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A machine learning approach to classification of gas entrainment and impeller wear in centrifugal pumps Bohn, Bryan 2021

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     A MACHINE LEARNING APPROACH TO CLASSIFICATION OF GAS ENTRAINMENT AND IMPELLER WEAR IN CENTRIFUGAL PUMPS  by BRYAN BOHN B.S., University of Connecticut, 2010    A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE AND DOCTORAL STUDIES (Mechanical Engineering)    THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)     February, 2021 © Bryan Bohn, 2021ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, a thesis entitled:   A MACHINE LEARNING APPROACH TO CLASSIFICATION OF GAS ENTRAINMENT AND IMPELLER WEAR IN CENTRIFUGAL PUMPS _____________________________________________________________________________________  submitted by _____________Bryan Bohn_________________ in partial fulfillment of the requirements for  the degree of ________ Master of Applied Science_____________________________________________ in ___________ Mechanical Engineering_____________________________________________________  Examining Committee: Boris Stoeber, Professor, Mechanical Engineering and Electrical and Computer Engineering, UBC_________ Supervisor Bhushan Gopaluni, Professor, Chemical and Biological Engineering, UBC____________________________ Supervisory Committee Member Hongshen Ma, Professor, Mechanical Engineering, UBC_________________________________________ Supervisory Committee Member Minkyun Noh, Assistant Professor, Mechanical Engineering, UBC__________________________________ Supervisory Committee Member     iii  ABSTRACT Centrifugal pumps are a fundamental part of fluid transport around the world.  Consequently, they are also one of the world’s dominant energy consumers.  The impacts of inefficient operation and undiagnosed wear are widely documented and can be disastrous environmentally, financially, and logistically.  Though commercial tools and methods for monitoring pump performance are abundant, they are used infrequently in practice.  This phenomenon derives from several factors, including monitoring systems’ poor scalability to function with large numbers of pumps, acquisition costs, the necessity for additional technical personnel, and stringent policies constraining process downtime. This thesis describes the development of an affordable, adaptable sensing method for classifying two conditions detrimental to centrifugal pump operation; gas entrainment and radial impeller wear.  The method utilizes dynamic pressure measurements, collected at the pump discharge using a solitary, conventional pressure transducer.  Decomposing these pressure fluctuations into a novel array of statistical features yields characteristic trends correlated to the target phenomena.  These features are then used to train a series of machine learning algorithms, including multilayer perceptrons (MLP), support vector machines (SVM), and random forests, which are in turn used to characterize the target conditions using binary, multi-class, and regression methods.   Dynamic pressure data for training and testing the classification algorithms is generated using simulated and experimental methods.  The binary MLP model predicts gas entrainment exceeding a 2% void fraction of air with 90% accuracy, and radial wear exceeding 1.5% of the impeller diameter with 97% accuracy.  The multi-class MLP classifies gas entrainment and radial impeller wear into severity classes spanning 1% increments with 62% and 82% success rates, respectively.  The random forest regression model achieves a median prediction error of 0.44% for gas entrainment and 0.16% for impeller wear.  The diagnostic system presented in this research is unique in that it is not conceived as a standalone tool for pump users, but rather a shared process to be trained and configured by the pump manufacturer, then implemented by the operators.  In its envisioned application, the scope of the classified phenomena would be augmented by the manufacturer to capture a wide variety of pump performance characteristics.    iv  LAY SUMMARY Centrifugal pumps are one of the world’s most important machines.  They convert electrical power into fluid movement to transport liquids.  In doing so, they also consume a lot of energy.  When pumps wear down, or the fluid they are transporting behaves unexpectedly, they may start operating poorly, which wastes energy.  There are a variety of commercial devices available to measure whether or not a centrifugal pump is operating well, but, in practice, they are not widely used because of their cost and difficulty to implement.   We developed a more practical method to quantify two kinds of problems that would cause a centrifugal pump to waste energy: mechanical wear and air bubbles in the fluid flow.  Our technique uses a single sensor to measure pressure fluctuations in the fluid.  After measuring fluctuations in many different conditions, we created machine learning algorithms to identify how severe the problem is.                 v  PREFACE This thesis is an original work by the author, Bryan Bohn, at The University of British Columbia, Vancouver, Canada (UBC).  The research herein was performed in its entirety by the author, under the supervision and guidance of Boris Stoeber, Professor of Mechanical Engineering and Electrical and Computer Engineering, and Bhushan Gopaluni, Professor of Chemical and Biological Engineering.  The work presented in this thesis was conducted as part of the Energy Reduction in Mechanical Pulping (ERMP) program at UBC.  The Principal Investigator for the ERMP is James Olson, Professor and Dean of the Faculty of Applied Science.  Dean Olson conceived the original concept for the project and provided guidance and advising throughout.      A portion of this research was presented at the 2019 IEEE Sensors conference in Montreal, QC, Canada and published in the proceedings: B. Bohn, J. Olson, B. Gopaluni, and B. Stoeber, “Sensing Concept for Practical Performance-Monitoring of Centrifugal Pumps,” in 2019 IEEE SENSORS, Montreal, QC, Canada, Oct. 2019, pp. 1–4, doi: 10.1109/SENSORS43011.2019.8956559.  The publication discusses the motivation for this work and proposes a concept for pump performance monitoring using a multi-sensor approach.  The remainder of the work in this thesis is unpublished, as of its date of submission.          vi  TABLE OF CONTENTS  ABSTRACT ..................................................................................................................................................... iii LAY SUMMARY ............................................................................................................................................. iv PREFACE ........................................................................................................................................................ v TABLE OF CONTENTS .................................................................................................................................... vi LIST OF TABLES ............................................................................................................................................ viii LIST OF FIGURES ........................................................................................................................................... ix LIST OF SYMBOLS ......................................................................................................................................... xiii ACKNOWLEDGEMENTS ............................................................................................................................... xvi DEDICATION ............................................................................................................................................... xvii 1. INTRODUCTION ......................................................................................................................................... 1 1.1 Motivation ........................................................................................................................................... 1 1.2 Pump Performance.............................................................................................................................. 3 1.3 Performance-monitoring Systems ....................................................................................................... 4 1.4 Phenomena of Interest ...................................................................................................................... 15 1.5 Pressure Dynamics in a Rotating Pump ............................................................................................. 21 1.6 Objectives .......................................................................................................................................... 29 2. MATERIALS AND EXPERIMENTAL METHODS ........................................................................................... 30 2.1 Simulation.......................................................................................................................................... 31 2.2 Experimental Apparatus .................................................................................................................... 38 2.3 Gas-entrainment Study ..................................................................................................................... 41 2.4 Impeller Wear Study .......................................................................................................................... 43 2.5 Combined Study ................................................................................................................................ 44 3. PRINCIPLES AND DESIGN ......................................................................................................................... 45 3.2 Signal Decomposition ........................................................................................................................ 46 3.3 Feature Space .................................................................................................................................... 54 3.4 Classification Models ......................................................................................................................... 57 4. RESULTS AND DISCUSSION ...................................................................................................................... 61 4.1 Gas Entrainment Study ...................................................................................................................... 61 4.3 Impeller Wear Study .......................................................................................................................... 77 4.4 Combined Study ................................................................................................................................ 87 vii  5. CONCLUSIONS AND FUTURE WORK ........................................................................................................ 90 5.1 Summary of Results ........................................................................................................................... 90 5.2 Applications and Limitations ............................................................................................................. 91 5.3 Conclusions ....................................................................................................................................... 92 5.5 Future Work ...................................................................................................................................... 93 BIBLIOGRAPHY ............................................................................................................................................. 94 APPENDIX A – SUPPLEMENTARY CALCULATIONS ........................................................................................ 99 A1 Turbulence Parameters ...................................................................................................................... 99 A2 Pearson Correlation Coefficient ....................................................................................................... 100 APPENDIX B – SUPPLEMENTARY FIGURES ................................................................................................. 101 B1 Virtual Probe Placement Dependency .............................................................................................. 101 B2 Gas Entrainment Feature Optimization Plots (Experimental) .......................................................... 102 B3 Impeller Wear Feature Optimization Plots (Simulated) .................................................................... 109           viii  LIST OF TABLES Table 2.1: Sensor Parameters……………………………………………………………………………………………………………………41 Table 4.1: Experimental Gas Entrainment States (Training)…………………………………………………………………….…61 Table 4.2: Truth Table for Binary MLP Classification of Gas Entrainment (Testing)……………………………………70 Table 4.3: Gas Entrainment States Misclassified by Binary MLP Model (Testing)………………………………………70 Table 4.4: Gas Entrainment States Severely Misclassified by Multi-class MLP Model (Testing)…………………71 Table 4.5: Gas Entrainment States Severely Misclassified by Regression Model (Testing)………………………..72 Table 4.6: Comparison of Gas Entrainment Classification Performance Before and After Feature Optimization....................................................................................................................................................75 Table 4.7: Simulated Radial Impeller Wear States…………………………………………………………………………………..…77 Table 4.8: Truth Table for Binary MLP Classification of Impeller Wear (Testing)……………………………………….84 Table 4.9: Impeller Wear States Misclassified by Binary MLP Model (Testing)………………………………………….84 Table 4.10: Impeller Wear States Misclassified by Multi-class MLP Model (Testing)…………………………………85 Table 4.11: Impeller Wear States Severely Misclassified by Regression Model (Testing)…………………………..85 Table 4.13: Simulated States for Combined Study………………………………………………………………………………….…87    ix  LIST OF FIGURES Figure 1.1: A 1.5 kW centrifugal pump and its components (left).  Blue arrows indicate the direction of fluid flow.  The red arrow specifies the direction of 𝜔.  Removing the housing reveals the impeller (right), which transfers kinetic energy to the working fluid…………………………………………………………….....................................1 Figure 1.2: The performance curve chart for a Gorman-Rupp model VG1 centrifugal pump [11].  The highlighted curves are the efficiency profiles.  Operating the pump at the flow (x-axis), pressure (y-axis), and impeller speeds corresponding to the regions within these profiles will yield the denoted efficiency.  For this pump, the BEP is 56%...........................................................................……………………………………………..3 Figure 1.3: A piezoresistive pressure transducer (left) and a negative temperature coefficient (NTC) thermistor (right).  Pressure and temperature sensing are two of the most common methods to monitor pump operation……………………………………………………………………………………………………………………………………….…6 Figure 1.4: The normalized frequency spectrum of a 30 kW centrifugal pump with a two-blade impeller, collected using a single-axis accelerometer on the axial face of the pump volute.  Here, the pump is operating at a rotating frequency (RF) of 15.4 Hz, which generates a small peak at the corresponding frequency. Likewise, the blade-passing frequency (BPF), equal to twice the RF, exhibits a modest peak.  However, the spectrum is dominated by a vibration mode occurring at three times the RF (47.3 Hz).  This is the undesirable influence from the close-coupled driving motor, which uses three-phase power delivery, transmitting vibrations through the driveshaft into the pump’s resonant structure.  Frequency content beyond 800 Hz has been removed using a low-pass filter……………………………………………………………........….…10 Figure 1.5: Distributed gas bubbles entrained in liquid flow through a pipe…………………………………………….…15 Figure 1.6: Erosive wear on the tips of a three-blade impeller……………………………………………………………………18 Figure 1.7: A two-dimensional schematic of a simplified centrifugal pump with a two-blade (𝑁imp = 2) impeller.  The inset shows a perspective view of a comparable physical pump for clarity.    Fluid flow direction is indicated by the blue arrows.  The operating fluid enters the pump through the port at the center of the impeller (from out of plane) and exits vertically.  A scroll-shaped funnel called the volute houses the impeller and guides the working fluid from the area of the rotating impeller to the discharge port. The notch at intersection of the curves of the volute scroll is the “cutwater.”..........………………………….21 Figure 1.8: Time-varying, experimental pressure measurements at the intake (𝑝in, red line) and discharge (𝑝out, blue line) of a centrifugal pump operating at 𝜔 = 16.4𝐻𝑧.  In this example, 𝑁imp = 2.  The respective sample averages 𝑝in̅̅ ̅̅ = 9.2𝑘𝑃𝑎 and 𝑝out̅̅ ̅̅ ̅ = 83.9𝑘𝑃𝑎 are indicated.  The difference  ∆𝑝 = 74.7𝑘𝑃𝑎.  Both measurements exhibit a cyclic fluctuations, but the behavior is most apparent in 𝑝out…………..…………….…22 Figure 1.9: 𝑝out as a function of impeller position over two rotations at 𝜔 = 16.4𝐻𝑧.  At point (1), the tip of the leading impeller blade has just reached the cutwater.  This defines 𝜃 = 0.  As the impeller rotates to (2), the discharge pressure drops abruptly, only starting to increase again once the second impeller blade starts to force fluid toward the discharge.  The discharge pressure increases until the second impeller reaches the cutwater at (3).  A pressure drop follows until the leading impeller blade reaches (4), where the pressure again rises and the cycles continue.  Over this time scale, it can be observed that the x  fluctuations of 𝑝out are not purely sinusoidal, but could be more appropriately generalized as a triangle or sawtooth wave…………………………………………………………………………………………………………………………………………23 Figure 1.10: a) Experimental 1 s samples of 𝑝out at 𝜔 = 16.4𝐻𝑧 for normal pump operation (blue) and a gas entrainment condition 𝜑air = 5.0% (green); b) A detail view of the corresponding fluctuations over two impeller rotations.  The impeller positions are synchronized for clarity…………………………………………….25 Figure 1.11: Geometry of radial impeller wear (wear region indicated in red).  ∆D is exaggerated for clarity...............................................................................................................................................................26 Figure 1.12: a) A simulated 0.5 s sample of 𝑝out for normal pump operation (blue) and an impeller wear condition 𝛼imp = 3.0% (orange) at ω = 18.75Hz; b) A detail view of the corresponding fluctuations over two impeller rotations.  The impeller positions are synchronized for clarity......................................……….…28 Figure 2.1: Workflow of the preliminary studies for a) gas entrainment and b) impeller wear.  The initial simulated data is used to evaluate signal decomposition techniques and propose machine learning methods……………………………………………………………………………………………………..……………………………………………30 Figure 2.2: Workflow of the primary studies for a) gas entrainment and b) impeller wear.  Experimental data is used to validate gas entrainment classification, whereas numerical data is used for impeller wear………30 Figure 2.3: Geometry of the simulated centrifugal pump.  The virtual pressure probe (greed dot) is positioned in the center of the discharge, 50 mm from the pump outlet……..……………………………………………31 Figure 2.4: a) Contrasting experimental (dotted) and simulated (solid) measurements of 𝑝out for normal pump operation at 𝜔 = 16.4𝐻𝑧.  Each sample is 0.5 s total and normalized to its maximum value.  The first 0.2 s are shown for clearer interpretation of the waveforms; b) Contrasting experimental (dotted) and simulated (solid) measurements of 𝑝out for the gas entrainment condition 𝜑air = 5.0%.  Each gas entrainment condition is normalized to the maximum value of its corresponding healthy condition from Figure 2.4a to show the relative reduction in variance…..............................................................................…37 Figure 2.5: The 30 kW centrifugal pump used as a testbed throughout the experimental studies………………38 Figure 2.6: A schematic of the fluid loop in the pilot-scale pulp refining plant at UBC…………..………………….…39 Figure 2.7: A Honeywell MLH050PGB06A pressure transducer………………………..…………………………………….…40 Figure 3.1: a) Simulated pressure signals of a pump at 𝜔 ≤ 18.75𝐻𝑧 with increasing gas entrainment; b) Simulated pressure signals of a pump at 𝜔 ≤ 18.75𝐻𝑧 with increasing impeller wear……….…………….…46 Figure 3.2: 𝐸s as a function of a) 𝜑air and b) 𝛼imp.  The magnitudes are normalized for comparison.  The blue squares, orange diamonds, and maroon inverted triangles correspond to the states with low (𝜔 =8.75𝐻𝑧), intermediate (𝜔 = 13.75𝐻𝑧), and high (𝜔 = 18.75𝐻𝑧) impeller speeds, respectively.  The trend in each speed range is illustrated using a second-order polynomial best-fit line.  Additionally, the trend for all the states is indicated by the black dotted line.  These trend lines are included to illustrate the cases where the feature has a dual dependence on 𝜔, as well and the target phenomenon.  This labeling convention is employed in each of the subsequent feature plots………………………………………………………….…48 Figure 3.3: 𝑉s as a function of a) 𝜑air and b) 𝛼imp.  The magnitudes are normalized for comparison………..49 xi  Figure 3.4: 𝐺s as a function of a) 𝜑air and b) 𝛼imp.  The magnitudes are normalized for comparison…….....50 Figure 3.5: 𝐾s as a function of a) 𝜑air and b) 𝛼imp.  The magnitudes are normalized for comparison……...…51 Figure 3.6: 𝐸R as a function of a) 𝜑air and b) 𝛼imp.  The magnitudes are normalized for comparison…….…52 Figure 3.7: 𝐸FFT as a function of a) 𝜑air and b) 𝛼imp.  The magnitudes are normalized for comparison……..54 Figure 3.8: PCA of the preliminary gas entrainment simulations.  a) The corresponding scree plot with an inset table of loadings for the first two principal components (PCs).  The loadings are the coefficients by which the input features can linearly combined to construct PCs.  They can be interpreted as the strength of influence a given feature has on the orientation of a particular PC.  The loadings are ordered by their contribution to PC1; b) The distribution of the simulated states across PC1 and PC2.  Each state is labeled with the magnitude of 𝜑air…………………………………………………………………………………….…………………………….…55 Figure 3.9: PCA of the preliminary impeller wear simulations. a) The corresponding scree plot with an inset table of loadings for PC1 and PC2.  The loadings are ranked by their contribution to PC1; b) The distribution of the simulated states across the first two PCs.  Each state is labeled with magnitude of 𝛼imp……..…………56 Figure 3.10: Schematic of the MLP architecture used for two-class classification [71]…………………………..…57 Figure 3.11: Schematic of MLP architecture used for multi-class classification [71].  The gas entrainment model has five output classes.  For impeller wear, three classes are used………………………………….………………59 Figure 4.1: Samples of discharge pressure fluctuations across a range of rotating frequencies, during operation without gas entrainment…………………………………………………….……………………………………………………62 Figure 4.2: Samples of discharge pressure fluctuations across a range of rotating frequencies, each with an air void fraction exceeding 4.5%......................................................................................................................63 Figure 4.3: The fluctuations of 𝑝out at 𝜔 = 14.9𝐻𝑧 for normal pump operation (blue) and an air entrainment condition with 𝜑air = 6.0% (green)…………………………..……………………………………………………..…64 Figure 4.4a-c: The normalized values of a) 𝑝out̅̅ ̅̅ ̅, b) 𝐸s, and c) 𝑉s as a function of 𝜑air……………..……………….…65 Figure 4.4d-f: The normalized values of d) 𝐺s, e) 𝐾s, and f) 𝐸𝑅 as a function of 𝜑air……………………..………….…66 Figure 4.4g: The normalized values of 𝐸FFT as a function of 𝜑air…………………………………………………………..…67 Figure 4.5: a) A scree plot of the PCs of gas entrainment and the loading for each variable with respect to PC1 and PC2, listed in order of their contribution to PC1; b) The distribution of the experimental states across the first two PCs……………………………………………………………………………………………………………………….……69 Figure 4.6: 𝑉s as a function of 𝜔.  The blue and green trend lines correspond to the states with 𝜑air = 0 and 𝜑air ≥ 5, respectively.  The dotted black curve and best fit equation reflect the trend of all the states….............................................................................................................................................................73 Figure 4.7: 𝑉s′ as a function of 𝜑air.  The blue, orange, and red trend lines correspond to 𝜔 = 8.9, 12.9, and 16.9𝐻𝑧, respectively.  The dashed black line reflects the trend of all the data points…........................…74 Figure 4.8: The average magnitudes of each input feature at 𝜔 = 14.9𝐻𝑧, with 𝜑air at 0% (blue bars), 2.8% (yellow bars) and 6.0% (red bars).  These correspond to states 1M, 6M, and 12M from Table 4.1.  Each xii  magnitude is determined by averaging ten repeated 1 s pressure measurements of the same state.    The error bars indicate an interval of two standard deviations…………………………………………………………………….…76 Figure 4.9: Simulated discharge pressure fluctuations across and range of rotating frequencies during a) normal operation and b) operation with 𝛼imp = 3.0%...........................................................................