Abstract ... I Contents ... III List of figures ... VII List of tables ... XV

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III

List of figures

Figure 1.1 Typical phase diagrams. Brennen [2] ... 2

Figure 1.2 Tip vortex cavitation. Brennen [2] ... 3

Figure 1.3 Photographs of bubble collapse. A. Ellis [4] ... 3

Figure 1.4 Cavitation damage on the blades of a Francis turbine. Brennen [1] ... 3

Figure 1.5 Types of cavitation in pumps. Brennen [1] ... 4

Figure 1.6 The picture on the left shows bubble cavitation whereas on the right the fully developed cavity is visible. Brennen [1] ... 5

Figure 1.7 From left to right: formation, separation and collapse of a cavitation cloud on the suction surface of a hydrofoil. The flow is from left to right. Brennen [2] ... 6

Figure 1.8 Cascade under rotating cavitation conditions. Brennen [1] ... 6

Figure 1.9 Cascade presenting partial cavitation (left) and supercavitation (right). Brennen [1] ... 7

Figure 1.10 Cavity closure in partial cavitation (left). Development, growth and collapse of cloud cavitation (right). Franc [3] ... 7

Figure 1.11 Schematic representation of whirl motion. ... 9

Figure 2.1 Garrett TPE331-14 turboprop engine. [1] ... 11

Figure 2.2 Cross-section of a pump impeller. Brennen [2] ... 12

Figure 2.3 Development of the meridional surface (top-left) and velocity triangle (bottom-left) with definitions of incidence and deviation angles (right). Brennen [2] ... 13

Figure 2.4 Typical specific speeds in turbomachinery. Lakshminarayana [3] ... 15

Figure 2.5 Typical pump geometries for different design speeds. Brennen [2] ... 15

Figure 2.6 Maximum efficiencies for different type of pumps. Brennen [2] ... 16

Figure 2.7 Typical noncavitating performance characteristics for an axial flow pump. Brennen [2].... 17

Figure 2.8 Origin of a secondary flow in a single blade. Lakshminarayana [3] ... 19

Figure 2.9 Flow in an axial flow compressor. Lakshminarayana [3] ... 19

Figure 2.10 Secondary flows in a cross-section of an unshrouded axial flow impeller. Brennen [2].... 20

Figure 2.11 Flow in a centrifugal compressor. Lakshminarayana [3] ... 20

Figure 2.12 Secondary flow in a vaneless volute of a centrifugal pump with flow rate below (left) and above(right) the design condition. Brennen [2] ... 21

Figure 2.13 Backflow as a consequence of tip leakage flow. Brennen [2] ... 22

Figure 3.1 Pressure coefficient distribution along a streamline. Brennen [1] ... 26

Figure 3.2 Liquid equilibrium pressure in a isothermal and gas mass constant process. Franc [4] ... 28

Figure 3.3 Evolution of bubble collapse. Franc [4] ... 29

Figure 3.4 Solution of Rayleigh-Plesset equation. Brennen [1] ... 30

Figure 3.5 Pressure field around bubbles during collapse phase. Franc [4] ... 31

Figure 3.6 (a) linear oscillations - (b) non linear oscillations - (c) limiting case between bubble oscillation and unlimited growth - (d) unlimited growth. Franc [4]... 32

Figure 3.7 Two examples of gas bubble under oscillating pressure. In case (a) strong nonlinearities are

visible whereas in case (b) subharmonics appears. Franc [4] ... 33

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VIII

Figure 3.8 Plesset-Chapman’s numerical results of a collapsing bubble close to a solid wall. Franc [4]

... 34

Figure 3.9 Paths of the particles during growth and collapse of a vapor bubble. Franc [4] ... 34

Figure 3.10 Typical cavitating performance curve Ψ(Φ,σ) with the three special cavitation numbers. Brennen [2] ... 35

Figure 3.11 Cavitating performance curves at different flow coefficients for an axial flow pump. Brennen [2] ... 35

Figure 3.12 Thermodynamic function Σ for different working fluids. Franc [4] ... 37

Figure 3.13 Cavitation performance curves for a centrifugal pump operating at different water temperature. Brennen [2] ... 38

Figure 3.14 Critical cavitation number ratio for different liquids and pumps. Brennen [2] ... 39

