The determination of in-core parameters in a BWR is a very complicated and challenging task which has a strong impact on both reactor safety and reactor operation. Today, many methods exist which are able to provide useful information about various important quantities inside the reactor core, such as mass flow rate, core water level, etc. Among other methods, various computational methods are widely used in many problems of interest. However, such methods are usually not robust and accurate enough since most of them rely on different corrections and assumptions. As a result, measurements are needed to be performed in order to validate such calculations. In addition, experimental measurements also provide an opportunity to detect the deviation of those parameters from normal conditions (so-called on-line monitoring of parameters) which usually cannot be undertaken by existing computational methods.

In the BWR reactor vessel, fuel heats coolant to boiling, producing bubbles. Since water is not only the coolant but also the neutron moderator, the amount of bubbles — the void fraction — affects the reactivity of the core. Void fraction varies in BWRs depending on many factors. Generally speaking void fraction increases with height.

The void fraction in a BWR has been intensively studied in the past. However, even today, there are no direct methods available which can supply reliable information about the void content. The main reason for the absence of such methods is a highly radioactive environment of the reactor core where most measuring instrumentation fails to function properly.

Due to the fact that the local void fraction can only be calculated, there is a need, expressed by power companies such as in Sweden, for an experimental method for verification purposes. Further, since the calculation methods have their own uncertainty, an accurate method of determining the local void fraction by measurements could lead to decreasing operational margins and hence improved safety and economy. As a consequence, there is an on-going effort to develop methods of experimental determination of the local void fraction with various techniques (for example, [2],[4],[5]).

Indirect methods, such as neutron noise-based methods, can be one possible solution to the difficulty of measuring void fraction directly. Such methods have several advantages compared to other existing methods. The most important one is that these methods require measurement of only one single quantity, that is, the thermal neutron flux. Existing nuclear plants are usually well-equipped with thermal neutron detectors which are meant to perform such measurements and, thus, no additional installation of equipment is necessary. Moreover, one has also a possibility to set up online monitoring of void fraction based on neutron noise methods, and no reactor shutdown is required. The main principle of neutron noise methods is based on a strong correlation between different material properties of reactor internals and the corresponding neutron flux. If, for instance, due to some reason, these properties start changing, one can immediately identify those changes from neutron flux.

The theoretical dependence of neutron noise on void fraction was established as long ago as the mid-1970’s. A straightforward way to understand their relationship will be to calculate the neutron noise analytically. In order to do such a calculation one can, for example, set up a simple model of how the neutron noise (due to a boiling process) is created or induced. The simplest known model of a boiling process, which is usually used in reactor noise calculations, is so-called propagating density perturbation. In this model, it is often assumed that a fluctuation or perturbation is a white noise at the inlet of the reactor and it propagates upwards in axial direction through the whole core. In such a case, the induced neutron noise can be represented as a convolution between this perturbation and the corresponding transfer function of the reactor which characterizes the transfer properties of the reactor and usually depends only on the static cross-sections of the unperturbed system. More specifically, the transfer function tells one how the system (in our case a reactor) interacts with different perturbations entering the system. One can measure the result of this interaction by neutron detectors.

All of these manipulations can be done analytically, and one can obtain an analytical expression for the auto power spectral density (APSD) of the induced noise, and plot it. Such calculations (as well as many experiments) show that such an APSD consists of two components: the so-called global and local components. The first one usually characterizes the behaviour of the reactor globally (due to global perturbations) whereas the local one shows how the reactor behaves due to local perturbations such as a boiling process. In the simplest case (homogeneous reactor) one can show that the global solution has a sine shape and the local one has an exponential shape in space. The curvature of the global solution is described by what we call global root of the characteristic equation whereas the local by the local root. Both roots are functions of static cross-sections (in first-order perturbation theory) and frequency [1]. This is the local component of the neutron noise which we need to pay attention to if we want to investigate the dependence of the neutron noise on void fraction. The local root is basically equivalent to what we have called the detector field of view ‘I’. Since the local root is a function of cross-sections, it also becomes a function of void fraction (since cross-sections depend on void fraction).

Once one obtains an analytical expression for the local component of the neutron noise, one can immediately notice that it is constant up to a certain frequency, and then experiences a break which is proportional to the perturbation velocity and inversely proportional to the field of view (or curvature of the local component). In turn, the velocity is also a function of void fraction due to the mass conservation equation. Thus, the break frequency will be a function of the void fraction.