…78 Figure 4.10a-c: The normalized values of a) 𝐸s, b) 𝑉s, and c) 𝐺sas a function of 𝛼imp…………………….……….…80 Figure 4.10d-f: The normalized values of d) 𝐾s, e) 𝐸𝑅, and f) 𝐸FFT as a function of 𝛼imp………………….........…81 Figure 4.11: a) A scree plot of the PCs of radial impeller wear and the loadings for each variable with respect to PC1 and PC2, listed in order of their contribution to PC1; b) The distribution of the experimental states across the first two PCs…………………………………………………………………………………………….……………………….……..83 Figure 4.12: The optimization steps of 𝑉s:  a) 𝑉s as a function of 𝛼imp; b) 𝑉s as a function of 𝜔; c) 𝑉s′ as a function of 𝛼imp…………………………………………………………………………………………………………….…………………………86 Figure 4.13: a) A scree plot of the PCs of normal operation, severe impeller wear, severe air entrainment, and combined conditions.  The loadings for each variable with respect to PC1 is adjacent, ranked in descending order; b) The distribution of the various conditions across the first two PCs…………………….……..88    xiii  LIST OF SYMBOLS Symbol  Description        Units 𝐶𝑝o  Specific heat at constant pressure of the pump outlet   𝐽/(𝑘𝑔 ∙ 𝐾) 𝐶𝜀1  Turbulent production rate scalar     Dimensionless 𝐶𝜀2  Turbulent dissipation rate scalar      Dimensionless 𝐶𝜇  Turbulent eddy viscosity scalar      Dimensionless 𝑑  Pipe diameter        𝑚 𝐷𝑜  Diameter of the unworn impeller     𝑚 𝐷𝛼  Diameter of the actual impeller      𝑚 ∆𝐷  Wear magnitude of the impeller      𝑚 𝑒  Euler’s number `       Dimensionless 𝐸FFT  Energy of the frequency spectrum     Dimensionless 𝐸R  Energy of the autocorrelation signal     Dimensionless 𝐸s  Signal energy        Dimensionless 𝐹  Volume shear force       𝑁/𝑚3 𝑭  Feature vector        Dimensionless {𝑭}  Feature space        Dimensionless 𝑓s  Sampling frequency       𝐻𝑧 𝑓max  Maximum frequency of interest      𝐻𝑧 𝑓min  Minimum frequency of interest      𝐻𝑧 𝑓nyq  Nyquist frequency       𝐻𝑧 ∆𝑓  Frequency resolution       𝐻𝑧 𝑔  Acceleration of gravity       𝑚/𝑠2 𝒈  Gravitational field       𝑚/𝑠2 𝐺s  Signal skewness        Dimensionless 𝐻  Head         𝑚 𝑰  Identity tensor        Dimensionless 𝑘  Turbulent kinetic energy      𝐽/𝑘𝑔, 𝑚2/𝑠2 𝐾s  Signal kurtosis        Dimensionless 𝐿  Length         𝑚 ?̇?  Mass flow rate        𝑘𝑔/𝑠 𝑀air  Number of prediction classes for air entrainment   Dimensionless 𝑀wear  Number of prediction classes for air impeller wear   Dimensionless 𝑁  Number of sampled data points      Dimensionless 𝑁imp  Number of impeller blades      Dimensionless 𝑝  Fluid pressure        𝑃𝑎 𝑷  Prediction vector       Dimensionless 𝑝in  Fluid pressure at the pump inlet      𝑃𝑎 𝑃𝑘  Turbulent kinetic energy production     𝐽/(𝑚3 ∙ 𝑠) 𝑝out  Fluid pressure at the pump outlet     𝑃𝑎 𝑷out  Signal vector of discharge pressure measurements   𝑘𝑃𝑎 ?̂?out  Unbiased signal vector of discharge pressure measurements  𝑘𝑃𝑎 ?̃?out  Normalized vector of discharge pressure measurements   Dimensionless ∆𝑝  Average pressure across the pump     𝑃𝑎 ∆𝑝L  Pressure loss over a pipe of length 𝐿     𝑃𝑎 xiv  𝑄  Volumetric flow rate       𝑚3/𝑠 𝑅𝑙  Autocorrelation coefficient      Dimensionless 𝑅2  Coefficient of determination      Dimensionless 𝑆  Strain rate tensor       𝐽/(𝑚3 ∙ 𝑠) 𝑡  Time         𝑠 𝑇in  Temperature at the pump inlet      𝐾 𝑇out  Temperature at the pump outlet     𝐾 𝑡s  Sampling interval       𝑠 𝑇s  Sample time duration       𝑠 ∆𝑇  Temperature change across the pump     𝐾 ∆𝑇i  Temperature change across an ideal, isentropic pump   𝐾 𝒖  Flow field velocity vector      𝑚/𝑠 v  Specific volume        𝑚3/𝑘𝑔 𝑣in  Flow velocity at the pump inlet      𝑚/𝑠 𝑣out  Flow velocity at the pump outlet     𝑚/𝑠 𝑉s  Signal variance        Dimensionless 𝑤  Neural network weight       Dimensionless 𝑊fluid  Fluid power output from the pump     𝑊 𝑊i  Input power required to operate an ideal, isentropic pump  𝑊 𝑊in  Input power to the pump      𝑊 𝑋  Frequency magnitude       Dimensionless 𝑿  FFT spectrum        Dimensionless 𝑥n  Pressure magnitude at index n      𝑘𝑃𝑎 𝑥n  Unbiased pressure magnitude at index n     𝑘𝑃𝑎 ?̃?n  Normalized pressure magnitude at index n    𝑘𝑃𝑎 ?̅?  Weighted average vector      Dimensionless   𝛼imp  Impeller material loss ratio      Dimensionless 𝛾air  Volumetric flow rate of air of air into the pump    𝑚3/𝑠 𝛾w  Volumetric flow rate of air of water into the pump   𝑚3/𝑠 Γ  Prediction target value       Dimensionless 𝜀  Turbulent kinetic energy dissipation rate     𝐽/(𝑘𝑔 ∙ 𝑠) 𝜖  Level-set transition thickness parameter     Dimensionless 𝜁  Level-set tuning parameter      Dimensionless 𝜂conv  Thermal efficiency determined using the conventional method  Dimensionless 𝜂LR  Learning rate coefficient      Dimensionless 𝜂therm  Thermal efficiency determined using the thermodynamic method Dimensionless 𝜃  Angular position of the impeller      𝑑𝑒𝑔 𝜇  Dynamic viscosity       (𝑁 ∙ 𝑠)/𝑚2 𝜇T  Turbulent eddy viscosity      (𝑁 ∙ 𝑠)/𝑚2 𝜈l  Kinematic viscosity of the liquid phase     𝑚2/𝑠 𝜌  Average fluid density       𝑘𝑔/𝑚3 𝜌air  Average fluid density of air      𝑘𝑔/𝑚3 𝜌water  Average fluid density of liquid water     𝑘𝑔/𝑚3 𝜎𝑘  Adjustable constant       Dimensionless 𝜎𝜀  Adjustable constant       Dimensionless xv  𝜏air  Air entrainment classification threshold     Dimensionless 𝜏wear  Impeller wear classification threshold     Dimensionless 𝜙  Level-set variable       Dimensionless 𝜑air  Volume fraction of air       Dimensionless 𝜔  Rotating frequency of the impeller     𝐻𝑧 𝜔max  Maximum rotating frequency      𝐻𝑧 𝜔min  Minimum rotating frequency      𝐻𝑧     xvi  ACKNOWLEDGEMENTS  I would like to thank my supervisors, Professor Boris Stoeber and Professor Bhushan Gopaluni for allowing me the opportunity to undertake this research.  Your guidance, enthusiasm, good humor, and patience have made for an exceptional research experience.  My time working with both of you has been engaging and illuminating, and for that I am tremendously appreciative.  Thank you both for everything.   I also extend my gratitude to the additional members of my examining committee, Professor Hongshen Ma and Professor Minkyun Noh.  Thank you for contributing your time and constructive feedback.  I appreciate it immensely.  Additionally, I would like the thank Professor James Olson, Dean of the UBC Faculty of Applied Science and the originator of this research project, for contributing guidance, perspective, and positivity throughout.  It has been a pleasure working with you and the ERMP members.  Thank you for all your help.   My time at UBC has been outstanding.  That is no small praise, considering that a year of it has overlapped a devastating global pandemic.  I am thankful to the wonderful people in my lab for making the endeavor so gratifying, both academically and personally.  I would like to acknowledge Sam, Pranav, Crystal, Nick, Sajana, Martina, Robin, Hamed, Jorge, and all the Stoeber Lab members, past and present.  You have helped make a truly extraordinary and memorable experience for me.  I’m looking forward to seeing you all in person again soon.  There have been many minor heroes whose efforts have benefited my work and life at UBC (sometimes without their knowing).  Lest they go unrecognized, I would like to thank George Soong and Reanna Seifert at the Paper and Pulp Centre for their perpetual help resolving problems, the maintenance staff of the Fred Kaiser Building, the UBC Recreation employees at Thunderbird Arena, the serving staffs at the Gallery, JamJar, and Rain or Shine, the owners of Dentry’s, my friends in BC and beyond, and my teammates on the Rain City Pigeons hockey team.          Lastly, I would like to acknowledge the people at home who sent their love and support across the whole continent while I uprooted to BC; Jer, Nicole, Robin, Corey, Sean, Conor, and the dogs – thank you guys.  Finally, thank you mom and dad.  I love you all.    xvii  DEDICATION     To my most enthusiastic collaborator, Captain.   1  1. INTRODUCTION 1.1 Motivation Centrifugal pumps have been a cornerstone of fluid-moving processes for centuries [1].  They are essential to virtually all modern industries, including manufacturing, mining, civil infrastructure, textiles, paper and pulp production, petrochemical refining, construction, and power production.  At an applied level, they perform myriad functions, like moving cooling fluid through web servers, extracting drinking water from wells, or exchanging the contents of fermentation tanks at a brewery. Whether a low wattage pump draining soapy water from a washing machine or a multi-thousand-horsepower colossus processing slurry from a mine, the mechanics of centrifugal pump operation are fundamentally similar.  Kinetic energy is provided by a rotating shaft, which is typically driven by an electric motor or combustion engine.  The rotating frequency (RF) of the shaft is designated 𝜔.  The shaft connects to a rotor, called an impeller, that revolves within the pump housing, imparting hydrodynamic energy on the incoming working fluid in the form of increased kinetic energy, pressure, and temperature.  The impeller is comprised of one or more vanes, called “blades,” which may be straight, curved, or a more complex helical shape in three dimensions.  The total number of blades on a particular impeller is 𝑁imp.  Figure 1.1 shows the main components of a centrifugal pump system.  The overall size, shape, construction materials, and configuration of a centrifugal pump depend on its intended application.   Figure 1.1: A 1.5 kW centrifugal pump and its components (left).  Blue arrows indicate the direction of fluid flow.  The red arrow specifies the direction of 𝝎.  Removing the housing reveals the impeller (right), which transfers kinetic energy to the working fluid. Impeller Blades Motor Driveshaft Centrifugal Pump Intake Discharge 2  As with any energy conversion process, there are losses.  Understanding and minimizing these losses is central to the reliability, financial viability, and environmental sustainability of virtually every pumping process. Centrifugal pumps are one of the world’s primary consumers of electrical energy.  In industries with extensive bulk fluid processing, centrifugal pumps can account for between 20 and 60% of expended electrical motor energy [2].  A 2001 study by the European Commission on improving the efficiency of pumps determined that centrifugal pumps were the largest single consumer of industrial electricity in the European Union [3].  The massive scale of electrical energy consumption by centrifugal pumps is not inherently negative, but instead underscores the potentially global impact that minor improvements (or declines) in pumping performance can have.  The causes and impacts of inefficient operation of centrifugal pumps have been extensively studied from a variety of perspectives [4]–[6].  However, despite this understanding, inefficiency is pervasive.  A 2010 survey of pump manufacturers suggested that more than 40% of centrifugal pumps were incorrectly specified from the start and would suffer reduced efficiency throughout their existence as a result [2].  The European Commission efficiency study concluded, “[…] the largest energy savings are to be made through better design and control of pump systems.”  The study notes that inefficiently operating pumps are not strictly a consequence of inadequate system design, but also challenges in process management, such as timely access to technical expertise and inability to disrupt production to perform preventative maintenance [3].  The list could be expanded to include limited financial flexibility for process improvement and the tendency to oversize pumps to account for unanticipated changes in production volume.  The underlying implication is that better performance cannot be achieved exclusively through better pump and pump control technologies.  Improved performance is equally, or perhaps more dependent on how accessible and practical those technologies are to the end user. The ability to quantify performance comprises not only the technical tools to make the measurements, but also the means to implement the tools in the first place, manage the data, and interpret the results.  There are many commercial devices and measurement approaches for quantifying centrifugal pump performance, yet they are not widely implemented [7].  Understanding this phenomenon and giving consideration to the real-world barriers to implementing performance monitoring systems is paramount to creating a method that can traverse the divide between a novel research concept and an impactful industrial tool. 3  1.2 Pump Performance  The term “performance” can be defined in a variety of ways for centrifugal pumps.  Informally, it is often employed as a vague synonym for efficiency, owing to the practical weight of minimizing waste energy in a given fluid process.  However, more explicitly, performance can be considered as the conditions and phenomena that reflect the overall health of a pumping process.  The most prominent elements include efficiency, operating cost, reliability, process stability, and the health of the mechanical components.   Depending on the type, the thermal efficiency of an exceptionally well-configured centrifugal pump can surpass 80%, meaning that, of the power delivered to the motor, 80% is realized as productive work on the operating fluid [8].  More conventional centrifugal pump efficiencies fall in the range of 30-70% [5], [9], [10].  Poorly controlled pumping processes may have efficiencies as low as 10%.  The highest achievable efficiency for a given pump configuration is known as the Best Efficiency Point (BEP).  Pumping efficiency, the BEP, and other performance factors are typically depicted in a characteristic pump performance curve chart produced by the manufacturer.  The performance curve is a critical reference for centrifugal pump operators across industries.  An example chart is shown in Figure 1.2.      Figure 1.2: The performance curve chart for a Gorman-Rupp model VG1 centrifugal pump [11].  The highlighted curves are the efficiency profiles.  Operating the pump at the flow (x-axis), pressure (y-axis), and impeller 4  speeds corresponding to the regions within these profiles will yield the denoted efficiency.  For this pump, the BEP is 56%.      Centrifugal pumps degrade over time, which results in reduced efficiency and a departure from the BEP.  The rate and severity of the performance decline can be a combined effect of a variety of factors; the fluid media, flow dynamics, impeller rotating speed, process uptime, operating temperature, maintenance, and more.  Reductions in performance can be considered in two interrelated categories; losses due to mechanical wear and losses due to adverse fluid phenomena.  Mechanical wear performance losses are the result of structural changes to the pump components over time.  These changes commonly include material erosion on the impeller surfaces, bearing degradation, corrosion, physical damage caused by process anomalies and flow debris, haphazard structural modifications to the impeller, or, perhaps most frequently, a combination thereof [7].  Performance losses due to wear are expected, even for a well-configured pumping process, but the impact can be exacerbated by poor management and lack of monitoring.   Losses from adverse fluid phenomena occur when a centrifugal pump expends energy on the operating media in ways other than productive fluid work.  Typical scenarios include cavitation, gas entrainment, leaks, stalling, flow recirculation, and other volumetric losses [12].  Adverse fluid phenomena can cause, or be caused by, mechanical wear, improper flow configuration, flow anomalies, and process control problems.   If the mechanical degradation or adverse flow conditions are severe enough that the pumping process is profoundly altered, a pump’s behavior can depart from the original performance curve entirely, leading to an operating state without an established reference for performance.  In industry, pumps operating in reference-less states are common and notoriously difficult to manage [13].  When the performance curve of a centrifugal pump is rendered unusable, whether by mechanical wear or flow irregularities, additional means to measure and monitor performance are vital.  1.3 Performance-monitoring Systems Just as there are many phenomena that comprise a centrifugal pump’s overall performance, there are equally many tools and methodologies for measuring it.  Before engaging in a discussion of the present technical avenues for quantifying centrifugal pump performance, it must be noted that performance-5  monitoring should not be considered exclusively from the domain of the end-user (though, being that the end-user is typically the person implementing the system, it often is).  The previous section notes industry’s strong reliance on the manufacturer’s performance curve chart and the difficulty that can arise when that reference no longer suits a particular centrifugal pump.  The intuitive next step is to jump to consideration of secondary monitoring tools that can be used to supplant the lost performance reference.  However, a more thoughtful approach will consider both the end-user and the pump manufacturer.  This perspective expands the applied problem from an assessment of discrete performance-monitoring devices to a broader discussion of how pump performance data is collected and what additional references would improve its value.  More succinctly, if the pump manufacturer provided the end-user with something more robust and flexible than a static performance curve chart, what kind of impact could that have on the efficacy and popularity of performance-monitoring systems?  As data analytics and machine learning become more widely accessible, the question becomes more intriguing.   Efficiently controlling a centrifugal pump relies heavily on the operator’s means to understand the phenomena at hand.  To a degree, this understanding is based on expertise with the specific system and fluid process.  Centrifugal pump operators can often identify adverse conditions simply by listening to a pump, or consulting basic temperature or pressure measurements if they are obtainable [7], [13].  However, in the case of slowly progressing conditions or unfamiliar fluid phenomena, subjective observation becomes problematic.  In these cases, unless more accurate performance-monitoring tools are used, inefficient operating states can persist indefinitely.  This underscores the importance of having the ability to dependably measure and quantify centrifugal pump performance.            Despite their relative lack of widespread use, there are an abundance of sensor systems and monitoring approaches for observing the performance of centrifugal pumps.  The following sections summarize the operating bases of the foremost commercial methods and associated research.    Commercial products typically fall into four groups: conventional sensors for process measurement, efficiency monitoring devices, vibration analysis systems, and fault detection systems based on machine learning.  Beyond the commercial domain there are experimental sensors and methodologies that have been demonstrated in limited applications but have yet to reach wide commercialization.  These include magnetic wear sensing, hydrophone monitoring, motor phase current sensing, and dynamic pressure analysis.    6  1.3.1 Conventional Sensors Though many industrial centrifugal pumps operate without instrumentation [7], it is common to use conventional sensors to quantify basic elements of a pump’s operating state.  Direct measurements of fundamental process parameters can be used to infer the performance of centrifugal pumps without additional analysis or signal processing.  Widely used sensors include pressure transducers, thermometers, flow meters, tachometers, and wattmeters (Figure 1.3).  Implementations of conventional pump process monitoring can include intake and discharge pressure measurements, bearing temperature readings, flow rate monitoring, and rotational speed measurement.    Figure 1.3: A piezoresistive pressure transducer (left) and a negative temperature coefficient (NTC) thermistor (right).  Pressure and temperature sensing are two of the most common methods to monitor pump operation.   Conventional sensing is generally applied as real-time, steady-state process measurements.  This approach has the benefit of being affordable, reliable, and straightforward to implement and interpret.  When a drastic variation occurs, or a measurement deviates from a required threshold, it suggests to the operator that some element of the pump’s operation has changed and needs attention.  However, this reactive approach is in itself insufficient for classifying complex or slowly progressing adverse conditions.  Without a broader system for recording data, analyzing trends, and validating phenomena, conventional process measurement falls short in the depth of information it can provide.    7  1.3.2 Efficiency Monitoring Efficiency-based performance-monitoring systems provide the pump user with a measurement of how effectively a centrifugal pump is converting input energy into productive fluid work.  There are two main methodologies to quantify pump efficiency; conventional thermal efficiency and the thermodynamic efficiency method.   The conventional thermal efficiency method correlates the pump’s fluid power output 𝑊fluid to the input power 𝑊in.  The ratio 𝜂conv =  𝑊fluid𝑊in      (1) defines the thermal efficiency of the pumping process.  In the case of a centrifugal pump, 𝑊fluid = 𝜌𝑔𝐻𝑄, where 𝜌 is the average fluid density, 𝑔 is the acceleration of gravity, 𝐻 is the average pressure head across the pump, and 𝑄 is the average volumetric flow rate through the system.  Equivalently,  𝑊fluid = ∆𝑝𝑄 ,       (2) where ∆𝑝 is the difference between the average pressures at the pump intake 𝑝in̅̅ ̅̅  and discharge 𝑝out̅̅ ̅̅ ̅.  Substituting (2) into (1) yields the conventional determination for a centrifugal pump’s thermal efficiency, 𝜂conv =  ∆𝑝𝑄𝑊in .      (3) This efficiency determination method requires four sensors to implement; pressure sensors at the pump intake and discharge to measure ∆𝑝, a flow meter to determine 𝑄, and a wattmeter to measure 𝑊in.  Where performance monitoring is used, this approach is common.  It is the method specified by ISO 9906:2012 for hydraulic performance testing of rotodynamic pumps [14], and specific implementations have been proposed by many researchers [10], [15].  It is also used in commercial devices [16], [17]. For centrifugal pumps generating more than 1000 kPa, or pumping processes where attaining an accurate flow rate or power measurement is impracticable, it may be more suitable to use a thermodynamic approach and determine efficiency through direct measurement of waste energy, rather than as a ratio of output and input work [18].  The thermodynamic efficiency 𝜂therm =  𝑊i 𝑊in ,      (4) where 𝑊i is the power required to operate an ideal pump.  In an isentropic compression cycle of a fluid with mass flow rate ?̇? and average specific volume ?̅? from 𝑝in to 𝑝out, the input power 8  𝑊i = ?̇? ∫ 𝑣𝑑𝑝𝑝out𝑝in= ?̇??̅?∆𝑝 = ?̇?1𝜌∆𝑝 .     (5) Even in an ideal pump, there is a small temperature increase when the liquid is compressed.  If it can be reasonably assumed that heat loss from the pump to the atmosphere is negligible, temperature difference between the pump discharge to intake ∆𝑇 = 𝑇out − 𝑇in can be defined as ∆𝑇 = ∆𝑇i + ∆𝑇′ ,      (6) where ∆𝑇i is the expected temperature increase from ideal compression and ∆𝑇′ is the additional temperature rise resulting from pumping inefficiency.  The value of Δ𝑇i is determined from established property tables for the working fluid undergoing compression Δp.  Using the specific heat at constant discharge pressure 𝐶𝑝𝑜, wasted power can be equated to the increase in enthalpy beyond the ideal compression, such that 𝑊in − 𝑊i  = ?̇?𝐶𝑝𝑜∆𝑇′ .     (7) Combining (4) through (7), the thermodynamic efficiency determination 𝜂therm =11+𝜌𝐶𝑝𝑜(Δ𝑇−Δ𝑇i)/(Δp) .         (8) The thermodynamic approach to efficiency measurement also requires four sensors; pressure transducers at the pump intake and discharge to measure ∆𝑝, and temperature sensors at the intake and discharge to determine ∆𝑇.  It has a benefit in that it eliminates the need for flow rate and power sensing, which can be costly to integrate.  Applied in accordance with their respective constraints and assumptions, each efficiency measurement approach will yield an equivalent result.  This sensing technique is also prevalent in both commercial monitoring and academia [19]. Each efficiency determination approach has disadvantages.  For the conventional thermal efficiency determination method, the main criticism is in the need to measure flow rate and input power.  A. Whillier highlights the common case in chemical transport where a pump efficiency measurement is desired, but the diversity of working fluids makes reliable flow rate measurement with a single sensor unattainable [20].  Similarly, A. Cattaert describes the challenges in accurately determining input power in mining applications where a pump is not powered by an electric motor [18].  In addition, first-hand discussions with pulp and paper producers have made clear the economic infeasibility of integrating flow and power sensors when a single mill may have hundreds of unique centrifugal pumps in operation [13]. 9  The thermodynamic efficiency determination method, despite the simplicity of the sensors involved, also has drawbacks. The main concern is the accuracy by which Δ𝑇 can be measured.  