Figure 3.15 Superficial fatigue failure due to cavitation damage on a mixed flow pump impeller. Brennen [1] ... 40

Figure 3.16 Typical trend of mass loss rate. Franc [4] ... 41

Figure 4.1 Ideal system of a rotor with center of mass displaced with respect to axis of rotation. Jery [2] ... 44

Figure 4.2 Ideal system of a rotor with center of mass displaced with respect to the axis of rotation, in the presence of a damper. Jery [2] ... 45

Figure 4.3 Forces acting on a displaced rotor center with whirl motion. Jery [2] ... 46

Figure 4.4 Representation of the rotordynamic forces in absolute and rotating reference frames. Torre et al. [1] ... 47

Figure 4.5 Typical dimensionless normal and tangential forces for the Impeller X/Volute A configuration at 1000 rpm and ϕ = 0.092. Jery et al. [6] ... 49

Figure 4.6 Conditions of stability for normal and tangential unsteady forces in case of positive (top) or negative (bottom) whirl frequency ratios. Torre et al. [1] ... 51

Figure 4.7 Locus of zero of radial forces for the Impeller X/Volute A combination. Brennen [3] ... 53

Figure 5.1 The Cavitating Pump Rotordynamic Test Facility at ALTA S.p.A. A. Bonaguidi [4]... 55

Figure 5.2 Main elements of the CPRTF. Modified C.Bramanti et. Al [3] ... 56

Figure 5.3 Cut-off of the CPRTF test section. Torre et al. [1] ... 56

Figure 5.4 External (center) and cut-off (left) views of main tank with a picture as in working position (right), placed above a vibrating platform. A. Bonaguidi [4] ... 57

Figure 5.5 CAD sketch (left) and picture (right) of the vibrating platform. A. Bonaguidi [4] ... 58

Figure 5.6 Concrete columns for the elimination of the vibrations at the inlet and discharge pipes. A. Bonaguidi [4] ... 58

Figure 5.7 Schematic representation of the pressurization system. G. Pace [8] ... 59

Figure 5.8 Flow straightener with main dimensions. G. Pace [8] ... 60

Figure 5.9 Generation of turbulence in pipe’s curves. Modified from [5] ... 61

Figure 5.10 Rubber expansion joint DILATOFLEX K. A. Bonaguidi [4] ... 61

Figure 5.11 Electromagnetic flowmeter 8732E with integral transmitter 8732 C, produced by Fisher rosemount. A. Bonaguidi [4] ... 62

Figure 5.12 Electromagnetic field generated by the flowmeter with resulting molecules polarization. [6] ... 62

Figure 5.13 Auxiliary pump Grundfos TPE 100-390/2. A. Bonaguidi [4] ... 63

Figure 5.14 Cut-off of the silent throttle valve. A. Bonaguidi [4] ... 64

Figure 5.15 Main motor MOOG FASF3V8029. ... 65

Figure 5.16 Relative position of the whirl motor with respect to the main motor. A. Bonaguidi [4] .... 66

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Figure 5.17 CV joint. ... 66

Figure 5.18 Test section (external view). A. Bonaguidi [4] ... 67

Figure 5.19 Cut-off of test chamber with VAMPIRE pump mounted. ... 67

Figure 5.20 Rotating dynamometer: design (left, [7]) and picture (right) ... 68

Figure 5.21 Behavior of the semiconductors (A,B,C) and positioning on the posts (top-right, A. Bonaguidi [4]) ... 68

Figure 5.22 Semiconductor positions with respect to rotating reference of system. G. Pace [8] ... 69

Figure 5.23 Mechanical representation of a post. G. Pace [8] ... 69

Figure 5.24 Wheatstone bridge. G.Pace [8] ... 71

Figure 5.25 Schematic representation (left, modified from [7]) and mechanism design (right, Raposelli [9]) of the kinematic system. ... 72

Figure 5.26 Picture of the whirl mechanism. Raposelli [9] ... 72

Figure 5.27 Vector composition of the eccentricity. G. Pace[8] ... 73

Figure 5.28 Hollow of the main motor’s shaft for the dynamometer’s wires (left) and electric connection with slip rings (right). G.Pace [8] ... 73

