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Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation

Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation applied sciences Article Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation Ruijia Cao and Jianping Yuan * National Research Center of Pumps, Jiangsu University, Zhenjiang 212013, China; 2111811001@stmail.ujs.edu.cn * Correspondence: yh@ujs.edu.cn Received: 11 October 2020; Accepted: 12 November 2020; Published: 19 November 2020 Abstract: 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 di erent stage. These feature targets are transformed into the same numerical scale by standardization process. In the end, the final comprehensive coecient 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 e ective in establishing the relevance between NPSH and vibration. Keywords: centrifugal pumps; cavitation; vibration; grey slope correlation; weight entropy; selection strategy 1. 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 Appl. Sci. 2020, 10, 8190; doi:10.3390/app10228190 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 8190 2 of 13 the running of analysis code enable the scholar to pay more attention in digging valuable results from vast amount of data. Currently available results indicated that the suction pressure di erence might change the pump vibration states [7–9], although this sort of vibration feature has a di erent appearance in di erent working flow rates [10–16]. However, cavitation can be defined as the rupture of a liquid due to a pressure drop [17]. It can be identified by vibration feature due to bubble burst. In these cited works of signal process, the feature target is a key element which should be considered and determined. Thus, root mean square (RMS) might be the most popular and universal parameter since it can be extracted from a historical vibration signal and reflect the cavitation occurrence. For instance, Dong et al. [18] and Zhang et al. [19] selected the concept of energy developed from the RMS to establish the potential variation caused by suction pressure and working flow rates. Similarly, to acquire the situations in di erent net positive suction head (NPSH), the RMS of acceleration in di erent monitor points displaced on the pump was captured to mirror the cavitation states [20]. Moreover, the parameters like the mean, variance, standard deviation, skewness, kurtosis, and crest factor were also used to carry out this kind of work [21]. In the literature [22], empirical mode decomposition (EMD) method with these parameters was used to decompose original signals into a number of intrinsic mode functions (IMFs). In the process of diagnosing the flow instabilities [23], the most troubling and confusion things were that di erent feature target selection might disturb the evaluation while the pump is working in the cavitation or air injection mode. The result points out the fact that the feature target might have di erent sensitivity under various circumstances. As mentioned in the first part, all the feature target selections were based on perceptual cognizance and personal experience. The details of strict mathematical explanation about why these targets can be used to evaluate the vibration manner were not adequately captured in the literature. It is against this backdrop that this research was conceived to find a reliable way to execute a robust and reliable relation between variables which would give an absolute fact instead of ambiguous justification identified by Al-Obaidi [24]. On the condition of the small sample, variety and complexity of uncertain factors, multivariate analysis cannot be directly applied. Grey relation analysis, a branch of grey system theory, as an e ective method to evaluate the relevance between variants, has exhibited its charming and universality in the multifarious discipline. For instance, in the automatic driving [25], the parameters which a ect safe driving can be extracted and analyzed. In the iron austempering [26], it can be used to establish the relationship between temperature and machinability performance. In the architectural planning [27], it can be used to evaluate and ensure the substation site selection. Above all, the grey relation method would be applied in the pump field to determine what targets derived from the initial vibration signal have an intensity relation with the NPSH. Moreover, the information entropy method would be used in the research to sort out the relevant parameters for its importance degree. 2. Grey Relation Entropy Analysis Method 2.1. 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 dicult 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 of1 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 Appl. Sci. 2020, 10, 8190 3 of 13 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. 2.2. 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 di erence 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 di erence on some index, the entropy is large which indicates that this index can provide a tiny amount of e ective information and the weight should be small. 2.3. 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 ), N(P ), N(P ), , N(P ) as the reference sequence, which indicates the sequence 1 2 3 of tabulated data of NPSH, where P represents the real-time pressure on nth times. Vibration features are taken as the comparative sequence, that is V = V (P ) , V (P ) , V (P ) ,  , V (P ) , i = 1 , 2 ,  , k, i i 1 i 2 i 3 i which denotes the comparative sequence. Step 2: Make sequence be dimensionless Due to the values’ physical scale di erence, 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 = 1 , 2 , 3 ,  , m (1) maxfP g max N(P ) X( P ) =  , n = 1 , 2 , 3 ,  , m (2) max N(P ) max V (P ) i n Y ( P ) =  , n = 1 , 2 , 3 ,  , m (3) max V (P ) Step 3: Calculate the coecient of grey slope correlation For the unequal interval sequence, define the grey slope correlation coecient as max max max ( P ) = sgn(DX( P ),DY ( P )) Q (4) n n n i i where, max max max max > 1 , DX( P )DY ( P )  0 < n n sgn(DX( P ),DY ( P )) = (5) n i n max max 1 , DX( P )DY ( P ) < 0 n i n max DX( P ) 1 + max D P Q = (6) max max max DX( P ) DX( P ) DY ( P ) n n n 1 1 1 i 1 + + max max max X X D P D P D P n n n Appl. Sci. 2020, 10, 8190 4 of 13 max max max DX( P ) = X( P ) X(P ) > n n n1 max max max > (7) DY ( P ) = DY ( P ) DY (P ) , n  2 > n n i i i n1 max max max D P = P P n n n1 max Y = Y ( P ) (8) i i n=1 max X = X( P ) (9) n=1 max Step 4: Standardize the target matrix  ( P ) max As the uncertainty of positive and negative value exist in  ( P ), therefore, the transmitting to i n the same sign is necessary for this paper. max Define, R = ( ( P )) in i in If the sequence belongs to the larger-the-better type-like positive value, the comparable sequence (CS) is calculated as R minfR g in in R = (10) in maxfR g minfR g in in n n If the sequence belongs to the smaller-the-better type like negative value, the comparable sequence (CS) is expressed as maxfR g R in in R = (11) in maxfR g minfR g in in n n where R 2 [0 , 1] in Step 5: Calculate grey slope correlation entropy Define the entropy of the nth to be: H = f ln f (12) n in in ln m n=1 in where, f = , while R = 0, let R ln R = 0 in m P in in in in n=1 Then, the nth entropy coecient is: 1 H ! = (13) 1 H n=1 Step 6: Calculate final comprehensive coecient From the weight entropy, the final coecient can be expressed as = !