Development and Testing of a Methodology for Assessing of the Correlation Velocity Measurements’ Accuracy for the Hydrodynamic Investigations of the Turbulent Coolant Flow in Nuclear Reactor Elements

The correlation method of the coolant flow measuring is widely used in research practice including for studying of turbulent coolant flows in scale models of elements of nuclear power plants. The aim of this work was to develop a technique for assessing the effect of noise recorded by a measuring system on the flow rate readings obtained using the correlation method. A technique to assess the effect of noise as well as the relative position and acquisition period of sensors is presented. An insignificant concentration of a salt solution (NaCl or Na2SO4 ) is used as a passive impurity which creates a conductivity gradient of the medium recorded by a conductometric system. Turbulent pulsations at the interface between two concurrent isokinetic flows in a channel with a square cross section are used as the signal source for the correlational algorithm. Paper presents the values of the turbulence′s transport time between spatial conductometers, the results of estimating the spectral power density and band of the recorded signal and also the signalto-noise ratios of the measuring system obtained on their basis which are subsequently used to estimate the confidence interval of the transport time. As a result of measurements the relationship between the confidence interval value and the signal length were obtained. The measurements which were carried out at different relative positions of conductometers make it possible to make a conclusion about an increase in the spectral width of the signal and, as a consequence, a decrease in the length of the confidence interval with increasing of distance between sensors. The presented work is an approbation of this approach for its application as part of an experimental model of a nuclear reactor in order to determine per-channel flow rates in the channels of the core simulator using mesh conductometric sensors taking into account the effect of noise.

Ключевые слова: пространственная кондуктометрия, сетчатый датчик, кондуктометрический датчик, моделирование процессов в ЯЭУ. DOI: 10.21122/2220-9506-2021-12-1-67-74 Introduction Nowadays special attention is paid to experimental studies of the hydrodynamic characteristics of the coolant flow in scale models of elements of nuclear power plants which are aimed at validating the computational models necessary to substantiate the reliability and safety of structures [1]. The question about the control of the indicators of the coolant flow rate in the elements under study when carrying out such experiments is often arised.
At the moment the correlation method for measuring the flow rate of a coolant is widely known the main requirement of which is the presence of a certain passive scalar flow function convectively transferred along with the flowing medium. It allows one to implement correlation measurements using various methods for measuring the properties of the flow: temperature, content of radioactive isotopes, optically distinguishable impurities, etc. [2][3]. This approach assumes determination of the transit time of the disturbance of the measured quantity (the so-called turbulence transport time) between the sensitive elements of the system which are located at some distance from each other.
The paper [4] presents a method of using a conductometric measuring system for the correlation determination of the flow velocity [5][6][7] based on the determination of the cross-correlation function maximum as the time of turbulence transport between sensors. To improve the measurement accuracy and reduce the influence of the most intense pickups a digital low-pass filter was used in the specified work [4] when processing the measuring signal [8]. However the question about assessing the contribution of the "noise" passed by the low-pass filter to the relative error in determining the position of the cross-correlation function maximum remains open.
This problem directly affects the measuring accuracy of the coolant flow rate in the elements of nuclear power plants therefore in the case of research facilities it can have a significant impact on results of experimental data processing which are used to substantiate the reliability and safety of the structures under study.
In correlation measurements of the flow velocity the duration of the realization is very important according to which the cross-correlation function is calculated. Therefore the question about the methodology for choosing of the optimal sampling rate of measuring elements (in this case conductometric sensors) arises, which would satisfy the conditions for maintaining an acceptable measurement accuracy at the maximum sampling rate of readings.
From the above it follows that the purpose of this work was to determine the confidence interval when finding the turbulence transport time between the measuring elements and to estimate the dependence of the confidence interval length on the recording period of the flow rate readings.

Test facility
The general layout of the stand (Figure 1) assumes the organization of an experimental regime with isothermal mixing in an open circulation loop (for studies flows with different concentrations of impurities were used) and non-isothermal mixing when flows with different temperatures were used.
The equipment of the stand makes it possible to create both laminar and turbulent flow regimes (with Re up to 20·10 3 ) at different temperatures, flow rates and tracer concentrations in the coolant flow. The main parameters of the test facility are shown in Table 1.

Experiments
Measurements were carried out in a test model with a square cross section of 50 × 50 mm 2 , the general view of which is shown in Figure 2. The model repeats in general details of the test model geometry from [9]. Wire mesh sensors (WMS) were installed in a zone of intense mixing at different distances from each other.
Measurements were carried out in the flow range up to 2.1 m 3 / h (Re = 20·10 3 ).
The main parameters of the experimental modes are presented in Table 2.
For each mode "non-contrast measurements" were made in which there was no impurity tracer in the mixed flows creating a gradient of the medium conductivity. In this case reading signals′ fluctuations of the mesh sensors were determined only by the noises recorded by the measuring system. These measurements were used to estimate the noise level and to determine the signal-to-noise ratio.   Table 2 Main parameters of the experimental modes Measurements of the coolant velocities in the cells of the wire mesh sensors were carried out in accordance with the procedure described in [4]. The accepted cell numbering is shown on Figure 3.