To satisfy its highest precision class, ISO 5198: Centrifugal, Mixed Flow, and Axial Pumps Code for Hydraulic Performance Tests specifies this method as only suitable for centrifugal pumps generating at least 100m head (980 kPa) [21].  In his study of the required measurement uncertainties for this efficiency determination approach, A. Milne finds that a low pressure centrifugal pump generating 200 kPa would require temperature measurement within 4 mK to determine the efficiency within 5% – not reasonably achievable in a typical industrial setting [22]. In terms of broader performance classification, efficiency-based monitoring falls short from a variety of perspectives.  At a fundamental level, meaningfully interpreting an efficiency measurement requires an understanding of the expected efficiency.  In an application with hundreds of pumps in varying configurations, the challenge of maintaining a suitable reference for good efficiency for each is significant.  On the inverse, even when an inefficient operating state is truthfully detected, the cause of the loss cannot necessarily be extrapolated from the efficiency determination itself.  This diminishes the effectiveness of standalone efficiency monitoring in remedying unhealthy operating states.  A measure of efficiency is useful to have, but requires additional data if it is to be used to make substantive judgments about a centrifugal pumping process.   Finally, from a process management standpoint, the intent of measuring efficiency is to avoid inefficient operating conditions.  Yet, this approach to performance monitoring requires the pump to have already reached a degraded state before the inefficiency can be recognized.  This can be mitigated somewhat by establishing efficiency trends and preventative maintenance schedules, but maintaining such a record introduces its own practical challenges.  In addition, characterizing a pump’s performance exclusively on efficiency limits the operator’s ability to recognize serious conditions that may not correlate strongly to efficiency losses, such as minor cavitation or flow debris.  These shortcomings ultimately discourage the widespread use of standalone efficiency monitoring systems. 1.3.3 Vibration Monitoring  Vibration monitoring is the application of accelerometers for the purpose of evaluating characteristic vibrations that propagate through the pump’s mechanical structure.  In a typical 10  implementation, accelerometer data is collected from an external surface on the pump1 and converted to a frequency spectrum using fast Fourier transform (FFT).  The frequency features can be manually interrogated and correlated to known pump phenomena or integrated as parameters into a broader diagnostic system or machine learning algorithm.  An example of the frequency content of a centrifugal pump accelerometer measurement is shown in Figure 1.4.   Figure 1.4: The normalized frequency spectrum of a 30 kW centrifugal pump with a two-blade impeller, collected using a single-axis accelerometer on the axial face of the pump volute.  Here, the pump is operating at a rotating frequency (RF) of 15.4 Hz, which generates a small peak at the corresponding frequency. Likewise, the blade-passing frequency (BPF), equal to twice the RF, exhibits a modest peak.  However, the spectrum is dominated by a vibration mode occurring at three times the RF (47.3 Hz).  This is the undesirable influence from the close-coupled driving motor, which uses three-phase power delivery, transmitting vibrations through the driveshaft into the pump’s resonant structure.  Frequency content beyond 800 Hz has been removed using a low-pass filter.                                                                1 It is noted that airborne acoustic measurement is occasionally used in lieu of structure-borne vibration monitoring of centrifugal pumps.  In terms of analysis, the signals and features can be treated similarly.  However, when conducting acoustic emission measurements using a microphone, the influence of environmental sound can be a major detriment.  In many noise-filled industrial settings, acoustic pump monitoring is not practicable.  As such, it is not explicitly considered in this work.      10 100 10000.00.20.40.60.81.01.2Frequency (Hz)Normalized Vibration AmplitudeRF BPF Motor 11  The mechanical vibration characteristics of centrifugal pumps have been extensively studied [23]–[25].  Vibration monitoring in the frequency domain is particularly suited for distinguishing cyclic mechanical pump phenomena [26] and motor faults [27].  Behaviors such as a warped input shaft, bearing flat spots, lack of lubrication, and unbalanced components have distinct spectral manifestations that can be classified by a trained operator.  C. Scheffer characterizes sources and diagnosis methods for a variety of mechanical-fault-induced centrifugal pump vibrations with consideration to informing remedial actions and preventing downtime [28].  He also discusses the correlation between accelerometer placement and how prominently a particular behavior manifests in the acquired measurement.  M. Abd-Elaal et al. apply vibration monitoring to identify mechanical irregularities in a small centrifugal pump [29].  They demonstrate shaft imbalance, impeller damage, and pedestal looseness using features in the vibration frequency spectrum.         The characteristic vibrations of an operating centrifugal pump can also contain evidence of fluid phenomena.  A. Abdulaziz et al. investigate the frequency spectrum of vibration measurements to determine the presence and severity of cavitation [30].  They establish that the inception of cavitation can be detected through discrete vibration levels corresponding to the rotating frequency (RF) and blade passing frequency (BPF).  The BPF is equal to the RF multiplied by 𝑁imp.  Specifically, as cavitation evolves, the energy peak at the BPF weakens dramatically, and noise beyond twice the BPF becomes the dominant spectral feature.   An evident shortcoming of vibration-based monitoring systems is that some phenomena simply do not propagate vibrations strongly enough to be measured externally [31].  In the case of mechanical faults, despite the direct transmission path through the pump structure and into the accelerometer, the observable features in a particular sample are heavily dependent on physical placement of the sensor [38].  For fluid-borne phenomena, where the vibrational energy must first transfer from the fluid to the pump structure, the vibration transmission may be insufficient to make a reliable measurement.  Conversely, some sources of vibration can propagate disproportionately strongly through the pump structure, obscuring more subtle phenomena.  This is demonstrated in Figure 1.4, where vibrations caused by power delivery to the electric motor have a detrimental influence on the vibration spectrum.  Environmental structural noise generated by large equipment vibrating nearby could yield a similar outcome.  In these applications, vibration monitoring would lose its efficacy.    Another concern with vibration analysis is the relative lack of generalizability of accelerometer measurements between pumps.  Specifically, because each individual centrifugal pump is a unique 12  resonating structure, the vibrational energy transferred from a particular phenomenon to the transducer may not be consistent.  The generalizability problem worsens when considering pumps of dramatically different sizes and forms.  While the trends may be shared, the amplitudes, frequencies, and relative manifestations of performance features will likely be considerably different. 1.3.4 Machine Learning for Performance Monitoring and Fault Detection The scope of insights gained from centrifugal pump performance monitoring can be augmented by using the collected information as inputs for machine learning algorithms.  Commercial fault detection systems generally rely on vibration measurements, but multi-sensor learning models have also been demonstrated.    To classify flow blockages and impending cavitation, A. Panda et al. implement a Support Vector Machine (SVM) algorithm on a series of statistical features of vibration measurements taken on a centrifugal pump’s casing and bearing housing [32].  Using a feature vector comprised of mean, standard deviation, kurtosis, crest factor, and signal entropy, they develop and test binary and multi-class classifiers that aim to detect a) flow conditions ranging from unobstructed to 67% blockage across a range of pump rotating frequencies and b) normal flow versus an impending cavitation state (which is designated as just prior to the formation of visible vapor bubbles in the pump discharge).  The flow obstruction classifier demonstrates test accuracies ranging from 61.4% (achieved in the lowest speed, lowest blockage percentage scenario) to 100% (achieved in the highest speed, highest blockage percentage scenario), with most classification accuracies falling between 80 and 100%.  The cavitation classifier demonstrates near-perfect accuracy across operating speeds.  While the experiment yields a high classification accuracy across the sampled states, the feasibility of using this approach to detect these conditions in industry is low.  Flow obstructions, particularly significant ones, where the learning algorithm is shown to be most accurate, would be readily detected through conventional measurement with a pressure transducer.  In addition, regarding the classification of healthy or near-cavitating states, the conditions can be distinguished by manual observation of the frequency spectrum or standard deviation, skewness, or kurtosis trends.  As such, employing a machine learning algorithm to achieve classification of these extreme states is not particularly advantageous.                         V. Muralidharan et al. compare a series of discrete wavelets of accelerometer measurements using a decision tree algorithm for the purpose of identifying the most suitable signal features to employ in classification of bearing faults, impeller damage, and combined scenarios [33].  Using the Reverse 13  Biorthogonal 1.5 (rbio1.5) wavelet, they are able to classify the mechanical condition with 99.84% accuracy.  The same authors compare naïve Bayes and Bayes network classifier algorithms, using wavelets, for the same purpose and achieve similarly high classification accuracy [34].  However, as with the study by A. Panda et al., the phenomena being identified would likely be conspicuous through manual observation and would be ascertained by more direct methods in practice.  Direct classification of comparable faults through frequency analysis has been experimentally demonstrated by A. Daraz et al. (via acoustic emission measurements) among many others and analytically by M. Zhang et al. [35], [36].  It is a core function of most commercial vibration-based centrifugal pump fault detection systems [37], [38].    However, there are cases when mechanical and fluid phenomena cannot be readily classified through manual data analysis and machine learning is necessary.  M. Shervani-Tabar et al. employ a multi-class SVM algorithm for the purpose of characterizing the severity of cavitation in an axial-flow pump using vibration measurements [39].  Whereas the presence of cavitation is typically observable through conventional monitoring methods, the nature of the severity is less so.  Their work suggests that machine learning methods may have particular value in assessing the severity of pump phenomena, rather than simply the presence or absence.  1.3.5 Alternative Sensors and Monitoring  There are elements of centrifugal pump performance that cannot be easily characterized by conventional process measurement, efficiency monitoring, vibration-based analysis, or associated machine learning methods.  A ubiquitous example is the state of mechanical erosion on the impeller.  Several purpose-built sensors and methods have been developed for this case, yet none have achieved wide-scale implementation [7].  An externally mounted sensor that measures the extent of material loss from a centrifugal pump impeller has been developed [40].  The device uses an inductive coil to drive magnetic flux through the operating area of a centrifugal pump impeller.  As material erodes from the impeller with time, the fluid gap between the impeller and pump side plate grows.  This increasing gap raises the reluctance of the magnetic circuit, which is then measured and correlated to determine the extent of impeller material loss within 0.25 mm.  The sensor has been demonstrated in a controlled trial, however its adaptability to pumps of various sizes is limited by its physical dimensions.  Additionally, the high-magnetic-permeability material of sensor’s flux guide is relatively expensive, difficult to machine, and fragile, making it suitable for a demonstration of the concept, but not industrial implementation.  Significant refinement would be needed to employ the sensor on a wide scale. 14  T. Ahonen et al. demonstrate an unconventional monitoring approach that applies phase current measurements to extrapolate shaft power, flow rate, and efficiency of fixed-speed centrifugal pumps [41].  The method is intended to provide a rapid, non-invasive, coarse means of auditing a pump’s operating state, which would be used to inform the user if more thorough monitoring is warranted.  With a known nominal rotating speed, their method experimentally determines shaft power and flow rate within 3% and 16%, respectively.  In conjunction with a performance curve chart, these measurements can allow for an efficiency determination.  Most critically, this approach is only suitable for an approximation.  Its intended purpose is to alert the pump operator of severe performance deterioration, such as running dry or stalling.  The method is not suitable for characterizing less extreme adverse operating conditions.  In addition, it relies on an accurate performance curve chart to make efficiency determinations.  As discussed previously, this reference is frequently unavailable to the pump operator.  Finally, the method has no means to capture types of wear that do not have a conspicuous effect on the pump shaft power, such as misalignment, bearing wear, or minor entrainment of gases.   S. Yuan et al. experimentally contrast the application of hydrophones, accelerometers, and dynamic pressure transducers to monitor pump flow noise phenomena in a healthy centrifugal pump [42].  They distinguish the pump phenomena into two groups; discrete frequency noise caused by cyclic behaviors and broadband noise caused by chaotic fluid phenomena.  The frequency peaks of the resulting spectra from each of the three sensing methods are strongly correlated, but with varying magnitudes, likely due to the sensitivity and response of the respective sensors utilized, as well as the resonant behavior of the testbed pump.  Similar results are reported by A. Suhane [43].  Yuan successfully shows that each sensing approach can be used to monitor flow noise, but the presence of normal structural noise (from the motor, bearings, etc…) impacts each measurement differently, clouding the analysis.  The comparison also does not evaluate which method is most suitable for a given fluid phenomenon.  In addition, the study utilizes a pressure transducer with a rise time of 1 µs – nearly three orders of magnitude faster than a conventional liquid water pressure transducer [44], [45], making it operate, effectively, as a wall-mounted hydrophone.     Though uncommon commercially, dynamic pressure monitoring has also been demonstrated experimentally.  X. Li et al. evaluate the dynamic pressure signals of a centrifugal pump to characterize the onset of cavitation [46].  The relationship between cavitation and pressure fluctuations is captured by interrogating the probability density function (PDF) of the resulting pressure signal.  Observing the peak of the PDF as a function of the cavitation severity, which is determined by the available net positive suction 15  head (NPSHA), the study shows a dramatic increase as cavitation occurs.  This validates that dynamic pressure monitoring is an effective way to observe cavitation, but as discussed previously, cavitation is generally a severe phenomenon that can be captured using a variety of less intrusive methods.  E. Higham et al. perform a related experiment, using direct evaluation of the pressure transducer frequency content in their analysis [47].  In addition to cavitation, the Higham study characterizes flow obstructions and running-dry operating conditions.  However, as with cavitation, all the conditions evaluated are severe and readily observable through a variety of methods.  1.4 Phenomena of Interest Centrifugal pump performance is impacted by a variety of dynamic phenomena, many of which are well understood in both the academic and applied senses.  These include the mechanical issues like lack of lubrication, bearing wear, misalignment, impeller damage, acoustic emission, and shaft play, as well as fluid-borne behaviors like cavitation, recirculation, flow debris, flow obstruction, turbulence, and hydraulic shock.  The list could tangentially include phenomena related to the electric motor and power transmission as well.  The body of study on these behaviors is extensive, therefore they are not further discussed in this research.  Instead, this work explores two centrifugal pump phenomena that are persistently difficult to measure and characterize in a typical industrial setting; flow entrainment of dispersed gas bubbles and radial impeller wear.  This section describes the physical behavior and associated measurement techniques for each.   1.4.1 Gas Entrainment The centrifugal pumps considered for this study are designed for transporting only liquids.  Gas entrainment is the undesirable presence of two-phase gas/liquid flow entering the centrifugal pump (Figure 1.5).   Figure 1.5: Distributed gas bubbles entrained in liquid flow through a pipe. 16  There are two2 primary modes of gas entrainment: dispersed two-phase flow (i.e. “bubbly”) and swirling flow with a separated gas phase in the center (“vortexing”).  Vortexing is common in fluid systems where a pump pulls fluid from an insufficiently full reservoir, generating a column of air [49].  Distributed two-phase flow can occur by a variety of mechanisms, including intake flow obstructions (where the pressure at the pump intake is reduced and a suction created, turning vents into inlets for air) or a reservoir that contains a two-phase mixture from a previous process that has had inadequate time to separate before being pulled into the pump [9].   The mechanics and performance impacts of vortexing two-phase flow in centrifugal pumps have been characterized by P. Bardet et al. [50] and T. Schäfer et al. [51].  F. Gülich notes that the performance detriments and physical manifestations of the ingress of a separated column of entrained air into a centrifugal pump are conspicuous [9].  It is therefore considered to be a condition that can be sufficiently characterized with current performance monitoring approaches.  This research focuses instead on dispersed two-phase conditions. T. Schäfer et al. experimentally demonstrate the performance impacts of varying volumes of gas entrainment in a centrifugal pump moving liquid water [51].  Performance is quantified using the hydraulic power output from the pump, normalized to a healthy (non-entrainment) condition.  For the distributed two-phase case, the study demonstrates an approximately linear decrease in relative hydraulic power output from the healthy state to an entrained air void fraction of 5%, at which point the power has dropped to 25% of normal.  Additionally, they employ high-resolution gamma ray computed tomography to quantify the phase fractions and gas distribution patterns (i.e. holdup patterns) on the impeller surface.  The study demonstrates that gas entrainment can cause a major detriment to performance, but the reported trends apply to a singular pump configuration, operating speed, and pressure head.  It can be assumed that the impacts of gas entrainment would vary with configuration changes.  In pump configurations where the decrease is less severe (or not measurable), fluid power would not be suitable feature for characterizing the volume fraction of air.  In addition, to draw more thorough performance conclusions from the trends, a broader set of operating speeds and pressures is needed.                                                             2 In related research, more precise descriptors of dispersed gas entrainment conditions are sometimes necessary.  T. Xie et al. use the following terms, listed from the most distributed to most consolidated bubble formation: dispersed bubbly flow, layered bubbly flow, incipient plug flow, plug flow, churn-turbulent flow, and slug flow [48].  While each can be classified as a unique phenomenon, from the application perspective of this research, the distinction is not strictly necessary.    17  M. Stan et al. analytically and experimentally explore the impacts of gas entrainment on a multi-stage centrifugal pump’s vibration frequency spectrum, acoustic emission, and efficiency [52].  The study evaluates dispersed air-water mixtures between 0% and 11% volume fractions (normal and stalling conditions, respectively).  They demonstrate that the introduction of air into the fluid flow changes the pump’s characteristic performance curve, which implies that presence of two-phase flow can render the manufacturer’s performance curve chart inaccurate.  The study also shows that at higher flow rates, the respective change in vibration spectral amplitude, caused by increasing the void fraction of air in the fluid, diminishes.  This suggests that accelerometer measurement may not be sufficient for observing the severity of gas entrainment in some operating conditions.   Dynamic pressure sensing has been successfully applied to investigate multi-phase flows.  Q. Si et al. explore the frequency content resulting from pressure pulsations caused by varying degrees of air entrainment through two different pumps [53].  They report an increase in broadband low frequency noise in the fluid pressure fluctuations as the volume fraction of air increases.  Comparable spectral changes are reported by Y. Xu et al. [54].  The measured spectra do not vary significantly as the transducer position changes within the pump volute.  They also demonstrate a decrease in the frequency-domain energy peak corresponding to the BPF.  Si shows that dynamic pressure sensing has the potential to adequately capture the effects of gas entrainment, but significance of the associated statistical features of the signals must be more thoroughly characterized if this sensing method is to be used in the application of performance monitoring. T. Xie et al. apply dynamic pressure sensing in conjunction with an artificial neural network (ANN) to classify the bubble morphology of entrained air in three-phase water-air-pulp flows in two related studies [48], [55].  In the first experiment, three pressure sensors are located in steady flow downstream of an injection nozzle.  The transducers are placed sufficiently far from the pump discharge such that the dynamic pressure phenomena caused by the pump itself are not present.  A multi-layer perceptron (MLP) with a single, seven-node hidden layer is constructed using standard deviation, skewness, kurtosis, and several time-shift autocorrelation values from each pressure sensor as input features.  The outputs are classified into “bubbly,” “chug,” “churn,” and “slug” flow regimes.  The volume fraction of entrained air is reported only as “not more than a few percent.”  Experiments are performed at 0.5, 1.0, and 1.5% pulp suspension consistencies, for a total of 197 recorded states.  Using this method, they successfully classify the type of bubble formation in 90% of the training cases.  In the second study, they perform a similar experiment, instead using a single pressure sensor.  From the fluctuating pressure measurements, they compute: a) the 18  sum spectral energies in a range of frequency bands (0-3, 3-8, 8-13, 13-25, and 25-30 Hz), b) the mean frequency (i.e. spectral centroid), and c) variance.  These measures form the input features for a multi-class MLP.  In this configuration, they achieve correct bubble regime classification with 87% accuracy. The studies by T. Xie demonstrate the potential for using dynamic pressure sensing in conjunction with neural networks for classifying air entrainment.  However, these studies explore the morphology of air-entrainment, not the severity.  The experiments are also conducted on three-phase water-air-pulp mixtures, whose dynamic fluid behavior does not necessarily correlate to two-phase air-water mixtures or other fluid combinations.  Further study is required to determine if this method is suitable for a sufficiently wide scope of volume fractions of air and constituent fluids.  Additionally, Xie’s experiments do not consider the potentially dramatic dynamic influence of the pressure fluctuations caused by the centrifugal pump.  To apply either of the reported methods for classification of the severity of gas-entrainment in pumps, more research is necessary.   1.4.2 Impeller Wear Impeller wear refers to undesirable mechanical erosion from the surfaces of a centrifugal pump’s impeller (Figure 1.6).  It is a natural occurrence in pumps, but the rate of degradation can be aggravated by poor process maintenance.  