Reactor Kinetics Research Articles

Trigonometric Fourier-series solutions of the point reactor kinetics equations

In this paper, a new method based on the Fourier series is introduced to obtain approximate solutions to the systems of the point kinetics equations. These systems are stiff involving equations with slowly and rapidly varying components. They are solved numerically using Fourier series expansion over a partition of the total time interval. Approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high order derivatives of the dependent variables at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. The obtained algebraic system is similar to the Vandermonde system. Evaluation of the inverse of the Vandermonde matrix is required to determine the coefficients of the Fourier series. Because the obtained Vandermonde matrix has a special structure, due to the properties of the sine and cosine functions, a new formula is introduced to determine its inverse using standard computations. The new method solves the general linear and non-linear kinetics problems with six groups of delayed neutrons. The validity of the algorithm is tested with five different types of reactivities including step reactivity insertion, ramp input, oscillatory reactivity changes, a reactivity as a function of the neutron density and finally temperature feedback reactivity. Comparisons are made with analytical and conventional numerical methods used to solve the point kinetics equations. The results confirm the theoretical analysis and indicate the validity of the method for all applications considered.

Robust fault diagnosis for an exothermic semi-batch polymerization reactor under open-loop

An independent radial basis function neural network (RBFNN) is developed and employed here for an online diagnosis of actuator and sensor faults. In this research, a robust fault detection and isolation scheme is developed for an open-loop exothermic semi-batch polymerization reactor described by Chylla–Haase. The independent RBFNN is employed here for online diagnosis of faults when the system is subjected to system uncertainties and disturbances. Two different techniques to employ RBFNNs are investigated. Firstly, an independent neural network (NN) is used to model the reactor dynamics and generate residuals. Secondly, an additional RBFNN is developed as a classifier to isolate faults from the generated residuals. Three sensor faults and one actuator fault are simulated on the reactor. Moreover, many practical disturbances and system uncertainties, such as monomer feed rate, fouling factor, impurity factor, ambient temperature and measurement noise, are modelled. The simulation results are presented to illustrate the effectiveness and robustness of the proposed method.

Module for thermomechanical modeling of LWR fuel in multiphysics simulations

We have developed a new light-weight fuel behavior code FINIX, specifically designed for modeling of LWR fuel rods in multiphysics simulations. A thermomechanical description of the rod is required especially in transient conditions, where the heat transfer and changes in the rod’s physical dimensions are strongly coupled. In addition to the mechanical deformations, FINIX solves the temperature distribution in the rod and the heat flux from the cladding to the coolant, allowing two-way coupling of the fuel behavior simulation with both neutronics and thermal hydraulics simulations. In this paper, we describe the FINIX module and compare its performance with experimental data and FRAPTRAN-1.4, a widely used fuel behavior code. The comparison reveals good agreement in both cases. We also demonstrate how FINIX can be integrated into multiphysics simulations. Coupled with the Monte Carlo reactor physics code Serpent, we simulate a fast reactivity transient with the fuel temperature and fission power solved self-consistently. With the reactor dynamics codes TRAB-1D and TRAB3D/SMABRE, we simulate a fast power transient and a PWR main steam line break. The latter serves as an example of coupled fuel behavior, neutronics and system-level thermal hydraulics simulation.

Experiments and validation analyses of HTTR on loss of forced cooling under 30% reactor power

In a safety demonstration test involving the loss of both reactor reactivity control and core cooling, the high-temperature engineering test reactor (HTTR) demonstrates spontaneous stabilization of the reactor power. The test and analytical results of tripping one or two out of three gas circulators without reactor scram have already been reported. Moreover, the pre-analytical result of tripping all three gas circulators without reactor scram has been presented. On the other hand, the test and analytical results of tripping all three gas circulators without reactor scram are shown in this paper. About experiments, at an initial reactor power of 30% (9 MW), when all three gas circulators were tripped without reactor scram to reduce the coolant flow rate to zero, the fuel temperature did not show a large increase because the large heat capacity of the graphite core could absorb heat from the fuel in a short period. Moreover, the decay heat could be transferred through the graphite core and the reactor pressure vessel (RPV), emitted by thermal radiation from its outer surface and removed to the active vessel cooling system; therefore, the core at 9 MW was never exposed to the danger of a core melt, and the reactor power was stabilized spontaneously. About analyses, the reactivity performance is important for predicting the converging level of reactor power that affects the fuel temperature during a loss of forced cooling (LOFC) without reactor scram. With regard to thermal hydraulics, the performances of graphite heat conduction in the reactor core and thermal radiation from the RPV surface to the reactor cavity cooling system are crucial for predicting the temperature behavior of the fuel and RPV in the LOFC condition. It was confirmed that reactor kinetics coupled with heat transfer could be applied to reactor safety and accident analysis based on the comparison between the experiments and the analyses.

