Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation

The cavitation states among centrifugal pumps can be mirrored by corresponding vibration features. To select the vibration feature target scientifically and objectively for monitor the cavitation states in real time, the analysis method of grey slope correlation with weight entropy was proposed in this paper to explore the relevance between cavitation and vibration features. Thus, the net positive suction head (NPSH) and vibration signal from centrifugal pumps under multiple operation conditions were captured. Moreover, the universal feature targets were extracted from the vibration signal. The grey slope correlation method was applied in the analysis of the positive and negative relevance between NPSH and the multiple operation conditions in a different stage. These feature targets are transformed into the same numerical scale by standardization process. In the end, the final comprehensive coefficient can be attached after endowing power by weight entropy method. These methods can be used to determine the feature targets which have intensive relevance with NPSH. The analysis results indicate that the kurtosis factor, variance, absolute mean, and root mean square obtained from the vibration acceleration signal have stable relevance with NPSH. These feature targets can be used for the proper detection and evaluation of cavitation states in centrifugal pumps. Therefore, the analysis method of grey slope correlation with weight entropy can be used to pre-select the feature targets based on the calculated grey incidence. This method is effective in establishing the relevance between NPSH and vibration.


Introduction
Real-time monitoring based on centrifugal pumps [1] has become a trending research point in the hydraulic machine as a result of development in artificial intelligence and communication technology. Nowadays, the acceleration signal can be received by vibration transducer and processed by corresponding algorithm, such as wavelet packet transform (WPT) and empirical mode decomposition (EMD) [2]. The acquired data from hydraulic machine [3,4] can be used to identify pump working states and give a valid disposal scheme based on intelligence diagnosis [5]. These can considerably reduce contingency occurrence probability and prolong the pump life cycle. However, in the real experiments, an error could be found due to poor incidence such as background noise and vibration, flow rate setting error, the influence of reflecting surfaces around the instrument, even the distance between the pump and the instrument [6]. Meanwhile, it is important to ensure that the applied data are robust enough to give an accurate result and reduce the misjudgment ratio induced by the diagnostic algorithm. In the present study, the selection of feature parameters extracted from the acceleration signal is a random and tedious process for some scholars which usually leads to ideal output. Considering the relevance between the independent variable and dependent variable as a most fundamental task, ensuring the significant degree and priority among the feature targets before

Grey Slope Correlation Method
Although the traditional Deng's relation computation has perfectly solved issues like small samples and poor information, some limited applied conditions are worthy of discussion in this algorithm. Existing literature [28,29] about pump parameter assumption established the use of positive correlation between variables. Meanwhile, there is a potential risk that a negative correlation may exist between NPSH and vibration feature. This situation makes it difficult to completely rely on the traditional method to draw a conclusion since it could result in fatal errors. As a result, the improved algorithm and grey slope correlation can be more appropriate in solving these problems. In the improved algorithm method, the slope is used to establish the relevance of the relationship between the numerical interval of −1 and 1. While the absolute slope value is closer to one, the extracted feature is more sensitive to NPSH. On the contrary, the insensitivity between two variables due to the positive and negative sign convention is a reflection of its positive or negative characteristic. Hence, a new method needs to be established that can account for both positive and negative characteristics between cavitation and vibration. Thus, to evaluate the feature target, the acquired data need to be validated and transformed into a unified standard. The above process is an essential part in the assessment process which enables the application of the weight entropy method in order to solve the problem.

Weight Entropy Method
Weight entropy is an objective weighting method. This concept was originally introduced into information theory from thermodynamics by Shannon [30]. For this method, if the feature values of the research target have a tremendous difference on some index, the entropy is small which indicates that this index can provide massive valid information and the weight should be vast. On the contrary, if the feature values of research target have a small difference on some index, the entropy is large which indicates that this index can provide a tiny amount of effective information and the weight should be small.

