Signal model establishment and measurement accuracy evaluation of a non-vortex generator permanent magnet sodium flowmeter

Abstract

The present application is aimed at permanent magnet sodium flowmeter without vortex generator, experiments are carried out, and voltage fluctuation signals of multiple pairs of cross-correlation electrodes are collected; the signals are pretreated by using the method of multiple electrode same direction subtraction and low frequency suppression; then, cross-correlation analysis is carried out to obtain transit time sequence, and a sliding average model of the transit time sequence is established; the model of the transit time sequence is deeply analyzed, and is decomposed into steady component and fluctuation component; the steady component and the fluctuation component are respectively related to two indexes determining the accuracy of in-situ calibration measurement, namely indication error and repeatability error; the parameter law of the steady component and the fluctuation component is analyzed, and ideal values of instrument correction coefficients and mean values of transit time of each measurement are respectively obtained; according to the upper limit of indication error introduced by the correction coefficient and the upper limit of repeatability error of transit time, the upper limit of cross-correlation measurement accuracy is obtained through analysis.

Description

Technical Field

[0001] This invention relates to the field of flow detection, and in particular to the signal model establishment and measurement accuracy evaluation of a permanent magnet sodium flow meter without a vortex generator. Background Technology

[0002] Energy remains one of the most pressing global concerns. As human society becomes increasingly modernized and intelligent, the consumption and demand for energy naturally expand. Nuclear power, with its stable, reliable, high-quality, and high-density operation, is gaining increasing attention as a baseload energy source. Global nuclear power development has generally progressed in the form of thermal neutron reactors—fast neutron reactors—fusion reactors. Thermal neutron reactor technology is largely mature, fast neutron reactors are under development, and fusion reactors are still in the exploratory stage. Thermal neutron reactors use uranium-235 as fission fuel and pressurized water reactors as the main reactor type. However, natural uranium-235 reserves are limited, accounting for only 0.711% of natural uranium resources, and the utilization rate of uranium resources by thermal neutrons is extremely low, only 1% to 2%. Therefore, to ensure the sustainable development of nuclear power and improve the utilization rate of uranium resources, fourth-generation nuclear power technology—fast neutron breeder reactors (referred to as "fast reactors")—has emerged. Fast reactors use over 99% uranium-238, which absorbs neutrons and transforms into fissile plutonium-239. The fission of plutonium-239 then further transforms uranium-238, creating a continuous multiplication process that achieves a high utilization rate of 60%–70%. Among fast reactors, the sodium-cooled fast reactor is currently the fastest-developing and relatively most mature type, and it is also the type that various countries are vigorously constructing.

[0003] Sodium-cooled fast reactors employ a three-loop design: sodium-sodium-water. Liquid alkali metal sodium possesses excellent properties such as low melting point, high boiling point, low viscosity, good heat transfer, small neutron absorption cross-section, and non-toxicity, making it suitable as both a coolant and heat transfer agent in fast reactors. To ensure rapid and sufficient core cooling and timely heat exchange, and to guarantee the safe operation of the fast reactor, real-time monitoring of the liquid sodium flow rate in both the primary and secondary loops is necessary. Permanent magnet sodium flow meters are commonly used for real-time monitoring of liquid sodium flow rate due to their simple structure, wide measuring range, low pressure loss, and fast response speed. Permanent magnet sodium flow meters (also known as ordinary permanent magnet sodium flow meters) operate based on Faraday's law of electromagnetic induction and mainly consist of a flow tube, magnets (permanent magnet, yoke, magnetic poles), and induction electrodes. The electromotive force E output by the induction electrodes is positively correlated with the fluid velocity v, magnetic flux density B, and pipe inner diameter D, thereby measuring the flow velocity of the sodium fluid.

[0004] Under normal operating conditions, the magnetic flux density B remains constant, and the amplitude U of the DC voltage signal output through the induction electrode... V1 The average flow rate in the pipe can be accurately calculated, which is the displayed flow rate Q of the permanent magnet sodium flow meter. iHowever, liquid sodium operates at temperatures between 250 and 550°C year-round. Permanent magnet sodium flow meters, operating in this high-temperature, high-neutron-radiation environment, experience demagnetization of their permanent magnets, leading to a decrease in magnetic flux density and causing the displayed flow rate to deviate from the standard value. Therefore, periodic calibration is necessary to correct the instrument coefficient of the permanent magnet sodium flow meter to eliminate this deviation. However, permanent magnet sodium flow meters installed within the reactor core are not removable. For example, a permanent magnet sodium flow meter in a primary loop fuel pipeline exposed to neutron radiation cannot be removed for offline calibration. Therefore, permanent magnet sodium flow meters need to have in-situ calibration capabilities.

[0005] In-situ calibration refers to calibrating an instrument without altering its operating state—that is, without removing it from the measurement site—while maintaining its original working condition. This is achieved through specific technical means, including modifying its structural components and employing different signal processing methods, to determine its measurement accuracy. For permanent magnet sodium flow meters, in-situ calibration involves processing the output signals of two pairs of cross-correlation electrodes using the cross-correlation method, calculating their delay (transit time), and obtaining the cross-correlation velocity and flow rate. This is then corrected to a standard flow rate and compared with the displayed flow rate obtained from the amplitude of the electromagnetic induction signal output by a pair of sensing electrodes to determine the accuracy of the flow rate measured by the electromagnetic induction principle. This creates a permanent magnet sodium flow meter with in-situ calibration capabilities (different from the ordinary permanent magnet sodium flow meter mentioned earlier). Depending on whether there are fluid obstructions in the flow tube, permanent magnet sodium flow meters are specifically divided into those with embedded vortex generators and those without.

[0006] For small-diameter pipes (e.g., pipes with a diameter of less than 100 mm), since the pipe is not the main pipe for sodium flow, obstructions can be added to the flow tube to generate fluid disturbance in a certain direction, providing a cross-correlation signal with a certain signal-to-noise ratio for cross-correlation measurement without affecting the measurement of the induced signal. This is called a permanent magnet sodium flow meter with an embedded vortex generator.