Impeller wear limits the lifetime of a centrifugal pump and can cause inefficiency and serious damage if not addressed appropriately [56].   Figure 1.6: Erosive wear on the tips of a three-blade impeller. Material Loss 19  S. Krüger et al. differentiate the causes of impeller material loss into two categories; shock-type processes, where particulate matter in the flow impacts the impeller surface, and friction-type processes, where adverse flow patterns and turbulence cause erosion over time [6].  They describe shock-type wear as most prevalent in liquid/solid multi-phase pumping applications, such as in mining, paper production, and marine applications, where pumps are expected to move liquids with abrasive constituents.  The study shows that shock-type erosion patterns manifest most severely on the leading surface of the impeller, near the fluid intake.  Similar results are demonstrated by Y. Wang et al. [57] and A. Daraz et al. [35].  In the case of friction-type wear processes, which are present in all centrifugal pumps, regardless of the flow media, Krüger demonstrates that the greatest material loss occurs at the trailing tips of the impeller blades, which shrinks the effective diameter of the impeller, and at the impeller surface abutting the side plate, which can increase the incidence of recirculation within the pump.  Their work suggests that both shock-type and friction-type wear processes can degrade the performance of the centrifugal pump.   The primary focus of this thesis is in classifying erosion resulting from friction-type wear, due to its applicability to all centrifugal pumps, rather than only pumps transporting solid content.  Specifically, radial material loss from the impeller tips is considered, as it is experienced by all centrifugal pumps, regardless of impeller configuration.3  The significance is demonstrated by M. Matlaka et al., who use numerical methods to characterize the flow performance impacts resulting from material removal from the tips of impeller blades [58].  They demonstrate a 40% decline in pumping power after reducing the impeller diameter by 15%, suggesting that characterizing impeller tip erosion has fundamental importance in evaluating the operating health of a centrifugal pump. A. Suhane experimentally characterizes the changes in pressure fluctuations, structural vibration, and airborne noise resulting from increasing the clearance between a rotating impeller and stationary diffuser vanes [43].  The study employs a 7.5 kW centrifugal pump, operating at 1440 RPM.  Dynamic measurements are interpreted in the frequency domain using FFT.  For the pressure samples, the author demonstrates that increasing the nominal clearance between the impeller and stationary vanes by a factor                                                           3 Wear along the side plate surface occurs primarily in centrifugal pumps with open-type impellers.  Impellers come in “open,” “semi-open,” and “closed” configurations.  Open impellers have no shrouds on either face of the blades, and must maintain a close gap with the side plate to avoid recirculating flow losses.  This is the type of impeller wear investigated in [40].  Semi-open impellers have an integrated shroud on the face opposite the pump intake.  Closed impellers have shrouds on both faces of the impeller, turning the gaps between the blades into flow channels.  On pumps that use semi-open and closed impellers, this wearing surface is not present.     20  of 4.5 (1.5 mm to 6.8 mm) results in an approximate 50% amplitude decrease at the BPF.  Reductions in the amplitudes of the associated frequency in the structural vibration and airborne noise measurements are also observed, but to a lesser extent.  Frequency impacts outside the BPF and associated harmonics are not reported.  Though the study specifically explores the spectral impacts of clearance between a pump impeller and diffuser vanes, the experimental configuration is analogous to the growing clearance between a radially wearing impeller and the pump’s discharge port.  It suggests a correlation between erosion of the impeller diameter and the pump’s vibration, in both the fluid and structural sense.  Further, Suhane shows that the relationship is most prominent when observed as fluid pressure fluctuations, rather than through structural vibration or emitted noise.  Further study is needed to clarify this correlation.  A. Jami et al. employ a machine learning approach to classify a damaged impeller in a centrifugal pump based on accelerometer measurements [31].  They apply a MLP with binary classification to identify the presence of impeller cracking and unbalance using an accelerometer.  The machine learning method uses a feature space consisting of time-domain features, including kurtosis, root-mean-square (RMS), skewness, and variance, as well as frequency-domain features, including spectral frequency peaks and wavelet decomposition coefficients.  The MLP contains one hidden layer and achieves near perfect classification across the majority of the trials when using the entire feature set.  However, the experimental impeller exhibits wear that would be considered exceptionally severe in practice.  The peak MLP performances based strictly on time-domain or frequency-domain features are less exceptional, reported as 63% and 74%, respectively.  The authors report difficulty in identifying slight changes in impeller condition using this method, particularly when only considering time-domain statistical features.  This reasons one of two conclusions: that time-domain statistical features may not contain satisfactory evidence to identify minor changes in impeller wear, or that accelerometer measurement may not be the most suitable method for capturing subtle changes in the impeller structure.  The work of Suhane discussed above [43] suggests the latter, though further substantiation is necessary.    L. Cao et al. conduct a numerical investigation of changes in fluid pressure fluctuations resulting from subtle changes in the axial gap between an impeller and pump housing, consistent with an impeller wearing against its side plate [59].  They apply the k-ω turbulence modeling method, which they argue is most suitable for prediction of rotating, separating flow.  The study shows that pressure fluctuations in general are most significant near the volute discharge, decreasing with distance down the discharge pipe.  Cao confirms that increasing the axial impeller gap by as little as 1 mm results in observable changes in lower frequency pressure fluctuations.  The study does not show dramatic change in the energy peaks at 21  the dominant frequencies, but this is to be expected when considering minor axial wear, which contributes to turbulence and recirculation, but would not change the pressure fluctuation modes associated with the impeller radius.  Their work suggests that dynamic pressure measurement is suited for the subtle fluid phenomena associated with minor wear.  Additional study is needed to demonstrate that this modeling method is applicable for pressure fluctuations associated with radial, rather than axial wear.  1.5 Pressure Dynamics in a Rotating Pump A schematic highlighting the terms and positions of the primary mechanical elements of a centrifugal pump is shown in Figure 1.7.  Figure 1.7: A two-dimensional schematic of a simplified centrifugal pump with a two-blade (𝑵𝐢𝐦𝐩 = 𝟐) impeller.  The inset shows a perspective view of a comparable physical pump for clarity.    Fluid flow direction is indicated by the blue arrows.  The operating fluid enters the pump through the port at the center of the impeller (from out of plane) and exits vertically.  A scroll-shaped funnel called the volute houses the impeller and guides the working fluid from the area of the rotating impeller to the discharge port. The notch at intersection of the curves of the volute scroll is the “cutwater.”        Impeller Blades Cutwater Volute 𝜔 22  1.5.1 Ordinary Operation During normal operation, fluid enters the pump body through the intake port with pressure 𝑝in, average velocity 𝑣in̅̅ ̅̅ , and temperature 𝑇in.  Fluid exits through the discharge port with pressure 𝑝out, average velocity 𝑣out̅̅ ̅̅ ̅, and temperature 𝑇out.  When a centrifugal pump moving nominally incompressible fluid has reached steady state operation, 𝑄 is constant.  Consequently, in the case where the cross-sections of the pump intake and discharge are equal in area, 𝑣in̅̅ ̅̅ = 𝑣out̅̅ ̅̅ ̅.  In powerful pumps, where ∆𝑝 > 1𝑀𝑃𝑎, ∆𝑇 may be a measurable quantity, but for the pump configurations investigated in this study, it can be assumed that ∆𝑇 is negligible.  The magnitude of ∆𝑝 is a function of the pump’s configuration and the surrounding fluid system.  Though the pressure head imparted by the surrounding fluid system is typically constant at steady-state, the measured values of 𝑝in and 𝑝out are time-dependent.  The time-dependence is a combined effect of natural pressure fluctuations caused by the operation of the pump, turbulence in the fluid, and noise in the measurement.  Illustrative pressure measurements collected at the intake and discharge ports of a centrifugal pump with a two-blade, 12-inch (305 mm) diameter impeller during normal operation are shown in Figure 1.8.   Figure 1.8: Time-varying, experimental pressure measurements at the intake (𝒑𝐢𝐧, red line) and discharge (𝒑𝐨𝐮𝐭, blue line) of a centrifugal pump operating at 𝝎 = 𝟏𝟔. 𝟒𝑯𝒛.  In this example, 𝑵𝐢𝐦𝐩 = 𝟐.  The respective sample averages 𝒑𝐢𝐧̅̅ ̅̅ = 𝟗. 𝟐𝒌𝑷𝒂 and 𝒑𝐨𝐮𝐭̅̅ ̅̅ ̅̅ = 𝟖𝟑. 𝟗𝒌𝑷𝒂 are indicated.  The difference  ∆𝒑 = 𝟕𝟒. 𝟕𝒌𝑷𝒂.  Both measurements exhibit cyclic fluctuations, but the behavior is most apparent in 𝒑𝐨𝐮𝐭.    0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0020406080100120Discharge Mean (Discharge)Intake Mean (Intake)Time (s)Pressure (kPa)23  In practice, when constant pressure measurements are desired – for example, to implement the efficiency determination method in (3), time-averaging would be implemented and the dynamic data discarded.  However, the example measurements exhibit a relevant dominant cyclic mode, accompanied by higher frequency fluctuations.   It can be reasoned that the fluctuations are more significant in 𝑝out because it is measured downstream of the impeller.  This behavior parallels results of experimental studies by E. Higham et al. [47], Z. Yao et al. [60], and Tan et al. [61] .  Correlating the pressure excursions in 𝑝out to the angular position of the impeller 𝜃 yields a relative timeline of the pressure events that occur with each rotation of the impeller.  This is shown in Figure 1.9.  Figure 1.9: 𝒑𝐨𝐮𝐭 as a function of impeller position over two rotations at 𝝎 = 𝟏𝟔. 𝟒𝑯𝒛.  At point (1), the tip of the leading impeller blade has just reached the cutwater.  This defines 𝜽 = 𝟎.  As the impeller rotates to (2), the discharge pressure drops abruptly, only starting to increase again once the second impeller blade starts to 0 90 180 270 360 450 540 630 7205060708090100110Impeller Angular Position (deg)Pressure (kPa)(1)  (2)  (3)  (4)  (5)  (2) 𝜃 = 65° (1) 𝜃 = 0 (3) 𝜃 = 180° (4) 𝜃 = 245° (5) 𝜃 = 360° 24  force fluid toward the discharge.  The discharge pressure increases until the second impeller reaches the cutwater at (3).  A pressure drop follows until the leading impeller blade reaches (4), where the pressure again rises and the cycles continue.  Over this time scale, it can be observed that the fluctuations of 𝒑𝐨𝐮𝐭 are not purely sinusoidal, but could be more appropriately generalized as a triangle or sawtooth wave.     Direct interrogation determines that the primary cyclic mode corresponds to the BPF, which is expected.   Here, the discharge pressure signal acts as a measure of the fluid-borne vibration.  In effect, the pressure transducer functions as a rudimentary hydrophone through which information about the pump’s operating state can be deduced, even in the absence of more complex instrumentation. 1.5.2 Gas Entrainment Condition The gas and liquid constituents of the working fluid in this study are air and water, respectively.  The severity of gas entrainment is characterized by the dimensionless void fraction (or equivalently, volume fraction) of air 𝜑air =  𝑞air𝑞w+𝑞air ,     (9) where 𝑞air and 𝑞w are the average volumes of air and water passing through the pump over time, respectively.  The sum corresponds to the total flow rate 𝑄 = 𝑞w + 𝑞air. The distribution of the air phase in the incoming flow in this study can be described as “dispersed bubbly” or “layered bubbly” flow using the morphology conventions described by T. Xie [48].  However, it is assumed that the turbulent mixing contribution of the impeller exceeds the surface tension forces present in larger bubbles considerably, yielding an approximately homogeneous distribution of bubbles with diameters smaller than 5 mm exiting the pump discharge, regardless of the morphology at the intake.   A centrifugal pump’s design and surrounding fluid process will ultimately dictate the range of air volume fractions under which it may function, but in a typical application, normal operation coincides with the state of 𝜑air = 0 and very severe air entrainment may reach void fractions exceeding 0.05, or 5% [51], [52].    For centrifugal pumps designed to move only liquids, any presence of entrained gas is undesirable.  At volume fractions greater than 5%, the performance of a conventional centrifugal pump is likely to have degraded so severely that manual observation would readily perceive the aberration [51].  Thus, higher air volume fractions are not considered.     Adding a significant proportion of entrained air impacts the pressure fluctuations at the pump discharge, as well as the general operation of the centrifugal pump.  To illustrate this, the healthy discharge 25  pressure measurement from Figure 1.8 is overlaid with a corresponding measurement at 𝜑air = 5.0% in Figure 1.10.  In the testbed pump, a 5.0% void fraction approaches the threshold of a stall condition.  Increasing the volume fraction further creates a loss of pressure and a ceasing of flow.    Figure 1.10: a) Experimental 1 s samples of 𝒑𝐨𝐮𝐭 at 𝝎 = 𝟏𝟔. 𝟒𝑯𝒛 for normal pump operation (blue) and a gas entrainment condition 𝝋𝐚𝐢𝐫 = 𝟓. 𝟎% (green); b) A detail view of the corresponding fluctuations over two impeller rotations.  The impeller positions are synchronized for clarity.  Introducing entrained air causes a decrease in the variance between peak and minimum pressures in Figure 1.10a.  It can also be observed that the pressure peaks in the sample with entrained air are 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.050607080901001105.0% Air NormalTime (s)Pressure (kPa)0.00 0.02 0.04 0.06 0.08 0.10 0.1250607080901001105.0% Air NormalTime (s)Pressure (kPa)(a)  (b)  26  flattened, which, in turn, indicates that the skewness of the signal has likely changed.  Over the shorter time scale in Figure 1.10b, it becomes apparent that the main cyclic mode of pressure sample with air entrainment has deteriorated and high frequency fluctuations have increased.  The waveform has also lost much of its triangle-wave-like character in the presence of severe gas entrainment.  This preliminary contrast between discharge pressure measurements at 𝜑air = 0 and 𝜑air = 5.0% suggests that more rigorous investigation has the potential to yield substantive trends that can be used to characterize the severity of gas entrainment.  1.5.3 Worn Impeller Condition There are multiple mechanisms by which an impeller blade can erode.  This work focuses on material loss from the impeller tips, radially.  Mechanical erosion from the ends of the impeller blades can occur through natural friction wear, abrasion, sudden damage, or intentional manual removal.  The intentional case, referred to as impeller trimming, is a modification intended to reduce the fluid volume displaced by a centrifugal pump in the absence of a means to control the rotating frequency.  Impeller wear is characterized using the difference  ∆𝐷 = 𝐷o − 𝐷𝛼 ,     (10) where 𝐷𝛼 is the actual impeller’s operating diameter and 𝐷o is that of the original, unworn impeller.  A schematic of this geometry is shown in Figure 1.11.   Figure 1.11: Geometry of radial impeller wear (wear region indicated in red).  ∆𝐃 is exaggerated for clarity.    𝐷o 𝐷𝛼 ∆𝐷 27  The severity of the tip erosion is defined by impeller loss ratio 𝛼imp =  ∆𝐷𝐷o .      (11) Substituting (10) into (11) yields the final form, 𝛼imp =  𝐷o−𝐷𝛼𝐷o .     (12) 𝛼imp denotes the relative reduction in impeller diameter from its unworn state.  𝛼imp = 0 corresponds to that nominal state, where no material has been eroded from the impeller tips.     The configuration and process of a specific centrifugal pump dictate the range of impeller loss ratios for under which it may continue to operate reasonably normally.  In the case of deliberate impeller trimming, impeller loss ratios as high as 𝛼imp = 0.5, or 50% may be considered acceptable [5], though the design of the pump would be fundamentally altered, limiting the value of comparing it to its original state.  Excluding cases of intentional material removal, which are not the primary focus of this study, it is assumed that realistic impeller loss ratios between 0 and 3% should be considered for characterization, as they are potentially severe enough to cause a deviation in a pumping process, yet may not be readily detectable using more conventional means.  Impeller loss ratios exceeding 𝛼imp = 3% are assumed to be so severe that they would likely be observable through other measurement methods, as demonstrated by M. Matlaka et al. [58].  As such, loss ratios beyond 3% are not explicitly investigated in this work. Generating an illustrative measurement of the dynamic discharge pressure impacts of impeller wear is not trivial, considering it requires permanently damaging a centrifugal pump.  Instead, a 2D finite element simulation of a centrifugal pump is created using COMSOL Multiphysics software.  The development and validation of this numerical model are discussed in Chapter 2 of this thesis.  Figure 1.12 contrasts simulated discharge pressure measurements for an unworn impeller and the same pump with an impeller loss ratio of 𝛼imp = 3%.  The simulation emulates a two-blade, 4-inch (102 mm) diameter impeller rotating at 𝜔 = 18.75𝐻𝑧.     28   Figure 1.12: a) A simulated 0.5 s sample of 𝒑𝐨𝐮𝐭 for normal pump operation (blue) and an impeller wear condition 𝜶𝐢𝐦𝐩 = 𝟑. 𝟎% (orange) at 𝛚 = 𝟏𝟖. 𝟕𝟓𝐇𝐳; b) A detail view of the corresponding fluctuations over two impeller rotations.  The impeller positions are synchronized for clarity.  The variance between the peak and minimum discharge pressures is reduced in the severe wear condition.  The simulated signals show no clear increase in high frequency fluctuations.  This is consistent with experimental results reported by A. Suhane [43].  As with the preliminary measurements for air entrainment, the impeller wear simulation exhibits a triangle-wave-like behavior, rather than purely sinusoidal.   However, unlike the air entrainment case, the shape of the fluctuations is retained in the 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5013141516171819203% Wear NormalTime (s)Pressure (kPa)0.00 0.02 0.04 0.06 0.08 0.1013141516171819203% Wear NormalTime (s)Pressure (kPa)(a)  (b)  29  presence of radial impeller wear, suggesting that the changing shape of the waveform could be a telltale feature to differentiate between air entrainment or impeller wear.  1.6 Objectives The research discussed in the previous sections suggests a pressing need for affordable, accurate, and readily implementable methods to monitor gas entrainment and impeller wear in centrifugal pumps.  Measurement of fluid pressure fluctuations at the pump discharge, in conjunction with machine learning methods, offers potential in this respect.  Considering this, the objectives of this research are as follows: (1) Statistical Features: Propose a set of statistical measures to quantify the severity of the target phenomena using dynamic discharge pressure. (2) Correlation: Assess the relevance of these measures as related to detecting the target phenomena. (3) State Prediction: Develop a classification technique to detect with at least 90% accuracy the occurrence of a) air entrainment exceeding a 2% void fraction and b) radial wear exceeding 1.5% of the impeller diameter in an operating centrifugal pump. (4) Refined Characterization: Demonstrate refined characterization of the target phenomena using multiple severity classes and continuous-value regression. (5) Accessibility: Implement the detection method using a single pressure transducer that costs less than $200 and has a rise time no faster than 2 ms. (6) Implementation: Suggest protocols for industrial implementation of the method.  The following chapters describe the design, characterization, and evaluation of the system.  Chapter 2 defines the numerical modeling, experimental materials, and methods employed to generate discharge pressure measurements of the target phenomena.  Chapter 3 contains the analytical background for the statistical measures, analysis method, and classification models.  Chapter 4 contains the experimental results and corresponding analysis.  Finally, Chapter 5 draws conclusions and describes potential further applications of this work.   30  2. MATERIALS AND EXPERIMENTAL METHODS The studies in this work are conducted in two stages.  First, we employ numerical modeling to generate dynamic pressure measurements corresponding to normal, intermediate, and extreme states for each of the target phenomena.  Contrasting these disparate conditions provides a basis from which to derive relevant measures to quantify the target conditions and propose classification algorithms.  The workflow of these initial studies is shown in Figure 2.1 and discussed in Chapter 3.    Figure 2.1: Workflow of the preliminary studies for a) gas entrainment and b) impeller wear.  The initial simulated data is used to evaluate signal decomposition techniques and propose machine learning methods.   In the second stage (Figure 2.2), we generate comprehensive sets of dynamic pressure measurements for each target condition.  Air entrainment data is generated using experimental methods.  Impeller wear pressure data is simulated using the numerical model. These complete sets of dynamic pressure measurements are then used to demonstrate and validate the classification methods in Chapter 4.   Figure 2.2: Workflow of the primary studies for a) gas entrainment and b) impeller wear.  Experimental data is used to validate gas entrainment classification, whereas numerical data is used for impeller wear. (a)  (b)  (a)  (b)  31  2.1 Simulation A numerical model of a rotating centrifugal pump is developed using COMSOL Multiphysics software (v. 5.5, b. 359, COMSOL Inc. 2019) to assess the pressure dynamics associated with gas entrainment and impeller wear.  The model applies finite element analysis (FEA) to perform time-dependent simulations of each of the target conditions, across a range of operating states, to generate an array of characteristic dynamic pressure signals at the pump discharge.   2.1.1 Model Configuration The simulated pump is based on an Ampco AC114 centrifugal pump, shown in Figure 1.1.  To reduce the computational complexity to a practical scale, the modeled geometry requires a series of simplifications.  First, being that centrifugal pumps contribute kinetic energy to the working fluid in an inherently radial manner, in the plane of the impeller, it is assumed that the dominant modes of pressure fluctuation and turbulence will also be in-plane.  The system is therefore modeled in 2D.  Secondly, to minimize the required quantity of finite elements, detailed structures that are not strictly essential to pump operation are removed or simplified.  This includes a) reducing the number of impeller blades from three to two, b) rounding sharp corners at the impeller ends and the cutwater, c) generalizing the impeller blade contours as radial, rather than a more complex spline, and d) minimizing the length of the discharge channel, to the extent that its boundary condition does not interfere with the pressure fluctuations.  The geometry of the simulated pump is shown in Figure 2.3.  Figure 2.3: Geometry of the simulated centrifugal pump.  The virtual pressure probe (green dot) is positioned in the center of the discharge, 50 mm from the pump outlet. 115𝑚𝑚 25.4𝑚𝑚 100𝑚𝑚 (to outlet) 38.1𝑚𝑚 Virtual Pressure Probe 32  In the nominal model configuration, 𝐷o = 10.2𝑚𝑚.  The minimum clearance between the impeller and cutwater is 0.75 mm.   The operating area of the simulation is composed from two domains.  First is the dynamic domain, which comprises the fluid inlet and the area of the rotating impeller blades.  Second is the static domain, which encompasses the volute wall and discharge channel.  The rotational frequency 𝜔 is applied as the input to the dynamic domain, therefore the position of the impeller throughout the simulation is time-dependent.  The circular intersection of the outer surface of the rotating domain and inner surface of the static domain is designated as an “identity pair,” meaning the solver treats the junction as a coincident surface across which continuity and fluid transport are to be maintained, even as the impeller domain rotates. Flow through the simulated pump is bounded by the conditions at the pump intake and discharge, which, in a physical system, are a function of 𝜔.  At the discharge, a fully developed flow condition with a specified average pressure is applied.   