Signal model establishment and measurement accuracy evaluation of permanent magnet sodium flowmeter with embedded vortex generator

Abstract

The voltage fluctuation signals of the cross-correlation electrodes of a permanent magnet sodium flowmeter with vortex shroud were collected by experiment. The generalized cross-correlation was used to estimate the transit time. The amplitude fluctuation of the transit time was analyzed and a moving average model was established. The model of the transit time was analyzed in depth and was decomposed into a steady component and a fluctuation component. The steady component and the fluctuation component were related to the two indicators of the measurement accuracy, namely the indication error and the repeatability error. The parameter law of the steady component was analyzed and the ideal value of the instrument correction coefficient was obtained. The upper limit of the accuracy of the cross-correlation measurement was obtained according to the indication error introduced by the correction coefficient. It was pointed out that the poor linearity of the instrument was the reason for the limited accuracy. According to the characteristics that the fluctuation component was a linear combination of Gaussian white noise, a method of two-stage moving average filtering was proposed to weaken the fluctuation component and reduce the repeatability error, so as to improve the measurement accuracy and make it reach the upper limit of the accuracy.

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 with an embedded vortex generator. Background Technology

[0002] Fast neutron reactors are the mainstay of the world's fourth-generation advanced nuclear energy systems. They have small cores and high power density, using liquid metallic sodium instead of water or other coolants to meet their rapid heat exchange requirements. To ensure adequate cooling and timely heat exchange in fast reactors, permanent magnet sodium flow meters are commonly used to monitor sodium flow in real time. A typical permanent magnet sodium flow meter consists of a flow tube, a magnet, and an induction electrode. The magnet generates a magnetic field, and the induction electrode picks up the electromotive force induced by the liquid metallic sodium fluid passing through the magnetic field and cutting magnetic lines of force. In other words, the permanent magnet sodium flow meter operates based on Faraday's law of electromagnetic induction. The amplitude of the output signal from the induction electrode is positively correlated with the magnetic flux density and flow velocity, i.e., E = KBDV, where K is the instrument coefficient, B is the magnetic flux density, D is the pipe inner diameter, and V is the flow velocity. Under normal operating conditions, the magnetic flux density remains constant, and the flow velocity in the pipe can be accurately calculated from the signal amplitude. However, permanent magnets exposed to high temperatures for extended periods will demagnetize, causing deviations in flow velocity measurement. Therefore, the instrument coefficient of the permanent magnet sodium flow meter needs to be corrected periodically, i.e., periodic calibration. However, permanent magnet sodium flow meters in the closed environment of the reactor are not removable. For example, a permanent magnet sodium flow meter in the primary loop fuel pipeline is exposed to neutron radiation and cannot be removed for offline calibration. Therefore, the permanent magnet sodium flow meter itself needs to have the function of in-situ calibration.

[0003] For permanent magnet sodium flow meters with pipe inner diameters of 100mm (DN100) and below, due to the small flow rate within the pipe, obstructions and cross-correlation electrodes are typically installed at appropriate locations and angles within the flow tube to form a permanent magnet sodium flow meter with in-situ calibration capabilities. This is called a permanent magnet sodium flow meter with an embedded vortex generator. The obstruction creates vortices in the fluid, and the cross-correlation electrodes pick up the voltage signal generated by these vortices. The cross-correlation method is used to calculate the propagation time (transit time τ) of the fluid flowing between two pairs of cross-correlation electrodes, thereby calculating the average flow velocity v in the pipe, i.e., v = L / τ (where L is the distance between the two pairs of cross-correlation electrodes). Theoretically, the flow velocity and flow rate obtained through this cross-correlation method (hereinafter referred to as cross-correlation velocity and cross-correlation flow rate) do not depend on the signal amplitude and are not affected by environmental factors such as radiation and high temperatures. Therefore, it can be used to calibrate the flow velocity obtained from the sensing electrodes.