[0007] For large-diameter pipes (e.g., pipes with diameters of 150mm, 200mm, and 300mm), since these pipes are the main pipelines for sodium flow and have large flow rates, configuring semi-circular or other obstructions to enhance fluid disturbance would cause severe pressure loss, increase dynamics, and pose a risk of obstruction detachment. Therefore, in-situ calibration can only be performed using the fluid's own disturbance, a technique known as a vortex-free permanent magnet sodium flow meter. Due to the absence of a vortex generator, the signals from the two pairs of cross-correlation electrodes are weak; simultaneously, due to magnetohydrodynamic phenomena, the flow velocity is high, resulting in a high magnetic Reynolds number R0. m When the value is greater than 1, the magnetic field and the flow field will interact strongly, and the magnetic field and the flow field will be distorted, resulting in severe nonlinearity and large repeatability error in the measurement results.

[0008] Chinese Invention Patent 1 (Xu Kejun, Yu Xinlong, Huang Ya, Wu Wenkai. A Nonlinear Correction Method for In-situ Calibration of Permanent Magnet Sodium Flowmeter Based on Signal Frequency Band Selection, Application No.: 202011085046.X, Application Date: 2020.10.12) addresses the nonlinear characteristics between cross-correlation flow rate and standard flow rate during in-situ calibration of a large-diameter permanent magnet sodium flowmeter without vortex generator in a long straight pipe section. It theoretically analyzes the influence of the magnetic field on the flow field, obtaining an "M-shaped" velocity distribution within the pipe. Combining the phase frequency characteristics of the cross-power spectral density of the two signals, the cross-correlation flow rate distribution of each frequency signal is calculated, leading to a turbulence distribution model of the pipe cross-section. This model shows that the cross-correlation flow rate of signals in certain frequency bands is close to the standard flow rate. Therefore, a nonlinear correction method based on frequency band selection is proposed, using signals from different frequency bands at different flow rates to correct the nonlinear characteristics. For ease of engineering application, only two signal frequency bands are selected for nonlinear correction across the entire flow range; one correction frequency band is selected for larger flow rates, and another for smaller flow rates.

[0009] Chinese Invention Patent 2 (Xu Kejun, Yu Xinlong, Huang Ya, Wu Wenkai. A Method for In-situ Calibration of Permanent Magnet Sodium Flowmeter Based on Error Correction using Cross-correlation Method, Patent No.: 202110187793.2, Application Date: 2021.02.18) addresses the repeatability and nonlinearity errors of cross-correlation flow in in-situ calibration of large-diameter permanent magnet sodium flowmeters without vortex generators in long straight pipe sections, proposing a method for in-situ calibration based on error correction. Starting from the perspective of repeatability error correction, the influence of low-frequency signals on repeatability error is qualitatively analyzed, and the repeatability error under different low-frequency bands is quantitatively calculated. It is found that as the filter cutoff frequency increases, the repeatability error is greatly improved. Based on this, three key technical points for repeatability error correction are given to optimize the correction. After repeatability error correction, nonlinear error can be corrected using a simple quadratic function model, thereby achieving in-situ calibration. This method uses signals within the same frequency band for in-situ calibration over a large flow range, and the implementation steps are simple and highly operable. Summary of the Invention

[0010] Chinese Invention Patent 1, starting from the interaction between the magnetic field and the flow field, established a turbulent average velocity model and proposed a nonlinear error correction method based on frequency band selection; Chinese Invention Patent 2 established a turbulent pulsating velocity model and proposed a repeatability error correction method based on low-frequency suppression. This invention will start with the quality of the output signal from the cross-correlation electrode itself, establish a mathematical model of the processed signal, reveal the characteristics of the processed signal, reflect the quality of the processed signal, and determine its relationship with measurement accuracy in order to predict the upper limit of the accuracy of in-situ calibration and provide direction for in-situ calibration.

[0011] The specific technical solution is as follows:

[0012] Because the cross-correlation electrode output signal of the vortex generator-free permanent magnet sodium flowmeter has complex composition and an extremely wide frequency range, and because the cross-correlation signal consists of two delayed signals, it is impossible to directly model the cross-correlation electrode output signal. Therefore, cross-correlation analysis is first performed on the two pairs of cross-correlation signals to obtain the transit time, and then the transit time series is modeled.

[0013] This invention targets a permanent magnet sodium flowmeter without a vortex generator. Experiments are conducted to collect voltage fluctuation signals from cross-correlation electrodes. Cross-correlation analysis is performed on the output signals of two pairs of cross-correlation electrodes to obtain a transit time series, and a moving average (MA) model of the transit time series is established. Alternatively, after collecting voltage fluctuation signals from multiple pairs of cross-correlation electrodes, preprocessing is performed using multi-electrode in-direction subtraction and low-frequency suppression methods. Then, cross-correlation analysis is performed on the two preprocessed signals to obtain a transit time series, and a moving average (MA) model of the transit time series is established. The transit time series model is further analyzed, decomposing it into steady-state and fluctuation components. The steady-state and fluctuation components are related to two indicators that determine the accuracy of in-situ calibration measurements—indication error and repeatability error. The parameter patterns of the steady-state and fluctuation components are analyzed to obtain the ideal value of the instrument correction coefficient and the average transit time for each measurement. Based on the upper limit of the indication error introduced by the correction coefficient and the upper limit of the repeatability error of the transit time, the upper limit of the cross-correlation measurement accuracy is obtained through analysis.