(n)R (n) (14) in n=1 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. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 13 From the weight entropy, the final coefficient can be expressed as = ω() nR (n) (14) ξ  i in n=1 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 Appl. Sci. 2020, 10, 8190 5 of 13 influence on the reference column will be. 3. Signal Capture and Pretreatment 3. Signal Capture and Pretreatment In order to verify the scientific feasibility of the proposed method as described earlier in In order to verify the scientific feasibility of the proposed method as described earlier in Sections Sections 2.1 and 2.2, a handle process was adopted as shown in Figure 1. The experiments were 2.1 and 2.2, a handle process was adopted as shown in Figure 1. The experiments were conducted in conducted in multiple suction pressure under three flow rate points. The vibration features were multiple suction pressure under three flow rate points. The vibration features were extracted from its extracted from its vibration acceleration signal. vibration acceleration signal. Figure 1. Flow chart of process. Figure 1. Flow chart of process. 3. 3.1. 1. TT est est Ri Rig g The experiments were carried out on a closed test rig located within Jiangsu University as The experiments were carried out on a closed test rig located within Jiangsu University as presented presentedin inFig Figur ure e2. 2 In t . Inhthe e cyclic proc cyclic process, ess, the fl theui fluid d from the ta from thenk enters tank enters into the pu into themp through the pump through the soft pipe by the rotational e ect of the impeller. The impeller transfers the fluid back to the tank 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 Appl. Sci. 2020, 10, 8190 6 of 13 (for adjust flow rate). Figure 2. Test rig. Figure 2. Test rig. T Table able 11 sho shows ws the important geometric and oper the important geometric and operational ational parameters of the prototype pump parameters of the prototype pump under investigation. under investigation. Table 1. Main parameters of the prototype pump. Table 1. Main parameters of the prototype pump. Name Symbol Name Symbol Value Value Designed flow rate Qd 50 m /h Designed flow rate Q 50 m /h Designed head H 37 m Designed head Hd 37 m Rated rotational speed n 3000 r/min Rated rotational speed n 3000 r/min Impeller inlet diameter D 74 mm Impeller inlet diameter D1 74 mm Impeller outlet diameter D 174 mm Impeller outlet diameter D2 174 mm Impeller outlet width b 12 mm Impeller outlet width b2 12 mm Blades Z 6 Blades Volute diameter D Z 184 mm 6 Rated Power p 5 kW Volute diameter D3 184 mm Rated Power p 5 kW 3.2. Experiment Instrument 3.2. Experiment Instrument In this experiment, vibration acceleration and suction pressure data were monitored and recorded in detail. In this experiment, v The vertical vibration ibration acceleration acceleration an signalsd ofsuction pressure suction pipe ektexine data were were monitor monitored ed using and a recorded computer in d and etail these . The vert signals ical wer vibe rat saved ion accel under erat di io ner sient gnaoperating ls of suctio conditions n pipe ekteof xin pr e w essur ere m e. o Figur nitore ed 3 shows using a com the monitor puter and th location ese sign on theals w tested ere saved pump. under different operating conditions of pressure. Figure The 3 shows the moni sensor used into this r loca experiment tion on the tested pump. is a high frequency sensor (PCB 352A60 series) with a sensitivity value of 10 mv/g and the frequency response range of500 g/Hz. A pressure transmitter (WIKA S-10) with 0.2% accuracy in full scale was used to record the pressure di erence. 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 [31–33]. Appl. Sci. 2020, 10, 8190 7 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13 Outlet pressure tapping Inlet pressure tapping Vibration monitor (a) (b) Figure 3. Instrument layout: (a) test site; (b) monitor points. Figure 3. Instrument layout: (a) test site; (b) monitor points. 3.3. Experiment Method 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 In this experiment, the pressure and vibration must be recorded simultaneously. At the given flow (rate, WIKA S- multiple 10) wi data, th ±0 captur .2% ac ed cura bycy the insuction full scalpr e w essur as us e,ewer d to reco e used rd th to study e pressure the vibration difference with . In or pressur der to e capture the relative signals accurately, the sampling frequency and time used were 16,000 Hz and1 s variation at a constant rotation speed of 3000 rpm. Firstly, the deflation valve was fully open and the respectively ball valve was [22]. For closed. furth Afteremeasuring r details ab the out the experi data under mental this condition, method, please the deflation refer to the valve was author’s closed previous work [31–33]. 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 3.3. Experiment Method 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 In this experiment, the pressure and vibration must be recorded simultaneously. At the given pump cannot take away any atmosphere from the tank or the test rig cannot provide the foreseeable flow rate, multiple data, captured by the suction pressure, were used to study the vibration with 3 3 dangers. The same steps would be repeated in the flow rate of 40 m /h and 60 m /h to guarantee the pressure variation at a constant rotation speed of 3000 rpm. Firstly, the deflation valve was fully open robust of algorithm. 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 3.4. Data Pretreatment at the suction side of the pump until cavitation occurred. After the emergence of cavitation, the Transforming the suction pressure into NPSH and the vibration acceleration signal would vacuum pump and ball valve were opened and observed over a period of time until there was a drop convert into fifteen (15) types of feature target which contain the maximum, minimum, mean, peak, in pressure at the inlet of the pump. At this point, the data acquisition process was put on hold until absolute mean, variance, standard deviation, kurtosis, skewness, root mean square, shape factor, the vacuum pump cannot take away any atmosphere from the tank or the test rig cannot provide the 3 3 crest factor, kurtosis factor, impulse factor, and margin factor. The specific mathematical function and foreseeable dangers. The same steps would be repeated in the flow rate of 40 m /h and 60 m /h to steps can be found in Appendix A from the literature [24]. guarantee the robust of algorithm. 