A unified theory of zero power and power reactor noise via backward master equations (original) (raw)
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Theory of Neutron Noise in a Temporally Fluctuating Multiplying Medium
Nuclear Science and Engineering, 2007
Neutron fluctuations in a constant multiplying medium (zero power noise) and those in a fluctuating medium (power reactor noise) have been traditionally considered as two separate disciplines that exist in two opposing limiting areas of operation (low and high power, respectively). They have also been treated by different mathematical methods, i.e., master equations and Langevin equation, respectively. In this paper we develop a theory of neutron fluctuations in a medium randomly varying in time, based on a forward-type master equation approach. This method accounts for both the zero power and the power reactor noise simultaneously. Factorial moments and related quantities (variance, power spectrum, etc.) of the number of the neutrons are calculated in subcritical systems with a stationary external source. It is shown that the pure zero power and power reactor noise results can be reconstructed in the cases of vanishing system fluctuations and high power, respectively, the latter being a nontrivial result. Further, it is shown that the effect of system fluctuations on the zero power noise is retained even in the limit of vanishing neutron number (reactor power). The results have thus even practical significance for lowpower systems with fluctuating properties. The results also have a bearing on other types of branching processes such as evolution of biological systems, germ colonies, epidemics, etc., which take place in a time-varying environment.
Some properties of zero power neutron noise in a time-varying medium with delayed neutrons
Annals of Nuclear Energy, 2008
ABSTRACT The temporal evolution of the distribution of the number of neutrons in a time-varying multiplying system, producing only prompt neutrons, was treated recently with the master equation technique by some of the present authors. Such a treatment gives account of both the so-called zero power reactor noise and the power reactor noise simultaneously. In particular, the first two moments of the neutron number, as well as the concept of criticality for time-varying systems, were investigated and discussed. The present paper extends these investigations to the case when delayed neutrons are also taken into account. Due to the complexity of the description, only the expectation of the neutron number is calculated. The concept of criticality of a time-varying system is also generalized to systems with delayed neutrons. The temporal behaviour of the expectation of the number of neutrons and its asymptotic properties are displayed and discussed.
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Annals of Nuclear Energy, 1989
~Neutron transport problems in a random medium are considered by defining a joint Markov process describing the fluctuations of neutron population and the random changes in the medium. Backward Chapman-Kolmogorov equations are derived which yield an adjoinI transport equation for the average neutron density. It is shown that this average density also satisfies the direct transport equation as given by the phenomenological model.
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Annals of Nuclear Energy, 2000
The general theory of linear reactor kinetics and that of the induced neutron noise is developed for systems with varying size, i.e. in which the position of the boundary fluctuates around a stationary value. The point kinetic and adiabatic approximations are defined by a generalisation of the flux factorisation, and the full solution of the general problem with an arbitrarily fluctuating boundary is given by the Green's function technique. The correctness of the general solution is proven both generally and also by considering the simple case of a 2-D cylindrical reactor with a fluctuating radius, in which case a direct compact solution is possible.
A MOC-based neutron kinetics model for noise analysis
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A 2-D noise model is implemented in the deterministic reactor code APOLLO3 R to simulate a periodic oscillation of a structural component. The Two/Three Dimensional Transport (TDT) solver, using the Method of Characteristics, is adopted for the calculation of the case studies, constituted by a moving detector and control-rod bundle. The periodic movement is built by properly linking the geometries corresponding to the temporal positions. The calculation is entirely performed in the real time domain, without resorting to the traditional frequency approach. A specifically defined dynamic eigenvalue is used to renormalize in average the reactivity over a period. The algorithm is accelerated by the DP N synthetic method. For each cell of the domain, the time values of fission rates are analysed to determine the noise extent. Moreover we propose a systematic approach to the definition of the macroscopic cross sections to be used in dynamical calculations starting from library data. As an aside of our work we have found that even in static calculation this approach can produce significant changes.
Calculation of the Parameters of Stochastic Neutron Kinetics in Zero Power Nuclear Reactors
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. SERIES: NUCLEAR AND REACTOR CONSTANTS, 2019
The majority of neutron-physical problems of nuclear power plants design can be solved on the basis of various approximations to the Boltzmann transport equation in terms of the averaged characteristics of the reactor: the effective multiplication factor, neutron flux, average neutron lifetime, etc. However, the neutron chain reaction itself is always stochastic. There are situations in which the stochastic nature of the chain reaction cannot be ignored. This is the so-called “blind” start-up problem with a weak external neutron source, the work of physical assemblies of “zero” power, the analysis of the reactivity noise of such assemblies, etc. Despite the well-developed theoretical basis for the stochastic description of the behavior of neutrons in a nuclear reactor, there are still not enough calculation algorithms and programs for stochastic kinetics analysis. The paper presents two computational algorithms for point reactor model, which are developed on the basis of the theory ...
1985
The fundamental features of thermohydraulic fluctuations characterising the subcooled boiling in the PWR core have been investigated with the aim of boiling detection via neutron noise analysis. The relationship between different formulations of the model for thermohydraulic fluctuations and the inportant contributing phenomena has been exanine6 along the time scale of interest. A quasi-adiabatic one-channel coolant nodel has been derived for the description of fluctuations of flow variables. ±he effects of thermal nonequilibriuF feature of the two-phase flow on the ther~nydraulic fluctuations and the relationship between their statistical characteristics and engineering parameters cf the flow have been analysed. KEYIqORDS PWR, subcooled boiling, neutron noise, nonequilibriun~ two-phase flow, therzohy~raulic fluctuations, one-channel n odel.
On the Zero-Neutron Density in Stochastic Nuclear Dynamics
Dynamics
In this short paper, we compare the deterministic model for the nuclear reactor dynamic (Hetrick, 1993) with the stochastic model (Kinard and Allen, 2004). Our numerical results show coincidences between the deterministic model and the mean of the stochastic paths, although, as already observed by other authors, there is alarge amount of dispersion between the individual paths. Notably, we always observe that the neutron density approaches zero within a short time. In this paper, we investigate this question; more concretely, we study the mean-extinction of the neutron density. The technique used here first builds the backward Kolmogorov differential equation and then solves it numerically using the finite-element method with FreeFem++. Our results confirm that in a very short time the neutrons disappear although later they recover probably due to the external source.
Modelling of a vibrating reactor boundary and calculation of the induced neutron noise
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Three different models of a moving (vibrating) reactor boundary in time-dependent diffusion theory are investigated. The models are: (a) a localized absorber of variable strength at the boundary (equivalent to a perturbational treatment); (b) a time-varying extrapolation length; (c) explicit treatment of the moving boundary with a new transformation technique. The induced neutron noise was calculated in first order of the perturbation parameter both exactly and in the adiabatic approximation. All three models lead to equivalent results, confirming the applicability of perturbation techniques in treating moving perturbations (e.g. vibrating control rods). Application of the adiabatic approximation in model (c) required the extension of the Henry formalism, i.e. the use of orthogonality relations expressed as integrals over the system, to cases with non-constant system volume. The incentives for investigating a time-varying boundary arose from problems related to vibrating control rods; however, the results have some general relevance for systems with a varying volume such as gaseous core or liquid fuel reactors.