[0014] The advantages of this invention are: it establishes a model of the transit time series, a direct variable determining measurement accuracy, reveals the characteristics of the signal, predicts the upper limit of measurement accuracy for in-situ calibration of a vortex-free permanent magnet sodium flowmeter, and indicates the direction and scope for improving and enhancing measurement accuracy. Specifically, the mathematical model consists of steady-state and fluctuation components, directly reflecting the accuracy of in-situ calibration. The steady-state component indicates the upper limit of the indication error introduced by the instrument correction coefficient; the fluctuation component indicates the upper limit of the repeatability error. Attached Figure Description

[0015] Figure 1 This is the main content and concept of the present invention;

[0016] Figure 2 This is a schematic diagram of the sensor for a permanent magnet sodium flow meter;

[0017] Figure 3 These are cross-sectional and side views of the sensor for a permanent magnet sodium flow meter without vortex generators.

[0018] Figure 4 It is a time series that spans across time;

[0019] Figure 5 It is a histogram of time travel;

[0020] Figure 6 It is a probability density distribution diagram for each flow rate;

[0021] Figure 7(a) shows 250m 3 Autocorrelation and partial autocorrelation coefficients at a flow rate of / h;

[0022] Figure 7(b) shows 100m 3 Autocorrelation and partial autocorrelation coefficients at a flow rate of / h;

[0023] Figure 8 It consists of the actual sequence and the fitted sequence that have passed through the time frame;

[0024] Figure 9(a) shows a comparison between the probability density function of the transit time series output by the model and the actual series;

[0025] Figure 9(b) shows a comparison between the cumulative distribution function of the transit time series output by the model and the actual series;

[0026] Figure 10 It is a correction curve determined by the steady-state components;

[0027] Figure 11 These are the fluctuation components and mean values ​​of three sets of transit times. Specific implementation methods

[0028] The present invention will be further described below with reference to the accompanying drawings.

[0029] Figure 1 This is the main content and concept of the invention. Cross-correlation analysis is performed on the output signals of two pairs of cross-correlated electrodes to obtain the transit time series, and a moving average (MA) model of the transit time series is established. Alternatively, multi-electrode in-direction subtraction and low-frequency suppression methods are used to preprocess the multi-channel cross-correlated signals. Then, cross-correlation analysis is performed on the two preprocessed signals to obtain the transit time series, and a moving average model of the transit time series is established. The transit time series model is analyzed in depth, decomposing it into steady-state and fluctuating components. The steady-state and fluctuating components are related to two indicators that determine the accuracy of in-situ calibration measurements—indication error and repeatability error. The parameter patterns of the steady-state and fluctuating components are analyzed to obtain the ideal value of the instrument correction coefficient and the mean (base value) of the transit time for each measurement. Based on the upper limit of the indication error introduced by the correction coefficient and the upper limit of the repeatability error of the transit time, the upper limit of the cross-correlation measurement accuracy is analyzed.

[0030] Figure 2This is a schematic diagram of the sensor structure of a typical permanent magnet sodium flow meter. The permanent magnet sodium flow meter (also known as a standard permanent magnet sodium flow meter) operates based on Faraday's law of electromagnetic induction and mainly consists of a flow tube, a magnet (permanent magnet, yoke, magnetic poles), and induction electrodes. Induction electrodes V1-1 and V1-2 are a pair of differential electrodes, abbreviated as V1, mounted on the surface of the flow tube, with their connection forming the central axis perpendicular to the flow field and magnetic field directions. Liquid metallic sodium flows through the magnetic field, cutting magnetic field lines, and inducing an electromotive force (EMF) at the induction electrode V1. The induced EMF E output by the induction electrode is positively correlated with the fluid velocity v, magnetic flux density B, and pipe inner diameter D.

[0031] E = BDv (1)

[0032] In the formula, the units of E, B, D, and v are V, T, m, and m / s, respectively.

[0033] Figure 3 These are cross-sectional and side views of the sensor for a vortex-free permanent magnet sodium flow meter. The sensor uses a cast aluminum-nickel-cobalt permanent magnet alloy with high magnetic energy, high stability, high temperature resistance, and radiation resistance as the permanent magnet. Electromagnetic pure iron is used for the yoke and poles, and the surface is treated with anti-oxidation. The pole face length of the permanent magnet is 2D (D is the inner diameter of the pipe). In the vortex-free permanent magnet sodium flow sensor, electrodes 1-1 and 1-2 used for in-situ calibration are a pair of cross-correlation electrodes (hereinafter referred to as electrode C1; other electrodes are similar), measuring one signal, called the cross-correlation signal C1; other electrodes are similar. The combination of electrodes C1 and C2 is used for cross-correlation measurement, hereinafter referred to as electrode C1&2; other electrode combinations are similar. The cross-correlation electrodes are located at a 45° angle to the flow tube (within a certain range). Figure 2 The horizontal line in the middle side view (serving as the reference line for the angle) is vertically welded to the pipe wall, mainly because the signal strength is greater at 45°.

[0034] To predict the accuracy of in-situ calibration of a vortex-free permanent magnet sodium flowmeter, the characteristics of the transit time series obtained by cross-correlation estimation, a direct variable that determines measurement accuracy, are analyzed, and a more accurate mathematical model is established.

[0035] The specific modeling steps are as follows:

[0036] 1. Cross-correlation analysis was used to calculate the transit time series.

[0037] The specific steps for estimating the transit time series are as follows: (1) Subtract the three cross-correlation signals C2, C4 and C5 in the same direction to obtain two new cross-correlation signals C2-4 and C4-5; (2) Remove the mean from the two newly obtained cross-correlation signals; (3) Bandpass filtering in the frequency domain: Perform a 32768-point FFT (Fast Fourier Transform) on the signal after removing the mean to transform the signal to the frequency domain, set the signal amplitude in the 0-4Hz frequency band to zero, retain the signal amplitude in the 4-40Hz frequency band, and then perform an IFFT (Inverse Fast Fourier Transform) to transform the frequency domain signal in the passband to the time domain to complete the signal filtering; (4) Segmented sliding cross-correlation calculation: Perform a cross-correlation calculation every 65536 points, update 8192 points each time the data slides, remove the values ​​greater than 0 from the estimated transit time series, and sequentially connect the transit time series calculated from the three sets of collected data to obtain the final sequence. When the flow rate is 250m 3 At / h, the estimated transit time series is as follows: Figure 4 As shown.