[0004] Chinese invention patent discloses an online calibration method and system for a permanent magnet sodium flow meter with an embedded vortex generator (Xu Kejun, Yan Xiaoxue, Xu Wei, Yu Xinlong, Xiong Wei. An online calibration method and system for a permanent magnet sodium flow meter with an embedded vortex generator, Patent No.: ZL201811353018.4, Application Date: 20181114, Authorization Announcement Date: 20200731). It describes how to generate vortices on small-diameter (e.g., DN40 and DN65) pipes using a semi-circular baffle, and how point electrodes installed parallel to the magnetic field pick up cross-correlation signals. Different signal preprocessing methods are used to achieve cross-correlation measurement. The patent indicates that the cross-correlation electrodes mainly pick up vortex signals in the Z-axis direction. A biased generalized cross-correlation method is used to achieve in-situ calibration with 3% accuracy.

[0005] A Chinese invention patent discloses an online calibration method for sodium flowmeters based on phase frequency characteristics (Xu Kejun, Yan Xiaoxue, Yu Xinlong, Xu Wei, Wu Jianping, Xiong Wei. An online calibration method for sodium flowmeters based on phase frequency characteristics, Patent No.: ZL201910318801.5, Application Date: 20190419, Authorization Announcement Date: 20200731). The method proposes using phase frequency characteristics to calculate the transit time, estimate the system's transfer function model, obtain the frequency response function, and use the phase of the frequency response function to solve for the transit time. It also proposes methods for fitting the slope of the phase frequency characteristic curve and for achieving cross-correlation measurement by selecting the average transit time of the frequency band using the amplitude spectrum. The measurement accuracy is 3%.

[0006] Wu Wenkai, Xu Kejun, Yu Xinlong, et al. proposed an in-situ method for permanent magnet sodium flowmeters based on adaptive filtering (W.-K.Wu,K.-J.Xu,X.-L.Yu,et al.,Adaptive filtering based sodium flowmeasurement method for in-situ calibration,Ann.Nucl.Energy,vol.150,Jan.2021,Art.no.107865.doi:10.1016 / j.anucene.2020.107865.). This method uses an adaptive filtering method based on minimum mean square error (LMS) to continuously approximate one signal with another through filtering, transforming the transit time estimation problem into a finite impulse response (FIR) filter parameter estimation problem. The measurement accuracy is 2%. Summary of the Invention

[0007] The above literature studies the in-situ calibration problem of permanent magnet sodium flowmeters with embedded vortex generators from the perspective of signal processing, that is, improving the accuracy of in-situ calibration by studying different signal processing methods. They do not study this problem from the perspective of the quality of the processed signal itself, nor do they establish a signal model for the permanent magnet sodium flowmeter with embedded vortex generators. The error in in-situ calibration consists of two parts: one is the error introduced by the processed signal itself, such as the error introduced by the signal bandwidth and noise; the other is the error introduced by the signal processing method, such as the finite integration time T introduced by cross-correlation calculation, i.e., the calculation error caused by signal truncation. Therefore, it is necessary to establish a mathematical model of the processed signal to study the in-situ calibration of permanent magnet sodium flowmeters with embedded vortex generators more directly and comprehensively. Specifically, establishing a signal model can reveal the characteristics of the processed signal, so as to adopt appropriate processing methods, and is also the basis for evaluating signal processing methods; the signal model can reflect the quality of the processed signal, establish its relationship with measurement accuracy, so as to assess the upper limit of the measurement accuracy of in-situ calibration, and also point out the direction for in-situ calibration and determine the room for improvement.

[0008] The specific technical solution is as follows:

[0009] Due to the complex composition and extremely wide frequency range of the cross-correlation electrode output signal in the permanent magnet sodium flowmeter with embedded vortex generator, and because the cross-correlation electrode signal consists of two delayed signals, it is impossible to directly model the cross-correlation signal. Therefore, cross-correlation analysis is performed on the output signals of the two pairs of cross-correlation electrodes to obtain the transit time. This transit time series is then modeled. It is directly related to the cross-correlation flow rate and is a key variable reflecting the accuracy of in-situ calibration measurements.

