Space-dependent Transfer Function Research Articles

Reactor space-dependent transfer function at low frequency approximated by neutron diffusion theory perturbation

The space-dependent transfer function defines the response of the neutron flux in reactor core to a periodic perturbation at a point in the core. To compute it at low excitation frequency for the ZED-2 zero power reactor, we introduce a perturbation method. The ZED-2 reactor is used to investigate the properties of CANDU® lattices. In CANDU®, the low frequency noise (0.1–5 Hz) allows one to study mechanical oscillations of structures in the core, the reactor control systems, as well as delayed neutron effects. The method for computing the ZED-2 transfer function was inspired by the results of measurements in the ZED-2 reactor. They showed that, at low excitation frequency, the imaginary part of the space-dependent transfer function is, up to normalization by the steady-state neutron flux, the same over all the core, while the real part varies significantly with the position of the flux detectors or the perturber in the core. This method was then found useful to compute the response of the ZED-2 reactor core to low frequency perturbations where other methods were found hard to apply. This opens the path to generalization of this method to CANDU® reactors. A rough calculation shows for which types of thermal reactors this perturbation theory is not worth applying. Also, the perturbation theory introduced here can be applied to other problems described by linear operators having a second smallest eigenvalue larger in absolute value compared to the perturbation. A further development, using the perturber induced reactivity, was found to make the core response normalization unnecessary. The theoretical basis of that method is outlined.

Squeeze free space with nonlocal flat optics

There has been substantial interest in miniaturizing optical systems by flat optics. However, one essential optical component, free space, fundamentally cannot be substituted with conventional local flat optics with space-dependent transfer functions, since the transfer function of free space is momentum-dependent instead. Overcoming this difficulty is important to achieve the utmost miniaturization of optical systems. In this work, we show that free space can be substituted with nonlocal flat optics operating directly in the momentum domain. We derive the general criteria for an optical device to replace free space and provide two concrete designs of photonic crystal slab devices. Such devices can substitute much thicker free space. Our work paves the way for the utmost miniaturization of optical systems using a combination of local and nonlocal flat optics.

Image formation in track-etch detectors—III. The space-dependent transfer function in thick detectors

Abstract A model of the imaging process in autoradiography with solid state nuclear track detectors has been employed to determine the space-dependent transfer function (optical density vs distance). The dependence of line-, plane- and edge-spread functions on the particle range, critical angle of etching, removed detector layer, track size and track density for thin detector foils was studied. Asymmetry of the edge-spread function at high exposure is revealed and explained.

Image formation in track-etch detectors—II. The space-dependent transfer function in thin detectors

A model of the imaging process in autoradiography with solid state nuclear track detectors has been employed to determine the space-dependent transfer function (optical density vs distance). The dependence of line-, plane- and edge-spread functions on the particle range, critical angle of etching, removed detector layer, track size and track density for thin detector foils was studied. Asymmetry of the edge-spread function at high exposure is revealed and explained.

Investigation of the space-dependent global component of the neutron-to-inlet coolant temperature fluctuations transfer function

By using the one-dimensional linearized stochastic neutronic equation in the diffusion approximation for an axially finite, homogeneous unreflected core with the slowing down of fast neutrons taken into account by an appropriately chosen slowing-down kernel, we have managed to split the thermal neutron noise field into a global and local component. By applying our theory to water-cooled and moderated cores and by disregarding feedback effects, we derive explicit expressions for the global component of the neutron-to-inlet coolant temperature fluctuations transfer function, both in the exact, space-dependent model and in the point model approximation. By numerically evaluating the gains and phases of the transfer functions for different core heights and coolant velocities, we conclude that considerable differences arise between the space-dependent and the point model, even for small cores. In particular, in the case of relatively large cores, we show that the phase-to-frequency relationship of the space-dependent transfer function can be used to calculate the velocity of the coolant through the core.

Investigations of the influence of feedback and coupling effects on neutron noise in a nuclear reactor

The neutron power spectral density of power reactors is influenced by many parameters. These can be summarized into two groups. The first being reactivity perturbations whose statistical characteristics are unknown in most cases, and the second being various feedback and spatially dependent transfer functions. In the case of all effects being simultaneously present, which is the case for power reactors, it is extremely difficult to understand the spectra in detail. However some of the reactivity perturbations and their influence on the neutron noise in large power reactors can be simulated in a zero power reactor. In this way the effects can be studied more flexibly without the interference of other effects. This paper describes investigations of the influence of a known feedback—namely a control loop with known transfer function—on the spectra of neutron chamber signals. Theoretical formulas for the spectra are derived using the point reactor model. These formulas were verified by noise measurements in a zero power reactor. In addition, special attention is given to the noise generated by the control loop. The influence of this feedback noise on the spectra is verified experimentally. In large reactors space dependent transfer functions must be taken into account. As a first approximation to handle the spatial dependence, the afore-mentioned investigations were extended to the two-point reactor model. Corresponding experimental work was done for the Argonaut Reactor Karlsruhe (ARK) with a symmetrical two-slab core loading. As an application to a more realistic situation coolant boiling in a BWR has been investigated. The boiling must be considered as a feedback mechanism as well as an external reactivity perturbation. In order to simulate the steam bubble content, nitrogen gas was injected into the water moderator of the ARK. By modulating the total gas flow according to the instantaneous reactor power the feedback effect was simulated. The gas flow produced a bandlimited, white reactivity noise. The upper break frequency could be used to determine the travelling time of the bubbles through the core.