4. Analysis and Methodology 3.4. Data Pretreatment On the foundation of Step 1, the above data in ever flow rate point would be turned into the Transforming the suction pressure into NPSH and the vibration acceleration signal would reference sequence (RS) and comparative sequence (CS) as the following matrix expresses: 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, RS = NPSH (p ), NPSH (p ), NPSH (p ) NPSH (p ), NPSH (p ) r 1 r 2 r 3 r n1 r n crest factor, kurtosis factor, impulse factor, and margin factor. The specific mathematical function 8 9 and steps can be found in Appendix A from the literature [24]. > > Maximum(p ) Maximum(p )  Maximum(p ) Maximum(p ) > 1 2 n1 > > > > > > > > Minimum(p ) Minimum(p )  Minimum(p ) Minimum(p ) > 1 2 n1 n > > > > 4. Analysis and Methodology > > < = . . . . . . . . CS = > > > . . . . > > > > > On the foundation of Step 1, the above data in ever flow rate point would be turned into the > > > > Impulse f actor(p ) Impulse f actor(p )  Impulse f actor(p ) Impulse f actor(p ) > > 1 2 n1 n > > > > reference sequence (RS) and comparative sequence (CS) as the following matrix expresses: : ; Margin(p ) Margin(p )  Margin(p ) Margin(p ) 1 2 n1 n 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. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 R{ S = NPSH( p), NPSH( p), NPSH( p) NPSH( p ), NPSH( p)} rr12 r3 rn−1 rn  Maximum() p Maximum( p )  Maximum( p ) Maximum( p )  12 nn −1   Minimum() p Minimum( p )  Minimum( p ) Minimum( p ) 12 nn −1     CS =        Impulse factor() p Impulse factor( p )  Impulse factor( p ) Impulse factor( p ) 12 nn −1    M arginp ( ) M arginp ( )  M argin(pM)argin(p)   12 nn −1  For the normalization processing of the data from a matrix by the maximum way according to Appl. Sci. 2020, 10, 8190 8 of 13 Step 2, the tackled data are drawn on Figure 4. Figure 4. Normalization value: (a) NPSH, (b) Maximum, (c) Minimum, (d) Mean, (e) Peak, (f) Figure 4. Normalization value: (a) NPSH, (b) Maximum, (c) Minimum, (d) Mean, (e) Peak, Absolute mean, (g) Variance, (h) Standard deviation, (i) Kurtosis, (j) Skewness, (k) Root mean (f) Absolute mean, (g) Variance, (h) Standard deviation, (i) Kurtosis, (j) Skewness, (k) Root mean square, square, (l) Shape factor, (m) Crest factor, (n) Kurtosis factor, (o) Impulse factor, (p) Margin factor (l) Shape factor, (m) Crest factor, (n) Kurtosis factor, (o) Impulse factor, (p) Margin factor. From Figure 4, in the NPSH decreasing process, the corresponding fifteen (15) vibration feature From Figure 4, in the NPSH decreasing process, the corresponding fifteen (15) vibration feature value in the test interval coexist in the situation of increase and decrease instead of monotonous value in the test interval coexist in the situation of increase and decrease instead of monotonous relations. Meanwhile, as the vacuum pump starts working, positive and negative relevance coexists relations. Meanwhile, as the vacuum pump starts working, positive and negative relevance coexists between the NPSH and fifteen (15) vibration feature. On the other hand, some values might be between the NPSH and fifteen (15) vibration feature. On the other hand, some values might be abnormal since the negative values exist in the original signal and the potential unknown factors are abnormal since the negative values exist in the original signal and the potential unknown factors are distributing. For instance, Figure 4d shows the mean value in 40 m /h. However, as weight entropy distributing. For instance, Figure 4d shows the mean value in 40 m /h. However, as weight entropy states, the rationale and credible value can be acquired based on the calculated value of grey relation 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 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 decided whether it is related or not. Furthermore, the relevance matrix  can be acquired with the data data in Figure 4b–p through the Step 3 calculation, the consequence of which can be seen in Figure 5. 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  expresses the grey slope coecient of the corresponding feature target in di erent 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 coecient 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 di erent 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 di erence. 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, 8190 9 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13 Figure 5. Grey slope correlation coefficient: (a) Maximum, (b) Minimum, (c) Mean, (d) Peak, (e) Figure 5. Grey slope correlation coecient: (a) Maximum, (b) Minimum, (c) Mean, (d) Peak, Absolute mean, (f) Variance, (g) Standard deviation, (h) Kurtosis, (i) Skewness, (j) Root mean (e) Absolute mean, (f) Variance, (g) Standard deviation, (h) Kurtosis, (i) Skewness, (j) Root mean square, square, (k) Shape factor, (l) Crest factor, (m) Kurtosis factor, (n) Impulse factor, (o) Margin factor (k) Shape factor, (l) Crest factor, (m) Kurtosis factor, (n) Impulse factor, (o) Margin factor. In Figure 5, n denotes the numbers of the calculated slope, and ζ expresses the grey slope Due to the existence of positive and negative value in the feature target, the relevance of the coefficient of the corresponding feature target in different stages. From Figure 5, the trend of all feature target cannot be judged directly. Therefore, transforming the negative and positive value into targets except the Kurtosis factor basically considered has a positive or negative relevance with NPSH Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13 the same positive interval by Step 4 as Figure 6 depicted. 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. (a) (b) (c) 3 3 3 Figure 6. Heat map of translated value: (a) 40 m /h, (b) 50 m /h, (c) 60 m /h. 3 3 3 Figure 6. Heat map of translated value: (a) 40 m /h, (b) 50 m /h, (c) 60 m /h. 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. Table 2. Final comprehensive coefficient. Target 40 50 60 Average Maximum 0.5084 0.8431 0.5090 0.6398 Minimum 0.4455 0.5725 0.5108 0.5130 Mean 0.7145 0.3881 0.8119 0.6131 Peak 0.4465 0.8429 0.5086 0.6187 Absolute mean 0.9254 0.9343 0.4888 0.8102 Variance 0.9472 0.9541 0.5519 0.8424 Standard deviation 0.9237 0.9344 0.4899 0.8099 Kurtosis 0.4068 0.4199 0.8836 0.5416 Skewness 0.7447 0.5816 0.8498 0.7093 Root mean square 0.9238 0.9345 0.4899 0.8100 Shape factor 0.3774 0.2944 0.8706 0.4790 Crest factor 0.4453 0.4200 0.5117 0.4534 Kurtosis factor 0.6869 0.9923 0.9162 0.8690 Impulse factor 0.4463 0.4201 0.5126 0.4541 Margin factor 0.44 0.4201 0.5130 0.4543 For the above calculating consequence, the value closer to 1 means the relevance is more intense. On the contrary, when the value is closer to 0, it depicts a weaker relevance. By ranking the feature target in the principle of small to large, the recommended ordering of vibration feature target in different flow rate is as follows: 1. 