[0010] This invention focuses on a permanent magnet sodium flowmeter with an embedded vortex generator (e.g., a semi-circular obstruction). Experiments were conducted to collect voltage fluctuation signals from the cross-correlation electrodes. The transit time was estimated using generalized cross-correlation, and the amplitude fluctuation of the transit time was analyzed to establish a moving average (MA) model for the transit time. The model was further analyzed, decomposing it into steady-state and fluctuating components. These components are related to two indicators determining measurement accuracy: indication error and repeatability error. The parameter patterns of the steady-state component were analyzed to obtain the ideal value of the instrument correction coefficient. Based on the indication error introduced by the correction coefficient, the upper limit of the accuracy of the cross-correlation measurement was obtained, indicating that poor instrument linearity is the cause of accuracy limitations. Based on the characteristic that the fluctuating component is a linear combination of Gaussian white noise, a two-stage moving average filtering method was proposed to weaken the fluctuating component, reduce repeatability error, and thus improve measurement accuracy, reaching the upper limit of accuracy.

[0011] The advantages of this invention are as follows: Previous literature has largely focused on improving signal processing methods to enhance the signal-to-noise ratio of cross-correlation signals, thereby improving the accuracy of in-situ calibration. However, the error in cross-correlation measurement consists of two parts: the error introduced by the original cross-correlation signal itself (errors introduced by signal bandwidth and noise) and the error introduced by cross-correlation calculation (the finite integration time T, i.e., the calculation error caused by signal truncation). Therefore, error analysis based on the transit time obtained from cross-correlation estimation is more direct and comprehensive. This invention starts with the transit time series to study the reasons for the limited accuracy of in-situ calibration of permanent magnet sodium flowmeters; simultaneously, it takes effective measures based on the characteristics of the signal to further improve the calibration accuracy. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of the primary instrument structure of a permanent magnet sodium flow meter with an embedded vortex generator;

[0013] Figure 2 It is a time series that spans across time;

[0014] Figure 3 It is a histogram of the transit time series;

[0015] Figure 4 It is a probability density distribution diagram of transit time under various traffic flows;

[0016] Figure 5(a) shows a flow rate of 8.1 m³ / s. 3 / h is the autocorrelation and partial autocorrelation coefficient;

[0017] Figure 5(b) shows a flow rate of 1.2 m³ / s. 3 / h is the autocorrelation and partial autocorrelation coefficient;

[0018] Figure 6 It consists of the actual sequence and the fitted sequence that have passed through the time frame;

[0019] Figure 7(a) shows a comparison of the probability density distributions of the analog signal and the original signal;

[0020] Figure 7(b) shows a comparison of the cumulative distribution functions of the analog signal and the original signal;

[0021] Figure 8 Correction curves for determining steady-state components. Specific implementation methods

[0022] The present invention will be further described below with reference to the accompanying drawings.

[0023] Figure 1This is a schematic diagram of the primary instrument structure of a permanent magnet sodium flow meter with an embedded vortex generator. The upstream system incorporates a semi-circular obstruction to enhance disturbance and improve the accuracy of the cross-correlation method measurement. The semi-circular obstruction is symmetrically installed along the central axis of the pipe. Downstream of the obstruction, two magnetic systems are arranged. Each magnetic system contains DC signal amplitude measurement electrodes (induction electrodes) and AC cross-correlation signal measurement electrodes (cross-correlation electrodes). V1-1 and V1-2 are a pair of induction electrodes, installed perpendicular to the magnetic field direction (x-axis) and the flow field direction (y-axis), collecting the induced electromotive force generated by the fluid cutting magnetic field lines in the y-axis direction. V2-1 and V2-2 are similarly arranged. H1-1 and H1-2 are a pair of cross-correlation electrodes, referred to as electrode H1, installed parallel to the magnet, picking up the broadband, approximately sinusoidal signal generated by the vortex cutting magnetic field lines in the z-axis direction; electrode H2 (including H2-1 and H2-2) picks up the signal generated by the downstream vortex cutting magnetic field lines in the z-axis direction. Cross-correlation calculations are performed on the signals from electrodes H1 and H2 to determine the transit time.

[0024] To determine and further improve the accuracy of in-situ calibration of permanent magnet sodium flowmeters with embedded vortex generators, 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.

[0025] The specific modeling steps are as follows:

[0026] 1. Calculate the transit time series using cross-correlation analysis.