Calculation Process
The concrete steps of grey slope correlation with weight entropy methods are listed as follows.
Step 1: Define reference sequence (RS) and comparative sequence (CS) Suppose N = N(P 1 ), N(P 2 ), N(P 3 ), · · · , N(P n ) as the reference sequence, which indicates the sequence of tabulated data of NPSH, where P n represents the real-time pressure on nth times. Vibration features are taken as the comparative sequence, that is which denotes the comparative sequence.
Step 2: Make sequence be dimensionless Due to the values' physical scale difference, the maximum value treatment can be used to normalize the data. This mathematical process can enable us to obtain more accurate results in the grey correlation analysis. The preprocessing can express as: max P n = P n max{P n } , n = 1 , 2 , 3 , · · · , m (1) Step 3: Calculate the coefficient of grey slope correlation For the unequal interval sequence, define the grey slope correlation coefficient as where, Appl. Sci. 2020, 10, 8190 Step 4: Standardize the target matrix ξ i ( max P n ) As the uncertainty of positive and negative value exist in ξ i ( max P n ), therefore, the transmitting to the same sign is necessary for this paper.
Define, R in = (ξ i ( max P n )) i×n If the sequence belongs to the larger-the-better type-like positive value, the comparable sequence (CS) is calculated as If the sequence belongs to the smaller-the-better type like negative value, the comparable sequence (CS) is expressed as where R * in ∈ [0 , 1] Step 5: Calculate grey slope correlation entropy Define the entropy of the nth to be: Then, the nth entropy coefficient is: Step 6: Calculate final comprehensive coefficient From the weight entropy, the final coefficient can be expressed as Accordingly, the ranking rule of the grey slope correlation sequence is obtained. The higher the entropy correlation degree of the comparison column and the reference column is, the greater the influence on the reference column will be.

Signal Capture and Pretreatment
In order to verify the scientific feasibility of the proposed method as described earlier in Sections 2.1 and 2.2, a handle process was adopted as shown in Figure 1. The experiments were conducted in multiple suction pressure under three flow rate points. The vibration features were extracted from its vibration acceleration signal.
Accordingly, the ranking rule of the grey slope correlation sequence is obtained. The higher the entropy correlation degree of the comparison column and the reference column is, the greater the influence on the reference column will be.

Signal Capture and Pretreatment
In order to verify the scientific feasibility of the proposed method as described earlier in Sections 2.1 and 2.2, a handle process was adopted as shown in Figure 1. The experiments were conducted in multiple suction pressure under three flow rate points. The vibration features were extracted from its vibration acceleration signal.

Test Rig
The experiments were carried out on a closed test rig located within Jiangsu University as presented in Figure 2. In the cyclic process, the fluid from the tank enters into the pump through the soft pipe by the rotational effect of the impeller. The impeller transfers the fluid back to the tank

Test Rig
The experiments were carried out on a closed test rig located within Jiangsu University as presented in Figure 2. In the cyclic process, the fluid from the tank enters into the pump through the soft pipe by the rotational effect of the impeller. The impeller transfers the fluid back to the tank through the elbow sections, electromagnetic flowmeter (for monitor flow rate) and magnetic valve (for adjust flow rate).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 13 through the elbow sections, electromagnetic flowmeter (for monitor flow rate) and magnetic valve (for adjust flow rate).  Table 1 shows the important geometric and operational parameters of the prototype pump under investigation.

Experiment Instrument
In this experiment, vibration acceleration and suction pressure data were monitored and recorded in detail. The vertical vibration acceleration signals of suction pipe ektexine were monitored using a computer and these signals were saved under different operating conditions of pressure. Figure 3 shows the monitor location on the tested pump.  Table 1 shows the important geometric and operational parameters of the prototype pump under investigation.