40 m /h: variance > absolute mean > root mean square > standard deviation > skewness > mean > kurtosis factor > maximum > margin factor> peak > impulse factor> minimum > crest factor > kurtosis > shape factor. 2. 50 m /h: kurtosis factor > variance > root mean square > standard deviation > absolute mean > maximum > peak > skewness > minimum > margin factor > impulse factor > crest factor > kurtosis > mean >shape factor. 3. 60 m /h: kurtosis factor > kurtosis > shape factor > skewness > mean > variance > margin factor > impulse factor > crest factor > minimum > maximum > peak > standard deviation > root mean square > absolute mean. Appl. Sci. 2020, 10, 8190 10 of 13 According to Step 5, the corresponding entropy weight can be attached under di erent pressure stages in a corresponding flow rate. The final relevant coecient 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. Table 2. Final comprehensive coecient. Target 40 50 60 Average Maximum 0.5084 0.8431 0.5090 0.6398 Minimum 0.4455 0.5725 0.5108 0.5130 Mean 0.7145 0.3881 0.8119 0.6131 Peak 0.4465 0.8429 0.5086 0.6187 Absolute mean 0.9254 0.9343 0.4888 0.8102 Variance 0.9472 0.9541 0.5519 0.8424 Standard deviation 0.9237 0.9344 0.4899 0.8099 Kurtosis 0.4068 0.4199 0.8836 0.5416 Skewness 0.7447 0.5816 0.8498 0.7093 Root mean square 0.9238 0.9345 0.4899 0.8100 Shape factor 0.3774 0.2944 0.8706 0.4790 Crest factor 0.4453 0.4200 0.5117 0.4534 Kurtosis factor 0.6869 0.9923 0.9162 0.8690 Impulse factor 0.4463 0.4201 0.5126 0.4541 Margin factor 0.44 0.4201 0.5130 0.4543 For the above calculating consequence, the value closer to 1 means the relevance is more intense. On the contrary, when the value is closer to 0, it depicts a weaker relevance. By ranking the feature target in the principle of small to large, the recommended ordering of vibration feature target in di erent flow rate is as follows: 1. 40 m /h: variance > absolute mean > root mean square > standard deviation > skewness > mean > kurtosis factor > maximum > margin factor> peak > impulse factor> minimum > crest factor > kurtosis > shape factor. 2. 50 m /h: kurtosis factor > variance > root mean square > standard deviation > absolute mean > maximum > peak > skewness > minimum > margin factor > impulse factor > crest factor > kurtosis > mean >shape factor. 3. 60 m /h: kurtosis factor > kurtosis > shape factor > skewness > mean > variance > margin factor > impulse factor > crest factor > minimum > maximum > peak > standard deviation > root mean square > absolute mean. 4. Average: kurtosis factor > variance > absolute mean > root mean square > standard deviation > skewness > maximum > peak > mean > kurtosis > minimum > shape factor > margin factor > impulse factor > crest factor. From the calculated results, the relevance coecient might have diversity under di erent operating conditions. However, the relevance coecient of the kurtosis factor, variance, absolute mean and root mean square above all along which recommend applying priority. The shape factor, margin factor, impulse factor, and peek factor always below 0.5 means that the low sensitive with NPSH. This explains why the summary feature targets from the literature [21,23] have good e ects in detecting and monitoring the cavitation in terms of mathematics. From the physical concept, such as the kurtosis factor, it is a quantity indicating how sharply a probability distribution increases and decreases around the distribution mean. As one sort of dimensionless coecient, it has great sensitivity to the impulse signal and is nearly independent of the rotation speed, size, and load with machine. The numerical value uncovers the fact that this feature target has an intensity relation with NPSH which can put the vibration signal induced by the bubble burst into the range of the impulse signal. Thus, this feature target is especially appropriate to establish the relation between the vibration and Appl. Sci. 2020, 10, 8190 11 of 13 cavitation. This further establishes how feasible the application of grey slope correlation and entropy weight method is in the selection of centrifugal pump. 5. Conclusions In this research, the vibration acceleration signal is captured under pressure and flow rate variation and extracted fifteen (15) common feature targets from it to establish the relevance issues between cavitation and vibration. The grey slope correlation is proposed to quantitatively evaluate the relevance between feature the target and cavitation. The new established method has successfully solved the problem of positive and negative relations which cannot be solved by the traditional Deng’s grey relation. In addition, with the entropy weight method applied, the feature target can be evaluated on the same scale. The numerical calculation shows that the kurtosis factor, variance, absolute mean, root mean square of vibration acceleration signal has intensity relevance with NPSH. The cavitation states of the centrifugal pump can be monitored by using these parameters. This paper provides an objective selection strategy of a vibration feature target in evaluating the cavitation based on the numerical value. In terms of feature target selection, the universal and specific mathematical standard is established in the research. Author Contributions: Conceptualization, R.C.; Methodology, R.C.; Software, R.C.; Validation, J.Y. Formal Analysis, R.C.; Investigation, R.C.; Resources, J.Y.; Data Curation, R.C.; Writing-Original Draft Preparation, R.C.; Writing-Review & Editing, R.C.; Supervision, J.Y.; Project Administration, J.Y.; Funding Acquisition, J.Y. Both authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Key Research and Development Program of China (No. 2018YFB0606103) and Jiangsu Key Research and Development Plan of China (No. BE2018085). Acknowledgments: The authors are sincerely grateful to editor Tracy Yu and anonymous reviewers for providing valuable comments and reviewing the manuscripts. This study is supported by National Key Research and Development Program of China (No. 2018YFB0606103) and Jiangsu Key Research and Development Plan of China (No. BE2018085). Conflicts of Interest: The authors declare no conflict of interest. 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Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation

Cao, Ruijia; Yuan, Jianping
Applied Sciences , Volume 10 (22) – Nov 19, 2020
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applied sciences Article Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation Ruijia Cao and Jianping Yuan * National Research Center of Pumps, Jiangsu University, Zhenjiang 212013, China; 2111811001@stmail.ujs.edu.cn * Correspondence: yh@ujs.edu.cn Received: 11 October 2020; Accepted: 12 November 2020; Published: 19 November 2020 Abstract: 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 di erent stage. These feature targets are transformed into the same numerical scale by standardization process. In the end, the final comprehensive coecient 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 e ective in establishing the relevance between NPSH and vibration. Keywords: centrifugal pumps; cavitation; vibration; grey slope correlation; weight entropy; selection strategy 1. 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 Appl. Sci. 2020, 10, 8190; doi:10.3390/app10228190 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 8190 2 of 13 the running of analysis code enable the scholar to pay more attention in digging valuable results from vast amount of data. Currently available results indicated that the suction pressure di erence might change the pump vibration states [7–9], although this sort of vibration feature has a di erent appearance in di erent working flow rates [10–16]. However, cavitation can be defined as the rupture of a liquid due to a pressure drop [17]. It can be identified by vibration feature due to bubble burst. In these cited works of signal process, the feature target is a key element which should be considered and determined. Thus, root mean square (RMS) might be the most popular and universal parameter since it can be extracted from a historical vibration signal and reflect the cavitation occurrence. For instance, Dong et al. [18] and Zhang et al. [19] selected the concept of energy developed from the RMS to establish the potential variation caused by suction pressure and working flow rates. Similarly, to acquire the situations in di erent net positive suction head (NPSH), the RMS of acceleration in di erent monitor points displaced on the pump was captured to mirror the cavitation states [20]. Moreover, the parameters like the mean, variance, standard deviation, skewness, kurtosis, and crest factor were also used to carry out this kind of work [21]. In the literature [22], empirical mode decomposition (EMD) method with these parameters was used to decompose original signals into a number of intrinsic mode functions (IMFs). In the process of diagnosing the flow instabilities [23], the most troubling and confusion things were that di erent feature target selection might disturb the evaluation while the pump is working in the cavitation or air injection mode. The result points out the fact that the feature target might have di erent sensitivity under various circumstances. As mentioned in the first part, all the feature target selections were based on perceptual cognizance and personal experience. The details of strict mathematical explanation about why these targets can be used to evaluate the vibration manner were not adequately captured in the literature. It is against this backdrop that this research was conceived to find a reliable way to execute a robust and reliable relation between variables which would give an absolute fact instead of ambiguous justification identified by Al-Obaidi [24]. On the condition of the small sample, variety and complexity of uncertain factors, multivariate analysis cannot be directly applied. Grey relation analysis, a branch of grey system theory, as an e ective method to evaluate the relevance between variants, has exhibited its charming and universality in the multifarious discipline. For instance, in the automatic driving [25], the parameters which a ect safe driving can be extracted and analyzed. In the iron austempering [26], it can be used to establish the relationship between temperature and machinability performance. In the architectural planning [27], it can be used to evaluate and ensure the substation site selection. Above all, the grey relation method would be applied in the pump field to determine what targets derived from the initial vibration signal have an intensity relation with the NPSH. Moreover, the information entropy method would be used in the research to sort out the relevant parameters for its importance degree. 2. Grey Relation Entropy Analysis Method 2.1. 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 dicult 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 of1 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 Appl. Sci. 2020, 10, 8190 3 of 13 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. 2.2. 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 di erence 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 di erence on some index, the entropy is large which indicates that this index can provide a tiny amount of e ective information and the weight should be small. 2.3. 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 ), N(P ), N(P ), , N(P ) as the reference sequence, which indicates the sequence 1 2 3 of tabulated data of NPSH, where P represents the real-time pressure on nth times. Vibration features are taken as the comparative sequence, that is V = V (P ) , V (P ) , V (P ) ,  , V (P ) , i = 1 , 2 ,  , k, i i 1 i 2 i 3 i which denotes the comparative sequence. Step 2: Make sequence be dimensionless Due to the values’ physical scale di erence, 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 = 1 , 2 , 3 ,  , m (1) maxfP g max N(P ) X( P ) =  , n = 1 , 2 , 3 ,  , m (2) max N(P ) max V (P ) i n Y ( P ) =  , n = 1 , 2 , 3 ,  , m (3) max V (P ) Step 3: Calculate the coecient of grey slope correlation For the unequal interval sequence, define the grey slope correlation coecient as max max max ( P ) = sgn(DX( P ),DY ( P )) Q (4) n n n i i where, max max max max > 1 , DX( P )DY ( P )  0 < n n sgn(DX( P ),DY ( P )) = (5) n i n max max 1 , DX( P )DY ( P ) < 0 n i n max DX( P ) 1 + max D P Q = (6) max max max DX( P ) DX( P ) DY ( P ) n n n 1 1 1 i 1 + + max max max X X D P D P D P n n n Appl. Sci. 2020, 10, 8190 4 of 13 max max max DX( P ) = X( P ) X(P ) > n n n1 max max max > (7) DY ( P ) = DY ( P ) DY (P ) , n  2 > n n i i i n1 max max max D P = P P n n n1 max Y = Y ( P ) (8) i i n=1 max X = X( P ) (9) n=1 max Step 4: Standardize the target matrix  ( P ) max As the uncertainty of positive and negative value exist in  ( P ), therefore, the transmitting to i n the same sign is necessary for this paper. max Define, R = ( ( P )) in i in If the sequence belongs to the larger-the-better type-like positive value, the comparable sequence (CS) is calculated as R minfR g in in R = (10) in maxfR g minfR g in in n n If the sequence belongs to the smaller-the-better type like negative value, the comparable sequence (CS) is expressed as maxfR g R in in R = (11) in maxfR g minfR g in in n n where R 2 [0 , 1] in Step 5: Calculate grey slope correlation entropy Define the entropy of the nth to be: H = f ln f (12) n in in ln m n=1 in where, f = , while R = 0, let R ln R = 0 in m P in in in in n=1 Then, the nth entropy coecient is: 1 H ! = (13) 1 H n=1 Step 6: Calculate final comprehensive coecient From the weight entropy, the final coecient can be expressed as = !