[0038] 2. Use probability density analysis to determine the distribution of random fluctuations.

[0039] The estimated transit time results show significant fluctuations, with a maximum fluctuation of 12.6%. These fluctuations are even more pronounced at low flow rates, for example, at 100m. 3 The rate even reached 40% at / h. Therefore, we calculated its probability density distribution from a statistical perspective, observed the distribution of its fluctuations near the mean, and explored the changing pattern of transit time.

[0040] Since there is no prior knowledge of the transit time, the classic non-parametric estimation method—kernel density estimation—is used to estimate the probability density distribution of the transit time. Kernel density estimation uses a continuous kernel function to smoothly fit the samples within each window of the dataset, then linearly superimposes and normalizes the results to obtain the overall probability density distribution. Its estimation expression is:

[0041]

[0042] In the formula, x i Let be a random variable, i.e., a transit-time series; n is the sample size; h is the window width; K(u) is the kernel function, satisfying ... The choice of kernel function and window width determines the quality of the estimation. An inappropriate kernel function will result in a large discrepancy between the estimated and actual values. After determining the kernel function, an excessively large window width will lead to an overly smooth probability density, resulting in a large estimation bias; an excessively small window width will cause spikes in the observed value region, while the estimated probability density value at the cutoff point will be very small, leading to a large probability density variance. The optimal window width is often determined by the Mean Integrated Square Error (MISE).

[0043]

[0044] In the formula, f(x) is the true probability density distribution of the sample.

[0045] The transit time fluctuates randomly around the mean, and its histogram is as follows: Figure 5 As shown, the distribution approximates a Gaussian distribution. Therefore, a classic Gaussian kernel is used. The probability density is smoothed. Under these conditions, the optimal solution for the window width h is:

[0046] h = 1.06σn -1 / 5 (4)

[0047] In the formula, σ is the standard deviation of the sample.

[0048] The probability density function of transit time under various flow rates is as follows: Figure 6 As shown by the solid line, the fitted Gaussian distribution is represented by the dotted line. At 100m... 3 At a flow rate of / h, the probability density function exhibits heavy tails. The Kolmogorov–Smirnov test shows it does not conform to a Gaussian distribution. However, graphically, the probability density appears to approximate a Gaussian distribution. The probability density functions at all other flow rate points conform to a Gaussian distribution.

[0049] 3. Establish a MA model for the transit time series.

[0050] According to probability density analysis, the random fluctuations in the amplitude of the transit time follow an approximately Gaussian distribution. To understand the amplitude fluctuation characteristics of the transit time in more detail, the amplitude fluctuations are decomposed into components, a mathematical model of the transit time series is established, and the existence of an inherent correlation in its amplitude fluctuations is explored.

[0051] (1) Before establishing a time series model, it is necessary to confirm whether the time series is stationary.

[0052] The stationarity of a series is determined using the Augmented Dickey-Fuller (ADF) unit root test from the Eviews econometric software package. For a p-order autoregressive (AR) model AR(p) with an intercept term c, its expression is:

[0053]

[0054] In the formula, φ j For the coefficient term, ε t With zero mean and variance σ 2 White noise. Equation (5) can be rewritten as:

[0055]

[0056] In the formula,

[0057]

[0058] Then its unit root test formula is:

[0059]

[0060] In the formula, γ = ρ⁻¹. The condition for the stationarity of AR(p) is its characteristic equation.

[0061] 1-φ1z-φ2z 2 -…-φ p z p =0 (9) has roots outside the unit circle, i.e., 1-φ1-φ2-…-φ p >0, γ<0; the non-stationary condition is that the roots of the characteristic equation lie on the unit circle, i.e., 1-φ1-φ2-…-φ p =0, γ=0. Therefore, the null hypothesis of ADF for AR(p) with intercept term is: the time series has a unit root, i.e., H0: c=0, r=0; the alternative hypothesis is: H1: c≠0, r<0. The probability of the null hypothesis being true in the test is P, and its t-statistic is calculated by equation (10):

[0062]

[0063] In the formula, Let γ be an estimate, and σ be an estimate. The standard deviation.

[0064] When the flow rate is 250m 3 At / h, the verification results show that the t-statistic is -3.5584, which is less than the critical values ​​of the test statistic under the 1%, 5%, and 10% confidence intervals (-3.4464, -2.8685, and -2.5705); the p-value is 0.007, which is less than 0.05, indicating that the null hypothesis is not valid and the series is stationary. At the point of greatest flow fluctuation at 100m... 3 The verification results for / h are as follows: the t-statistic is -6.2646, which is less than the critical values ​​(-3.4461, -2.8684, and -2.5704); the p-value is 0. At other flow points, the p-values ​​of the t-statistics are all 0. Therefore, the transit time series is a stationary series.

[0065] (2) Autocorrelation (AC) and partial autocorrelation (PAC) functions are used to analyze the transit time series in order to explore the variation law of the transit time series itself and determine the structure and order of the model.

[0066] The autocorrelation function uses the autocovariance to represent the transit time τ. t Its k-th lag term τ t-k Correlation between them:

[0067]

[0068] In the formula, E represents the expected value. However, this correlation is affected by the intermediate k-1 terms. Therefore, a partial autocorrelation function is introduced, and τ is calculated after removing the intermediate k-1 terms. t With τ t-k Correlation:

[0069]

[0070] In the formula, Eτ t =E[τ t |τ t-1 ,…,τ t-k+1 That is, the middle k-1 terms were removed, Eτ t-k Similarly, φ in equation (12) k This problem cannot be solved independently; it involves a linear regression process and is typically solved using the Yule-Walker method. (250m) 3 / h and 100m 3 The autocorrelation and partial autocorrelation coefficients for / h are shown in Figure 7. In the figure, the dashed line represents the range of 2 standard deviations. For high flow rates, the autocorrelation coefficient is truncated at the seventh order (autocorrelation coefficients of orders higher than seventh are less than 2 standard deviations; therefore, sequences with lags of seven or higher are considered to have no autocorrelation). The partial autocorrelation coefficient exhibits tailing, exceeding the range of 2 standard deviations for sixth and ninth order lags. For low flow rates, the autocorrelation coefficient is generally truncated at the sixth order, and the partial autocorrelation coefficient also exhibits tailing. Therefore, a preliminary approach is to use a seventh-order moving average (MA) model or an autoregressive model with seventh, eighth, or ninth-order terms to find a model that can describe the transit time series across the entire flow rate range.