The average pressure increase ∆𝑝 is chosen to be a fixed hydraulic head of 1.5 m (14.7 kPa). The flow rate 𝑄 is estimated using the manufacturer’s data sheet for the Ampco AC114 centrifugal pump [62].  Flow rates of 50 l/min at the maximum impeller speed of 𝜔max = 28.75𝐻𝑧 and 15 l/min at the minimum impeller speed 𝜔min = 8.75𝐻𝑧 are assumed, with linear behavior between.  The flow rates are considered only as a representative approximation, given that the configuration of the impeller blades has been simplified for modeling.  The average outlet velocity  vout̅̅ ̅̅ ̅ is determined by dividing 𝑄 by the outlet area.   At the pump inlet, a normal inflow velocity vin̅̅ ̅̅  at atmospheric pressure is specified.  The magnitude of vin̅̅ ̅̅  is determined by multiplying vout̅̅ ̅̅ ̅ by the ratio between the diameter of the pump outlet and the circumference of the model’s inlet (rather than their respective areas, because in 2D, the outlet flow is in plane and the inlet is not).  With 𝑝in̅̅ ̅̅ = 0𝑘𝑃𝑎 (gauge), ∆𝑝 = 𝑝out̅̅ ̅̅ ̅.     Samples of 𝑝out are collected at a virtual probe point 50 mm below the model’s outlet boundary, as indicated in Figure 2.3.  The probe is positioned to avoid the turbulent effects in the immediate area of the rotating impeller, but sufficiently far from the fluid exit boundary condition to avoid damping effects from the boundary condition.  The positional impact of the probe placement is shown in Appendix B1.  To attain quasi-steady state operation, 𝜔, ∆𝑝, and 𝑄 are linearly ramped up from rest over a buffer period of 1.5 s.  Simulated pressure data from this transient period is not considered.   33  Across all the simulations, dynamic pressure samples are collected over a total time duration 𝑇s =0.5𝑠 at a sampling frequency 𝑓s = 103𝐻𝑧.  This yields 𝑁 = 500 pressure data points within each measurement.  The sampling interval 𝑡s = 1 𝑓𝑠⁄ = 10−3𝑠.  The duration of 𝑇𝑠 is selected to provide frequency resolution ∆𝑓 = 1/𝑇𝑠 = 2𝐻𝑧, down to the minimum frequency of interest 𝑓min = 2𝐻𝑧.     The dynamics of a physical pressure transducer are not reflected in the virtual probe measurement.  To conservatively avoid the potential influence of any fluctuations exceeding what would be experimentally observable using a pressure sensor with a rise time of 2 ms, frequencies exceeding 400 Hz are filtered from the simulated signal with a low-pass filter (LPF).  400 Hz is the maximum frequency of interest 𝑓max.  Any potential fluctuations resulting from coupling changes – that is, whether the transducer diaphragm is in contact with the liquid, gas, or both phases at a given instant – cannot be replicated using the numerical model.              2.1.2 Numerical Methods The centrifugal pump simulation incorporates two physics models; a flow modeling method and a multi-phase method.   Turbulence Model Flow through a centrifugal pump is assumed to have regions with a high Reynolds number, so a numerical model that considers turbulence is employed.  The Realizable k-ε turbulence model has been applied successfully in related studies [63] and demonstrated to be particularly suited for applications with rotating flows [64].  It is the turbulence model selected for this research.   The Realizable k-ε approach solves transport equations for the turbulent kinetic energy of the liquid phase 𝑘l and the turbulent energy dissipation rate of the liquid phase 𝜀l using the realizability conditions 𝑢𝑖2̅̅ ̅̅ ≥ 0       (13) and 𝑢𝑖𝑢𝑗̅̅ ̅̅ ̅̅2𝑢𝑖2̅̅ ̅̅ ̅ 𝑢𝑗2̅̅ ̅̅ ̅≤ 1      (14) where 𝑢𝑖 and 𝑢𝑗 are the unfiltered components of the time-averaged flow velocity vector 𝒖.  The transport equation for 𝑘l, 34  𝜌𝜕𝑘l𝜕𝑡+ 𝜌(𝒖 ∙ 𝛻)𝑘l = 𝛻 ∙ [(𝜇 +𝜇Tl𝜎𝑘) ∇𝑘l] + 𝑃𝑘 − 𝜌𝜀l   (15) is equivalent to that of the standard k-ε method [64].   In this steady-state application, the time rate of change of the turbulent kinetic energy of the liquid phase 𝜕𝑘l𝜕𝑡= 0.  The dissipation rate 𝜀l is calculated by the transport equation  𝜌𝜕𝜀𝑙𝜕𝑡+ 𝜌(𝒖 ∙ ∇)εl = ∇ ∙ [(𝜇 +𝜇Tlσϵ) ∇εl] + Cε1𝜌𝑆εl − Cε2𝜌ε2𝑘l+√𝜈𝑙εl ,  (16) where the time rate of change of the turbulent dissipation rate of the liquid phase 𝜕𝜀l𝜕𝑡= 0.   In (15) and (16), the total fluid density 𝜌 and dynamic viscosity 𝜇 are determined by the two-phase model (see (21) and (22)).  Kinematic viscosity νl corresponds to the liquid fluid phase (water).  𝑃𝑘 represents the turbulent kinetic energy production term.  𝑆 is the strain-rate tensor.  The adjustable constants 𝜎𝑘 = 1.0, 𝜎𝜖 = 1.2, and 𝐶𝜀2 = 1.9 are used [63].  The turbulent eddy viscosity of the liquid phase 𝜇Tl = 𝐶𝜇𝜌l𝑘l2𝜀l      (17) contains a scalar term 𝐶𝜇 that is not constant.  The computations for 𝐶𝜇, 𝑃𝑘, 𝑆, the turbulent production rate scalar 𝐶𝜀1, and their associated calculations are shown in Appendix A1.   Continuity is determined using the Reynold’s Averaged Navier-Stokes (RANS) turbulence equations 𝜌𝜕𝒖𝜕𝑡+ 𝜌(𝒖 ∙ 𝛻)𝒖 = 𝛻 ∙ {−𝑝𝑰 + 𝜇[𝛻𝒖 + (𝛻𝒖)𝑇]} + 𝐹 − 𝜌𝒈   (18)          and  ∇ ∙ ρ𝒖 = 0 ,      (19)          where the fluid is approximated to be incompressible and Newtonian.  𝑰 is the identity tensor, 𝐹 is the volume shear force, and 𝒈 is the gravitational field.  The wall boundaries of the simulated pump, including the rotating surfaces of the impeller blades, are assumed to be non-slip. Multi-phase Model In the multi-phase portion of the numerical model, it is assumed that the dynamic pressure contribution of air bubble coalescing and breakup is non-trivial and must be accounted for, despite the non-trivial computational cost of calculating individual phase boundaries in a time-dependent study.  Simpler two-phase mixing models, such as the Euler-Euler method used by J. Zhang et al. [8] and the 35  distributed bubbly flow technique applied by K. Minemura and T. Uchiyama [65] are not considered suitable.  The Euler-Euler method assumes that the fluid phases are interpenetrating and calculates pressures using mixture averaging, which is unlikely to generate satisfactory resolution of pressure fluctuations around the gas phase.  The bubbly flow technique by Minemura requires that the volume of entrained gas be small and homogeneously distributed enough for coalescing and fragmentation to be neglected.  Dynamic pressure considerations notwithstanding, it is not clear that this is a justifiable assumption in flows with 𝜑air exceeding a few percent.   To achieve detailed spatial and temporal resolution of the air-water interface while keeping computational cost manageable, a level set two-phase flow model is employed [66].  The level set method is intended for simulating the boundary between two immiscible phases (e.g. liquid water and air).  The technique defines a level-set variable 0 ≤ 𝜙 ≤ 1, where 𝜙 = 0 and 𝜙 = 1 correspond to the pure water and pure air domains, respectively, with a transition region of thickness 𝜖 between them.  Here, a quadratic transition is employed and 𝜖 is defined as one half the length of the largest mesh element in the simulation.  The fluid interface is defined as the iso-contour 𝜙 = 0.5.  The motion of the interface is calculated by solving  𝜕𝒖𝜕𝑡+ 𝒖 ∙ 𝛻𝜙 = 𝜁𝛻 ∙ [𝜖𝛻𝜙 − 𝜙(1 − 𝜙)𝛻𝜙|𝛻𝜙|] ,   (20) where ζ is a dimensionless tuning parameter to control the numerical stability of the function.  The total fluid density  𝜌 = 𝜌water + (𝜌air − 𝜌water)𝜙      (21) and the total dynamic viscosity 𝜇 = 𝜇water + (𝜇air − 𝜇water)𝜙      (22) apply to transport equations (15) and (16).  The void fraction of air entering the system is specified by the inlet condition.  Surface tension is considered as well.       2.1.3 Mesh The simulation mesh is optimized to balance dynamic accuracy and computational time.  Element size is controlled using COMSOL Multiphysics’ integrated meshing function, which increases the density in areas with complex geometry.   36  Mesh independence is validated by simulating the condition 𝜑air = 5.0% and 𝜔 = 18.75𝐻𝑧 with successively increasing element density, until minimal influence is observed.  The influence of mesh density becomes negligible at 104 elements.  However, at this density, generating the corresponding 0.5 s pressure sample takes approximately seven hours.  For a single simulation, this is not insurmountable, but for building extensive sets of simulated conditions, it is problematic.  Reducing the mesh density to 103 elements shortens the corresponding computation time by a factor of five, while retaining the prominent fluctuation characteristics (as validated in Section 2.1.4).  Thus, this density is selected for the model.  2.1.4 Validation The numerical model is validated for ordinary operation and gas entrainment conditions against the preliminary experimental measurements from Figure 1.10.  Overlaying the simulated data onto the experimental samples yields the correlations in Figure 2.4. 37   Figure 2.4: a) Contrasting experimental (dotted) and simulated (solid) measurements of 𝒑𝐨𝐮𝐭 for normal pump operation at 𝝎 = 𝟏𝟔. 𝟒𝑯𝒛.  Each sample is 0.5 s total and normalized to its maximum value.  The first 0.2 s are shown for clearer interpretation of the waveforms; b) Contrasting experimental (dotted) and simulated (solid) measurements of 𝒑𝐨𝐮𝐭 for the gas entrainment condition 𝝋𝐚𝐢𝐫 = 𝟓. 𝟎%.  Each gas entrainment condition is normalized to the maximum value of its corresponding healthy condition from Figure 2.4a to show the relative reduction in variance.   The experimental and simulated discharge pressure signals in Figure 2.4a exhibit similarities.  Both waveforms are approximately triangular with a dominant fluctuation mode at the BPF and contribution from higher frequencies.  Calculating their Pearson correlation coefficient yields an agreement of 0.83 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2Normal (exp) Normal (sim)Time (s)Relative Pressure Magnitude0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.25.0% Air (exp) 5.0% Air (sim)Time (s)Relative Pressure Magnitude(a)  (b)  38  (see (A10) in Appendix A2 for calculation).  In both the experimental and simulated signals, the inclusion of a 5.0% void fraction of air causes a decrease in the variance of the pressure fluctuations and disruption of the main fluctuation modes (Figure 2.4b).  Together, these advocate that the numerical model provides a satisfactory representation of the primary characteristic elements of a physical centrifugal pump.     2.2 Experimental Apparatus Dynamic discharge pressure data from a physical centrifugal pump is used to validate the simulation dynamics and provide reference data by which to evaluate classification methods.  Physical pressure data is collected for normal and air entrainment conditions only. 2.2.1 Pump and Fluid System Experimental measurements are collected on the centrifugal pump powering the pilot-scale pulp refining loop (Pilot Plant) in the Paper and Pulp Centre (PPC) at The University of British Columbia (UBC), shown in Figure 2.5.   Figure 2.5: The 30 kW centrifugal pump used as a testbed throughout the experimental studies.  The testbed pump is a 40 HP (30 kW) model manufactured by Westcan Industries Ltd.  The pump contains a 2-blade open impeller with a diameter of 12 in (305 mm).  Driving power is delivered by a close-Pump Motor 39  coupled three-phase electric motor, connected to a variable-frequency drive (VFD) with integrated power and rotating frequency sensing.  The pump is capable of displacing nearly 6,000 liters per minute at 3250 RPM, though the construction of the surrounding fluid system limits safe operation to approximately 1,400 liters per minute, which corresponds to 1,125 RPM (𝜔 = 18.75𝐻𝑧) under minimal pressure load.  The minimum impeller speed required to propel fluid through the system, from stationary, is approximately 525 RPM (𝜔 = 8.75𝐻𝑧).     A schematic of the adjoining pump loop is shown in Figure 2.6.   Figure 2.6: A schematic of the fluid loop in the pilot-scale pulp refining plant at UBC. The loop comprises a 5,000 liter water reservoir, leading to a 4 m horizontal segment of pipe which feeds the pump intake.  The pump discharge feeds into a 5 m section of vertical pipe, which then traverses back and down to the water tank.  The water re-enters the tank below the surface, creating a closed fluid loop.      Air is injected into the fluid flow at the base of the reservoir using an extended compressed air nozzle with a perforated tip attachment.  The reservoir is necessarily large to allow time for the entrained Water Reservoir Flow Sensor Pump Discharge Pressure Sensor Air Injection Nozzle Intake Pressure Sensor 1m 40  air to dispel to the atmosphere prior to recirculation, preventing accumulation over time.  The air flow rate is measured using a rotameter and controlled by a manual gate valve. 2.2.3 Sensors As discussed in the opening sections of this thesis, it is required that pressure measurements be collected using a transducer that would be affordable and readily attainable in industrial practice.  While there exist higher-performance sensors that would potentially provide more refined pressure measurements, the intent is to demonstrate characterization of the target phenomena using a sensor available to any centrifugal pump operator.  The objectives in Section 1.6 set this threshold to be under $200, with a rise time no faster than 2 ms.    A Honeywell MLH series piezoresistive pressure transducer (model MLH050PGB06A, Figure 2.7) [44] is employed to measure 𝑝out.  The sensor is mounted through the wall of the discharge pipe, 0.5 m from the pump outlet.  The transducer has an operating pressure range up to 345 kPa and an accuracy of ±0.25% of full-scale.  Full-scale output is 4.5 V and the excitation voltage is 5 V.  The response time is specified by the manufacturer as 2 ms.  The cost per transducer is approximately $115 USD.  Figure 2.7: A Honeywell MLH050PGB06A pressure transducer. A series of secondary sensors are used to control the apparatus and provide supplementary data for the analysis.  The intake pressure 𝑝in is measured by an additional Honeywell MLH050PGB06A pressure transducer at the pump intake.  Flow rate 𝑄 is determined using a Rosemount 8700 Series magnetic flowmeter [67].  The values of 𝑊in and 𝜔 are determined using the wattmeter integrated into the pump’s 41  variable frequency drive (VFD), which is a Baldor VS1PF-NM1E.  𝑞air is measured using an Omega FL-203 rotameter.  The operating parameters and associated accuracies for each sensor are shown in Table 2.1. Table 2.1: Sensor Parameters Measurand Symbol Manufacturer Model No. Type Range Accuracy Discharge Pressure 𝑝out Honeywell MLH050PGB06A Piezoresistive 0-350 kPa (gauge) ±0.25% at f.s. Intake Pressure 𝑝in Honeywell MLH050PGB06A Piezoresistive 0-350 kPa (gauge) ±0.25% at f.s. Rotating Frequency 𝜔 Baldor VS1PF-NM1E Integrated 0-3500 Hz Unspecified Input Power 𝑊in Baldor VS1PF-NM1E Integrated 0-30 kW Unspecified Flow Rate 𝑄 Rosemount 8707 Magnetic 3500 l/min ±0.5% at f.s. Air Flow Rate 𝑞air Omega FL-203  Rotameter 11.3-140 l/min ±5% at f.s.  2.2.4 Data Collection  Experimental measurements of 𝑝out are collected using a Measurement Computing USB-1208HS data acquisition (DAQ) device [68].  The DAQ is controlled by a custom program, written in Python 3.5, utilizing functions from the Measurement Computing Python library.  In each measurement, the data collection interval 𝑇s = 1𝑠, allowing for scrutiny of frequencies as low as 1 Hz, which is below 𝑓𝑚𝑖𝑛.  This yields ∆𝑓 = 1𝐻𝑧.  The sampling frequency 𝑓s = 104𝐻𝑧 and 𝑡s = 10−4𝑠.  A total of 𝑁 = 104 data points are generated within each experimental measurement.  To exclude any potentially dubious dynamic behavior attained near the transducer’s response limit of 500 Hz, each measurement is filtered by a LPF with a cutoff at 400 Hz prior to analysis.  Oversampling is employed to resolve the wave shape and peak values of the pressure fluctuations.      The static measurements of 𝑄, 𝑊in, 𝜔, and 𝑞air are manually recorded for each sample.  Where employed, the dynamic measurements of 𝑝in are collected in the same manner as 𝑝out.   2.3 Gas-entrainment Study The goals of the gas entrainment study are to a) evaluate the associated pressure phenomena using simulation, b) generate the phenomenon experimentally to validate the analysis from the simulated phase and provide training/testing data for classification, and c) apply the classification algorithms to characterize the severity of gas entrainment in the physical system.  The primary objective is to differentiate experimental operating states into “acceptable” or “severe” conditions (i.e. binary classification) using the condition 𝜑air > 2% as the criterion for “severe.”  For this basic characterization, the threshold for a 42  satisfactory classification success rate is 90%.  The second objective is to categorize the operating states into multiple classes, corresponding to 0 ≤ 𝜑air < 1%, 1% ≤ 𝜑air < 2%, etc. and appraise the accuracy.  The final objective is to implement a prediction model that yields a continuous-value prediction of 𝜑air and evaluate the prediction error. The study is performed in two portions.  First, the numerical model is used to generate preliminary measurements on which an analysis method and classification approach can be established.  Following that, a comprehensive set of experimental state measurements is collected, allowing for validation and refinement of the analysis and classification methods.               2.3.1 Preliminary Simulation The ranges of air volume fractions investigated by Schäfer (𝜑air = 0 − 5%) [51] serve as an initial reference for defining a realistic range of gas entrainment conditions.  It is assumed that void fractions of 0, 2.5%, and 5.0% reflect normal, moderate, and severe gas entrainment, respectively.  Referencing the range of safe impeller speeds from the physical testbed pump (8.75𝐻𝑧 ≤ 𝜔 ≤ 18.75𝐻𝑧), simulations are conducted at 𝜔 = 8.75𝐻𝑧, 13.75𝐻𝑧, and 18.75𝐻𝑧 for healthy, moderate, and severe states, yielding a total of nine initial reference states. 2.3.2 Classification States An extensive map of experimental operating conditions is necessary to generate a data set suitable for the application of machine learning.  To achieve this, the operable4 range of 𝜔 values for the testbed pump is divided into 0.5 Hz increments, from 𝜔 = 8.9 − 16.9𝐻𝑧5 (17 impeller speeds in total).  Airflow into the system is divided into 24 steps at fixed flow rates, from 𝑞air = 0 − 68.0𝑙/𝑚𝑖𝑛.  This yields 408 total experimental air entrainment states.                                                            4  A small safety margin is added at the high end of the operable values of 𝜔.  This is done to conservatively avoid any potentially damaging behavior that might result from the addition of gas bubbles to the fluid.     5  Arbitrary non-integer values of 𝜔 are used to minimize the likelihood of overlap with 60 Hz electrical noise or any of its associated harmonics during analysis.    43  2.4 Impeller Wear Study The impeller wear study is conducted using a method similar to the gas entrainment study.  The intent is to generate a small initial set of simulated wear conditions, evaluate the relevant behaviors, and then generate a comprehensive set of dynamic pressure measurements, corresponding to a variety of impeller wear conditions, on which to apply machine learning.  However, to circumvent the need to damage the physical centrifugal pump to produce this comprehensive set, the pressure samples are instead generated using the numerical model. The primary objective is to distinguish radial impeller erosion into “acceptable” and “severe” states using the threshold 𝛼imp > 1.5% to delineate “severe.”  The performance target for this binary classification is 90%.  The second objective is to differentiate the conditions into multiple classes, corresponding to 0 ≤ 𝛼imp < 1%, 1% ≤ 𝛼imp < 2%, etc.  As with the air entrainment study, the final objective is to demonstrate a model that yields a continuous-value prediction of 𝛼imp, and assess its prediction accuracy. The impeller wear study is performed in two segments.  The numerical model is initially used to generate a small set of pressure measurements to permit development of an analytical basis for classifying radial impeller erosion.  After, an extensive set of simulated conditions is generated to be used for training, testing, and validation of the classification technique.      2.4.1 Preliminary Simulation The work by A. Suhane suggests that the adverse impacts of radial impeller clearance become conspicuous when the nominal gap between the impeller and cutwater grows by a factor of five [43].  Applying this factor to the nominal clearance in the simulated pump (𝐷o = 0.75𝑚𝑚) yields a radial loss of approximately 3 mm.  From (10), the corresponding impeller loss ratio can be calculated as 𝛼imp = 3%.  This is used as the initial reference for severe radial impeller erosion.  Impeller loss ratios for normal, moderate, and severe radial erosion are defined as 0, 1.5%, and 3.0%, respectively.  Evaluating each of these wear conditions at 𝜔 = 8.75𝐻𝑧, 13.75𝐻𝑧, and 18.75𝐻𝑧 yields nine initial reference states. 2.4.2 Classification States Samples from a wide range of states are necessary to apply machine learning to classify radial impeller wear.  In this simulated case, the range of impeller speeds are not bounded by the applied safety constraints of the physical system, allowing for a wider range of 𝜔 values to be investigated.  Incrementing 44  in 5 Hz steps, a collection of simulated measurements is compiled for 8.75𝐻𝑧 ≤ 𝜔 ≤ 28.75𝐻𝑧 (five speeds total).  The impeller loss ratio is incremented in 0.25% steps, from 0 ≤ 𝛼imp ≤ 3.0% (thirteen wear states total).  This yields a total of 65 simulated states.    2.5 Combined Study  A preliminary investigation is conducted to examine the feasibility of distinguishing gas entrainment, impeller wear, or the simultaneous presence of both using dynamic pressure measurements.  The following simulated states are considered at 𝜔 = 8.75𝐻𝑧, 13.75𝐻𝑧, and 18.75𝐻𝑧, a) normal operation, b) severe gas entrainment (𝜑air = 5.0%), severe radial erosion (𝛼imp = 3.0%), and the simultaneous presence of both conditions in severe form.  The objective is to provide initial context on whether the proposed diagnostic method is potentially suitable for classifying such a condition.      45  3. PRINCIPLES AND DESIGN This chapter comprises the analytical theory and design of the classification method.  Here we discuss the techniques used to reduce dynamic discharge pressure measurements into more comprehensible sets of statistical measures.  Justification for these measures is provided using data from the simulations described in Sections 2.3.1 and 2.4.1.  Following that, three classification architectures are discussed, each with increasing refinement in how precisely it predicts the severity of the corresponding phenomenon.    3.1 Reference Signals  An array of discharge pressure measurements 𝑷out is simulated for each of the conditions in Sections 2.3.1 and 2.4.1.  Each 𝑷out comprises a list of 𝑁 discrete measurements of 𝑝out at the given operating state, each having a pressure magnitude 𝑥𝑛 at index 𝑛, where 1 ≤ 𝑛 ≤ 𝑁.  A portion of the reference simulations for gas entrainment and impeller wear is shown in Figure 3.1.  46   Figure 3.1: a) Simulated pressure signals of a pump at 𝝎 ≤ 𝟏𝟖. 𝟕𝟓𝑯𝒛 with increasing gas entrainment; b) Simulated pressure signals of a pump at 𝝎 ≤ 𝟏𝟖. 𝟕𝟓𝑯𝒛 with increasing impeller wear.     3.2 Signal Decomposition To quantify the signal differences resulting from changing severity of gas entrainment or impeller wear, each 𝑷out is decomposed into a set of characteristic statistical values called features.  Each feature 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.2013.013.514.014.515.015.516.016.517.017.518.05.0% air (sim) 2.5% air (sim) Normal (sim)Time (s)Pressure (kPa)0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.2013.013.514.014.515.015.516.016.517.017.518.03% wear (sim) 1.5% wear (sim) Normal (sim)Time (s)Pressure (kPa)(a)  (b)  47  is determined solely through numerical manipulation of 𝑷out.  These features form the group of inputs used to classify air entrainment or impeller wear at a given state.   The proposed features for both phenomena are discussed in parallel below, grouped according to the domain in which they are extrapolated from.  To permit interpretation and validation, only features with justifiable physical significance6 are considered in the analysis.  Ideally, each feature is expected to exhibit a meaningful change in the signals across the range of measured states.  Each description contains the calculation method and the behavior it is intended to quantify.  Their influence with regards to classification performance (or lack thereof) is validated in Chapter 4.  3.2.1 Time Domain Features (a) Mean: The arithmetic mean of the discharge pressure signal   𝑝out̅̅ ̅̅ ̅ =1𝑁 ∑ 𝑥𝑛𝑁𝑛=1      (23) is calculated by summing each discrete pressure magnitude and dividing by the total number of data points in the measurement.  The mean is necessary for two purposes.  First, it serves as a feature in its own right, representing the average pressure being generated at the pump discharge, which is a fundamental parameter of pump operation that should be accounted for.  Second, it is employed in calculations of other features that require removal of the steady-state component (i.e. the DC bias) of 𝑷out.  Each “unbiased” pressure magnitude 𝑥𝑛 = 𝑥𝑛 − 𝑝out̅̅ ̅̅ ̅ .     (24) The unbiased signal is denoted ?̂?out.  Where it is additionally necessary to scale ?̂?out to fall between -1 and 1, each normalized data point  ?̃?𝑛 =𝑥𝑛max(|?̂?out|) ,     (25) where max(|?̂?out|)is the maximum value of |𝑥𝑛| in ?̂?out.  The normalized pressure signal is denoted ?̃?out.                                                           6 It is noted that using features with plausible physical meaning is not strictly necessary for this method.  