(n)R (n) (14) in n=1 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. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 13 From the weight entropy, the final coefficient can be expressed as = ω() nR (n) (14) ξ  i in n=1 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 Appl. Sci. 2020, 10, 8190 5 of 13 influence on the reference column will be. 3. Signal Capture and Pretreatment 3. Signal Capture and Pretreatment In order to verify the scientific feasibility of the proposed method as described earlier in In order to verify the scientific feasibility of the proposed method as described earlier in Sections Sections 2.1 and 2.2, a handle process was adopted as shown in Figure 1. The experiments were 2.1 and 2.2, a handle process was adopted as shown in Figure 1. The experiments were conducted in conducted in multiple suction pressure under three flow rate points. The vibration features were multiple suction pressure under three flow rate points. The vibration features were extracted from its extracted from its vibration acceleration signal. vibration acceleration signal. Figure 1. Flow chart of process. Figure 1. Flow chart of process. 3. 3.1. 1. TT est est Ri Rig g The experiments were carried out on a closed test rig located within Jiangsu University as The experiments were carried out on a closed test rig located within Jiangsu University as presented presentedin inFig Figur ure e2. 2 In t . Inhthe e cyclic proc cyclic process, ess, the fl theui fluid d from the ta from thenk enters tank enters into the pu into themp through the pump through the soft pipe by the rotational e ect of the impeller. The impeller transfers the fluid back to the tank 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 Appl. Sci. 2020, 10, 8190 6 of 13 (for adjust flow rate). Figure 2. Test rig. Figure 2. Test rig. T Table able 11 sho shows ws the important geometric and oper the important geometric and operational ational parameters of the prototype pump parameters of the prototype pump under investigation. under investigation. Table 1. Main parameters of the prototype pump. Table 1. Main parameters of the prototype pump. Name Symbol Name Symbol Value Value Designed flow rate Qd 50 m /h Designed flow rate Q 50 m /h Designed head H 37 m Designed head Hd 37 m Rated rotational speed n 3000 r/min Rated rotational speed n 3000 r/min Impeller inlet diameter D 74 mm Impeller inlet diameter D1 74 mm Impeller outlet diameter D 174 mm Impeller outlet diameter D2 174 mm Impeller outlet width b 12 mm Impeller outlet width b2 12 mm Blades Z 6 Blades Volute diameter D Z 184 mm 6 Rated Power p 5 kW Volute diameter D3 184 mm Rated Power p 5 kW 3.2. Experiment Instrument 3.2. Experiment Instrument In this experiment, vibration acceleration and suction pressure data were monitored and recorded in detail. In this experiment, v The vertical vibration ibration acceleration acceleration an signalsd ofsuction pressure suction pipe ektexine data were were monitor monitored ed using and a recorded computer in d and etail these . The vert signals ical wer vibe rat saved ion accel under erat di io ner sient gnaoperating ls of suctio conditions n pipe ekteof xin pr e w essur ere m e. o Figur nitore ed 3 shows using a com the monitor puter and th location ese sign on theals w tested ere saved pump. under different operating conditions of pressure. Figure The 3 shows the moni sensor used into this r loca experiment tion on the tested pump. is a high frequency sensor (PCB 352A60 series) with a sensitivity value of 10 mv/g and the frequency response range of500 g/Hz. A pressure transmitter (WIKA S-10) with 0.2% accuracy in full scale was used to record the pressure di erence. 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 [31–33]. Appl. Sci. 2020, 10, 8190 7 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13 Outlet pressure tapping Inlet pressure tapping Vibration monitor (a) (b) Figure 3. Instrument layout: (a) test site; (b) monitor points. Figure 3. Instrument layout: (a) test site; (b) monitor points. 3.3. Experiment Method 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 In this experiment, the pressure and vibration must be recorded simultaneously. At the given flow (rate, WIKA S- multiple 10) wi data, th ±0 captur .2% ac ed cura bycy the insuction full scalpr e w essur as us e,ewer d to reco e used rd th to study e pressure the vibration difference with . In or pressur der to e capture the relative signals accurately, the sampling frequency and time used were 16,000 Hz and1 s variation at a constant rotation speed of 3000 rpm. Firstly, the deflation valve was fully open and the respectively ball valve was [22]. For closed. furth Afteremeasuring r details ab the out the experi data under mental this condition, method, please the deflation refer to the valve was author’s closed previous work [31–33]. 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 3.3. Experiment Method 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 In this experiment, the pressure and vibration must be recorded simultaneously. At the given pump cannot take away any atmosphere from the tank or the test rig cannot provide the foreseeable flow rate, multiple data, captured by the suction pressure, were used to study the vibration with 3 3 dangers. The same steps would be repeated in the flow rate of 40 m /h and 60 m /h to guarantee the pressure variation at a constant rotation speed of 3000 rpm. Firstly, the deflation valve was fully open robust of algorithm. 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 3.4. Data Pretreatment at the suction side of the pump until cavitation occurred. After the emergence of cavitation, the Transforming the suction pressure into NPSH and the vibration acceleration signal would vacuum pump and ball valve were opened and observed over a period of time until there was a drop convert into fifteen (15) types of feature target which contain the maximum, minimum, mean, peak, in pressure at the inlet of the pump. At this point, the data acquisition process was put on hold until absolute mean, variance, standard deviation, kurtosis, skewness, root mean square, shape factor, the vacuum pump cannot take away any atmosphere from the tank or the test rig cannot provide the 3 3 crest factor, kurtosis factor, impulse factor, and margin factor. The specific mathematical function and foreseeable dangers. The same steps would be repeated in the flow rate of 40 m /h and 60 m /h to steps can be found in Appendix A from the literature [24]. guarantee the robust of algorithm. 