Intuitively, such a dependence of break frequency on void fraction becomes obvious if one takes a closer look at how such neutron noise due to boiling is created and measured. What we measure with a neutron detector is the fluctuations of neutrons in the reactor core. Since the neutron population strongly depends on the moderator/coolant properties, any change in the density of the moderator (in particular, due to boiling which can be represented as the creation and collapse of bubbles) will be seen in the changes of neutron population. The latter means that with a neutron detector, we actually measure the fluctuations in the bubble population around the detector or, strictly speaking, the number of bubbles which pass by the detector. Since the bubbles travel with a certain finite speed (which is a function of void fraction) one can register those bubbles only with a certain limited frequency (limited by the bubble velocity) and cannot register something above this frequency (assuming that there are no other noise sources in this frequency region). This leads to a drastic decrease in the APSD and thus creates the above-mentioned break.

However, this is only one part of the whole story. There is a second process/mechanism which also affects the break frequency and is related to the detector field of view (or local root of the neutron noise). The size of this field of view is strongly defined by the static cross-sections (primarily by the removal cross section) which are also functions of void fraction. As result, the higher one goes upwards in the reactor, the greater is void fraction (less moderation) and the smaller ‘I’ (more localized and fewer bubbles are covered by the detector field of view). The main consequence of this process is that the break of APSD will be shifted to higher frequencies. One can also make a similar conclusion if one explores an analytical expression for the local component [1].

This process is simulated numerically as follows. As a starting point, 1320-second long signals imitating the time-history of the bubbly flow in a BWR heated channel are generated. The corresponding normalized neutron noise is generated by spatial convolution between the void fraction signals and the local component of the transfer function. The measurements are simulated in eight different axial levels within the same radial position. The corresponding spectral densities are calculated by using a standard Fast Fourier Transform. The result is given in Figure 2, where one can clearly observe the characteristic break frequency of the APSD which monotonically increases with increasing axial elevation. In addition, the break frequencies for all eight APSDs as well as their amplitudes were numerically estimated by fitting the corresponding curves to a so-called Lorentzian coefficient, that is, a function showing a break frequency.

The estimated break frequencies are substituted into an equation [elaborated in 4] to calculate the void fraction in eight detector locations. The result is plotted in Fig. 3 (red line). For comparison, the ‘true’ void profile (the output from Monte Carlo simulation calculated by time averaging over the corresponding set of void fraction signals) is also given in Fig. 3 (blue line). From the two figures, one can conclude that the void fraction calculated from the simulation qualitatively reproduces the behaviour of the true one. Quantitatively, there are systematic deviations between the two curves, which is not surprising, since this is a first attempt for void fraction determination in a simple model.

To get realistic results (or results which are applicable to real cases) one should estimate the local component in a slightly different way than what we did in our simulations. In the method, a relationship between the local void fraction and the local void velocity is required. For reasons of simplicity, in the studied case a simple correlation based on the mass conservation equation was used, and the same velocity for both liquid and gaseous phases was assumed. From a practical point-of-view, such an approximation is far from being realistic and, thus more advanced and complex modelling is necessary.

However, to further improve the method, more adequate and realistic correlation between the void fraction and the local component, rather than simple homogeneous CASMO calculations, is needed and will be accomplished in the future work in 2013. In view of the complicated geometry of a real BWR core structure at the fuel assembly and pin level, the local component of the neutron physical transfer of the core can be estimated by Monte Carlo methods [6].

## References

1. G. Kosály, "Noise investigations in boiling-water and pressurized water reactors", Progress in Nuclear Energy, vol.23, pp.145-199, 1980.

2. J. Loberg, M. Österlund, J. Blomgren and K.-H. Bejmer, "Neutron Detection Based Void Monitoring in Boiling Water Reactors", Nuclear Science and Engineering, Vol. 164, pp. 69-79 (2010).

3. I. Pázsit, and C. Demazière, Noise Techniques in Nuclear Systems, chapter in Handbook of Nuclear Engineering- Reactors of Generation II, Springer, New York & USA (2010).

4. V. Dykin C. Montalvo Martín, H. Nylen and I. Pázsit, Ringhals Diagnostics and Monitoring Annual Research Report 2012, CTH-NT-269/RR-17, December 2012.

5. V. Dykin and I. Pázsit, "Simulation of in-core neutron noise measurements for axial void profile reconstruction in Boiling Water Reactors", Nuclear Technology, Vol. 183 September 2013, pages 354-366.

6. T. Yamamoto, "Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise", Annals of Nuclear Energy, vol. 58, pp. 72-79, 2013.

Victor Dykin, Chalmers University of Technology Department of Applied Physics, Division of Nuclear Engineering, Fysikgården 4, SE-412 96, Gothenburg, Sweden