[0027] To ensure that the transit time series obtained from the cross-correlation estimation accurately reflects the characteristics of the original wave signal, the generalized cross-correlation method is used to estimate the transit time. The discrete expression for calculating the cross-correlation of real numbers is:

[0028]

[0029] Where x(n) and y(n) are two fluctuating signals, m is the shift value of x(n) during cross-correlation calculation, and N is the number of cross-correlation calculation points. The transit time is calculated from the x-axis of the maximum cross-correlation value. Therefore, in order to eliminate outliers in the cross-correlation results at the data truncation point, a biased estimation method is used.

[0030] In the cross-correlation estimation process, only the two signals are subjected to mean removal and low-pass filtering to preserve the originality of the signals as much as possible. The cutoff frequency of the fourth-order low-pass filter is the same as the hardware cutoff frequency. Cross-correlation calculations are performed in segments, with each data point being 65536 points long and the next data update being 8192 points long. During the cross-correlation calculation, the upstream sensor signal is used as the first signal and the downstream sensor signal as the second signal. Since the theoretical transit time obtained from each cross-correlation estimation is negative, values ​​with a transit time greater than 0 are removed in each estimation. When the flow rate is 8.1 m³ / s...3 At / h, the estimated transit time series is as follows: Figure 2 As shown.

[0031] 2. Use probability density analysis to determine the distribution of random fluctuations.

[0032] The estimated transit time results show significant fluctuations, with a maximum fluctuation of 2.6%. These fluctuations are even more pronounced at low flow rates, for example, at 1.2m. 3 The value even reached 12% at / h. Therefore, we estimated 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.

[0033] 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:

[0034]

[0035] 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 peaks 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).

[0036]

[0037] Where f(x) is the true probability density distribution of the sample.

[0038] The transit time fluctuates randomly around the mean, and its histogram is as follows: Figure 3 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:

[0039] h = 1.06σn -1 / 5 (4)

[0040] In the formula, σ is the standard deviation of the sample.

[0041] The probability density function of transit time under various flow rates is as follows: Figure 4 As shown by the solid line, the fitted Gaussian distribution is represented by the dotted line. At 1.2m... 3 At / h, the probability density function exhibits a heavy-tailed characteristic. The Kolmogorov–Smirnov test shows it does not conform to a Gaussian distribution. However, graphically, the probability density appears to approximate a Gaussian distribution. And 2.6m 3 The probability density function of high flow rate points above / h all conforms to a Gaussian distribution.

[0042] 3. Establish a MA model for the transit time series.

[0043] 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 intrinsic correlation of its amplitude fluctuations is explored.

[0044] Modeling a transit-time series is equivalent to building a time series model of its dynamic error. Time series models are often used as mathematical models to describe stochastic processes and have important applications in fields such as dynamic data modeling.

[0045] (1) Before establishing a time series model, it is necessary to confirm whether the time series is stationary.

[0046] The stationarity of the series was determined using the Augmented Dickey-Fuller (ADF) unit root test from the Eviews econometric software package. The flow rate was 8.1 m³ / s. 3 At a flow rate of / h, the validation results showed that the test statistic was -6.3311, which is much smaller than the critical values ​​of the test statistic under the 1%, 5%, and 10% confidence intervals (-3.4492, -2.8697, and -2.5712), and the p-value was 0. At the point of greatest flow fluctuation at 1.2m... 3 The test statistic for the h-hour flow rate was -6.1045, which is less than the critical values ​​(-3.4541, -2.8719, and -2.5724), and the p-value (a parameter used to determine the hypothesis test result) was 0. The same result applies to other flow rates. Therefore, the transit time series is a stationary series.

[0047] (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.

[0048] The autocorrelation function uses the autocovariance to represent the transit time τ. t Its k-th lag term τ t-kThe correlation between the terms is determined, but this correlation is affected by the intermediate k-1 terms. Therefore, a partial autocorrelation function is introduced to remove the intermediate k-1 terms before calculating τ. t With τ t-k The correlation. The flow rate is 8.1m. 3 / h and 1.2m 3 The autocorrelation and partial autocorrelation coefficients at / h are as follows: Figure 6 As shown, at low flow rates, the autocorrelation coefficient is truncated at the fourth order, and the partial autocorrelation coefficient is truncated at approximately the second order. At high flow rates, the autocorrelation coefficient is truncated at approximately the fifth order, and the partial autocorrelation coefficient exhibits a tailing phenomenon, exceeding the confidence interval at the fourth, seventh, and ninth orders. Therefore, using an MA model or an AR (autoregressive) model with higher-order terms such as seventh, eighth, and ninth orders can provide a model that describes the signal across the entire flow rate range.