[0071] (3) The transit time series is described using a seventh-order MA model over the entire flow range.

[0072] Through experimentation, a suitable AR or ARMA (Autoregressive Moving Average) model could not be found to describe the characteristics of the transit time series. However, the MA model, with its highest-order term being seventh, can well describe the transit time series. Its mathematical model is as follows:

[0073]

[0074] In the formula, τ s The mean term; ε t The noise is random noise, which is the actual transit time series τ. twith fitted sequence The residual; B is the delay operator; θ is the coefficient term.

[0075] When the flow rate is 250m 3 / h, τ s =14.3369; θ = [0.8991, 0.8460, 0.7358, 0.7456, 0.7180, 0.6438, 0.5356], the actual transit time series τ output by Eviews software. t with fitted sequence For example Figure 8 As shown.

[0076] (4) The model established using the white noise test of the residual series was used to test whether the residuals contained unextracted variables. The p-value of the Q statistic was used for the test. The Q statistic Q was calculated using the autocorrelation coefficient of the residual series. st :

[0077]

[0078] In the formula, N is the number of observations, q is the lag order, and r j This is the j-th order autocorrelation coefficient of the residual sequence. The null hypothesis of the Q statistic is that the sequence does not exhibit q-th order autocorrelation. After testing, the p-value of the Q statistic for the residual sequence is much greater than 0.05, meaning that the residual sequence does not exhibit autocorrelation, and the model is effective. The residual distribution conforms to Gaussian white noise with zero mean and a standard deviation of 0.1276, which can be used to describe the model's input ε. t .

[0079] According to the MA model, the transit time series consists of a linear combination of the mean term and Gaussian white noise. The mean term represents a stable quantity in the cross-correlated signal. After removing the mean term, its fluctuations are only linearly correlated with the random disturbance term, and there is no correlation between them. In other words, the dynamic error is random, and the error source is also relatively random. There is no specific error source that causes the fluctuations to exhibit trends, periods, or seasonality.

[0080] 4. Verify the accuracy of the model.

[0081] To further verify the accuracy of the model, based on the mathematical expression of the model (13), Gaussian white noise with a mean of 0 and a standard deviation of 0.1276 was generated using the randn function in MATLAB as the input quantity ε of the model. tAdding the mean of the original transit time series, we obtain the transit time series output by the model. Since the model is described using random noise, its correctness can only be verified through model statistics. The probability density function and cumulative distribution function of the model output signal are estimated, where the cumulative distribution function is the integral of the probability density function. The comparison between the model output signal and the original signal is shown in Figure 9.

[0082] Figure 9 shows that the probability density function and cumulative distribution function of the transition time series output by the model are basically consistent with those of the actual series. The correlation coefficient is used to quantitatively represent the similarity of the probability density functions. The correlation coefficient is given by the covariance of the function:

[0083]

[0084] In the formula, ρ xy Let be the correlation coefficient, cov(x,y) be the covariance, and σ be the variance. The correlation coefficient between the probability density function of the transition time series output by the model and the actual series is 0.984, which is very close to 1, indicating a strong correlation between the two.

[0085] Mean squared error is used to quantitatively calculate the difference between the cumulative distribution function of the model's output transit time series and the actual series:

[0086]

[0087] In the formula, C A and C M These are the cumulative distribution functions of the model's output transit time series and the actual series, respectively. The mean squared error between them is 3.28 * 10^- ... -4 The value approaches 0. Therefore, the established MA model can effectively reflect the characteristics of transit time.

[0088] Based on the transit time series model, the accuracy of in-situ calibration can be assessed, predicted, and improved. The steps are as follows:

[0089] 1. Decompose the model into steady-state components and fluctuation components.

[0090] The MA model is used to describe the transit time series, as shown in Equation (13). The transit time series can be decomposed into two parts: the first part is the mean τ of the time series. s The first part, called the steady-state component of the transit time, represents the ideal value of the transit time obtained from cross-correlation measurements. It determines the indication error and is related to the instrument's correction coefficient. The second part is the fluctuation of the time series. It is a white noise sequence, called the fluctuation component of the transit time, which determines the repeatability error of the transit time.

[0091] Before leaving the factory, flow measurement instruments must first undergo calibration experiments to determine their correction coefficients. For in-situ calibration of permanent magnet sodium flow meters without vortex generators, this involves adjusting the average cross-correlation flow rate Q obtained using cross-correlation techniques. c Corrected to reference flow Q r Q r =kQ c +b, thus determining the instrument coefficients k and b; then, a calibration experiment is conducted by measuring the flow rate Q. m Determine the accuracy of in-situ calibration of a permanent magnet sodium flow meter. The accuracy index mainly consists of indication error and repeatability error. Therefore, the upper limit of accuracy for in-situ calibration of a vortex-free permanent magnet sodium flow meter can be explored by examining the steady-state and fluctuation components.

[0092] 2. Use steady-state components to reflect the indication error.