Dynamic biogas upgrading based on the Sabatier process: Thermodynamic and dynamic process simulation

This study aimed to investigate the feasibility of substitute natural gas (SNG) generation using biogas from anaerobic digestion and hydrogen from renewable energy systems. Using thermodynamic equilibrium analysis, kinetic reactor modeling and transient simulation, an integrated approach for the operation of a biogas-based Sabatier process was put forward, which was then verified using a lab scale heterogenous methanation reactor. The process simulation using a kinetic reactor model demonstrated the feasibility of the production of SNG at gas grid standards using a single reactor setup. The Wobbe index, CO2 content and calorific value were found to be controllable by the H2/CO2 ratio fed the methanation reactor. An optimal H2/CO2 ratio of 3.45–3.7 was seen to result in a product gas with high calorific value and Wobbe index. The dynamic reactor simulation verified that the process start-up was feasible within several minutes to facilitate surplus electricity use from renewable energy systems.

Generalization of the analytical solution of neutron point kinetics equations with time-dependent external source

Point reactor kinetics equations with one group of delayed neutrons in the presence of the time-dependent external neutron source are solved analytically during the start-up of a nuclear reactor. Our model incorporates the random nature of the source and linear reactivity variation. We establish a general relationship between the expecta- tion values of source intensity and the expectation values of neutron density of the sub-critical reactor by ignoring the term of the second derivative for neutron density in neutron point kinetics equations. The results of the analytical solution are in good agreement with the results obtained with numerical solution.

Calculation of reactor kinetics parameters with Monte Carlo differential operator sampling

The methods to calculate the reactor kinetics parameters of βeff and Λ with the differential operator sampling have been reviewed. The comparison of the results obtained with the differential operator sampling and iterated fission probability approaches has been performed. It is shown that the differential operator sampling approach gives the same results as the iterated fission probability approach within the statistical uncertainty. In addition, the prediction accuracy of the evaluated nuclear data library JENDL-4.0 for the measured βeff/Λ and βeff values is also examined. It is shown that JENDL-4.0 gives a good prediction except for the uranium-233 systems. The present results imply the need for revisiting the uranium-233 nuclear data evaluation and performing the detailed sensitivity analysis.

Safety analysis of MTR type research reactor during postulated beam tube break inducing positive reactivity

MTR type research reactors are commonly equipped with beam tubes which functions as experimental facilities out off the core. When used for irradiation, the tubes are filled with air, in contrary they are flooded with water when idle. Should the break occur during irradiation, the water comes in and the air goes out. Beam tubes break should be viewed as potential accident to occur during reactor life time, caused, for instance by an earth quake. The break was assumed to occur inside the reactor pool and no water leaked out of the pool as the tube isolation valves worked properly. When being used for irradiation, the change from voided to flooded condition due to the break, inserts positive reactivity as more neutrons are moderated and reflected back to the core by the water. Functioning as moderator, water also reflects neutrons back to the core. More neutrons in the beam tube slowed down to a lower energy. For the above reason, safety analysis needs to be performed for beam tube break and its influence on reactor dynamics. The radial beam tubes gave higher positive reactivity feedback than that of tangential ones. The calculations on reactivity induced by beam tubes break were performed with MCNP5 code. It was treated as reactivity initiated accident analyzed using PARET/ANL code. Beam tube S5 gave the highest reactivity feedback of 0.062%dk/k and was treated as the initiating event. The maximum fuel and clad temperature reached 175.28, and 136.75°C, respectively. All safety parameters were confirmed under acceptable limits.