Experiment Instrument
In this experiment, vibration acceleration and suction pressure data were monitored and recorded in detail. The vertical vibration acceleration signals of suction pipe ektexine were monitored using a computer and these signals were saved under different operating conditions of pressure. Figure 3 shows the monitor location on the tested pump.
The sensor used in this experiment is a high frequency sensor (PCB 352A60 series) with a sensitivity value of 10 mv/g and the frequency response range of ±500 g/Hz. A pressure transmitter (WIKA S-10) with ±0.2% accuracy in full scale was used to record the pressure difference. In order to capture the relative signals accurately, the sampling frequency and time used were 16,000 Hz and 1 s respectively [22]. For further details about the experimental method, please refer to the author's previous work [31][32][33]. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13 (a) (b) The sensor used in this experiment is a high frequency sensor (PCB 352A60 series) with a sensitivity value of 10 mv/g and the frequency response range of ±500 g/Hz. A pressure transmitter (WIKA S-10) with ±0.2% accuracy in full scale was used to record the pressure difference. In order to capture the relative signals accurately, the sampling frequency and time used were 16,000 Hz and1 s respectively [22]. For further details about the experimental method, please refer to the author's previous work [31][32][33].

Experiment Method
In this experiment, the pressure and vibration must be recorded simultaneously. At the given flow rate, multiple data, captured by the suction pressure, were used to study the vibration with pressure variation at a constant rotation speed of 3000 rpm. Firstly, the deflation valve was fully open and the ball valve was closed. After measuring the data under this condition, the deflation valve was closed and the ball valve and vacuum pump were opened gradually in order to reduce the pressure at the suction side of the pump until cavitation occurred. After the emergence of cavitation, the vacuum pump and ball valve were opened and observed over a period of time until there was a drop in pressure at the inlet of the pump. At this point, the data acquisition process was put on hold until the vacuum pump cannot take away any atmosphere from the tank or the test rig cannot provide the foreseeable dangers. The same steps would be repeated in the flow rate of 40 m 3 /h and 60 m 3 /h to guarantee the robust of algorithm.

Data Pretreatment
Transforming the suction pressure into NPSH and the vibration acceleration signal would convert into fifteen (15) types of feature target which contain the maximum, minimum, mean, peak, absolute mean, variance, standard deviation, kurtosis, skewness, root mean square, shape factor, crest factor, kurtosis factor, impulse factor, and margin factor. The specific mathematical function and steps can be found in Appendix A from the literature [24].

Analysis and Methodology
On the foundation of Step 1, the above data in ever flow rate point would be turned into the reference sequence (RS) and comparative sequence (CS) as the following matrix expresses:

Outlet pressure tapping
Inlet pressure tapping Vibration monitor

Experiment Method
In this experiment, the pressure and vibration must be recorded simultaneously. At the given flow rate, multiple data, captured by the suction pressure, were used to study the vibration with pressure variation at a constant rotation speed of 3000 rpm. Firstly, the deflation valve was fully open and the ball valve was closed. After measuring the data under this condition, the deflation valve was closed and the ball valve and vacuum pump were opened gradually in order to reduce the pressure at the suction side of the pump until cavitation occurred. After the emergence of cavitation, the vacuum pump and ball valve were opened and observed over a period of time until there was a drop in pressure at the inlet of the pump. At this point, the data acquisition process was put on hold until the vacuum pump cannot take away any atmosphere from the tank or the test rig cannot provide the foreseeable dangers. The same steps would be repeated in the flow rate of 40 m 3 /h and 60 m 3 /h to guarantee the robust of algorithm.

Data Pretreatment
Transforming the suction pressure into NPSH and the vibration acceleration signal would convert into fifteen (15) types of feature target which contain the maximum, minimum, mean, peak, absolute mean, variance, standard deviation, kurtosis, skewness, root mean square, shape factor, crest factor, kurtosis factor, impulse factor, and margin factor. The specific mathematical function and steps can be found in Appendix A from the literature [24].