[0093] To analyze the accuracy of an instrument from its steady-state components, it is first necessary to analyze its parameter patterns. Based on the MA model of transit time, the steady-state component τ of the transit time is approximately considered... s This is the ideal value of the cross-correlation measurement at a fixed flow rate point; the actual transit time obtained each time fluctuates around the ideal value. Therefore, the cross-correlation flow rate (denoted as Q) can be obtained by fitting the steady-state component. cs Q cs =πD 2 L / 4τ s ) and reference (standard) flow Q r To determine the ideal correction factors k and b for the instrument, the steady-state components of the transit time and their cross-correlation flow rates are shown in Table 1. The flow measurement relationship during the instrument's calibration and measurement process is then given by Q. m =kQ c +b, measure flow rate Q m The indication error can be expressed as:

[0094]

[0095] Q is not considered in equation (17) for the time being. c The repeatability error is considered to be Q. c =Q cs This refers to the indication error introduced by the ideal instrument coefficient determined only by the steady-state component. This error represents the upper limit of the indication error for this measurement method under current conditions. To reiterate, Q... c It is the directly obtained cross-correlation flow, Q cs It is the cross-correlation flow obtained by fitting the steady-state components.

[0096] Linear fitting typically uses the least squares method, Q cs With Q rThe fitted instrument coefficients are: k = 0.7301, b = 7.8473, and the fitted curve is as follows: Figure 10 As shown. The flow rate measured during the calibration process is expressed as: Q m =0.7301Q cs +7.8473. For ease of representation, this invention uses the least squares method to calculate the measured flow rate Q. m Let Q1 be the value of the value, and its indication error be e1. The calculation results are shown in Table 1.

[0097] The measured flow rate Q1 calculated from the steady-state component can be considered as the ideal value of the flow rate obtained by the cross-correlation technique. However, based on its indication error, the indication error at the minimum flow point is the largest, at 2.25%. This means that the linearity of the instrument coefficient obtained by the least squares fitting method is poor, and the upper limit of the accuracy that the instrument can achieve is 2.5%, i.e., a measurement accuracy of 2.5%.

[0098] When fitting coefficients using the least squares method, the minimum sum of squared errors and the minimum sum of squared residuals are used. In the formula, Let y be the fitted value and y be the actual value; this is equivalent to calculating the sum of squares of absolute errors. For smaller flow rates, the fitted value results in a smaller absolute error compared to larger flow rates. Therefore, when fitting instrument coefficients, there is a bias towards reducing the residual for larger flow rates. However, instrument measurement error is expressed as a relative value of indication error. Poor instrument linearity leads to a larger indication error for smaller flow rates than for larger flow rates, resulting in lower instrument measurement accuracy.

[0099] Table 1 Steady-state components of the transit time series

[0100]

[0101] To address this problem, this invention uses the principle of minimizing the sum of squared indication errors to fit the instrument correction coefficient, thus distributing the error more widely across large flow rates and reducing measurement errors at small flow rates. This method is called the indication error fitting method. The result obtained using the indication error fitting method is: Q m =0.7348Q cs +6.7723, the measured flow rate is denoted as Q2, and its indication error is e2. The calculation results are shown in Table 1.

[0102] Based on the fitting results of the indication error, the error at the low flow rate point was significantly improved, and the maximum indication error of -1.45% was located at the intermediate flow rate point of 150m. 3The fitted flow rate and indication error at this point are close to the results obtained by the least squares method. This is because the indication error fitting method distributes the error from small flow rate points to large flow rate points without changing the error at intermediate flow rate points. The indication error fitting method improves the overall linearity of the instrument, keeping the indication error within 1.5%, achieving a measurement accuracy of 1.5%. However, it should be noted that the steady-state component is considered an ideal value for cross-correlation measurements. Under this condition, the indication error resulting from the fitting correction coefficient will be close to 1.5%. In other words, from the perspective of indication error, using the whole-segment fitting method, under current conditions, the upper limit of the in-situ calibration measurement accuracy of a vortex-free permanent magnet sodium flow meter is 1.5%.

[0103] 3. Use fluctuation components to reflect repeatability error

[0104] Another decisive indicator of instrument accuracy is repeatability error. Repeatability error refers to the error obtained by performing multiple consecutive measurements of the same input value under the same working conditions and in the same direction. Repeatability error is given by the relative standard deviation, as shown in equation (18).

[0105]

[0106] It can be seen that repeatability error is determined by the fluctuation of the sample. And according to the formula Q... c =πD 2 L / 4τ, the transit time is positively correlated with the cross-correlation flow rate, and both transit time and cross-correlation flow rate have the same repeatability error. Therefore, the fluctuation component of the transit time determines the repeatability error of the cross-correlation measurement. According to the MA model of the transit time, the fluctuation component of the transit time is a linear combination of Gaussian white noise. Description. For noise with Gaussian distribution characteristics, it can theoretically be reduced or even eliminated by infinitely repeated moving averages. Therefore, the fluctuation component of the transit time can be reduced to near its mean (base value).

[0107] Repeatability error verification requires multiple measurements. In the in-situ calibration of a vortex-free permanent magnet sodium flowmeter, each measurement acquires a signal of approximately 100 seconds, and cross-correlation is performed in segments to obtain a transit time series. Each transit time series fluctuates around its own mean. The stable component of the transit time at that flow point is obtained by merging multiple transit time series and calculating their mean. Due to the Gaussian nature of the transit time fluctuations, they can be filtered out; however, they will ultimately fall near their mean. Therefore, the mean of each measurement result can be considered the ideal value of the transit time for that measurement. Repeatability error is a measure of the magnitude of fluctuations in multiple measurement results; therefore, the limiting factor for repeatability error is only the ratio of the mean of the fluctuation component of each measurement result to the value of the stable component.

[0108] When the flow rate is 250m 3 At / h, the mean of the three transit time series is as follows: Figure 11 As shown, the symbol 'x' represents the mean transit time of the first group, and its region represents the transit time fluctuation component; the symbol 'o' represents the second group; and the symbol '△' represents the third group. It can be seen that there is a certain difference between the mean transit times of the three groups, and they are not completely equal to the stable component. This difference does not decrease with increasing filter depth; it is an inherent error between the three groups of data, possibly originating from random errors during the experiment, such as flow fluctuations. This leads to a certain upper limit on repeatability error.