The raw time series data, spectral histogram, or other larger feature sets could also be suitable inputs for machine learning.  Here, the decision to investigate a set of targeted statistical measures is intended to a) retain a degree of physical interpretability for validation, b) lessen computational load, and c) advise and promote generalizability for the diagnostic method across a variety centrifugal pumps.    48  (b) Energy: Signal energy 𝐸s = ∑ |𝑥𝑛|2𝑁𝑛=1      (26) is calculated by summing the squares of the unbiased pressure magnitudes in ?̂?out over the length of the sample. The term “signal energy” is not applied in the physics sense, but by the signal processing convention, where it quantifies the magnitude of the periodic portion of the signal.  It is applied to generalize overall amplitude of the signal – “loudness,” to use the acoustic analogy – but also correlates to the frequency and waveform shape itself (i.e. a sine wave, square wave, and triangle wave of the equivalent amplitudes and frequencies will have different signal energies).  Applying (26) to the pressure signals in the preliminary simulations and normalizing the resulting trends yields the following trends for gas entrainment (Figure 3.2a) and radial impeller wear (Figure 3.2b).   Figure 3.2: 𝑬𝐬 as a function of a) 𝝋𝐚𝐢𝐫 and b) 𝜶𝐢𝐦𝐩.  The magnitudes are normalized for comparison.  The blue squares, orange diamonds, and maroon inverted triangles correspond to the states with low (𝝎 = 𝟖. 𝟕𝟓𝑯𝒛), intermediate (𝝎 = 𝟏𝟑. 𝟕𝟓𝑯𝒛), and high (𝝎 = 𝟏𝟖. 𝟕𝟓𝑯𝒛) impeller speeds, respectively.  The trend in each speed range is illustrated using a second-order polynomial best-fit line.  Additionally, the trend for all the states is indicated by the black dotted line.  These trend lines are included to illustrate the cases where the feature has a dual dependence on 𝝎, as well and the target phenomenon.  This labeling convention is employed in each of the subsequent feature plots.        The magnitude of 𝐸s tends to decrease slightly as gas entrainment worsens.  Here, the influence of impeller speed is also apparent, causing a vertical spread at each condition.  This is expected, yet is still undesirable with regards to classification.  The correlation is less present at higher impeller speeds, which correspond to the top points (maroon) across Figure 3.2a.  In Section 4.1.4, a method is discussed to 0.0 1.0 2.0 3.0 4.0 5.00.00.20.40.60.81.01.2Air Volume Fraction (%)Signal Energy - Es0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Signal Energy - Es(a)  (b)  49  mitigate the influence from 𝜔.  Figure 3.2b suggests a positive correlation between 𝐸𝑠 and impeller wear, across impeller speeds.  (c) Variance: The signal variance 𝑉s can be described using central moments, where 𝑚𝑖 =1𝑁 ∑ (𝑥𝑛)𝑖𝑁𝑛=1       (27) is the 𝑖th moment about the signal mean.  Variance is defined as the second central moment.  Substituting 𝑖 = 2 into (27) yields 𝑉s = 𝑚2 =1𝑁 ∑ (𝑥𝑛)2𝑁𝑛=1 .    (28) Variance characterizes the overall spread of the data points in the signal about their average value.  Applying (28) to the initial simulations yields the trends in Figure 3.3.   Figure 3.3: 𝑽𝐬 as a function of a) 𝝋𝐚𝐢𝐫 and b) 𝜶𝐢𝐦𝐩.  The magnitudes are normalized for comparison. Both gas entrainment and impeller wear exhibit a decrease in variance as their severity increases.  This infers that a reduction in variance may be a useful indication of both conditions.  The simulations again demonstrate a secondary correlation between variance of 𝑝out and 𝜔, as indicated by the vertical spacing at each respective trend line.  (d) Skewness: The signal skewness 𝐺s is calculated using the Fisher-Pearson coefficient of skewness as the ratio of moments in (29) [69]. 𝐺s =𝑚3𝑚23 2⁄       (29) 0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Variance - Vs0.0 1.0 2.0 3.0 4.0 5.00.00.20.40.60.81.01.2Air Volume Fraction (%)Variance - Vs(a)  (b)  50  Substituting (27) into (29) with the respective values of 𝑖 and simplifying using (28) yields the calculation for the skewness coefficient. 𝐺s =1𝑁 ∑ (?̂?𝑛)3𝑁𝑛=1[1𝑁 ∑ (𝑥𝑛)2𝑁𝑛=1 ]3 2⁄ =1𝑁 ∑ (𝑥𝑛)3𝑁𝑛=1𝑉s3 2⁄     (30) Skewness describes the asymmetry of the signal about its mean.  Physically, this reflects the relative orientation of the peaks and valleys of the pressure fluctuations during each blade-passing event.  A signal with a positive skewness will have the majority of data points fall below the signal average and the reverse for a negative skewness coefficient.  Calculating the skewness of the initial simulations generates the trends in Figure 3.4.  Figure 3.4: 𝑮𝐬 as a function of a) 𝝋𝐚𝐢𝐫 and b) 𝜶𝐢𝐦𝐩.  The magnitudes are normalized for comparison. Figure 3.4a shows a perceptible decrease in 𝑮𝐬, suggesting the sampled pressures tend to increase with respect to their signal mean as 𝜑air becomes more severe.  The same calculation for simulated wear (Figure 3.4b) does not yield an obvious change, signifying that skewness may not be of particular value in characterizing impeller wear. (e) Kurtosis: Signal kurtosis 𝐾s is calculated using the Fisher7 convention [69] as the ratio of moments                                                           7  The alternative kurtosis calculation method is the Pearson convention, which is identical to (45), but with 3 subtracted from the right hand side.  This corresponds to removing the kurtosis of a standard normal distribution, leaving the “excess” kurtosis, which 0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Skewness - Gs0.0 1.0 2.0 3.0 4.0 5.00.00.20.40.60.81.01.2Air Volume Fraction (%)Skewness - Gs(a)  (b)  51  𝐾s =𝑚4𝑚22 .       (31) Substituting (27) into (31) with the respective values of 𝑖 and simplifying using (28) yields  𝐾𝑠 =1𝑁 ∑ (𝑥𝑛)4𝑁𝑛=1[1𝑁 ∑ (𝑥𝑛)2𝑁𝑛=1 ]2 =1𝑁∑ (?̂?𝑛)4𝑁𝑛=1𝑉s2  .    (32) The magnitude of 𝐾s describes the tendency of the signal to contain data points far removed from its mean.  Physically, it is correlated to the magnitudes and breadths of the pressure peaks and valleys.  The kurtosis values for the preliminary simulations are shown in Figure 3.5.  Figure 3.5: 𝑲𝐬 as a function of a) 𝝋𝐚𝐢𝐫 and b) 𝜶𝐢𝐦𝐩.  The magnitudes are normalized for comparison. The preliminary states do not exhibit a strong correlation between kurtosis and the target conditions.  However, experimental pressure measurements reported by T. Xie [48] indicate that severe entrainment can introduce broadband fluctuations, which would influence the signal shape dramatically, affecting the kurtosis.  It is plausible that the lack of correlation in Figure 3.5 is the byproduct of the trend being concealed by the stronger correlation to impeller speed, but more data is necessary to make that determination.  This is evaluated further in Chapter 4.  (f) Autocorrelation Energy:  Autocorrelation is a method for quantifying the periodicity of a signal.  The discrete autocorrelation coefficients 𝑅𝑙 of the signal are calculated by correlating ?̃?out to a duplicate                                                           can be useful for characterizing signals in some cases.  However, in this research, each feature is ultimately scaled to fall between 0 and 1, so the inclusion of the -3 term does not provide any benefit.      0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Kurtosis - Ks0.0 1.0 2.0 3.0 4.0 5.00.00.20.40.60.81.01.2Air Volume Fraction (%)Kurtosis - Ks(a)  (b)  52  of itself ?̃?′out, which is identical, but incorporates a time-shift 𝑙.  The correlation is equal to dot product of the signals at the given time shift, or 𝑅𝑙 = ?̃?out  ∙ ?̃?′out = ∑ ?̃?𝑛?̃?𝑛+𝑙𝑁𝑛=1  .    (33) Computing 𝑅𝑙 for 0 ≤ 𝑙 ≤ 𝑁 at steps equal to 𝑡s, assembling the coefficients into an array, and normalizing8 yields the autocorrelation signal 𝑹s.   For an unbiased periodic signal, 𝑹s will oscillate between positive and negative correlation, decaying as the time overlap between the signals shrinks.  The addition of noise will hasten the decay.  Calculating the energy of 𝑹s using the same method as (26) yields the autocorrelation energy 𝐸R.  The autocorrelation signals for the preliminary simulations are shown in Figure 3.6.  Figure 3.6: 𝑬𝐑 as a function of a) 𝝋𝐚𝐢𝐫 and b) 𝜶𝐢𝐦𝐩.  The magnitudes are normalized for comparison. It is observed in Figure 3.6a that the addition of air to the fluid decreases the autocorrelation, which indicates that the introduction of air causes the periodicity of 𝑝out to degrade.  This is supported by the experimental measurements in Figure 1.10.  Increasing radial impeller wear creates the opposite effect, as shown in Figure 3.6b.  The eroding impeller diameter increases autocorrelation energy, suggesting the fluctuations are more periodic.  This is notable because it implies that 𝐸R may be an effective feature to differentiate between wear and air entrainment.                                                           8  The autocorrelation signal is normalized by dividing each discrete coefficient by the maximum value in the array.  This maximum value occurs at the first value, where the time-shift 𝑙 = 0.  Here the correlation is defined to be 1 because the signals ?̃?𝑜𝑢𝑡 and 𝑷′̃𝑜𝑢𝑡 are equivalent. 0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Autocorr. Energy - Er0.0 1.0 2.0 3.0 4.0 5.00.00.20.40.60.81.01.2Air Volume Fraction (%)Autocorr. Energy - Er(a)  (b)  53  3.2.2 Frequency Domain Features Features from the frequency domain are also employed.  Each signal is converted to a spectrum in the frequency domain using the Fast Fourier Transform (FFT) algorithm.  This spectrum 𝑿 comprises a list of magnitudes 𝑋𝑚, each at a corresponding discrete frequency 𝑓𝑚 with index 𝑚.  The frequency spectrum has a total of 𝑀 data points.  The peak frequency in the spectrum 𝑓𝑀 = 𝑓nyq, where 𝑓nyq = (𝑓𝑠 2⁄ ) – the Nyquist frequency.  However, since the raw signals are filtered using a 400 Hz LPF prior to analysis, frequencies beyond 400 Hz are not considered, even though they fall below 𝑓nyq.   Studies by Yuan [42] and Suhane [43] show that the dominant frequency peak in the discharge pressure signal is the BPF.  Noting this, the BPF can inversely be determined by extracting the frequency corresponding to the peak magnitude in 𝑿.  This is critically important because it permits use of the BPF in the analysis and classification.  Its determination requires no external input beyond the dynamic pressure measurement.  Were it otherwise, its incorporation would not be justified in this method.  Normalizing the frequencies in 𝑿 to the BPF allows to content of each spectrum to be compared independent of its individual RF/BPF, simplifying the analysis.   (g) Energy: The total energy of the frequency spectrum 𝐸FFT is calculated using a method similar to (26).  However, instead of the unbiased data points, the calculation is done using the magnitudes from the frequency spectrum, with each frequency scaled to a multiple of the BPF, with the spectral peak at the BPF normalized to 1.   𝐸FFT = ∑ |𝑋𝑚|2𝑀𝑚=1      (34) Parseval’s theorem states that the energy of a signal is equal to the energy of its frequency spectrum (being a linear transformation).  However, by the scaling the frequencies using the BPF of the respective signal, then normalizing against this peak magnitude, the linearity is not upheld.  Here, as a result, 𝐸FFT characterizes the minor spectral peaks with respect to the main frequency peak, rather than the total energy of the signal itself.  It can be influenced by a number of factors, including the noise floor, growth or decay at specific fluctuation frequencies, and changing waveforms (which can add or detract frequency peaks at their associated harmonics). Applying (34) to the preliminary simulations yields the trends shown in Figure 3.7.   54   Figure 3.7: 𝑬𝐅𝐅𝐓 as a function of a) 𝝋𝐚𝐢𝐫 and b) 𝜶𝐢𝐦𝐩.  The magnitudes are normalized for comparison.  The simulations demonstrate that 𝐸FFT has slight positive and negative correlations to 𝜑air and 𝛼imp, respectively.  However, the vertical spread at each condition, caused by the increasing impeller speed, obscures the trend.  A larger pool of measurements is needed to more firmly corroborate the relationship between 𝐸FFT and the target conditions.  3.3 Feature Space  The seven features from Section 3.2 comprise a feature vector 𝑭 = [𝑝out̅̅ ̅̅ ̅, 𝐸s, 𝑉s, 𝐺s, 𝐾s, 𝐸R, 𝐸FFT]    (35) for each measured state.  The feature vectors are employed as inputs for classification models.  Each feature vector is assigned a classification label coinciding with its operating condition – either 𝜑air or 𝛼imp – to be referenced as target values during training and testing.  The aggregate of the feature vectors across all sampled states forms the feature space {𝑭}. 0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)FFT Energy - Efft0.0 1.0 2.0 3.0 4.0 5.00.00.20.40.60.81.01.2Air Volume Fraction (%)FFT Energy - Efft(a)  (b)  55   Applying principal component analysis (PCA) to {𝑭} of the preliminary gas entrainment simulations yields the trends shown in Figure 3.8.  Figure 3.8: PCA of the preliminary gas entrainment simulations.  a) The corresponding scree plot with an inset table of loadings for the first two principal components (PCs).  The loadings are the coefficients by which the input features can linearly combined to construct PCs.  They can be interpreted as the strength of influence a given feature has on the orientation of a particular PC.  The loadings are ordered by their contribution to PC1; b) The distribution of the simulated states across PC1 and PC2.  Each state is labeled with the magnitude of 𝝋𝐚𝐢𝐫.   The initial gas entrainment data suggests that there is a principal component (PC) across which 68% of the feature variance manifests (Figure 3.8a, PC1).  Plotted across PC1 and PC2, the preliminary states demonstrate a degree of linear separability, with the most severe gas entrainment states trending toward the top left quadrant and the healthy states trending to the opposite. (a)  (b)  56   Applying PCA to {𝑭} of the preliminary impeller wear simulations yields the trends shown in Figure 3.9.  Figure 3.9: PCA of the preliminary impeller wear simulations. a) The corresponding scree plot with an inset table of loadings for PC1 and PC2.  The loadings are ranked by their contribution to PC1; b) The distribution of the simulated states across the first two PCs.  Each state is labeled with magnitude of 𝜶𝐢𝐦𝐩.   The initial impeller wear data demonstrates that 52% of the feature variance manifests across PC1 (Figure 3.9a).  Correlating PC1 and PC2, Figure 3.9b shows a relatively good linear separability in the target condition, with the most severe wear states trending toward the top right quadrant and the healthy states trending to the opposite.   Figures 3.8 and 3.9 show that the both conditions may be linearly separable using the proposed feature vector.  However, this assessment is based on a limited number of states and must be considered as an estimation until it can be validated on a more extensive pool of conditions.  Therefore, in selecting prediction models for binary, multi-class, and continuous-value regression, it is assumed that the target conditions will not be perfectly linearly separable for all conditions.   The preliminary features discussed in Section 3.2 reveal characteristic trends across the feature space for both gas entrainment and impeller wear.  The correlations between the features and target conditions suggests that classification can be achieved through the use of conventional supervised machine learning methods, in lieu of more complex deep learning methods.     (a)  (b)  57  3.4 Classification Models   Three methods are employed to categorize gas entrainment and radial impeller wear; binary (i.e. two-class), multi-class, and continuous-value classification.  Each makes a successively more specific prediction of the target condition.  The respective learning models employ supervised learning to classify the target phenomena.  Each model is developed in Python 3.5.  The binary and multi-class neural network algorithms are programmed largely from scratch, whereas the support vector machines (SVM) and random forest regression methods are implemented using the scikit-learn library [70]. Binary Classification The binary classification model predicts the severity of the gas entrainment and impeller wear with respect to acceptability thresholds 𝜏air and 𝜏imp, respectively.  Below the threshold, the centrifugal pump is assumed to be operating in a satisfactory condition.  The nominal thresholds for gas entrainment and impeller wear are selected to be 𝜏air = 2.0% and 𝜏imp = 1.5%, respectively.  The binary classification target label “0” is appended to the feature vectors for states that fall below their respective severity threshold.  For states that have reached or exceeded their respective threshold, the condition is regarded as degraded and the target label “1” is instead appended. A multilayer perceptron (MLP) is applied for binary classification.  In this application, an MLP with a single hidden layer is used.  The structure is shown in Figure 3.10.   Figure 3.10: Schematic of the MLP architecture used for two-class classification [71].  𝐸s 𝑉s 𝑝out̅̅ ̅̅ ̅ 𝐺s 𝐾s 𝐸R 𝐸FFT Input Layer Hidden Layer Output Layer Sigmoid Sigmoid 𝑤𝑖𝑗 𝑤𝑗𝑘 ?̅? 𝑷 58  The model comprises three layers; and input layer, one hidden layer, and the output.  The input layer has seven nodes, each corresponding to a feature in 𝑭.  The hidden layer contains four nodes.9  The output layer is a single node, corresponding to the binary prediction.   In the feedforward stage, the features from a randomly selected state in {𝑭} are input.  The nodes are connected by associated weights 𝑤𝑖𝑗 and 𝑤𝑗𝑘, with indices i, j, and k corresponding to the input, hidden and output node index, respectively.  The weights are assigned random values between -1 and 1, initially.  The nodes in the hidden layer compute a weighted average of the incoming values, and output a local prediction using a sigmoid activation function, which tends to 0 for very negative numbers and 1 for very positive numbers.  The predictions from the hidden nodes are then weighted again and passed to the output node, which computes a weighted average ?̅? and generates a final prediction 𝑷 based on the sigmoid function applied to ?̅?. The model learns by backpropagating the prediction error and adjusting the network weights.  This is done using gradient descent of the mean squared error (MSE)10 loss function of the prediction, in which the weights are shifted in the direction opposite their respective gradient to find a local minimum that minimizes the MSE (i.e. a least mean squares (LMS) algorithm).  The MSE of target prediction Γ𝑚 at index 𝑚 is calculated as MSE =1𝑀∑ (Γ𝑚 − 𝑷𝑚)2𝑀𝑚=1  ,     (36) where 𝑀 is the total number of output predictions (which is equivalent to the number of nodes in the output layer).  When 𝑀 = 1, as in the model in Figure 3.10, (36) simplifies to 𝑀𝑆𝐸 = (Γ − 𝑃)2.  As the MSE reduces with successive iterations, the values of the weights converge, yielding the configured prediction model.  The rate at which the weights are adjusted is controlled by the dimensionless tunable learning rate parameter 𝜂𝐿𝑅 = 0.02.  The accuracy of the model over training iterations is determined by rounding the                                                           9  An appropriate number of nodes in the hidden layer is initially determined through trial and error using an estimated data set, fabricated from the trends of the preliminary simulations.  It is later validated as the optimum using experimental data. 10  Ordinarily, MSE would be used as a loss function for regression problems, not classification.  However, its application here is warranted because gas entrainment and impeller wear are fundamentally regression-like phenomena, even if the predictions are eventually grouped, using a threshold, into a particular class.  C. Beletes et al. argue that this is particularly justified in classification problems where some states may be simply be ambiguous based on their features [72].    59  predictions greater than or equal to 0.5 up to 1 and outputs less than 0.5 to 0, then comparing the rounded prediction to the target classification labels representing the true condition.  For contrast and validation of the MLP’s performance, a two-class, soft-margin SVM model is employed in parallel.  The applied SVM algorithm uses a polynomial kernel, with a regularization parameter 𝐶𝑆𝑉𝑀 = 5 and kernel coefficient 𝛾SVM = 5. Multi-class Classification The multi-class classification model categorizes the target phenomenon into a group called a class, which corresponds to a designated severity range.  Gas entrainment conditions are divided into 𝑀air = 6 classes; 0 ≤ 𝜑air < 1%, 1% ≤ 𝜑air < 2%, etc., up to 𝜑air > 5%.  Radial impeller wear is divided into 𝑀wear = 3 classes, corresponding to 0 ≤ 𝛼imp < 1%, 1% ≤ 𝛼imp < 2%, and 2% ≤ 𝛼imp.  For each input state, 𝑭 is appended with a 1xM array of labels, with each value corresponding to a severity class, from least to most severe.  This label array is one-hot encoded, with all values equal to zero, except for the entry corresponding to the true severity class, which is 1 (for example: the target label for 𝜑air = 0.6% would be [1 0 0 0 0 0] and the label for 𝛼imp = 2.6% would be [0 0 1]).  These labels are used as the target when calculating the prediction accuracy The MLP shown in Figure 3.10 can be reconfigured to suit multi-class classification.  The structure is shown in Figure 3.11.  Figure 3.11: Schematic of MLP architecture used for multi-class classification [71].  The gas entrainment model has six output classes.  For impeller wear, three classes are used.  𝐸s 𝑉s 𝑝out̅̅ ̅̅ ̅ 𝐺s 𝐾s 𝐸R 𝐸FFT Input Layer Hidden Layer Output Layer Sigmoid Softmax 𝑤𝑖𝑗 𝑤𝑗𝑘 ?̅? 𝑷 ● ● ● 60  The multi-class MLP has 𝑀 output nodes, one for each severity class.  As with the binary classification model, the hidden layer uses the sigmoid activation function.  However, here the softmax function  softmax(?̅?𝑘) =𝑒?̅?𝑘∑ 𝑒?̅?𝑚𝑀𝑚=1 ,     (37) is applied to ?̅? at each output node index 𝑘, yielding 𝑷, a 1xM array of real valued probabilities for each severity class, the sum of which is 1.  The element of 𝑷 with the highest value corresponds to the most likely class.  As with the two-class prediction model, gradient descent is used during backpropagation, adjusting the weights so as to minimize the MSE loss function in (36).  As with the binary case, the multi-class MLP is contrasted to a multi-class SVM in parallel.  A polynomial kernel is employed again, with 𝐶SVM = 5.  The kernel coefficient 𝛾SVM is increased to 20 to provide better compensation for potential non-linearities between classes. Regression In the final classification step, a random forest algorithm is employed to generate a real-valued prediction of the severity of the target phenomena.  The random forest algorithm creates a series of regression trees containing a randomly sampled (“bootstrapped”) subset of the training data set.  In doing so, the risk of overfitting is curtailed.  In this model, the random forest is populated with 103 decision trees.  The bootstrapped data sets are sampled from the entire feature space, rather than a subset.  61  4. RESULTS AND DISCUSSION In this chapter, we present the results and analysis from the gas entrainment and radial impeller wear studies.  The entrainment study employs experimental measurements, whereas the wear study uses simulated conditions.  In each, we discuss the characteristics of the dynamic pressure measurements, evaluate the relevance of the statistical features put forth in Section 3.2, and demonstrate classification of the target conditions.  An analysis of measurement and classification errors follows, after which we propose a method for refining the feature sets to improve classification accuracy.  4.1 Gas Entrainment Study Dynamic pressure measurements of the discharge flow are collected at 310 states of varying air entrainment, ranging from normal operation to 𝜑air = 6.0%.  The samples are divided equally into two sets of 155 states, one for training the learning algorithms and another for testing, with a similar variety of severities in each.  The values of 𝜔 and 𝜑air for each training state are shown in Table 4.1.  The states are indexed by row and column for reference.  Table 4.1: Experimental Gas Entrainment States (Training)    A B C D E F G H I J K L M N O P Q   𝝎 (Hz) → 8.9 9.4 9.9 10.4 10.9 11.4 11.9 12.4 12.9 13.4 13.9 14.4 14.9 15.4 15.9 16.4 16.9 1 𝝋𝒂𝒊𝒓 (%)→  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 1.6 1.6 1.5 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1.0 1.0 1.0 0.9 0.9 0.9 0.9 3 2.5 2.3 2.2 2.1 2.0 1.9 1.8 1.8 1.7 1.6 1.6 1.5 1.5 1.4 1.4 1.3 1.3 4 3.3 3.1 3.0 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 2.0 1.9 1.8 1.8 1.7 5 4.2 3.9 3.7 3.5 3.4 3.3 3.1 3.0 2.8 2.7 2.6 2.5 2.4 2.4 2.3 2.2 2.1 6 5.3 4.6 4.4 4.2 4.0 3.8 3.6 3.5 3.3 3.2 3.1 2.9 2.8 2.7 2.7 2.6 2.5 7   5.3 5.1 4.8 4.6 4.3 4.2 4.0 3.9 3.7 3.6 3.4 3.3 3.2 3.1 3.0 8      5.2 5.0 4.8 4.6 4.5 4.2 4.1 3.9 3.8 3.7 3.6 3.4 9         5.2 5.0 4.8 4.6 4.4 4.2 4.1 4.0 3.9 10            5.1 4.9 4.8 4.6 4.5 4.3 11            5.7 5.5 5.3 5.2 4.9 4.8 12             6.0 5.9 5.7 5.4 5.2  Each row corresponds to a fixed value of 𝑞air.  The states without entries are conditions where the void fraction was great enough to cause the pump to stall, preventing a reliable pressure measurement.  This occurrence reinforces the initial assumption that entrained gas void fractions exceeding 5% can be 62  considered exceptionally severe.  