[0049] The AIC (Akaike Information Criterion) is used to determine the optimal model. The AIC criterion uses the principle of maximizing the likelihood function estimate to determine the optimal model order.

[0050] AIC=-2lnL(θ)+2k (5)

[0051] In the formula, L(θ) is the maximum likelihood function, and k is the number of model parameters. The smaller the AIC value, the better the model. The AIC values ​​of the MA model and AR models with seventh, eighth, and ninth order terms are shown in Table 1.

[0052] Table 1. AIC values ​​for MA and AR models

[0053]

[0054] Based on the AIC value, the MA model's value is smaller than the AR model's, and the AR model shows an increasing number of higher-order terms as the flow rate increases. Therefore, the MA model is more suitable for describing the fluctuations in transit time.

[0055] (3) Determine the MA model

[0056] The MA model for crossing time series is as follows:

[0057]

[0058] In the formula, τ s The mean; ε t For random noise, the actual transit time series τ t with fitted sequence The residual; B is the delay operator; θ is the coefficient term.

[0059] When the flow rate is 8.1m 3 / h, τs =161.4848; θ = [0.814996 0.707379 0.718435 0.719731 0.651121 0.609600 0.422941], the actual transit time series τ output by Eviews software. t with fitted sequence For example Figure 6 As shown.

[0060] (4) White noise test

[0061] The established model also needs to undergo a white noise test on the residual series to check whether the residuals contain unextracted variables. The p-value of the Q statistic is used for this test. The Q statistic Q is calculated using the autocorrelation coefficient of the residual series. st :

[0062]

[0063] 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 0.401, which can be used to describe the model's input ε. t .

[0064] 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.

[0065] 4. Model Validation

[0066] To further verify the accuracy of the model, based on the mathematical expression of the model (6), Gaussian white noise with a mean of 0 and a standard deviation of 0.401 was generated using the randn function in MATLAB as the model input ε. t Adding 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. We estimate the probability density function and cumulative distribution function of the model output signal, 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 7.

[0067] As shown in Figure 7, the probability density function and cumulative distribution function of the model output signal are basically consistent with those of the original signal. 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:

[0068]

[0069] In the formula, cov(x,y) represents the covariance, and σ represents the variance. The correlation coefficient between the model output signal and the probability density function of the original signal is 0.986, which is very close to 1, indicating a strong correlation between the two.

[0070] Mean squared error is used to quantitatively calculate the difference between the cumulative distribution function of the model output signal and the original signal:

[0071]

[0072] In the formula, C A and C M These are the cumulative distribution functions of the original signal and the model output signal, respectively. The mean square error between them is 3.73 * 10^- ... -4 The value approaches 0. Therefore, the established MA model can effectively reflect the characteristics of the original signal.

[0073] Based on the transit time series model, the accuracy of in-situ calibration can be assessed, predicted, and improved. The steps are as follows:

[0074] 1. Decompose the model into steady-state components and fluctuation components.

[0075] The transit time series described by the MA model 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 each cross-correlation measurement. 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.

[0076] 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, this involves applying the average correlated flow rate Q obtained through cross-correlation technology. c Corrected to reference flow Q r Q r =kQ s +b, thus determining the instrument coefficients k and b; then, a calibration experiment is conducted to determine the accuracy of the in-situ calibration of the permanent magnet sodium flowmeter. The accuracy index mainly consists of indication error and repeatability error. The accuracy of the in-situ calibration of the permanent magnet sodium flowmeter is discussed below, focusing on both steady-state and fluctuating components.

[0077] 2. Use steady-state components to reflect the indication error.

[0078] 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. In reality, the transit time obtained each time fluctuates around the ideal value. Therefore, the cross-correlation flow rate Q can be obtained by fitting the steady-state component of the transit time. cs =πD 2 L / (4τ s The ideal correction factors k and b for the instrument are determined using the reference flow rate. Note: Q c The average cross-correlation flux Q is obtained by cross-correlation technique. cs The cross-correlation flux is obtained by fitting the steady-state component of the transit time. The least squares method is used to fit Q. cs With Q r The functional relationship between them, and the fitting result is: Q r =0.9271Q cs -0.3059, such as Figure 8 As shown.

