may affect the accuracy of these computed quantities for the SCWR core: one is the two-energy-group neutron-diffusion theory; the other is the generation of lattice-homogenized properties with the lattice code based on the single-lattice-cell model without considering the effects of the environment. It has been illustrated that the single-lattice-cell method is not sufficiently accurate for heterogeneous core configurations when adjacent channels experience significant spectrum interaction. To ensure the qualification of these computed quantities for the SCWR core, a 2-D SCWR benchmark problem was setup (with the reference solution provided by the continuous energy Monte-Carlo code SERPENT) to assess the traditional neutron-diffusion core-analysis method. The assessment shows that the traditional two-group neutron-diffusion theory with the single-lattice-cell-based lattice properties is not sufficient to capture either the spectral change or the environment effect for the SCWR core. The solution of the eight-group neutron-diffusion equation by using lattice-homogenized cross sections calculated with the multicell model is considered appropriate for the analysis of the PT-based SCWR core.

On the Method ASCM for the Solution of the External Source Problem in a Slightly Subcritical Reactor

Numerical evaluation of the steady-state neutron flux distribution in a slightly subcritical nuclear reactor due to the presence of a fixed external source is considered by using neutron diffusion theory. It has been shown in the literature that in the particular case when keff is very close to unity (say, within 1 mk), many solution techniques face severe convergence problems. In this context, an acceleration method called Accelerated SubCritical Multiplication (ASCM), originally suggested in the well-known neutron transport code TORT, is investigated in this paper specifically for such cases. The studies are based on a realistic heavy water reactor test case analyzed by two-group diffusion theory. ASCM is found to work very well. It is useful even when the distributions of the external source and the fission source are vastly different. ASCM is based on iterative scaling of the overall flux level in the reactor. An alternative way to evaluate the “scaling factor” is discussed. A somewhat new ASCM-like scheme is suggested to accelerate the Jacobi and Gauss-Seidel iterations needed for the within-group calculations. Conditions for the effectiveness of this scheme are discussed. Implications of the present work in reactor kinetics and some other fields are indicated.

Application of Time-Dependent Neutron Transport Theory to High-Temperature Reactors of Pebble Bed Type

We introduce a new coupled neutronics/thermal-hydraulics code system for analyzing transients of high-temperature gas-cooled reactors (HTGRs), based on a neutron transport theory approach. At the heart of the coupled code system resides the DORT-TD code, a time-dependent extension of the well-known DORT discrete ordinates code. DORT-TD uses a fully implicit time integration scheme and is coupled via its generalized thermal-hydraulics interface to the THERMIX-DIREKT code, an HTGR-specific heat conduction/convection code for pebble bed-type reactor cores. Feedback is accounted for by interpolating multigroup cross sections from libraries pregenerated with appropriate spectral codes. These libraries are structured for user-specified discrete sets of thermal-hydraulic parameters, e.g., fuel and moderator temperatures. The coupled code system is applied to a pebble bed HTGR model case, i.e., the PBMR 268 MW design. Steady-state studies are performed, and several design-basis and beyond-design-basis transients are simulated in an effort to assess the adequacy of using neutron diffusion theory against the more accurate but yet computationally more expensive neutron transport approach. Relatively small but significant differences arise from using either theoretical approach, from which it is concluded that transport theory as the more versatile tool should be used as reference to quantify the effects of the approximations inherent in diffusion and to gain confidence in its predictive power, especially in safety analyses. In an effort to validate the DORT-TD/THERMIX code system, the neutronics stand-alone solver is benchmarked against available transient benchmark exercises, and the coupled code system is applied to the Organisation for Economic Co-operation and Development/Nuclear Energy Agency/Nuclear Science Committee PBMR 400 MW Coupled Neutronics Thermal Hydraulics Transient Benchmark, demonstrating its remarkable viability for a wide range of safety cases. The final product is a high-fidelity, highly flexible, and well-validated state-of-the-art computer code system, with multiple capabilities to analyze HTGR safety-related transients in an accurate and efficient manner.