It also parallels experimental outcomes reported by Schäfer [51] and Stan [52].          The pressure waveforms of the ordinary operating states from the training set (Table 4.1, Row 1) are shown in Figure 4.1.    Figure 4.1: Samples of discharge pressure fluctuations across a range of rotating frequencies, during operation without gas entrainment.   As 𝜔 increases, so does the discharge pressure.  This is the byproduct of the increasing hydraulic resistance caused by faster flow through the system.  At the slowest impeller speed, the main oscillating mode of the pressure varies within a band of approximately 7 kPa.  At the highest, the fluctuations span 37 kPa.  The magnitudes of the higher frequency pressure excursions – those not correlated to blade-passing events – remain consistent across rotating frequencies and are predominantly less than 3 kPa. The fluctuations in 𝑝out also show unexpected alternating depths in the pressure minima following blade-passing events (most perceptible in the highest speed sample in Figure 4.1).  This behavior is independent of gas entrainment.  It is likely the result of slightly unequal geometry between the two impeller blades.  Though it is not within the scope of this work, this phenomenon could potentially be leveraged to detect impeller asymmetry, even if the impeller weight itself is not unbalanced.  However, in 63  evaluating the dynamic pressure frequency content, this phenomenon is anticipated to generate a spectral peak at half the BPF (i.e. the RF) that should otherwise not be present.   The inclusion of a substantial void fraction of gas bubbles has a visible influence on the discharge pressure fluctuations.  The waveforms of 𝑝out at the highest achievable 𝜑air for each rotating frequency (bottom box of each column in Table 4.1) are shown in Figure 4.2.  Figure 4.2: Samples of discharge pressure fluctuations across a range of rotating frequencies, each with an air void fraction exceeding 4.5%. Across frequencies, the main fluctuating mode is not well defined.  In all cases, the pressure peaks corresponding to blade passing events are diminished, causing the overall span of the oscillations to decrease.  At the lowest speed, the pressure fluctuates within a 5 kPa band, which is approximately 30% narrower than the same margin during normal operating condition.  At the highest rotating frequency, the decrease is more substantial.  The fluctuation band narrows to 20 kPa, which is a 45% decrease from the normal condition.  This reinforces the correlation between variance and the severity of gas entrainment inferred from the initial simulations.  The pressure measurements also exhibit an increase in higher frequency fluctuations, which was also demonstrated in the initial simulations.  However, there are additional chaotic spikes and drops, particularly in the intermediate rotating frequencies, which were not present in the preliminary simulated measurements.  This is plausibly an effect of air bubbles momentarily adhering to the transducer diaphragm then separating or collapsing, similar to behavior reported by 64  Nakatani [73].  However, validating this hypothesis exceeds the scope of this research.  A detail comparison of two discharge pressure signals with varying gas entrainment is shown in Figure 4.3 (states 1M and 11M, from Table 4.1).  Figure 4.3: The fluctuations of 𝒑𝐨𝐮𝐭 at 𝝎 = 𝟏𝟒. 𝟗𝑯𝒛 for normal pump operation (blue) and an air entrainment condition with 𝝋𝐚𝐢𝐫 = 𝟔. 𝟎% (green). At 𝜑air = 6.0%, the main fluctuation mode is substantially degraded. The variance between the local pressure peaks and minima has reduced, and disordered pressure spikes are frequent.  4.1.1 Feature Correlation  Using the methods described in Section 3.2, the measurements of 𝑝out are decomposed into statistical features for each state in Table 4.1.  The normalized values of each feature are plotted in Figures 4.4a to 4.4g.  Each plot also contains four second-order polynomial trend lines corresponding to the features associated with i) the lowest impeller speed, 𝜔 = 8.9𝐻𝑧 (blue line), ii) the median speed, 𝜔 =12.9𝐻𝑧 (orange line), iii) the highest speed, 𝜔 = 16.9𝐻𝑧 (maroon line), and iv) all the data points (black dashed line).  These are included to better illustrate the cases where the correlation between the feature and gas entrainment severity has an additional dependence on 𝜔.  0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.204045505560657075806.0% air NormalTime (s)Pressure (kPa)65   Figure 4.4a-c: The normalized values of a) 𝒑𝐨𝐮𝐭̅̅ ̅̅ ̅̅ , b) 𝑬𝐬, and c) 𝑽𝐬 as a function of 𝝋𝐚𝐢𝐫. 0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)Mean - p_out_bar0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)Signal Energy - Es0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)Variance - Vs(a)  (b)  (c)  66   Figure 4.4d-f: The normalized values of d) 𝑮𝐬, e) 𝑲𝐬, and f) 𝑬𝑹 as a function of 𝝋𝐚𝐢𝐫. 0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)Skewness - Gs0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)Kurtosis - Ks0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)Autocorrelation Energy - Er(d)  (e)  (f)  67    Figure 4.4g: The normalized values of 𝑬𝐅𝐅𝐓 as a function of 𝝋𝐚𝐢𝐫.  Figure 4.4a demonstrates that average pressure 𝑝out̅̅ ̅̅ ̅ has a poor correlation to gas entrainment for all of the experimental states.  Neither the high, median, nor low speed measurements indicate a meaningful positive or negative association with increasing void fraction.  Instead, the vertical spacing of the respective trend lines shows that the average discharge pressure is highly correlated to the impeller speed.  This suggests that 𝑝out̅̅ ̅̅ ̅ may instead function to orient input features with respect to their rotating frequency during classification.   From 4.4a, it can also be inferred that the slopes of the trend lines indicate a correlation to 𝜑air (or, for the case of 𝑝out̅̅ ̅̅ ̅, an absence of correlation) and their respective deviations from each other (whether vertical spread, dissimilar slopes, etc.) suggest a dependence on 𝜔.           The samples of the signal energy 𝐸s in Figure 4.4b are an example of mutual dependence on both 𝜑air and 𝜔.  At the lowest impeller speed, the 𝐸s does not exhibit any correlation to 𝜑air.  However, in both the median and high-speed states, as well as the overall average, 𝐸s decreases.  This agrees with the simulated states shown in Figure 3.2a.  Physically, this suggests that the average amplitudes of the pressure fluctuations diminish as gas entrainment becomes more severe, but only if the impeller is rotating fast enough to exhibit the effect.  If the impeller is moving slowly, the correlation weakens.  Figure 4.4c shows that variance 𝑉s has a negative correlation to 𝜑air at high impeller speeds.  This closely parallels the simulated results in Figure 3.3a and agrees with the observations of Figure 4.2.  The 0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)FFT Energy - Efft(g)  68  correlation at intermediate and slow impeller speeds is unclear, due to the dominating influence of the high speed 𝑉s values.  In Section 4.2.6, a method is discussed for normalizing out the influence of 𝜔, clarifying the relationship.  In Figure 4.4d, a negative correlation between skewness 𝐺s and 𝜑air is observed at intermediate and high speeds.  This matches the trends of the initial simulations in Figure 3.4a.  A reduction in 𝐺s indicates that the values of 𝑝out have decreased with respect to the signal mean.  Physically, this means a) the pressure peaks associated with a blade passing event have been diminished, or b) the proceeding pressure drops have been accentuated.  The waveforms in Figures 4.1 and 4.2 corroborate the former.  The pressure peaks at each blade-passing event are truncated when air is introduced, skewing the measurements downward without significantly affecting the mean.  The correlation is less perceptible at the slowest impeller speed.  This again suggests that the value of 𝐺s as a feature for classification diminishes as the impeller slows.     The relationship between Kurtosis 𝐾s and 𝜑air (Figure 4.4e) is comparable to that of 𝐺s, though with an opposite sign.  Intermediate and high speeds exhibit a positive correlation, signifying that the individual values of 𝑝out tend to become focused about the mean as gas entrainment worsens.  However, the association is again weakened as the impeller slows.  Figure 4.4f shows that autocorrelation energy 𝐸R has a strong negative correlation across impeller speeds, but particularly at the highest speed.  This matches the preliminary simulated results from Figure 3.6a and substantiates the observation that increasing 𝜑air contributes disorder to the pressure fluctuations, reducing the periodicity of the measurement.  The vertical spacing of the trend lines suggests an additional dependence on 𝜔.   The energy of the FFT spectrum 𝐸FFT increases with 𝜑air (Figure 4.4g), owing to the increase in higher frequency pressure fluctuations.  This agrees with the simulated trends in Figure 3.7a.  The association lessens with reduced impeller speed.  PCA is applied to evaluate the correlations between these seven input features.  The scree plot is shown in Figure 4.5a, with the loadings for PC1 and PC2 adjacent.  The first two PCs are plotted with respect to each other in Figure 4.5b.  69   Figure 4.5: a) A scree plot of the PCs of gas entrainment and the loading for each variable with respect to PC1 and PC2, listed in order of their contribution to PC1; b) The distribution of the experimental states across the first two PCs. In Figure 4.5a, it can be observed that PC1 and PC2 account for more than 80% of the variation across the features.  Plotting the experimental states over these two PCs (Figure 4.5b) reveals a degree of grouping, with the states with lower 𝜑air tending to fall to the right, but there are significant regions of overlap.  This reinforces the initial assessment that the gas entrainment states should not be treated as linearly separable and justifies the application of non-linear machine learning algorithms for gas entrainment classification. 4.1.2 Binary Classification To evaluate the performance of the binary classification model, the MLP is trained and tested, independently, 20 times.  Training performance is determined by averaging the errors over the final 5·103 -6 -4 -2 0 2 4 6 8-4-3-2-1012340-1% 1-2% 2-3% 3-4% 4-5% >5%PC1PC2PC1 PC2 PC3 PC4 PC5 PC6 PC70102030405060Percentage of Explained Variation (%)(a)  (b)  70  iterations in each training cycle.  With the acceptability threshold for gas entrainment set at 𝜏air = 2.0% the binary classifier correctly predicts the gas entrainment severity category in 90% of the training states.11  The testing accuracy is 84%.  The difference of 6% between training and testing suggests that slight overfitting may be occurring, but the impact is minimal.  The binary SVM classifier achieves 88/83% training/testing accuracy using the same states. The truth table for the binary MLP is shown in Table 4.2. Table 4.2: Truth Table for Binary MLP Classification of Gas Entrainment (Testing) True Healthy: 31 False Healthy: 5 False Degraded: 20 True Degraded: 99   Of the 25 misclassified states in Table 4.2, 80% overestimate the entrainment severity, suggesting the condition had eclipsed the threshold when it has not.  Table 4.3 shows the specific conditions misclassified in the testing data set (underlined) by the binary MLP.  The true healthy states are shaded gray and the degraded red. Table 4.3: Gas Entrainment States Misclassified by Binary MLP Model (Testing)                                                            11 The configured MLPs are not perfectly identical each time due to the random starting weights.  As such, one or two states may be correctly classified in one configuration and not in another, leading to (negligible) variability in the classification performance.   A B C D E F G H I J K L M N O P Q   𝝎 (Hz) → 8.9 9.4 9.9 10.4 10.9 11.4 11.9 12.4 12.9 13.4 13.9 14.4 14.9 15.4 15.9 16.4 16.9 1 𝝋𝒂𝒊𝒓 (%)→ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 1.4 1.3 1.2 1.2 1.1 1.1 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.6 1.5 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.1 4 3.0 2.9 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.8 1.7 1.7 1.6 1.6 5 3.9 3.7 3.5 3.3 3.2 3.0 2.9 2.8 2.6 2.5 2.5 2.4 2.3 2.2 2.1 2.1 2.0 6 5.1 4.5 4.3 4.1 3.9 3.7 3.5 3.4 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 7   5.1 4.9 4.6 4.4 4.1 4.0 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 2.9 8      5.0 4.8 4.6 4.4 4.3 4.0 3.9 3.8 3.7 3.5 3.4 3.3 9         5.0 4.9 4.6 4.5 4.3 4.1 4.0 3.9 3.7 10            5.0 4.8 4.6 4.5 4.3 4.2 11            5.5 5.3 5.2 5.0 4.8 4.7 12             5.9 5.8 5.6 5.3 5.1 71  It is observed that the erroneously classified states are predominantly near the acceptability threshold (2.0%, the gray/red interface), which is to be expected.  Only one very severe entrainment condition is misclassified (10Q).  The results also suggest that the classification performance degrades at the lowest impeller speeds.  This aligns with the observations of the feature trends in Figure 4.4, many of which show weak correlations at the slowest impeller speed.  It is proposed that this can be mitigated by normalizing the influence of impeller speed.  Though the binary classification model performs reasonably well using the raw feature set, it does not meet the 90% accuracy threshold stated in the research objectives, and therefore must be improved.  4.1.3 Multi-class Classification The multi-class classification model is evaluated on the same training and testing sets as the binary model.  Its performance is likewise evaluated using a similar process.  The network is trained and tested 20 separate times, starting from random weights, to determine its accuracy and repeatability.  Training accuracy is determined by averaging the prediction errors over the final 5·103 iterations in each training cycle.  Using classes spanning 1%, the multi-class model places each training state in the correct group with 82% accuracy.  The classification accuracy on the testing set is 59%.  Of the misclassified testing states, approximately 70% are within one class of the correct prediction.  Those that are not are underlined in Table 4.4, with the respective classes color coded. Table 4.4: Gas Entrainment States Severely Misclassified by Multi-class MLP Model (Testing)     A B C D E F G H I J K L M N O P Q   𝝎 (Hz) → 8.9 9.4 9.9 10.4 10.9 11.4 11.9 12.4 12.9 13.4 13.9 14.4 14.9 15.4 15.9 16.4 16.9 1 𝝋𝒂𝒊𝒓 (%)→ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 1.4 1.3 1.2 1.2 1.1 1.1 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.6 1.5 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.1 4 3.0 2.9 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.8 1.7 1.7 1.6 1.6 5 3.9 3.7 3.5 3.3 3.2 3.0 2.9 2.8 2.6 2.5 2.5 2.4 2.3 2.2 2.1 2.1 2.0 6 5.1 4.5 4.3 4.1 3.9 3.7 3.5 3.4 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 7   5.1 4.9 4.6 4.4 4.1 4.0 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 2.9 8      5.0 4.8 4.6 4.4 4.3 4.0 3.9 3.8 3.7 3.5 3.4 3.3 9         5.0 4.9 4.6 4.5 4.3 4.1 4.0 3.9 3.7 10            5.0 4.8 4.6 4.5 4.3 4.2 11            5.5 5.3 5.2 5.0 4.8 4.7 12             5.9 5.8 5.6 5.3 5.1 72  As with the binary MLP prediction model, it is observed that the majority of the misclassifications occur at the slower impeller speeds, where the correlations between the input features and target condition are weakest.  The multi-class SVM model yields training and testing prediction accuracies of 80% and 60%, respectively.  These prediction results reinforce the need to improve the relevance of the features for the low speed states.   4.1.4 Regression  The random forest regression model is implemented on the gas entrainment training and testing sets.  The median prediction error on the training set is a void fraction within 0.20 of the true 𝜑air percentage value (standard deviation of 0.26).  The median prediction error on the testing set is within 0.47 of the true severity (standard deviation of 0.73).  Considering the testing predictions miss the true void fraction by more than 1.25, it can be observed that they occur with much greater frequency below 𝜔 =12.9𝐻𝑧.  These states are underlined in Table 4.5. Table 4.5: Gas Entrainment States Severely Misclassified by Regression Model (Testing)   Though the prediction accuracy of the regression model is adequate for an estimate of gas entrainment, this result reiterates that the input features cannot precisely quantify the target condition in low speed states and must be improved.       A B C D E F G H I J K L M N O P Q   𝝎 (Hz) → 8.9 9.4 9.9 10.4 10.9 11.4 11.9 12.4 12.9 13.4 13.9 14.4 14.9 15.4 15.9 16.4 16.9 1 𝝋𝒂𝒊𝒓 (%)→ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 1.4 1.3 1.2 1.2 1.1 1.1 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.6 1.5 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.1 4 3.0 2.9 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.8 1.7 1.7 1.6 1.6 5 3.9 3.7 3.5 3.3 3.2 3.0 2.9 2.8 2.6 2.5 2.5 2.4 2.3 2.2 2.1 2.1 2.0 6 5.1 4.5 4.3 4.1 3.9 3.7 3.5 3.4 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 7   5.1 4.9 4.6 4.4 4.1 4.0 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 2.9 8      5.0 4.8 4.6 4.4 4.3 4.0 3.9 3.8 3.7 3.5 3.4 3.3 9         5.0 4.9 4.6 4.5 4.3 4.1 4.0 3.9 3.7 10            5.0 4.8 4.6 4.5 4.3 4.2 11            5.5 5.3 5.2 5.0 4.8 4.7 12             5.9 5.8 5.6 5.3 5.1 73  4.1.5 Feature Optimization  The correlations between the input features and gas entrainment severity are improved by minimizing their respective feature variation caused by impeller speed.  The objective is to improve the coefficient of determination 𝑅2 of the input features’ trends with respect to gas entrainment.  To do so, 𝜔 is first extracted from the FFT spectrum at each state by isolating the frequency corresponding to the highest peak (the BPF) and dividing that value by 𝑁imp. Plotting each input feature against its 𝜔 values reveals its association to impeller speed.  Figure 4.6 demonstrates this correlation for 𝑉s.   Figure 4.6: 𝑽𝐬 as a function of 𝝎.  The blue and green trend lines correspond to the states with 𝝋𝐚𝐢𝐫 = 𝟎 and 𝝋𝐚𝐢𝐫 ≥ 𝟓, respectively.  The dotted black curve and best fit equation reflect the trend of all the states. In this example, the correlation between 𝑉s and can be suitably described by a third-degree polynomial.  After determining the best fit function for the states, a refined variance 𝑉𝑠′ = 𝑉𝑠/𝑉𝑠(𝜔) can be calculated and renormalized. 8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2Rotating Frequency (Hz)Variance - Vs𝑉𝑠(𝜔) ≈ 9.3 ∙ 10−4𝜔3 − 2.9 ∙ 10−2𝜔2 + 0.3𝜔 − 1.1 74  Figure 4.7 shows the trend of 𝑉s′ with respect to 𝜑air.       Figure 4.7: 𝑽𝐬′ as a function of 𝝋𝐚𝐢𝐫.  The blue, orange, and red trend lines correspond to 𝝎 = 𝟖. 𝟗, 𝟏𝟐. 𝟗, and 𝟏𝟔. 𝟗𝑯𝒛, respectively.  The dashed black line reflects the trend of all the data points. With the effects of 𝜔 reduced, it can be more clearly inferred that variance decreases universally with worsening gas entrainment.  In contrast with the original feature, 𝑉s′ has increased variation across the experimental states and closer grouping about the overall trend (𝑉s exhibits 𝑅2 = 0.16 about a third-order polynomial, whereas 𝑉s′ has 𝑅2 = 0.40).  This refinement process is applied to every input feature except for 𝑝out̅̅ ̅̅ ̅, which is retained specifically for its relationship to 𝜔.  The individual plots are shown in Appendix B2.  Those that have correlations to both 𝜔 and 𝜑air are improved by varying degrees (e.g. 𝑉s, 𝐸R, and 𝐸FFT).  Those that have minimal correlation to 𝜔 are effectively unchanged (e.g. 𝐸s, 𝐺s, , and 𝐾s).  In the case where a feature appears poorly correlated to 𝜑air – namely, 𝑝out̅̅ ̅̅ ̅ – performing this calculation functions as a validation for that observation.  Computing 𝑝𝑜𝑢𝑡̅̅ ̅̅ ̅̅ (𝜔) yields a near perfect linear correlation, and dividing that linear function out of 𝑝out̅̅ ̅̅ ̅ to produce 𝑝out̅̅ ̅̅ ̅′ reduces the feature to only noise.  As such, it is omitted from the optimization.   0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2Air Volume Fraction (%)Refined Variance - Vs'75  With the refined set of input features, the binary, multi-class, and regression prediction models can be reevaluated. The results are shown in Table 4.6. Table 4.6: Comparison of Gas Entrainment Classification Performance Before and After Feature Optimization           The impacts of the refined features are most prevalent in the binary prediction models.  With the improved feature set, the binary MLP testing accuracy is improved to 90%, which meets the 90% threshold set in the research objectives.  The binary SVM model improves to meet this threshold as well.  The multi-class MLP model improves as well, placing 62% of the test states into the correct severity class.  If its performance margin is relaxed to allow states that were misclassified by one class up or down, the testing accuracy rises to 88%.  The multi-class SVM model exhibits no significant improvement in testing accuracy.  Finally, the regression model demonstrates a minimal, but nonetheless positive change with the refined feature set.  It predicts the gas entrainment severity within 0.44 of the true severity percentage in half of the testing states, and within 1.25 in 90%.      4.1.6 Error and Range of Application While the prediction models achieve their intended purpose, there are additional methods that should be employed to improve their accuracy.  The first proposed improvement is in how the data sets are assembled.  In this study, the 310 training and testing states are aggregated manually, using a single dynamic pressure sample taken over one second at the given state.  However, repeated measurements at the same operating condition reveal variability within each feature.  This is shown in Figure 4.8. Prediction Model Original Train/Test Accuracy (%) Refined Train/Test Accuracy (%) Binary MLP 90/84 94/90 Binary SVM 88/83 92/90 Multi-class MLP 82/59 85/62 Multi-class SVM 80/60 84/60 Regression Half within 0.20/0.47 Half within 0.18/0.44 76   Figure 4.8: The average magnitudes of each input feature at 𝝎 = 𝟏𝟒. 𝟗𝑯𝒛, with 𝝋𝐚𝐢𝐫 at 0% (blue bars), 2.8% (yellow bars) and 6.0% (red bars).  These correspond to states 1M, 6M, and 12M from Table 4.1.  Each magnitude is determined by averaging ten repeated 1 s pressure measurements of the same state.    The error bars indicate an interval of two standard deviations.    This variability is due to the flow phenomena being somewhat inconsistent over the sample duration, rather than measurement error.12  In the experimental data, the variability causes outliers.  Though these outliers do not significantly skew the overall correlations between the features and gas entrainment, the classification algorithms can assign them undue weight, leading to erroneous predictions.  To minimize this, the data should be composed of state measurements from significantly longer samples.  This would have an averaging effect of the features in the training and testing sets, diminishing the influence of any transitory flow phenomena.    The second potential improvement is related to range of application.  Across every model, states with lower impeller speeds are misclassified more frequently.  The operating range is selected based on the viable 𝜔 values for the testbed system.  However, in practice, a centrifugal pump operator is unlikely to use a centrifugal pump near its minimum output.  It would be defensible to specify a narrower operating                                                           12 This can be inferred because each feature is calculated from the same measurement.  If the variability were due primarily to errors in the transducer or data acquisition, each feature would exhibit similar variation.   Mean Sig. Energy Variance Skewness Kurtosis Ac. Energy FFT Energy0.000.100.200.300.400.500.600.700.800.901.00Normal 2.8% Air6.0% AirNormalized Magnitude77  band for the classification method, near the upper operating range of the pump, in which the gas entrainment predictions are more precise.  4.3 Impeller Wear Study Radial impeller wear is simulated over 65 states, ranging from normal operation to 𝛼imp = 3.0%.  The training and testing sets contain 35 and 30 states, respectively.  The values of 𝜔 and 𝛼imp for each state are shown in Table 4.7, with the training states highlighted in gray.  The states are indexed by row and column for reference.     Table 4.7: Simulated Radial Impeller Wear States   A B C D E   𝝎 (Hz) → 8.75 13.75 18.75 23.75 28.75 1 𝜶𝐢𝐦𝐩 (%) →  0.0 0.0 0.0 0.0 0.0 2 0.2 0.2 0.2 0.2 0.2 3 0.5 0.5 0.5 0.5 0.5 4 0.7 0.7 0.7 0.7 0.7 5 1.0 1.0 1.0 1.0 1.0 6 1.2 1.2 1.2 1.2 1.2 7 1.5 1.5 1.5 1.5 1.5 8 1.7 1.7 1.7 1.7 1.7 9 2.0 2.0 2.0 2.0 2.0 10 2.2 2.2 2.2 2.2 2.2 11 2.5 2.5 2.5 2.5 2.5 12 2.7 2.7 2.7 2.7 2.7 13 3.0 3.0 3.0 3.0 3.0   Figure 4.9 shows the simulated waveforms of 𝑝out at 𝛼imp = 0 and 𝛼imp = 3.0% (rows 1 and 13 from Table 4.7).     78   Figure 4.9: Simulated discharge pressure fluctuations across and range of rotating frequencies during a) normal operation and b) operation with 𝜶𝐢𝐦𝐩 = 𝟑. 𝟎%. The simulated pressure waveforms in Figure 4.9a mirror discharge pressure fluctuations reported by Cao [59], S. Chu et al. [74], and Higham [47].  In the normal operating condition, at 𝜔 = 8.75𝐻𝑧, the (a)  (b)  79  main mode of the discharge pressure fluctuation varies within a band of 1 kPa.13  When 𝜔 is increased to 28.75 Hz, the signal variation grows to approximately 7 kPa.  