Power distribution control of CANDU reactors based on modal representation of reactor kinetics

Modal synthesis representation of a neutronic kinetic model for a CANDU reactor core has been utilized in the analysis and synthesis for reactor control systems. Among all the mode shapes, the fundamental mode of the power distribution, which also coincides with the desired reactor power distribution during operation, is used in the control system design. The nonlinear modal models are linearized around desired operating points. Based on the linearized model, linear quadratic regulator (LQR) control approach is used to synthesize a state feedback controller. The performance of this controller has been evaluated by using the original nonlinear models under load-following conditions. It has been demonstrated that the proposed reactor control system can produce more uniform power distribution than the traditional reactor regulation systems (RRS); in particular, it is more effective in compensating the Xenon induced transients.

Efficient Multiphysics Coupling for Fast Burst Reactors in Slab Geometry

An efficient neutron transport-based multiphysics coupling algorithm is developed to model fast-burst reactor dynamics. It extends the previously developed tightly coupled implicit-explicit (IMEX) multiphysics algorithm based on neutron diffusion and linear mechanics to follow the dynamic time scale of the problem. The diffusion model is not expected to adequately represent the neutronic behavior of the system due its small size. The transport effects are incorporated using the moment-based acceleration concept. This concept enables us to isolate the angular flux from the coupled multiphysics system by using a consistently discretized lower-order system for the scalar flux in the coupling equations, thus helping to mitigate the increased numerical burden associated with a neutron transport model. Solution of a representative slab model demonstrates the difference between diffusion and transport models and also demonstrates second-order convergence in the coupled simulation with the combination of the moment-based acceleration concept and the IMEX algorithm.

Mechanistic modeling of atomic layer deposition of alumina process with detailed surface chemical kinetics

A mathematical model of atomic layer deposition of alumina is developed using a TMA + H2O ALD operation as a physical base. The model coherently bridges the process dynamics of ALD reactor with the detailed surface chemical kinetics, where the time-dependent variations of surface species are modeled during each precursor pulse. The model is validated by the experimental data. Upon the validation, continuous ALD cycling operation is simulated under different operating conditions. Results show that temperature rising can reduce the surface reaction time in two half cycles, and meanwhile intensify the desorption rate of precursors and backward surface reactions. Suitably increasing the system pressure can improve the completion of surface reactions and film formation in the cycling. The sensitivity analysis shows that surface species variations of |–OH, |–Al(CH3)2 and |–Al(CH3)2 have different degree of sensitivities to the surface reaction steps, and reveals the rate-limiting steps in two precursor treatment cycles.

Ignition dynamics of dry-etched vertical cavity single-hole microdischarge reactors in ac regime operating in noble gases

Silicon-based multi-cavity microdischarge reactors allow the generation of large-area uniform glow discharges over a wide pressure range up to atmospheric pressure. These devices, fabricated using micro electro-mechanical system technology, have shown complex interactions between the individual cavities. To discriminate these interactions, devices with only one shallow vertical cavity are studied here. Operation characteristics are investigated using electrical and optical analysing techniques. The spatial and temporal dynamics of the discharge are investigated for positive and negative voltage polarity of the applied ac voltage ramp by phase-resolved imaging. Within each voltage half-period, emission from the single cavity shows repetitive pulsing features and distinct spatial distributions. In the positive half-period, ring-shaped structures develop, while the negative half-period is distinguished by a bell-shaped intensity distribution. Effects of pressure and operation frequency on the spatial and temporal intensity profiles are discussed.

Fractional neutron point kinetics equations for nuclear reactor dynamics – Numerical solution investigations

This paper presents results concerning numerical solutions to a fractional neutron point kinetics model for a nuclear reactor. The paper discusses and expands on results presented in (Espinosa-Paredes et al., 2011). The fractional neutron point kinetics model with six groups of delayed neutron precursors was developed and a numerical solution using the Edwards’ method was proposed (Edwards et al., 2002). The mathematical model was implemented in the Matlab environment and tested using typical step input change. Experimental evaluation of numerical stability was complemented by an analysis of the effect selected model parameters have on obtained results.