[0109] The upper limit of the repeatability error caused by this inherent error is calculated below. In order to reduce the error introduced by the cross-correlation calculation, the cross-correlation integration time T is increased in this invention, and the number of calculation points is changed to 131072 points, with 65536 points updated each time. In addition, when calculating the mean of the transit time, the direct averaging and sorting to take the mean of the intermediate parameters are used respectively to prevent the error introduced by outliers. The specific calculation steps are as follows: (1) Subtract the three cross-correlation signals C2, C4 and C5 in the same direction to obtain two new cross-correlation signals C2-4 and C4-5; (2) Remove the mean from the two newly obtained cross-correlation signals; (3) Frequency domain bandpass filtering: Perform a 32768-point FFT on the signal after removing the mean, transform the signal to the frequency domain, set the signal amplitude in the 0-4Hz frequency band to zero, retain the signal amplitude in the 4-40Hz frequency band, and then perform IFFT to transform the frequency domain signal in the passband to the time domain, and complete the signal Filtering; (4) Segmented sliding cross-correlation calculation: cross-correlation is performed every 131072 points, and 65536 points are updated each time the data slides. Values ​​greater than 0 are removed from the estimated transit time series; (5) Calculate the mean of transit time: the mean of each measurement is calculated by direct averaging and by taking the mean of the middle nine points, respectively, and is used as the ideal value of the transit time for this measurement; (6) Calculate the upper limit of repeatability error: the mean of the three transit times is used, and the repeatability error is calculated according to formula (18). The mean of each measurement transit time and the repeatability error between the three sets of mean are shown in Table 2. Among them, transit time 1 is obtained by direct averaging; transit time 2 is obtained by taking the mean of the middle nine points.

[0110] Table 2. Repeatability error caused by the mean transit time for each measurement.

[0111]

[0112] As shown in Table 2, the repeatability errors between the average transit times of the three measurements are not significantly different under both methods, both approaching 0.5%. This means the inherent repeatability error of the signal is 0.5%, i.e., the upper limit of repeatability error is 0.5%. According to the electromagnetic flowmeter calibration procedure, the best measurement accuracy achievable under this repeatability error condition is 1.5%. Therefore, considering repeatability error, the upper limit of measurement accuracy for in-situ calibration of the vortex-free permanent magnet sodium flowmeter is 1.5%. Taking into account the indication error caused by the instrument coefficient determined by the steady-state component, it can be determined that, under the current conditions, the upper limit of accuracy for in-situ calibration of the vortex-free permanent magnet sodium flowmeter is 1.5%.

Claims

1. A method for establishing a signal model and evaluating the measurement accuracy of a permanent magnet sodium flowmeter without vortex generators; For a permanent magnet sodium flowmeter without vortex generators, experiments are conducted to collect voltage fluctuation signals output from cross-correlation electrodes; after collecting voltage fluctuation signals output from multiple pairs of cross-correlation electrodes, preprocessing is performed using multi-electrode in-direction subtraction and low-frequency suppression methods; then, cross-correlation analysis is performed on the two preprocessed signals to obtain a transit time series, and a moving average model of the transit time series is established; the model of the transit time series is analyzed in depth, decomposing it into steady-state components and fluctuation components; the steady-state components and fluctuation components are respectively related to two indicators that determine the accuracy of in-situ calibration measurement—indication error and repeatability error; the parameter laws of the steady-state components and fluctuation components are analyzed to obtain the ideal value of the instrument correction coefficient and the mean value of each measurement transit time; based on the upper limit of the indication error introduced by the correction coefficient and the upper limit of the repeatability error of the transit time, the upper limit of the cross-correlation measurement accuracy is obtained through analysis; characterized in that: To predict the accuracy of in-situ calibration of a vortex-free permanent magnet sodium flowmeter, the characteristics of the transit time series obtained from cross-correlation estimation, a direct variable determining measurement accuracy, are analyzed, and an accurate mathematical model is established. The steps are as follows: 1) Cross-correlation analysis was used to calculate the transit time series. The specific steps for estimating the transit time series are as follows: (1) Subtract the three cross-correlation signals C2, C4 and C5 in the same direction to obtain two new cross-correlation signals C2-4 and C4-5; (2) Remove the mean from the two newly obtained cross-correlation signals; (3) Frequency domain bandpass filtering: Perform a 32,768-point FFT on the signal after removing the mean to transform the signal to the frequency domain, set the signal amplitude in the 0-4Hz frequency band to zero, retain the signal amplitude in the 4-40Hz frequency band, and then perform an IFFT to transform the frequency domain signal in the passband to the time domain to complete the signal filtering; (4) Segmented sliding cross-correlation calculation: Perform a cross-correlation calculation every 65,536 points, update 8,192 points each time the data slides, remove the values ​​greater than 0 from the estimated transit time series, and sequentially connect the transit time series calculated from the three sets of collected data to obtain the final sequence; 2) Use probability density analysis to determine the distribution of random fluctuations. The probability density distribution of transit time is estimated using the classic non-parametric estimation method—kernel density estimation. 3) Establish a MA model for the transit time series. Based on probability density analysis, the random fluctuations in the amplitude of the transit time conform to an approximate Gaussian distribution; the amplitude fluctuations are decomposed into components to establish a mathematical model of the transit time series. (1) Before establishing a time series model, it is necessary to first confirm whether the time series is stationary. Use the ADF unit root test to determine the stationarity of a sequence; Upon examination, the transit time series was determined to be a stationary series. (2) Perform autocorrelation and partial autocorrelation function analysis on the transit time series to explore the variation law of the transit time series itself and determine the structure and order of the model; After using the autocorrelation function and partial autocorrelation function, we initially used the seventh-order moving average model or the autoregressive model with seventh, eighth, and ninth-order higher-order terms to find a model that can describe the transit time series across the entire flow range. (3) The transit time series is described using a seventh-order MA model over the entire flow range; Its mathematical model is: (1) In the formula, This is the mean term; It is random noise, representing the actual transit time series. with fitted sequence The residual; For delay operators; For coefficient terms; (4) The model established by the white noise test of the residual sequence is used to test whether there are unextracted variables in the residual; The p-value of the Q statistic was used for testing; After testing, the residual sequence showed no autocorrelation, indicating the model was effective; the residual distribution conformed to Gaussian white noise with zero mean and a standard deviation of 0.1276, which was used to describe the model's input. ; According to the MA model, the transit time series consists of a linear combination of the mean term and Gaussian white noise; the mean term represents the stable quantity in the cross-correlation signal; after removing the mean term, its fluctuations are only linearly correlated with the random disturbance term, and there is no correlation between them. 4) Verify the accuracy of the model To further verify the accuracy of the model, Gaussian white noise with a mean of 0 and a standard deviation of 0.1276 was generated using the randn function in MATLAB as the input to the model. Adding the mean of the original transit time series, we obtain the transit time series output by the model; we then estimate the probability density function and cumulative distribution function of the model output signal. The correlation coefficient is used to quantitatively represent the similarity between the probability density function of the model output signal and the original signal; the mean square error is used to quantitatively calculate the difference between the cumulative distribution function of the model output transit time series and the actual series.