Each sample in Figure 4.9a exhibits a degree of high frequency fluctuation, which is interpreted as a combined effect of fluid turbulence and numerical noise generated by the necessarily coarse mesh.    In Figure 4.9b, the effects of radial erosion of the impeller manifest primarily as reduced variance in 𝑝out.  Unlike gas entrainment, impeller wear does not appear to dramatically alter the shape of the pressure pulses, other than reducing their amplitude.   However, at the highest frequencies (orange and blue trends), it is observed that the pressure peaks have increasing variability from peak to peak, which suggests that low frequency fluctuations may become more prevalent.   4.3.1 Feature Correlation  As with the gas entrainment study, the signal decomposition methods from Section 3.2 are employed to reduce the simulated measurements of 𝑝out into the desired feature space.  The resulting feature trends are shown in Figures 4.10a through 4.10f.  The trend for  𝑝out̅̅ ̅̅ ̅ is excluded, as it is a function of the simulation’s boundary condition and not an external phenomenon.  Each figure displays four second-order polynomial best-fit lines corresponding to the features associated with i) the lowest impeller speed, 𝜔 = 8.75𝐻𝑧 (blue line), ii) the median speed, 𝜔 = 18.75𝐻𝑧 (orange line), iii) the highest speed, 𝜔 =28.75𝐻𝑧 (maroon line), and iv) all the data points (black dashed line).  These are intended to illustrate cases where the feature has a dual dependence on impeller wear and 𝜔.                                                                 13 Due the considerable geometric differences between the numerical model and physical testbed centrifugal pump, the raw pressure magnitudes between the entrainment and wear studies should be interpreted independently.   80   Figure 4.10a-c: The normalized values of a) 𝑬𝐬, b) 𝑽𝐬, and c) 𝑮𝐬as a function of 𝜶𝐢𝐦𝐩. 0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Signal Energy - Es0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Variance - Vs0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Skewness - Gs(a)  (b)  (c)  81   Figure 4.10d-f: The normalized values of d) 𝑲𝐬, e) 𝑬𝑹, and f) 𝑬𝐅𝐅𝐓 as a function of 𝜶𝐢𝐦𝐩. 0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)FFT Energy - Efft0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Autocorrelation Energy - Er0.0 1.0 2.0 3.00.00.20.40.60.81.01.2Impeller Loss Ratio (%)Kurtosis - Ks(d)  (e)  (f)  82  Figure 4.10 suggests that some of the features within the training data are less strongly correlated to radial impeller wear than gas entrainment.  Signal energy 𝐸s and skewness 𝐺s (Figures 4.10a and 4.10c), for example, demonstrate no coherent relationship overall, which is consistent with the results from the preliminary simulations.  Kurtosis 𝐾s (Figure 4.10d) appears to follow a slightly negative trend as 𝛼imp increases, but the correlation is marginal.  The training set is not extensive enough to conclusively state that these three features are universally uncorrelated to the target condition, but in the context of the presented feature space, it is unlikely that they will contribute significantly to classification.    In Figure 4.10b, 𝑉s follows a consistent, decreasing trend.  The additional correlation to impeller speed is also apparent, causing a vertical spread between the slow, intermediate, and high speed states.  This infers that the feature optimization described in Section 4.1.5 may be advantageous.  Autocorrelation energy 𝐸R (Figure 4.10e) and FFT energy 𝐸FFT (Figure 4.10f) also exhibit consistent variation with respect to 𝛼imp and may benefit from optimization as well.   The input features are further evaluated using PCA.  The resulting scree plot is shown in Figure 4.11a, along with the loadings for PC1 and PC2.  A comparison of PC1 and PC2 is shown in Figure 4.11b. 83   Figure 4.11: a) A scree plot of the PCs of radial impeller wear and the loadings for each variable with respect to PC1 and PC2, listed in order of their contribution to PC1; b) The distribution of the experimental states across the first two PCs.  Figure 4.11a shows that approximately 70% of the explained variation across the features is accounted for by PC1 and PC2.  The PC1 loadings show 𝐸s, 𝐸R, and 𝐸FFT, as the dominant contributors.  The scatterplot in Figure 4.11b establishes that the varying states of impeller wear are not linearly separable based on the feature space. 4.3.2 Binary Classification With the acceptability threshold 𝜏wear = 1.5%, the training and testing accuracies of the binary MLP are 93% and 87%, respectively.  This is achieved using a learning rate of 0.02 and 7·104 training PC1 PC2 PC3 PC4 PC5 PC6 PC70102030405060Percentage of Explained Variation (%)-3 -2 -1 0 1 2 3 4 5-4-3-2-1012340-1% 1-2% >2%PC1PC2(b)  (a)  84  iterations.  The binary SVM model achieves equivalent training and testing accuracies.  The resulting truth table for the binary MLP testing data is shown in Table 4.8. Table 4.8: Truth Table for Binary MLP Classification of Impeller Wear (Testing) True Healthy: 18 False Healthy: 2 False Degraded: 2 True Degraded: 8  Of the thirty training states, four are misclassified, with an even split between false healthy and false degraded predictions.  Each incorrect classification occurs near 𝜏wear, as shown (underlined) in Table 4.9. Table 4.9: Impeller Wear States Misclassified by Binary MLP Model (Testing)      The binary MLP and SVM models nearly achieve the 90% accuracy goal in the stated research objectives, despite minimal contribution from three of the six input features. However, there is still margin for improvement by reducing the influence of 𝜔 on the input features. 4.3.2 Multi-class Classification The multi-class MLP model places each training state in the correct group with 89% accuracy.  The corresponding accuracy in the testing data is 70%.  This discrepancy infers that the model may be on the threshold of overfitting the training data.  Testing error is exacerbated with more training iterations.  It is suspected that this is a consequence of the relatively small pool of points within each class in the training data set.  The multi-class SVM model achieves training/testing prediction accuracies of 87% and 73%, respectively.     A B C D E   𝝎 (Hz) → 8.75 13.75 18.75 23.75 28.75 1 𝜶𝐢𝐦𝐩 (%) → 0.2 0.2 0.2 0.2 0.2 2 0.7 0.7 0.7 0.7 0.7 3 1.2 1.2 1.2 1.2 1.2 4 1.7 1.7 1.7 1.7 1.7 5 2.2 2.2 2.2 2.2 2.2 6 2.7 2.7 2.7 2.7 2.7 85  The misclassified testing states from the multi-class MLP model are shown in Table 4.10. Table 4.10: Impeller Wear States Misclassified by Multi-class MLP Model (Testing)         Overfitting can likely be reduced using a larger training set and refining the input features to improve their correlation to the target condition. 4.3.3 Regression  The regression model performs moderately well, with a median training prediction error of 0.22 (standard deviation 0.20) of the true percentage value of 𝛼imp and a median testing prediction error of 0.30 (standard deviation 0.31).  The testing predictions with error greater than 0.6 from the true 𝛼imp are shown in Table 4.11. Table 4.11: Impeller Wear States Severely Misclassified by Regression Model (Testing)      4.3.4 Feature Optimization  Using the method described in Section 4.1.5, each input feature is refined to reduce its dependence on 𝜔.  The impact is most significant on 𝑉s, shown in Figure 4.12.   A B C D E   𝝎 (Hz) → 8.75 13.75 18.75 23.75 28.75 1 𝜶𝐢𝐦𝐩 (%) → 0.2 0.2 0.2 0.2 0.2 2 0.7 0.7 0.7 0.7 0.7 3 1.2 1.2 1.2 1.2 1.2 4 1.7 1.7 1.7 1.7 1.7 5 2.2 2.2 2.2 2.2 2.2 6 2.7 2.7 2.7 2.7 2.7   A B C D E   𝝎 (Hz) → 8.75 13.75 18.75 23.75 28.75 2 𝜶𝐢𝐦𝐩 (%) → 0.2 0.2 0.2 0.2 0.2 4 0.7 0.7 0.7 0.7 0.7 6 1.2 1.2 1.2 1.2 1.2 8 1.7 1.7 1.7 1.7 1.7 10 2.2 2.2 2.2 2.2 2.2 12 2.7 2.7 2.7 2.7 2.7 86   Figure 4.12: The optimization steps of 𝑽𝐬:  a) 𝑽𝐬 as a function of 𝜶𝐢𝐦𝐩; b) 𝑽𝐬 as a function of 𝝎; c) 𝑽𝐬′ as a function of 𝜶𝐢𝐦𝐩. 0.00 0.50 1.00 1.50 2.00 2.50 3.000.00.20.40.60.81.01.2R² = 5.50E-02Impeller Wear (%)V_s5 10 15 20 25 300.00.20.40.60.81.01.2f(x) = 2.58E-06 x 3^.72E+00R² = 9.53E-01Rotating Frequency (Hz)V_s0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2R² = 8.10E-01Impeller Loss Ratio (%)V_s(a)  (b)  (c)  87  The remaining feature plots are shown in Appendix B2.  The features 𝑝out̅̅ ̅̅ ̅, 𝐸s, 𝐺s, and 𝐾s are largely independent of 𝜔 and exhibit minimal change.  However, the correlations of 𝑉s, 𝐸R, and 𝐸FFT are strongly enhanced.  This impact is demonstrated when the new feature set is used for classification, as shown in Table 4.12. Table 4.12: Comparison of Impeller Wear Classification Performance Before and After Feature Optimization      With the optimized feature set, all classification methods have better accuracy.  For binary classification with the MLP, the training and testing accuracy jump to 100% and 97%, respectively. The training/testing accuracy of the multi-class MLP improve to 92/82%.  Notably, the median error of the regression model halves for both training and testing data, classifying half of the points in the testing set within 0.16 of their true wear extent.  These dramatic performance improvements reinforce the importance of employing input features with robust correlation to the target phenomenon.     4.4 Combined Study  A total of twelve states are simulated in the combined study.  The set comprises states from normal operation, severe impeller wear (𝛼imp = 3.0%), severe gas entrainment (𝜑air = 5.0%), and the simultaneous occurrence of both.  They are shown in Table 4.13.    Table 4.13: Simulated States for Combined Study   A B C   𝝎 (Hz) → 8.75 13.75 28.75 1 Normal - - - 2 𝜶𝐢𝐦𝐩 (%) → 3.0 3.0 3.0 3 𝝋𝐚𝐢𝐫 (%) → 5.0 5.0 5.0 4 Combined 𝜶𝐢𝐦𝐩/𝝋𝐚𝐢𝐫 (%) 3.0/5.0 3.0/5.0 3.0/5.0 Prediction Model Original Train/Test Accuracy (%) Refined Train/Test Accuracy (%) Binary MLP 93/87 100/97 Binary SVM 93/87 97/93 Multi-class MLP 87/70 92/82 Multi-class SVM 89/73 89/82 Regression Half within 0.22/0.30 Half within 0.11/0.16 88  The small pool of states precludes the application of machine learning to characterize the conditions.  PCA is applied to evaluate the correlation of the input features and the linear separability of the states.  The resulting scree plot and loadings for PC1 and PC2 are shown in Figure 4.13a.  The states are plotted across the first two PCs in Figure 4.13b.     Figure 4.13: a) A scree plot of the PCs of normal operation, severe impeller wear, severe air entrainment, and combined conditions.  The loading for each variable with respect to PC1 is adjacent, ranked in descending order; b) The distribution of the various conditions across the first two PCs.  Figure 4.13a shows that PC1 and PC2 represent approximately 70% of the explained variation between the states.  The loadings for PC1 suggest that 𝐸R, 𝐸FFT, and 𝐺s are the dominant contributors.  -3 -2 -1 0 1 2 3 4 5-4-3-2-101234Normal 3% Wear 1 5% Air CombinedPC1PC2PC1 PC2 PC3 PC4 PC5 PC6 PC70102030405060Percentage of Explained Variation (%)(a)  (b)  89  Projecting the states onto PC1 and PC2 reveals relatively poor separation between normal, gas entrainment, and impeller wear states.  This follows the results from the two previous studies, which suggested that the individual conditions are not linearly separable.  However, when the adverse conditions manifest simultaneously, the states fall in a distinct region (to the right of PC1).  This supports that the input features are satisfactory for distinguishing individual and combined phenomena in their severe states.  However, a significantly larger data set is required to validate their classification performance on intermediate combined-phenomena states.         90  5. CONCLUSIONS AND FUTURE WORK  This chapter recapitulates the results presented in this thesis, applications and limitations of the diagnostic method, the original research objectives, conclusions, and recommendations for future research related to centrifugal pump monitoring via dynamic pressure measurements.  5.1 Summary of Results This work describes a novel method for characterizing the severity of adverse operating conditions in a centrifugal pump using dynamic pressure measurements collected at the discharge.  The method is validated in respective studies against two pervasive phenomena; gas entrainment and radial erosion of the impeller blades.  First, simulations of the two target phenomena are used to generate representative pressure measurements.  These are, in turn, employed to propose a unique set of characteristic statistical features, with correlations to both phenomena.  The features include signal mean, energy, variance, skewness, kurtosis, energy of the autocorrelation function, and total spectral energy.  Based on the respective trends from the preliminary simulations, three machine learning methods are put forth, each with increasing resolution of the target phenomenon.  The methods are; binary classification using a multi-layer perceptron (MLP), multi-class classification using a MLP, and continuous-value prediction using a random forest regression algorithm.    The method is validated in two principal studies.  In the first, experimental pressure measurements are collected in 310 states of varying impeller speed and gas entrainment severity, up to 6% void fraction of air.  The states are divided in to two equal sets of 155 states for training and testing of the prediction algorithms.  Using the base feature set, the binary MLP successfully predicts gas entrainment exceeding 2% void fraction in the testing set with 84% accuracy.  The multi-class MLP places the testing states into the correct 1% severity band with 59% accuracy, with 70% of the misclassifications falling just one severity band up or down.  Binary and multi-class support vector machines (SVM) models, implemented in parallel for validation, yield comparable prediction accuracies.  The random forest regression model has a median error of 0.47 of the true gas entrainment severity percentage.   To improve the prediction performance, an optimization method is proposed that refines the correlations between the input features and gas entrainment by reducing their dual dependence on impeller speed.  In doing so, the testing performance of the binary and multi-class models is improved to 91  90% and 62%, respectively.  The median error of the regression model reduces to within 0.44 of the true entrainment severity.   In the second study, pressure measurements are simulated for 65 states of varying impeller speed and radial wear, up to 3% diametric loss.  The states are divided 35/30 to form the training and testing sets.  The binary MLP successfully predicts impeller wear exceeding 1.5% in 87% of the testing states, which improves to 97% after optimizing the input features.  The multi-class MLP places the testing samples into the correct 1% severity band with 70% accuracy, improving to 82% after optimizing.  The median prediction error of the regression model before and after refinement is 0.30 and 0.16 of the true wear severity percentage, respectively.   In addition, a preliminary study is presented concerning the ability of the diagnostic method to differentiate between simultaneously occurring gas entrainment and impeller wear.  The initial simulations advocate its feasibility, and a follow-up experiment is proposed.   5.2 Applications and Limitations As discussed in the introduction, improving centrifugal pump performance monitoring is as much a problem of practicability as outright technology.  Cost, adaptability, scalability, and long term accuracy are as fundamentally important as the sensing method by which performance is measured.  Existing commercial solutions address this balance poorly, yet the development of elaborate pump diagnostic tools which are impractical for wide scale use is perpetual.  The void of practical alternatives has led to over reliance on rudimentary, static performance references, such as pump performance curve charts, which can lose accuracy with process changes.  The end consequence is that many centrifugal pumps around the world operate in unknown, often inefficient states. Condition monitoring via dynamic pressure measurements and machine learning presents an opportunity to improve the state of the art in both accessibility and versatility.  A principal advantage of the method discussed in this research is that the process would fall primarily to the pump manufacturer to employ, rather than the individual user.  The classification models would be trained and configured by the manufacturer, then provided as a supplementary diagnostic service for the pump operator.  The operator need only invest in a single transducer per pump, make dynamic measurements following the manufacturer’s method, and observe the classification results.   92  Here, gas entrainment and radial impeller wear are studied because of their relative difficulty to characterize with existing systems.  However, in practice, there could be many target conditions to be classified using this method.  The main constraint would be in the variety of adverse conditions the pump manufacturer could generate to amass reliable training data.  The classification technique is not without limitations, however.  The studies in this thesis satisfactorily demonstrate classification of adverse pump phenomena via dynamic pressure sensing as a concept, but certain industrial processes may require resolution beyond what has been reported.  Achieving this classification accuracy would require substantially larger sets of longer state measurements, and potentially additional refinement and expansion of the feature set.  A second limitation is that, as a fluid-based sensing approach, this method is not well suited for classifying the many mechanical faults that can impact the performance of centrifugal pumps.  This could be remedied by employing a combined pressure-sensor/accelerometer and employing features based on structural vibrations.      5.3 Conclusions The original objectives for this research are presented in Section 1.6.  Based on the presented studies, we conclude the following:   (1) Statistical Features: We have proposed a set of statistical measures that can be used to quantify the target phenomena based solely on dynamic discharge pressure measurements.     (2) Correlation: The relevance of those measures, as associated to detecting the target phenomena, has been evaluated.  Further, that evaluation has also been leveraged to improve the correlation of the statistical features to the target phenomena. (3) State Prediction: Using a binary MLP, experimental gas-entrainment states exceeding a 2% volume fraction of air have been classified with 90% accuracy across varying impeller speeds.  Simulated radial wear exceeding 1.5% of the impeller diameter has been classified with 97% accuracy using the same algorithm. (4) Refined Characterization: We have demonstrated satisfactory prediction using multi-class classification via a MLP and continuous-value regression via random forest algorithm.  (5) Accessibility: The developed diagnostic method employs a solitary, conventional pressure transducer that costs $115 and has a rise time of 2 ms. 93  (6) Implementation: Approaches for industrial implementation have been discussed.   This research has established that dynamic measurement of the discharge pressure fluctuations from a centrifugal pump is a viable method to evaluate fluid phenomena.  We have shown that the severity of the target phenomena can be categorized using a set of characteristic statistical measures derived from the discharge pressure fluctuations.  Reducing these features’ dependence on extraneous input variables (specifically, impeller speed), improves their correlation to the target phenomena.  It has also been established that prediction performance is reliant on a suitably extensive and diverse set of training conditions. Finally, we have demonstrated the method using an affordable, universally accessible sensor.  5.5 Future Work Although this work has demonstrated the validity of characterizing adverse centrifugal pump phenomena using dynamic discharge pressure measurements, there are a variety of avenues by which the research could be augmented.  First, it is recommended that trials be conducted to validate the impeller wear classification method using experimental measurements.  Additionally, it would be beneficial to automate the experimental setup in order to expediently recreate the experiments using a considerably larger set of training states (i.e. thousands, rather than hundreds or tens), using longer samples of the target conditions.  This would allow for a more robust evaluation of the input features, and could potentially yield better optimization of the feature correlations or additional statistical measures beyond the set discussed herein.  Such an automated setup would also allow for a thorough investigation of combined-phenomena states, which are preliminarily discussed here. Our research focuses specifically on gas entrainment and impeller wear, but there is a potential to expand the approach to include a variety of other fluid phenomena, adverse or otherwise.  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(A8) In (A8), the mean rotation rate tensor 𝛀 =12[∇𝒖 − (∇𝒖)𝑇] .      (A9)    100   A2 Pearson Correlation Coefficient  For two signals, 𝑥 and 𝑦, each with N samples, the Pearson correlation coefficient 𝑟P =∑ (𝑥𝑖−?̅?)𝑁𝑖=1 (𝑦𝑖−?̅?)√∑ (𝑥𝑖−?̅?)2𝑁𝑖=1 √∑ (𝑦𝑖−?̅?)2𝑁𝑖=1 .     (A10)         101  APPENDIX B – SUPPLEMENTARY FIGURES B1 Virtual Probe Placement Dependency  Figure B1: Simulated pressure fluctuations as a function of virtual probe placement (in mm from the model outlet boundary) at 𝝎 = 𝟖. 𝟕𝟓𝑯𝒛.  The black line (50 mm) represents the selected position.   Figure B2: Simulated pressure fluctuations as a function of virtual probe placement (in mm from the model outlet boundary) at 𝝎 = 𝟏𝟖. 𝟕𝟓𝑯𝒛.  The black line (50 mm) represents the selected position. 0 0.05 0.1 0.15 0.2 0.2513.014.015.016.017.018.019.020.010mm 30mm 50mm 100mmTime (s)Pressure (kPa)0 0.05 0.1 0.15 0.2 0.2513.014.015.016.017.018.019.020.010mm 30mm 50mm 100mmTime (s)Pressure (kPa)102  B2 Gas Entrainment Feature Optimization Plots (Experimental)  The following figures show the respective correlation of each feature to 𝜔, then the resulting feature trend of its optimized form.   Figure B3: Optimization of 𝒑𝐨𝐮𝐭̅̅ ̅̅ ̅̅  for gas entrainment.    8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2Rotating Frequency (Hz)Pout_bar0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2R² = 1.18E-02Air Volume Fraction (%)Pout_bar103   Figure B4: Optimization of 𝑬𝐬 for gas entrainment.      8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2Rotating Frequency (Hz)E_s0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2R² = 4.28E-01Air Volume Fraction (%)E_s104   Figure B5: Optimization of 𝑽𝐬 for gas entrainment.      8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2Rotating Frequency (Hz)V_s0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2R² = 3.96E-01Air Volume Fraction (%)V_s105   Figure B6: Optimization of 𝑮𝐬 for gas entrainment.      8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2Rotating Frequency (Hz)G_s0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2R² = 4.73E-01Air Volume Fraction (%)G_s106   Figure B7: Optimization of 𝑲𝐬 for gas entrainment.      8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2Rotating Frequency (Hz)K_s0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2R² = 4.46E-01Air Volume Fraction (%)K_s107   Figure B8: Optimization of 𝑬𝐑 for gas entrainment.      8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2R² = 2.50E-01Rotating Frequency (Hz)E_r0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2R² = 4.72E-01Air Volume Fraction (%)E_r108   Figure B9: Optimization of 𝑬𝐅𝐅𝐓 for gas entrainment.      8 9 10 11 12 13 14 15 16 17 180.00.20.40.60.81.01.2R² = 1.94E-01Rotating Frequency (Hz)E_fft0.0 1.0 2.0 3.0 4.0 5.0 6.00.00.20.40.60.81.01.2R² = 3.19E-01Air Volume Fraction (%)E_fft109  B3 Impeller Wear Feature Optimization Plots (Simulated) The following figures show the respective correlation of each feature to 𝜔, then the resulting feature trend of its optimized form.          Figure B10: Optimization of 𝑬𝐬 for impeller wear.   0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2R² = 8.50E-04Impeller Loss Ratio (%)E_s5 10 15 20 25 300.00.20.40.60.81.01.2R² = 2.99E-01Rotating Frequency (Hz)E_s110   Figure B11: Optimization of 𝑽𝐬 for impeller wear.      5 10 15 20 25 300.00.20.40.60.81.01.2f(x) = 2.10E-06 x 3^.79E+00R² = 9.59E-01Rotating Frequency (Hz)V_s0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2R² = 8.18E-01Impeller Loss Ratio (%)V_s111   Figure B12: Optimization of 𝑮𝐬 for impeller wear.      5 10 15 20 25 300.00.20.40.60.81.01.2R² = 4.67E-01Rotating Frequency (Hz)G_s0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2R² = 1.52E-02Impeller Loss Ratio (%)G_s112   Figure B13: Optimization of 𝑲𝐬 for impeller wear.      5 10 15 20 25 300.00.20.40.60.81.01.2R² = 3.62E-01Rotating Frequency (Hz)K_s0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2R² = 6.49E-02Impeller Loss Ratio (%)K_s113   Figure B14: Optimization of 𝑬𝐑 for impeller wear.      5 10 15 20 25 300.00.20.40.60.81.01.2R² = 4.45E-02Rotating Frequency (Hz)E_r0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2f(x) = 6.16E-03x² + 1.59E-01x + 2.67E-01R² = 4.88E-01Impeller Loss Ratio (%)E_r114   Figure B15: Optimization of 𝑬𝐅𝐅𝐓 for impeller wear.     5 10 15 20 25 300.00.20.40.60.81.01.2R² = 7.01E-01Rotating Frequency (Hz)E_fft0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2R² = 4.54E-01Impeller Loss Ratio (%)E_fft

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