A novel mathematical model for two-energy groups of the point kinetics reactor dynamics

The point kinetics equations for reactor dynamic systems are normally described and treated for one-energy group, which modeled as stiff coupled differential equations, and their solution by the conventional explicit methods will give a stable consistent result only for very small time steps. A novel analytical formulation is constructed and converged to high accuracy from the merger of the piecewise constant functions over a partition in time into the fundamental matrix for the two-energy group of the point kinetics equations. The resulting system of stiff linear and/or nonlinear differential equations for an arbitrary number of delayed neutrons is solved exactly over each time step. Through analytical inversion technique of the fundamental matrix and the stability of the method, we demonstrate its high accuracy for a variety of imposed reactivity insertions found in the literature for three dimensional homogeneous reactors. From knowledge of how the error term behaves the computational results indicate that the method is efficient and accurate for multi-dimensional homogeneous reactors.

Microbial community composition and dynamics in high-temperature biogas reactors using industrial bioethanol waste as substrate.

Stillage, which is generated during bioethanol production, constitutes a promising substrate for biogas production within the scope of an integrated biorefinery concept. In this study, a microbial community was grown on thin stillage as mono-substrate in a continuous stirred tank reactor (CSTR) at a constant temperature of 55 °C, at an organic loading rate of 1.5 goTS/L*d and a retention time of 25 days. Using an amplicon-based dataset of 17,400 high-quality sequences of 16S rRNA gene fragments (V2-V3 regions), predominance of Bacteria assigned to the families Thermotogaceae and Elusimicrobiaceae was detected. Dominant members of methane-producing Euryarchaeota within the CSTR belonged to obligate acetoclastic Methanosaetaceae and hydrogenotrophic Methanobacteriaceae. In order to investigate population dynamics during reactor acidification, the organic loading rate was increased abruptly, which resulted in an elevated concentration of volatile fatty acids. Acidification led to a decrease in relative abundance of Bacteria accompanied with stable numbers of Archaea. Nevertheless, the abundance of Methanosaetaceae increased while that of Methanobacteriales decreased successively. These findings demonstrate that a profound intervention to the biogas process may result in persistent community changes and reveals uncommon bacterial families as process-relevant microorganisms.

Sorption-enhanced steam reforming of ethanol for continuous high-purity hydrogen production: 2D adsorptive reactor dynamics and process design

Sorption enhanced steam reforming of ethanol process for high-purity hydrogen production has been investigated in this work. A two-dimensional mathematical model has been developed to describe the coupled mass and heat transport phenomena within both the axial and radial directions of the adsorptive reactor for sorption enhanced reaction process. The product distribution from the numerical simulation matches well with experimental data obtained from a laboratory-scale fixed-bed adsorptive reactor. In addition, the performance of sorption enhanced reaction is found to be sensitive to the temperature gradient in the radial direction in a scale-up reactor. Afterwards, four reactors with a seven steps cycle operation scheme have been employed in a continuous hydrogen production process. It is found that a hydrogen stream (dry basis) with purity higher than 99mol %, carbon monoxide content 25ppm and 0.95molkg−1h−1 productivity of hydrogen can be produced under cyclic steady state operation. The ratio between energy output and input is around 1.7, and the estimated cost of energy is 0.71US$ per kilogram of hydrogen produced. Besides, high purity carbon dioxide (dry basis) can also be captured as a by-product during the sorbent regeneration step in this process.