2. The method for signal model establishment and measurement accuracy evaluation of a permanent magnet sodium flowmeter without vortex generator as described in claim 1, characterized in that: Based on the transit time series model, the steps for assessing, predicting, and improving the accuracy of in-situ calibration are as follows: 1) Decompose the model into steady-state components and fluctuation components. The MA model is used to describe the transit time series, which is decomposed into two parts: the first part is the mean of the time series. The first part, called the steady-state component of the transit time, represents the ideal value of the transit time obtained from cross-correlation measurements. It determines the indication error and is related to the instrument's correction coefficient. The second part is the fluctuation of the time series. This is a white noise sequence, called the fluctuation component of the transit time, which determines the repeatability error of the transit time. 2) Use steady-state components to reflect the indication error. Cross-correlation flow rate obtained by fitting steady-state components With reference flow Determine the ideal correction factor for the instrument and ; The flow measurement relationship during the instrument's calibration and measurement process is as follows: Measure flow rate The indication error is expressed as: (2) Equation (2) is not considered for the time being The repeatability error is considered That is, the indication error introduced by the ideal instrument coefficient determined only by the steady-state component, which represents the upper limit of the indication error of this measurement method under the existing conditions; It is the directly obtained cross-correlation traffic; When fitting coefficients using the least squares method, the minimum sum of squared errors and the minimum sum of squared residuals are used. In the formula, These are the fitted values. The actual value is equivalent to calculating the sum of squares of absolute errors. The principle of minimizing the sum of squares of indicated errors is used to fit the instrument correction coefficient, so that the error is more distributed over large flow rates, thereby reducing the measurement error of small flow rates. 3) Use fluctuation components to reflect repeatability error Another decisive indicator of instrument accuracy is repeatability error; repeatability error refers to the error obtained by performing multiple consecutive measurements of the same input value under the same working conditions and in the same direction; repeatability error is given by the relative standard deviation and is determined by the fluctuation of the sample; and according to the formula The transit time is positively correlated with the cross-correlation flow rate, and both transit time and cross-correlation flow rate have the same repeatability error. Therefore, the fluctuation component of the transit time determines the repeatability error of the cross-correlation measurement. According to the MA model of the transit time, the fluctuation component of the transit time is a linear combination of Gaussian white noise. Description: Reduce the fluctuation component of the transit time to near its mean; Multiple measurements are required in the repeatability error test. In the in-situ calibration of the permanent magnet sodium flow meter without vortex generator, a 100-second signal is collected for each measurement and cross-correlation is performed in segments to obtain a set of transit time series. Each set of transit time series fluctuates around its own mean. The stable component of the transit time can be obtained by merging multiple sets of transit time series and taking the mean. The upper limit of the repeatability error caused by this inherent error is calculated below; in order to reduce the error introduced by the cross-correlation calculation, the cross-correlation integration time T is increased, the number of calculation points is changed to 131072 points, and 65536 points are updated each time; in addition, when calculating the mean of the transit time, the direct averaging and sorting to take the mean of the intermediate parameters are used respectively to prevent the error introduced by outliers; the specific calculation steps are as follows: (1) Subtract the three cross-correlation signals C2, C4 and C5 in the same direction to obtain two new cross-correlation signals C2-4 and C4-5; (2) Remove the mean from the two newly obtained cross-correlation signals; (3) Frequency domain bandpass filtering: perform a 32768-point FFT on the signal after removing the mean to transform the signal Switch to the frequency domain, set the signal amplitude in the 0-4Hz frequency band to zero, retain the signal amplitude in the 4-40Hz frequency band, and then perform IFFT to transform the frequency domain signal in the passband to the time domain to complete signal filtering; (4) Segmented sliding cross-correlation calculation: cross-correlation is performed once every 131072 points, and 65536 points are updated each time the data slides. Values ​​greater than 0 are removed from the estimated transit time series; (5) Calculate the mean of transit time: the mean of each measurement is calculated by using the direct averaging method and the sorting and taking the mean of the middle nine points, respectively, and is used as the ideal value of the transit time for this measurement; (6) Calculate the upper limit of repeatability error: the mean of three sets of transit time is used to calculate the repeatability error.

3. The method for signal model establishment and measurement accuracy evaluation of a permanent magnet sodium flowmeter without vortex generator as described in claim 1, characterized in that: Because the cross-correlation electrode output signal of the vortex generator-free permanent magnet sodium flowmeter has complex components and an ultra-wide frequency range, and because the cross-correlation signal consists of two delayed signals, cross-correlation analysis is performed on it to obtain the transit time, and then the transit time series is modeled.

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