Analysis and Methodology
On the foundation of Step 1, the above data in ever flow rate point would be turned into the reference sequence (RS) and comparative sequence (CS) as the following matrix expresses: Impulse f actor(p 1 ) Impulse f actor(p 2 ) · · · Impulse f actor(p n−1 ) Impulse f actor(p n ) Margin(p 1 ) Margin(p 2 ) · · · Margin(p n−1 ) For the normalization processing of the data from a matrix by the maximum way according to Step 2, the tackled data are drawn on Figure 4.
For the normalization processing of the data from a matrix by the maximum way according to Step 2, the tackled data are drawn on Figure 4.  Figure 4d shows the mean value in 40 m 3 /h. However, as weight entropy states, the rationale and credible value can be acquired based on the calculated value of grey relation and entropy weight. In this way, the objective relation between NPSH and feature parameter can be decided whether it is related or not. Furthermore, the relevance matrix θ can be acquired with the data in Figure 4b-p through the Step 3 calculation, the consequence of which can be seen in Figure 5.  Figure 4d shows the mean value in 40 m 3 /h. However, as weight entropy states, the rationale and credible value can be acquired based on the calculated value of grey relation and entropy weight. In this way, the objective relation between NPSH and feature parameter can be decided whether it is related or not. Furthermore, the relevance matrix θ can be acquired with the data in Figure 4b-p through the Step 3 calculation, the consequence of which can be seen in Figure 5.
In Figure 5, n denotes the numbers of the calculated slope, and ζ expresses the grey slope coefficient of the corresponding feature target in different stages. From Figure 5, the trend of all targets except the Kurtosis factor basically considered has a positive or negative relevance with NPSH but the grey coefficient tends to 0 in the terminal. In mathematical terms, these parameters do not have strong relevance with NPSH in the terminal. However, from a physics perspective, this kind of description cannot satisfy common sense. According to the definition of grey slope correlation, using the slope in different stages reflects the relevance between the vibration feature target and NPSH. Mirrored in Figure 5, in the cavitation stage, the slope value of the feature target and vibration has a big difference. The physics states in the pump are changed and the corresponding physics meaning is the minimum variation in NPSH which would cause logarithmic leaps among the feature targets. Thus, these descriptions correspond to the fact of phase-change vibration caused by bubble burst.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13 In Figure 5, n denotes the numbers of the calculated slope, and ζ expresses the grey slope coefficient of the corresponding feature target in different stages. From Figure 5, the trend of all targets except the Kurtosis factor basically considered has a positive or negative relevance with NPSH but the grey coefficient tends to 0 in the terminal. In mathematical terms, these parameters do not have strong relevance with NPSH in the terminal. However, from a physics perspective, this kind of description cannot satisfy common sense. According to the definition of grey slope correlation, using the slope in different stages reflects the relevance between the vibration feature target and NPSH. Mirrored in Figure 5, in the cavitation stage, the slope value of the feature target and vibration has a big difference. The physics states in the pump are changed and the corresponding physics meaning is the minimum variation in NPSH which would cause logarithmic leaps among the feature targets. Thus, these descriptions correspond to the fact of phase-change vibration caused by bubble burst.
Due to the existence of positive and negative value in the feature target, the relevance of the feature target cannot be judged directly. Therefore, transforming the negative and positive value into the same positive interval by Step 4 as Figure 6 depicted. Due to the existence of positive and negative value in the feature target, the relevance of the feature target cannot be judged directly. Therefore, transforming the negative and positive value into the same positive interval by Step 4 as Figure 6 depicted.

According to
Step 5, the corresponding entropy weight can be attached under different pressure stages in a corresponding flow rate. The final relevant coefficient in the corresponding flow rate can be calculated by Step 6. The average value can be acquired by repeating Step 5 and Step 6. The calculation results are enumerated in Table 2.

According to
Step 5, the corresponding entropy weight can be attached under different pressure stages in a corresponding flow rate. The final relevant coefficient in the corresponding flow rate can be calculated by Step 6. The average value can be acquired by repeating Step 5 and Step 6. The calculation results are enumerated in Table 2.