[0079] According to the flow measurement relationship Q during the instrument calibration process m =kQ cs +b, the indication error can be expressed as

[0080]

[0081] In the formula, Q m For measuring flow rate. It can be seen that the indication error is related to the steady-state component.

[0082] The flow rate Q obtained through least squares fitting m Let Q1 be the value and its indication error e1 be as shown in Table 2.

[0083] Table 2 Flow rate measurement and indication error

[0084]

[0085] The flow rate Q1 calculated from the steady-state component can be considered as the ideal value of the flow rate obtained by cross-correlation technology. However, based on its indication error, the indication error at the minimum flow point is the largest, at 2.13%. In other words, 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 3%, which is a level three accuracy instrument.

[0086] When fitting the optimal coefficients using the least squares method, the minimum sum of squared errors and the 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 the instrument coefficients, there's a bias towards reducing the residual for larger flow rates. However, the measurement error of an instrument is expressed as a relative value of the indication error. Poor instrument linearity means that the indication error for smaller flow rates is greater than that for larger flow rates, thus reducing the accuracy of the instrument measurement.

[0087] To address this problem, this invention uses the principle of minimizing the sum of squares of indication errors to fit the instrument correction coefficient, thus distributing the error more widely across large flow rates and reducing the measurement error at small flow rates. This method is called the indication error fitting method. The result obtained using the indication error fitting method is: Q r =0.9232Q cs -0.2819. The flow rate Q2 and its indication error e2 fitted by the indication error are shown in Table 2.

[0088] 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 0.82% was located at the intermediate flow rate point of 5.025m. 3 At the / h point, the fitted flow rate and indication error are consistent with 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 linearity of the instrument overall, keeping the indication error within 1%, achieving the accuracy of a Class 1 meter. 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 caused by the correction coefficient obtained from the fitting will be close to 1%. In other words, from the perspective of indication error, using the whole-segment fitting method, the upper limit of the in-situ calibration accuracy of the permanent magnet sodium flow meter with embedded vortex generator is 1%.

[0089] Furthermore, based on the analysis of the upper limit of accuracy of in-situ calibration, it can be seen that the poor linearity of the instrument is the main reason for limiting the accuracy of the instrument. Therefore, the focus of the improvement direction for this instrument system is its linearity, making it more linear in structure.

[0090] 3. Use fluctuation components to reflect repeatability error

[0091] Another crucial indicator of instrument accuracy is repeatability error. Repeatability error refers to the random error obtained by performing multiple consecutive measurements of the same input value under identical operating conditions and in the same direction. Repeatability error is given by the relative standard deviation:

[0092]

[0093] In the formula, Let be the sample mean, and n be the number of measurements. Equation (11) shows that the repeatability error is determined by the sample fluctuation. According to 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 repeatability error of cross-correlation measurements can be reduced to a very low level. In other words, the lower limit of repeatability error is very low, and the accuracy of the instrument is basically not limited by repeatability error.

[0094] However, as mentioned earlier, the maximum fluctuation of transit time during low flow is 12%. If the transit time is not filtered, the repeatability error is large and cannot meet the requirements of first-level accuracy. Therefore, in view of the Gaussian white noise characteristics of the fluctuation component, a two-stage moving average filtering method is proposed to improve the repeatability error of cross-correlation measurement. The specific algorithm steps are as follows: (1) Perform mean removal and fourth-order Butterworth low-pass filtering on three consecutively collected data sets. The data collection time for each set is 100s; (2) Perform cross-correlation calculation on each set of data with 65536 points. Use a sliding method to cover 50% of the old data each time, remove transit time estimates greater than 0, and store the transit time estimates in an array; (3) Perform first-level moving average filtering: sort the array once when it reaches 10 points, and then... Then take the average of the middle 6 points and store it in a new array. The array slides one point at a time, that is, remove the initial value at the head and fill in the new value at the tail, and continue to sort and take the average; (4) Perform secondary filtering: sort the new transit time array and take the average of the middle 8 points (if the array is oddly long, take the middle 7 points) as the final transit time; (5) Calculate the repeatability error of the three transit times according to formula (11); (6) Calculate the cross-correlation flow rate by taking the average of the three transit times; (7) Fit the instrument correction coefficient by the indication error method according to the cross-correlation flow rate and the reference flow rate value, and calculate the indication error.