Iron oxidation and jarosite precipitation in a two-stage airlift bioreactor

Continuous iron oxidation and removal of jarosite precipitates from solution at low pH, ambient temperature and pressure was successfully demonstrated with a two-stage airlift bioreactor (ALBR) system. The ALBR system, comprising of two ALBRs, each with its own settler, was operated for iron oxidation and precipitation at room temperature with a mixed culture of mesophilic iron oxidisers. The two-stage reactor design allowed optimization of the overall reactor kinetics by facilitating the growth of two distinct groups of iron oxidizers at low (430mV vs. Ag/AgCl) and high (517mV) redox potential in the two reactors, respectively. The influent with pH1.5 contained (gL−1) 15 Fe2+, 1.5 Cu, 1.5 Ni, nutrients and trace elements. The hydraulic retention time (HRT) was decreased stepwise to evaluate process performance. With the lowest HRTs (8h in ALBR1 and 10h in ALBR2), the overall iron oxidation and precipitation rates of the two-stage system were 0.75±0.02gL−1h−1 and 0.15±0.01gL−1h−1, respectively and overall iron oxidation and precipitation of 94±3% and 18±1%, respectively. The percent of influent Fe, S, Cu and Ni removed as precipitates from settlers were 30.9%, 16.7%, 1.1% and 0.2%, respectively. The precipitates were predominately comprised of (>95%) jarosite with potassium jarosite being the dominant form, followed by hydronium, ammonium and sodium jarosites. In conclusion, the two-stage ALBR system allowed efficient iron oxidation and precipitation of the oxidised iron as well settling jarosite with only minor loss of Cu and Ni via co-precipitation. The process demonstrated performance that is promising for a variety of hydrometallurgical process flow sheets, where it allows iron removal from ferrous solutions without chemical addition and with negligible base metal co-precipitation losses.

Detailed bifurcation analysis with a simplified model for advance heavy water reactor system

The bifurcation analysis of fixed points and limit cycles with a simplified mathematical model representing system dynamics of a boiling water reactor has been carried out, specifically parameter values for AHWR is used. The lumped parameter model that includes point reactor kinetics equation for neutron balance in the reactor core and one node model for fuel and coolant thermal hydraulics is used in the analysis. The nonlinearity due to reactivity is considered in the present model; while other nonlinearities due to heat transfer process between fuel–clad and fuel–coolant has been neglected. The system loses its stability via Hopf bifurcation as the system parameters are varied. The continuations of subcritical and supercritical Hopf points show the existence of limit point bifurcations of limit cycles (LPC). The codimension one and codimension two bifurcations of fixed points for the system have been analyzed. The stability of observed limit cycles has been analyzed by Floquet multiplier as well as by Lyapunov coefficient. The pattern of limit cycles and envelopes of limit cycles over the fixed points have been studied by numerical integrations and depicted by time history graphs.

Implementation of a fast running full core pin power reconstruction method in DYN3D

This paper presents a substantial extension of the pin power reconstruction (PPR) method used in the reactor dynamics code DYN3D with the aim to better describe the heterogeneity within the fuel assembly during reactor simulations. The flexibility of the new implemented PPR permits the local spatial refinement of one fuel assembly, of a cluster of fuel assemblies, of a quarter or eight of a core or even of a whole core. The application of PPR in core regions of interest will pave the way for the coupling with sub-channel codes enabling the prediction of local safety parameters. One of the main advantages of considering regions and not only a hot fuel assembly (FA) is the fact that the cross flow within this region can be taken into account by the subchannel code. The implementation of the new PPR method has been tested analysing a rod ejection accident (REA) in a PWR minicore consisting of 3×3 FA. Finally, the new capabilities of DNY3D are demonstrated by the analysing a boron dilution transient in a PWR MOX core and the pin power of a VVER-1000 reactor at stationary conditions.

G4-STORK: A Geant4-based Monte Carlo reactor kinetics simulation code

In this paper we introduce G4-STORK (Geant4 STOchastic Reactor Kinetics), a new, time-dependent, Monte Carlo particle tracking code for reactor physics applications. G4-STORK was built by adapting and expanding on the Geant4 Monte Carlo toolkit. The toolkit provides the fundamental physics models and particle tracking algorithms that track each particle in space and time. It is a framework for further development (e.g.for projects such as G4-STORK).G4-STORK derives reactor physics parameters (e.g.keff) from the continuous evolution of a population of neutrons in space and time in the given simulation geometry. In this paper we detail the major additions to the Geant4 toolkit that were necessary to create G4-STORK. These include a renormalization process that maintains a manageable number of neutrons in the simulation even in very sub- or supercritical systems, scoring processes (e.g.recording fission locations, total neutrons produced and lost, etc.) that allow G4-STORK to calculate the reactor physics parameters, and dynamic simulation geometries that can change over the course of simulation to illicit reactor kinetics responses (e.g.fuel temperature reactivity feedback).The additions are verified through simple simulations and code-to-code comparisons with established reactor physics codes such as DRAGON and MCNP. Additionally, G4-STORK was developed to run a single simulation in parallel over many processors using MPI (Message Passing Interface) pipes.