Quantitative interpretation of pulsed neutron capture logs: Part 1 — Fast numerical simulation

Pulsed neutron capture (PNC) logs are commonly used for formation evaluation behind casing and to assess time-lapse variations of hydrocarbon pore volume. Because conventional interpretation methods for Σ logs assume homogeneous formations, errors may arise, especially in thinly bedded formations, when appraising petrophysical properties of hydrocarbon-bearing beds. There exist no quantitative interpretation methods to account for shoulder-bed effects on Σ logs acquired in sand-shale laminated reservoirs. Because of diffusion effects between dissimilar beds, Σ logs acquired in such formations do not obey mixing laws between the Σ responses of pure-sand and pure-shale end members of the sedimentary sequence. We have developed a new numerical method to simulate PNC rapidly and accurately logs. The method makes use of late-time, thermal-neutron flux sensitivity functions (FSFs) to describe the contribution of multilayer formations toward the measured capture cross section. It includes a correction procedure based on 1D neutron diffusion theory that adapts the transport-equation-derived, base-case FSF of a homogeneous formation to simulate the response of vertically heterogeneous formations. Benchmarking exercises indicate that our simulation method yields average differences smaller than two capture units within seconds of computer central processing unit time with respect to PNC logs simulated with rigorous Monte Carlo methods for a wide range of geometrical, petrophysical, and fluid properties.

Water Content Measurement in Expansive Soils Using the Neutron Probe

Abstract Capacitance-type methods for measuring soil water content are known to be unreliable in expansive soils, as cracking disrupts the intimate contact between the soil and the measuring device. The neutron probe, which infers water content from the thermalisation of a cloud of neutrons, is potentially less affected by cracking. The effect of cracking on neutron probe measurements was investigated by a series of numerical simulations using an axisymmetric finite element model based on seven-group neutron-diffusion theory. The simulations employed a consistent soil cracking model based on Maryland clay, in which crack volumes are determined from the changes in void ratio in the shrinking bulk soil. The results show that the presence of cracks in a clay soil affects the inferred water content and that measurements affected by air-filled cracking under-predict not only the water content in the uncracked soil peds but also the average water content in the larger cracked soil mass. The reason for this under-prediction is understood by considering the spatial distribution of the thermalised neutrons in the cracked and uncracked soils. The fast neutrons emitted from the source are seen to diffuse preferentially along air-filled cracks, traveling a large distance from the detector before they become thermalised, thus reducing their likelihood of being back-scattered to the detector where they can be counted. The proximity of the first crack to the probe in the ground also affects the measurement. Water-filled cracks are seen to have the opposite (but lesser) effect to air-filled cracks. A comparison of a simple uniform width crack model to a more realistic model in which crack width varies with changing water content shows that the model is sensitive to crack distribution and that the linear calibration expressions that are typically employed for neutron probes are likely to be unreliable in cracked clay soils.

Evaluation of dominant time-eigenvalues of neutron transport equation by Meyer’s sub-space iterations

The aim of this paper is to explore the use of Meyer’s sub-space iteration (SSI) method for the evaluation of dominant prompt time-eigenvalues of the neutron transport equation. The integro-differential form of the transport equation is considered. The SSI method is known to be an efficient technique to find the dominant eigenvalues of a non-symmetric matrix. It has been earlier used for eigenvalue problems in neutron diffusion theory. However, it does not seem to be tried in the transport theory case. Here, the use of SSI has been tested in transport theory for some 1-D mono-energetic homogeneous and heterogeneous benchmark problems. The space variable is discretised by finite differencing while neutron directions are discretised by discrete ordinates (S n -) method. The SSI method needs frequent multiplication of the relevant matrix operator with vectors. As known from earlier works in this area, this can be achieved in terms of external source calculations for which a 1-D programme was developed and used. With the availability of more versatile S n -method codes, it may perhaps be possible to extend use of SSI to more realistic cases.

OPTIMIZATION OF FUEL RELOADING PATTERNS FOR A RESEARCH REACTOR BY SIMULATED ANNEALING

This article presents results obtained from a research into an application of simulated annealing method to the in-core fuel reloading pattern optimization for a research reactor. The decision variable of the optimization problem is a fuel reloading pattern for the next cycle after the present cycle finishes. The objective function maximizes the effective multiplication factor keff at the beginning of cycle while it is established to include an important safety paramater – the power peaking factor, in search process. A procedure for searching the optimal solutions was formed and a computer code was developed in the Fortran language running on PCs. Nuclear safety parameters for the optimization problem are provided from the results of the multigroup neutron diffusion theory computation program CITATION. A sample calculation was performed to find the optimal fuel reloading patterns for the second cycle of the Dalat research reactor and the results are presented in this article.

A nodal formulation method relating assembly power and reactivity for a boiling water reactor

A second-order nodal method is derived for the one and a half-group version of neutron diffusion theory in X-Y geometry. Based on the linear reactivity model, two reactivity relations are developed to treat fresh and depleted loading batches as a way to account for the spatial variation of moderator properties due to the substantial fraction of steam void present as well as the effect of the control rod positioning. Based on these ideas a programme was developed to predict fuel batch burnup and fuel power sharing in a Boiling Water Reactor (BWR) operating cycle. The method was tested by assessing two consecutives current BWR operating fuel cycles.