Method of experimental data processing
In accordance with the method described in [10] estimation of the confidence interval for the turbulence transport time estimate between sensors is reduced to the determination of the normalized rootmean-square error of the cross-correlation function which for the frequency-limited "white noise" has the form [11]: where: R xy (τ) -the value of the mutual correlation function at the moment of time τ; Т -duration of realization, s; B -spectral width of white noise, Hz; M/S, N/S -the signal-to-noise ratio of the first and second WMS, respectively.
Assuming that the measurement signal S (t ) can be considered as frequency-limited "white noise" from expression (1) we obtain: In this case, the band of the measurement signal B can be defined as the frequency interval containing 95 % of the energy of the signal under investigation. The definition of B was made based on the analysis of the energy spectrum of the conductivity pulsations in the mixing zone obtained by the method of the modified Welch's periodogram using the expression: where: W -the power carried by the signal in the entire considered frequency range, W; P( f ) -the function of the spectral power density, W·s; f -frequency of pulsations of electrical conductivity in the measuring area, Hz.
The signal-to-noise ratio (Figure 4) was estimated individually for each sensor as the average quotient of spectral densities of the noise and signal in the band range B: where: S( f ), M( f ) -the functions of the spectral power density for contrast (signal) and non-contrast (noise) mixing, respectively, W·s; f -frequency of pulsations of specific electrical conductivity in the measuring area, Hz.

Analysis of results
As a result of processing the experimental data in accordance with the method described above, the cross-correlation functions of the realizations of the first and second WMS were obtained, the averaged values of the flow rates in the measu-ring cells of the sensors were obtained, and the confidence interval was evaluated when determining the position of the maximum of the crosscorrelation function. Figure 5 shows the graphs of the cross-correlation function with the designated confidence intervals for the mode Re = 20·10 3 at various distances between the sensors. 72 Figure 5 -Normalized electrical conductivities and cross-correlation functions in test with Re = 20·10 3 with different distances between wire mesh sensors for cell 28 Graphs show that with an increase of the distance between the measuring planes of the wire mesh sensor from 40 mm to 640 mm (approximately equal to the scale of the main energy-carrying vortices) a significant decrease in the level of signal correlation is observed.
Numerical estimates of the deviations in determining the position of the function maximum were obtained, as well as the variance of the normalized mean square error of the maximum position. Also average values of the measuring signal band and data on the average flow rate were obtained. The results are recorded in Table 3.
Studies of dependence of the confidence interval length on the duration of realization were carried out on the basis of dividing the initial signal realization (duration 120 s) into windows with durations ranging from 1 to 120 s (120 and 1 window, respectively). Further, in accordance with the technique (1)- (6), the confidence interval of the cross-correlation function maximum position was estimated for each window. Devices andMethods of Measurements 2021, vol. 12, no. 1, pp. 67-74 I.A. Konovalov et al. Приборы и методы измерений 2021. -Т. 12, № 1. -С. 67-74 I.A. Konovalov et al. The results of the experimental data processing for the mode with Re = 20·10 3 at a distance between the WMS of 40 mm are shown in Figure 6 as the dependence of the absolute interval length on the interrogation period of the WMS. The graph shows the exponential dependence of the confidence interval length and consequently, the measurement accuracy on the duration of the realization (time window).
The dispersion of τ max values was taken as a criterion for stability of determining of the crosscorrelation function maximum position for each window within one realization (Figure 7). The graph shows that the variance values do not exceed 10 -6 s 2 . This indicates that the readings of the first and second wire mesh sensors are correlated along all the considered window lengths.

Conclusion
The experimental data analysis made it possible to estimate the confidence interval of the crosscorrelation function maximum position for the correlation method of measuring of the flow rate of the coolant turbulence transport time based on the correlation methodlogy. The obtained value of the relative deviation in determining of the turbulence transport time between the measuring planes of the mesh sensors indicates a low contribution of the "noise" recorded by the conductometric measuring system to the signal under study.
A decrease in the distance between the sensors allows one to obtain a higher value of the correlation coefficient which, in turn, can affect the accuracy of determining of the position of the cross-correlation function maximum when measuring the flow velocity with a low degree of turbulence. However, this entails an increase in the relative error in the study of high-speed flows due to decrease in the total time of turbulence transport between mesh conductometric sensors.
The results of the work can be used to determine the optimal relative position and interrogation period of sensors designed to register changes in time of a certain scalar quantity (temperature, concentration, radioactivity, etc.) when using sensors to determine the flow rate by the correlation method of measuring the speed for correlation flow rate methodology.