4. Analysis and Methodology 3.4. Data Pretreatment On the foundation of Step 1, the above data in ever flow rate point would be turned into the Transforming the suction pressure into NPSH and the vibration acceleration signal would reference sequence (RS) and comparative sequence (CS) as the following matrix expresses: 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, RS = NPSH (p ), NPSH (p ), NPSH (p ) NPSH (p ), NPSH (p ) r 1 r 2 r 3 r n1 r n crest factor, kurtosis factor, impulse factor, and margin factor. The specific mathematical function 8 9 and steps can be found in Appendix A from the literature [24]. > > Maximum(p ) Maximum(p )  Maximum(p ) Maximum(p ) > 1 2 n1 > > > > > > > > Minimum(p ) Minimum(p )  Minimum(p ) Minimum(p ) > 1 2 n1 n > > > > 4. Analysis and Methodology > > < = . . . . . . . . CS = > > > . . . . > > > > > On the foundation of Step 1, the above data in ever flow rate point would be turned into the > > > > Impulse f actor(p ) Impulse f actor(p )  Impulse f actor(p ) Impulse f actor(p ) > > 1 2 n1 n > > > > reference sequence (RS) and comparative sequence (CS) as the following matrix expresses: : ; Margin(p ) Margin(p )  Margin(p ) Margin(p ) 1 2 n1 n 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. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 R{ S = NPSH( p), NPSH( p), NPSH( p) NPSH( p ), NPSH( p)} rr12 r3 rn−1 rn  Maximum() p Maximum( p )  Maximum( p ) Maximum( p )  12 nn −1   Minimum() p Minimum( p )  Minimum( p ) Minimum( p ) 12 nn −1     CS =        Impulse factor() p Impulse factor( p )  Impulse factor( p ) Impulse factor( p ) 12 nn −1    M arginp ( ) M arginp ( )  M argin(pM)argin(p)   12 nn −1  For the normalization processing of the data from a matrix by the maximum way according to Appl. Sci. 2020, 10, 8190 8 of 13 Step 2, the tackled data are drawn on Figure 4. Figure 4. Normalization value: (a) NPSH, (b) Maximum, (c) Minimum, (d) Mean, (e) Peak, (f) Figure 4. Normalization value: (a) NPSH, (b) Maximum, (c) Minimum, (d) Mean, (e) Peak, Absolute mean, (g) Variance, (h) Standard deviation, (i) Kurtosis, (j) Skewness, (k) Root mean (f) Absolute mean, (g) Variance, (h) Standard deviation, (i) Kurtosis, (j) Skewness, (k) Root mean square, square, (l) Shape factor, (m) Crest factor, (n) Kurtosis factor, (o) Impulse factor, (p) Margin factor (l) Shape factor, (m) Crest factor, (n) Kurtosis factor, (o) Impulse factor, (p) Margin factor. From Figure 4, in the NPSH decreasing process, the corresponding fifteen (15) vibration feature From Figure 4, in the NPSH decreasing process, the corresponding fifteen (15) vibration feature value in the test interval coexist in the situation of increase and decrease instead of monotonous value in the test interval coexist in the situation of increase and decrease instead of monotonous relations. Meanwhile, as the vacuum pump starts working, positive and negative relevance coexists relations. Meanwhile, as the vacuum pump starts working, positive and negative relevance coexists between the NPSH and fifteen (15) vibration feature. On the other hand, some values might be between the NPSH and fifteen (15) vibration feature. On the other hand, some values might be abnormal since the negative values exist in the original signal and the potential unknown factors are abnormal since the negative values exist in the original signal and the potential unknown factors are distributing. For instance, Figure 4d shows the mean value in 40 m /h. However, as weight entropy distributing. For instance, Figure 4d shows the mean value in 40 m /h. However, as weight entropy states, the rationale and credible value can be acquired based on the calculated value of grey relation 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 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 decided whether it is related or not. Furthermore, the relevance matrix  can be acquired with the data data in Figure 4b–p through the Step 3 calculation, the consequence of which can be seen in Figure 5. 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  expresses the grey slope coecient of the corresponding feature target in di erent 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 coecient 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 di erent 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 di erence. 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, 8190 9 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13 Figure 5. Grey slope correlation coefficient: (a) Maximum, (b) Minimum, (c) Mean, (d) Peak, (e) Figure 5. Grey slope correlation coecient: (a) Maximum, (b) Minimum, (c) Mean, (d) Peak, Absolute mean, (f) Variance, (g) Standard deviation, (h) Kurtosis, (i) Skewness, (j) Root mean (e) Absolute mean, (f) Variance, (g) Standard deviation, (h) Kurtosis, (i) Skewness, (j) Root mean square, square, (k) Shape factor, (l) Crest factor, (m) Kurtosis factor, (n) Impulse factor, (o) Margin factor (k) Shape factor, (l) Crest factor, (m) Kurtosis factor, (n) Impulse factor, (o) Margin factor. In Figure 5, n denotes the numbers of the calculated slope, and ζ expresses the grey slope Due to the existence of positive and negative value in the feature target, the relevance of the coefficient of the corresponding feature target in different stages. From Figure 5, the trend of all feature target cannot be judged directly. Therefore, transforming the negative and positive value into targets except the Kurtosis factor basically considered has a positive or negative relevance with NPSH Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13 the same positive interval by Step 4 as Figure 6 depicted. 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. (a) (b) (c) 3 3 3 Figure 6. Heat map of translated value: (a) 40 m /h, (b) 50 m /h, (c) 60 m /h. 3 3 3 Figure 6. Heat map of translated value: (a) 40 m /h, (b) 50 m /h, (c) 60 m /h. 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. Table 2. Final comprehensive coefficient. Target 40 50 60 Average Maximum 0.5084 0.8431 0.5090 0.6398 Minimum 0.4455 0.5725 0.5108 0.5130 Mean 0.7145 0.3881 0.8119 0.6131 Peak 0.4465 0.8429 0.5086 0.6187 Absolute mean 0.9254 0.9343 0.4888 0.8102 Variance 0.9472 0.9541 0.5519 0.8424 Standard deviation 0.9237 0.9344 0.4899 0.8099 Kurtosis 0.4068 0.4199 0.8836 0.5416 Skewness 0.7447 0.5816 0.8498 0.7093 Root mean square 0.9238 0.9345 0.4899 0.8100 Shape factor 0.3774 0.2944 0.8706 0.4790 Crest factor 0.4453 0.4200 0.5117 0.4534 Kurtosis factor 0.6869 0.9923 0.9162 0.8690 Impulse factor 0.4463 0.4201 0.5126 0.4541 Margin factor 0.44 0.4201 0.5130 0.4543 For the above calculating consequence, the value closer to 1 means the relevance is more intense. On the contrary, when the value is closer to 0, it depicts a weaker relevance. By ranking the feature target in the principle of small to large, the recommended ordering of vibration feature target in different flow rate is as follows: 1. 