[0095] The data processing results of the second-order moving average filter are shown in Table 3. The fitting function is: Q r =0.9218Q c-0.2756. The repeatability error results show that the two-stage moving average filtering method significantly improves the repeatability error at low flow points, with a maximum repeatability error of 0.20%. According to the electromagnetic flowmeter calibration procedure, the repeatability error of the instrument must not exceed 1 / 3 of the maximum permissible error, which refers to the upper limit of accuracy. For a first-stage flowmeter, the maximum permissible error is 1%, and the repeatability error must not exceed 0.33%. Therefore, the two-stage moving average filtering method meets the repeatability error requirements for a first-stage flowmeter. Furthermore, the indicated errors of the measured flow rate are all within ±1%. Both repeatability error and measurement error are improved compared to the aforementioned literature.

[0096] Table 3 Data Processing Results

[0097]

[0098] To further validate the two-stage moving average filtering method, a verification experiment was conducted again. In this verification experiment, it was unnecessary to refit the instrument correction coefficients; the function Q was used directly. r =0.9218Q c The indication error was calculated using -0.2756, and other data processing steps were the same as above. The experimental results are shown in Table 4.

[0099] Table 4. Results of the verification experiment

[0100]

[0101] The verification results show that both repeatability error and indication error meet the first-level accuracy requirements. Therefore, the second-level moving average filtering method can enable the in-situ calibration accuracy of the permanent magnet sodium flowmeter to reach its upper limit of 1%.

Claims

1. A method for establishing a signal model and evaluating the measurement accuracy of a permanent magnet sodium flowmeter with an embedded vortex generator, comprising: conducting experiments on a permanent magnet sodium flowmeter with an embedded semi-circular obstruction body; collecting voltage fluctuation signals output by cross-correlation electrodes; estimating the transit time through generalized cross-correlation; analyzing the amplitude fluctuation of the transit time; establishing a moving average model of the transit time; further analyzing the transit time model and decomposing it into steady-state components and fluctuation components, which are respectively related to two indicators that determine the measurement accuracy—indication error and repeatability error; By analyzing the parameter patterns of the steady-state components, the ideal value of the instrument correction coefficient is obtained. Based on the indication error introduced by the correction coefficient, the upper limit of the accuracy of the cross-correlation measurement is obtained. According to the characteristic that the fluctuation component is a linear combination of Gaussian white noise, the two-stage moving average filtering method is used to weaken the fluctuation component, reduce the repeatability error, and thus improve the measurement accuracy to reach the upper limit of accuracy. Its features are: To determine and further improve the accuracy of in-situ calibration of permanent magnet sodium flowmeters with embedded vortex generators, 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) Calculate the transit time series using cross-correlation analysis To ensure that the transit time series obtained by cross-correlation estimation can truly reflect the characteristics of the original wave signal, the generalized cross-correlation method is used to estimate the transit time; to eliminate outliers in the cross-correlation results at data truncation points, a biased estimation method is used. During the cross-correlation estimation process, only the two signals are subjected to mean removal and low-pass filtering to preserve the originality of the signals as much as possible; the cutoff frequency of the fourth-order low-pass filter is the same as the hardware cutoff frequency; cross-correlation calculation is performed in segments, with each data length being 65536 points and the next data update length being 8192 points; during the cross-correlation calculation, the upstream sensor signal is used as the first signal and the downstream sensor signal is used as the second signal. Since the theoretical transit time obtained from each cross-correlation estimation is negative, values ​​with a transit time greater than 0 are removed in each estimation. 2) Use probability density analysis to determine the distribution of random fluctuations. The probability density distribution of transit time is estimated using the kernel density estimation method; 3) Establish a MA model for the transit time series. Based on probability density analysis, the random fluctuation of the transit time amplitude conforms to an approximate Gaussian distribution; the amplitude fluctuation is decomposed into components to establish a mathematical model of the transit time. (1) Before establishing a time series model, it is necessary to first confirm whether the time series is stationary. The stationarity of the series was determined using the ADF unit root test; the test showed that the transit time series is 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; Based on the autocorrelation function and partial autocorrelation function, a model describing the signal across the entire flow range is used, either an MA model or an AR model with seventh, eighth, or ninth-order terms; the AIC criterion is used to determine the optimal model. (3) Determine the MA model Based on the AIC value, the MA model is more suitable for describing the fluctuations in transit time; the MA model for transit time series is as follows: (1) In the formula, The mean; The noise is random, derived from the actual transit time series. with fitted sequence Residual description; For delay operators; For coefficient terms; (4) White noise test The established model also needs to pass the white noise test of the residual sequence to check whether there are any unextracted variables in the residuals; the p-value of the Q statistic is used for the test; after the test, the p-value of the Q statistic of the residual sequence is much greater than 0.05, that is, there is no autocorrelation in the residual sequence, and the model is effective; the residual distribution conforms to Gaussian white noise with zero mean and a standard deviation of 0.401, which is used to describe the input of the model. ; 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. 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 show a trend, periodicity, or seasonality. 4) Model Validation To further verify the accuracy of the model, based on the mathematical expression of the model (1), Gaussian white noise with a mean of 0 and a standard deviation of 0.401 was generated using the randn function in MATLAB as the model input. The mean of the original transit time series is added to obtain the transit time series of the model output; the probability density function and cumulative distribution function of the model output signal are estimated; the correlation coefficient is used to quantitatively represent the similarity between the probability density function of the model output signal and the original signal; and the mean square error is used to quantitatively calculate the difference between the cumulative distribution function of the model output signal and the original signal.

