Rapport, 2006

This report gives an account of the work performed by the Department of Nuclear Engineering, Chalmers University of Technology, in the frame of a research contract with the Swedish Nuclear Power Inspectorate (SKI), contract No. 14.5-2004103-20040521 and contract No. 2005/1199-20050513. The present report is based on work performed by Carl Sunde, Christophe Demazière, Berit Dahl, Larisa Mileshina and Imre Pázsit, with the latter being the project leader. This report describes the results obtained during Stage 11 and 12 of a long-term research and development program concerning the development of diagnostics and monitoring methods for nuclear reactors. The long-term goals are elaborated in more detail in e.g. the Final Reports of stage 1 and 2. A brief proposal for the continuation of this program in Stage 13 is also given at the end of the report. The program executed in Stage 11 and 12 consists of fiveparts and the work performed in each part is summarized below.
Development of core calculational methods for calculating higher eigenvalues and eigenfunctions
Higher order eigenfunctions and eigenvalues of the diffusion equation, describing a static core, have lost their significance when doing calculations in realistic systems, since there are usually made by nodal methods or other direct numerical techniques. However, there are situations when knowledge of the higher order modes is still useful. Such case is the separation of the global and regional flux oscillations in the case of BWR instability. Another case is the investigation of the stability properties of large inhomogeneous cores, which is usually quantified with the so-called eigenvalue separation, ES=1/k1-1/k0 where k0 and k1 are the fundamental and first higher order eigenvalues, respectively. Numerical codes used for ICMF calculations usually do not have the option of calculating the higher order eigenvalues and eigenfunctions. In idealised systems, these can be calculated with semi-analytical methods (the eigenvalue is determined numerically from a transcendental equation, but the eigenfunctions are simple trigonometric or hyperbolic functions). In order to get insight into the characteristics of the higher order eigenmodes of the neutron flux and the adjoint, these were calculated in a reflected homogeneous system with two-group theory. The adjoint eigenmodes are necessary if an orthogonality property needs to be used, such as the separation of the modes from a flux shape which is a sum of several eigenmodes. At the same time, the so-called numerical noise simulator, developed at the Department, was extended such that it became suitable for the calculation of the higher order eigenmodes, both direct and adjoint ones. This simulator can treat real inhomogeneous cores, with an input deck compatible with that of SIMULATE. It had a static module from the beginning, because it is essential that the algorithm for the dynamic part works with data that belong to a critical core. This static module was now developed further such that it can calculate higher eigenmodes. The simulator was used to calculate the eigenfunctions and eigenvalues in the same model system as the ones in which the semi-analytical calculations were made. Excellent agreement was found between the two methods. Calculations were made in a large and a small system, and the decay of the higher eigenvalues with the order number could be compared. Since the benchmark showed the correct functioning of the simulator, it can be used in the continuation for treating real inhomogeneous systems.
Investigation of reactor kinetics and dynamics in a reflected 2-groups system
The validity and applicability of the point kinetic approximation in noise diagnostic applications was investigated thoroughly in the past. However, most of these investigations were performed in one-group theory and homogeneous, non-reflected systems. It is clear that in such systems the dynamical properties depend on the neutronic coupling in the core through the neutron chains. On the other hand, the conclusions drawn from such models are not necessarily valid in all situations. This is especially the case then the detector is situated in the reflector, where no neutron multiplication takes place, and the dynamics is not determined by the fission process. For example when measuring the reactivity in a core under loading, an extra detector can only be put outside the fuel assemblies already in place. The difference between the core behaviour and the detection in the reflector was observed for instance in the recently finished EU 5th Framework Program MUSE, which showed that the results of pulsed neutron measurements with detectors situated in the reflector could not be interpreted by the conventional theory, based on homogeneous system. It is therefore interesting to investigate the behaviour of reflected systems in two-group theory, and to compare it with the point kinetic behaviour. The investigations of such cases was hindered in the past by the fact that the analytical treatment is cumbersome even in a homogeneous reflected core, and prohibitively complicated in non-homogeneous cores. However, the noise simulator, mentioned above, is a very suitable tool to investigate this question. It was thus used in a 2-dimensional model of a realistic reactor, supposed to run in a subcritical state, driven by a source. It was meant to simulate a core under loading. For the sake of comparison, a small system was also investigated. The full space-frequency response of the system to fluctuations of the source strength were calculated, and compared to the point kinetic response, also calculated by the simulator. The so-called break frequency method of determining the reactivity was also investigated. It was found that the system behaviour deviated from the point kinetic one quite markedly even in the small system, and that the break frequency method showed a relatively large error, that depended on the position of the detector used.