40 m /h: variance > absolute mean > root mean square > standard deviation > skewness > mean > kurtosis factor > maximum > margin factor> peak > impulse factor> minimum > crest factor > kurtosis > shape factor. 2. 50 m /h: kurtosis factor > variance > root mean square > standard deviation > absolute mean > maximum > peak > skewness > minimum > margin factor > impulse factor > crest factor > kurtosis > mean >shape factor. 3. 60 m /h: kurtosis factor > kurtosis > shape factor > skewness > mean > variance > margin factor > impulse factor > crest factor > minimum > maximum > peak > standard deviation > root mean square > absolute mean. Appl. Sci. 2020, 10, 8190 10 of 13 According to Step 5, the corresponding entropy weight can be attached under di erent pressure stages in a corresponding flow rate. The final relevant coecient 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. Table 2. Final comprehensive coecient. Target 40 50 60 Average Maximum 0.5084 0.8431 0.5090 0.6398 Minimum 0.4455 0.5725 0.5108 0.5130 Mean 0.7145 0.3881 0.8119 0.6131 Peak 0.4465 0.8429 0.5086 0.6187 Absolute mean 0.9254 0.9343 0.4888 0.8102 Variance 0.9472 0.9541 0.5519 0.8424 Standard deviation 0.9237 0.9344 0.4899 0.8099 Kurtosis 0.4068 0.4199 0.8836 0.5416 Skewness 0.7447 0.5816 0.8498 0.7093 Root mean square 0.9238 0.9345 0.4899 0.8100 Shape factor 0.3774 0.2944 0.8706 0.4790 Crest factor 0.4453 0.4200 0.5117 0.4534 Kurtosis factor 0.6869 0.9923 0.9162 0.8690 Impulse factor 0.4463 0.4201 0.5126 0.4541 Margin factor 0.44 0.4201 0.5130 0.4543 For the above calculating consequence, the value closer to 1 means the relevance is more intense. On the contrary, when the value is closer to 0, it depicts a weaker relevance. By ranking the feature target in the principle of small to large, the recommended ordering of vibration feature target in di erent flow rate is as follows: 1. 40 m /h: variance > absolute mean > root mean square > standard deviation > skewness > mean > kurtosis factor > maximum > margin factor> peak > impulse factor> minimum > crest factor > kurtosis > shape factor. 2. 50 m /h: kurtosis factor > variance > root mean square > standard deviation > absolute mean > maximum > peak > skewness > minimum > margin factor > impulse factor > crest factor > kurtosis > mean >shape factor. 3. 60 m /h: kurtosis factor > kurtosis > shape factor > skewness > mean > variance > margin factor > impulse factor > crest factor > minimum > maximum > peak > standard deviation > root mean square > absolute mean. 4. Average: kurtosis factor > variance > absolute mean > root mean square > standard deviation > skewness > maximum > peak > mean > kurtosis > minimum > shape factor > margin factor > impulse factor > crest factor. From the calculated results, the relevance coecient might have diversity under di erent operating conditions. However, the relevance coecient of the kurtosis factor, variance, absolute mean and root mean square above all along which recommend applying priority. The shape factor, margin factor, impulse factor, and peek factor always below 0.5 means that the low sensitive with NPSH. This explains why the summary feature targets from the literature [21,23] have good e ects in detecting and monitoring the cavitation in terms of mathematics. From the physical concept, such as the kurtosis factor, it is a quantity indicating how sharply a probability distribution increases and decreases around the distribution mean. As one sort of dimensionless coecient, it has great sensitivity to the impulse signal and is nearly independent of the rotation speed, size, and load with machine. The numerical value uncovers the fact that this feature target has an intensity relation with NPSH which can put the vibration signal induced by the bubble burst into the range of the impulse signal. Thus, this feature target is especially appropriate to establish the relation between the vibration and Appl. Sci. 2020, 10, 8190 11 of 13 cavitation. This further establishes how feasible the application of grey slope correlation and entropy weight method is in the selection of centrifugal pump. 5. Conclusions In this research, the vibration acceleration signal is captured under pressure and flow rate variation and extracted fifteen (15) common feature targets from it to establish the relevance issues between cavitation and vibration. The grey slope correlation is proposed to quantitatively evaluate the relevance between feature the target and cavitation. The new established method has successfully solved the problem of positive and negative relations which cannot be solved by the traditional Deng’s grey relation. In addition, with the entropy weight method applied, the feature target can be evaluated on the same scale. The numerical calculation shows that the kurtosis factor, variance, absolute mean, root mean square of vibration acceleration signal has intensity relevance with NPSH. The cavitation states of the centrifugal pump can be monitored by using these parameters. This paper provides an objective selection strategy of a vibration feature target in evaluating the cavitation based on the numerical value. In terms of feature target selection, the universal and specific mathematical standard is established in the research. Author Contributions: Conceptualization, R.C.; Methodology, R.C.; Software, R.C.; Validation, J.Y. Formal Analysis, R.C.; Investigation, R.C.; Resources, J.Y.; Data Curation, R.C.; Writing-Original Draft Preparation, R.C.; Writing-Review & Editing, R.C.; Supervision, J.Y.; Project Administration, J.Y.; Funding Acquisition, J.Y. Both authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Key Research and Development Program of China (No. 2018YFB0606103) and Jiangsu Key Research and Development Plan of China (No. BE2018085). Acknowledgments: The authors are sincerely grateful to editor Tracy Yu and anonymous reviewers for providing valuable comments and reviewing the manuscripts. This study is supported by National Key Research and Development Program of China (No. 2018YFB0606103) and Jiangsu Key Research and Development Plan of China (No. BE2018085). Conflicts of Interest: The authors declare no conflict of interest. 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Published: Nov 19, 2020

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