2. The method for establishing a signal model and evaluating the measurement accuracy of a permanent magnet sodium flowmeter with an embedded 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 transit time series described by the MA model 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 each cross-correlation measurement. 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 the steady-state component of the transit time. Determine the ideal correction factor for the instrument using the reference flow rate. and ; For the steady-state component that transits through time; Based on the flow measurement relationship during the instrument calibration process The indication error is expressed as (2) In the formula, For measuring flow rate; It can be seen that the indication error is related to the steady-state component; The flow rate calculated from the steady-state component is considered an ideal value for the flow rate obtained by the cross-correlation technique. When fitting the optimal 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 the absolute errors; The instrument correction coefficient is fitted using the principle of minimizing the sum of squares of the indication error, 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 random 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; 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. describe; To address the Gaussian white noise characteristics of the fluctuation component, a two-stage moving average filtering method is adopted to improve the repeatability error of cross-correlation measurements. The specific algorithm steps are as follows: (1) Perform mean removal and fourth-order Butterworth low-pass filtering on three consecutively collected data sets, with each data set having a collection time of 100s; (2) Perform cross-correlation calculation on each data set with 65536 points, using a sliding method, covering 50% of the old data each time, removing transit time estimates greater than 0, and storing the transit time estimates in an array; (3) First-stage moving average filtering: perform a sorting operation once the array reaches 10 points. (3) Sort the new transit time array, and take the average of the middle 6 points and store it in a new array. The array slides one point at a time, that is, remove the initial value at the head and fill in the new value at the tail, and continue to sort and take the average; (4) Secondary filtering: Sort the new transit time array, take the average of the middle 8 points as the final transit time, and if the array is oddly long, take the middle 7 points; (5) Calculate the repeatability error of the three transit times; (6) Take the average of the three transit times to calculate the cross-correlation flow; (7) According to the cross-correlation flow and the reference flow value, fit the instrument correction coefficient by the indication error method and calculate the indication error.

3. The method for establishing a signal model and evaluating the measurement accuracy of a permanent magnet sodium flowmeter with an embedded vortex generator as described in claim 1, characterized in that: Because the output signals of the correlated electrodes of the permanent magnet sodium flowmeter with embedded vortex generator have complex components and an ultra-wide frequency band, and because the correlated electrode signals are two signals with delays, cross-correlation analysis is performed on the output signals of the two pairs of cross-correlation electrodes to obtain the transit time. This transit time series is modeled because it is directly related to the cross-correlation flow and is a key variable reflecting the accuracy of in-situ calibration measurement.

Images