Some considerations on neutron instrumentation requirements for malfunction diagnosis in power reactors using noise analysis techniques

To determine the response of a neutron detector to local perturbations in a typical fast breeder reactor detailed numerical calculations of space-dependent source transfer functions were performed for the German prototype reactor SNR 300. The small local reactivity perturbations considered here were replaced by a neutron source modulation for simplicity reasons. The source transfer function was obtained from the Laplace transform of the time-dependent inhomogeneous diffusion equation. The 26 groups KFK INR cross section set and one-dimensional cylinder geometry were used for the numerical solution of the diffusion equation in the frequency domain. The detector was treated as an integral part of the reactor. The detector materials B-10, U-235 and Pu-239 were chosen for comparison. The positions of source and detector were independently varied within the core and blanket of the reactor. From the space-dependent transfer function of the reactor-detector system and the general theory, relating power spectral densities of neutron detector signals and reactivity perturbations for the point reactor model, the neutron instrumentation requirements for detecting local perturbations in a fast reactor were deduced. It was found that in the present class of fast breeder prototype reactors no in-core neutron detectors will be needed for a continuous inspection using noise analysis techniques to detect local perturbations.

Spatial Effects in Transfer-Function Measurements of Large Nuclear Reactor Power Systems

The importance of spatial effects in proposed transfer-function measurements in large, fast reactors was investigated. A general theoretical formulation of space-dependent transfer-functions is derived which includes the effects of input and output device spatial characteristics. A quantitative indication of spatial effects is given by numerical simulation of measured transfer-functions in a Liquid Metal Fast Breeder Reactor (LMFBR). An approximate method of solution using the Galerlkin's procedure was developed in conjunction with the investigation.

Excitation of Higher Space Modes in Reactor Transfer Function Measurement (Part I)

An explicit time-dependent two-group flux, expressed by a series of space modes, is established when a forced oscillation is applied to a reactor. The self-consistent time-dependency method developed here minimizes necessary mathematical transformations and enables one to clearly visualize the physical reasons why the higher space modes are only excited at high frequencies. The conditions necessary for a particular higher space mode to be appreciably excited and detected are discussed in detail. The results show that the major factor is due to the increase of the input frequency as compared with the decay constants of several higher space modes at high frequencies. This method was applied to the NORA reactor for which the space-dependent transfer functions have been measured. Results of the calculations closely agree with the published experimental results as well as with theoretical gain and phase shift curves obtained by the conventional modal expansion-Laplace transform method. The relative amplitude of each higher space mode with respect to the fundamental mode shows the rate of convergence of the modal expansion method.

A New Synthesis Method for Pulsed-Neutron Experiments

A new method of synthesis of the time-dependent behavior of a neutron pulse or of the space-dependent transfer function is given which combines analytical time dependence with numerical (or analyti...

On the synthesis of space dependent transfer functions

A method to synthesize the space dependent transfer functions of a class of dynamic systems from the frequency response data is presented. The systems that give rise to space dependent transfer functions are those which may be represented by partial differential equations in which the space and time dependences of the dependent variables are not separable. The method consists of representing the system transfer function as a series of suitable space modes with frequency dependent coefficients. The resulting synthesis procedure is straightforward and is applicable to many practical cases of interest. The method is applicable to both minimum and nonminimum phase transfer functions.

Calculating Space-Dependent Reactor Transfer Functions Using Statics Techniques

A method has been developed by which statics techniques can be used to calculate source transfer functions in the multigroup, multidimensional approximation. With the flux resolved into steady and fluctuating components, the time-dependent neutron balance equations are satisfied by the fluctuating part alone. Assuming that the external source and the flux response are sinusoidal, the original time-dependent equations transform into a set of complex equations dependent on space and frequency but independent of time. Separating the equations into real and imaginary parts yields coupled, inhomogeneous differential equations (two for each group). These can be solved by well-known statics techniques for the real and imaginary components φR and φI of the complex amplitudes of the fluxes, in turn yielding the gain and phase shift for each frequency of interest.This method was applied to the NORA reactor for which the space-dependent transfer function had been determined experimentally. The two-group telegrapher's equations were programmed for one-dimensional cylindrical geometry and the difference equations solved by direct matrix inversion and also by interative techniques. Results of the calculations closely reproduce the reported experimental results for gain and phase shift.