Development of the theory of neutron fluctuations in a system varying randomly in time with the master equation approach
Zero power noise and power reactor noise are two different branches of the field of neutron noise. They depend on different underlying physical processes (branching in a static system, and fluctuations of the cross sections in the reactor, respectively); they are dominating in two different power regimes (low or "zero" power and high power, respectively); and last, but not least, they are treated with two completely different methods. The case of neutron noise in static low power systems is described by the master or Chapman-Kolmogorov equations for the probability distribution of the neutron number, whereas power reactor noise, induced by the fluctuations of the reactor material at high power, is treated by the Langevin technique, for the neutron flux as a stochastic process. For both the completeness of the description, and to describe the co-existence of zero power and power reactor noise in intermediate power systems, a model system with fluctuating parameters was treated by the master equation method. The first and second moments of the neutron distribution were calculated both for chains started by one single neutron, and by a stationary source. A number of new results were obtained, that are described in Section 3. First of all it was shown that the backward equation treatment is not applicable in systems with fluctuating parameters. The concept of criticality had to be generalised to "criticality in the mean", and it was shown that a system whose state fluctuates between a subcritical and a supercritical state, can be made critical with a given special combination of the system properties in the two states and the frequency of system state changes. For systems critical in the mean, the variance diverges exponentially, as opposed to the known linear divergence in static systems. It was also shown that in the case of low power, the fluctuations of the system parameters gives a contribution to the zero power noise, whereas at high power, the zero power noise component, arising from the branching (fission process) is indeed negligible and for small system changes (weak perturbations) the result obtained from the master equation approach agrees with that from the Langevin technique.
Simulations of Feynman- or Rossi-alpha-methods for reactivity measurements of fissile material using MCNP-PoliMi
Simulation of pulse trains of neutrons coming from a multiplying sample, or trains of detector counts caused by such particles, is very useful for studying stochastic reactivity measurement methods, such as the Feymnan- or Rossi-alpha methods, or problems in nuclear material control (safeguards). Such simulations can though not performed by the traditional Monte Carlo codes, since these do not sample the neutron histories with their full statistics. Besides, to speed up the calculations of the mean value, they use variance reduction techniques, which change the properties of the higher order moments, which are needed to the reactivity measurement or safeguards methods. Recently some Monte Carlo codes were extended with the possibility of performing simulations that reconstruct the full statistics of the neutron generation and transport. We have obtained and installed the code MCNP-PoliMi, which has the most general stochastic capability for doing time-dependent calculations for both neutrons and photons. The first application of the code was its use in a safeguards benchmark exercise. In the work performed in Stage 12, pulse trains were generated in systems close to criticality, such that the obtained pulse train was suitable for simulating a Feynman-alpha or Rossi-alpha experiment. This consisted of two steps. First a system had to be designed that was slightly subcritical. For this, the traditional MCNP had to be run, because there is no option of calculating keff with MCNP-PoliMi. In the second stage MCNP-PoliMi was run to generate the pulse train of detector counts. It was shown that with this method, both the Feynman- and Rossi-alpha curves can be reconstructed from a simulation. Hence the code will be used in the next Stage for the study of the accuracy of the Feynman- and Rossi-alpha methods in real inhomogeneous reactors.
Pilot experiment in a moderator with a Cf-252 source in order to test the Feynman- or the Rossi-alpha method
We have obtained, relatively recently, two so-called Cf-252 detectors from Japan. These are actually neutron sources, but constructed in such a way that the californium, whose spontaneous fission generates the neutrons, is put on the surface of one electrode of an ionisation chamber. The ionized fission products give a signal in the detector at each time, indicating thereby the neutron emission times. Such a source/detector is very suitable for neutron coincidence measurements in multiplying systems, such as for reactivity measurements, because one signal (the start signal) is obtained without the consumption of a neutron. Hence one of the detectors was transported to the Belgian nuclear research institute SCK.CEN, for reactivity measurement experiments. These will be described in a later Stage of the research project. The californium source was used in a pilot experiment to measure the time spectrum of neutrons measured in a pure moderator system. The signal from the detector was used as a start signal, and the detected neutron used as the stop signal. This way, the time decay of a neutron pulse in a pure moderator or absorbing material can be measured. Measurements were made with and without a moderator between the source and the detector, and a clear difference in the time spectrum was found.

core computational methods

reactor kinetics and dynamics

neutron noise analysis

reactivity measurements

Chalmers, Teknisk fysik, Nukleär teknik

Chalmers, Teknisk fysik, Nukleär teknik

Chalmers, Teknisk fysik, Nukleär teknik

Chalmers, Teknisk fysik, Nukleär teknik

Chalmers, Teknisk fysik, Nukleär teknik

Övrig annan teknik

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