Figure 5.29 Rotordynamic configuration for DAPROT3 in CPRTF facility. ... 74

Figure 5.30 CI

²

TF configuration for tests on hydrodynamic instabilities. G. Pace [8] ... 75

Figure 5.31 Piezoelectric effect. G. Pace [8] ... 77

Figure 5.32 Contrast sensors for main (left) and secondary (right) motor. ... 77

Figure 6.1 DAPROT3 impeller, frontal view. ... 82

Figure 6.2 DAPROT3 with diffuser mounted on CPRTF facility. ... 83

Figure 6.3 Noncavitating pumping performance and hydraulic efficiency of the DAPROT3 inducer. 84 Figure 6.4 Cavitating performance at zero eccentricity and 2 mm clearance of DAPROT3 inducer at T = 20 °C and Φ ₌ Φ

_D

= 0.065. ... 85

Figure 6.5 Hydraulic efficiency of DAPROT3 inducer (at zero eccentricity and 2 mm clearance) as a function of the cavitation number. ... 86

Figure 6.6 Cavitating performance at nonzero eccentricity (ε = 1.063 mm) of DAPROT3 inducer at T = 20 °C and Φ ₌ Φ

_D

= 0.065. ... 87

Figure 6.7 Nondimensional normal forces as a function of the cavitation number for different whirl speeds at T = 20 °C and Φ ₌ Φ

_D

= 0.065. ... 87

Figure 6.8 Nondimensional tangential forces as a function of the cavitation number for different whirl speeds at T = 20 °C and Φ ₌ Φ

_D

= 0.065. ... 88

Figure 6.9 Nondimensional rotordynamic force intensity as a function of the cavitation number for different whirl speeds at T = 20 °C and Φ ₌ Φ

_D

= 0.065. ... 88

Figure 6.10 Impeller of VAMPIRE. ... 89

Figure 6.11 VAMPDAP pump assembly without volute. ... 90

Figure 6.12 Volute design with discharge structural section (left) and picture of volute and centrifugal impeller during VAMPDAP assembly (right). ... 91

Figure 6.13 Cut-off of VAMPDAP configuration. ... 91

Figure 6.14 Cut-off rendering with VAMPDAP mounted. ... 92

Figure 6.15 Noncavitating pumping performance and hydraulic efficiency of the VAMPDAP pump with a clearance of 2 mm. ... 93

Figure 6.16 Noncavitating pumping performance and hydraulic efficiency of the VAMPIRE pump

with a clearance of 0.16 mm. ... 93

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Figure 6.17 Cavitating performance at zero eccentricity and 2 mm clearance of VAMPDAP pump at T

= 20 °C and Φ ₌ Φ

_D

= 0.092. ... 94

Figure 6.18 Hydraulic efficiency of VAMPDAP pump (at zero eccentricity and 2 mm clearance) as a function of the cavitation number. ... 95

Figure 6.19 Cavitating performance at nonzero eccentricity (ε = 1.160 mm) of VAMPDAP pump at Φ = Φ

_D

= 0.092 and Φ = 0.074 (T = 20 °C). ... 95

Figure 6.20 Nondimensional normal forces as a function of the cavitation number for different whirl speeds at T = 20 °C, and Φ ₌ Φ

_D

= 0.092 or Φ = 0.074. ... 96

Figure 6.21 Nondimensional tangential forces as a function of the cavitation number for different whirl speeds at T = 20 °C, and Φ ₌ Φ

_D

= 0.092 or Φ = 0.074. ... 96

Figure 6.22 Nondimensional rotordynamic force intensity as a function of the cavitation number for different whirl speeds at T = 20 °C, and Φ ₌ Φ

_D

= 0.092 or Φ = 0.074. ... 97

Figure 7.1 Schematic representation of the forces in rotating reference frame (x,y) with respect to the absolute reference frame (X,Y). L. Pecorari [1] ... 100

Figure 7.2 Schematic representation of rotordynamic force F

RD

= [A]ε(t). ... 102

Figure 7.3 Fundamental angles for main shaft. ... 103

Figure 7.4 Fundamental angles for whirl motion. ... 104

Figure 7.5 Sinusoidal force during offset test. ... 105

Figure 7.6 Forces during offset test. ... 109

Figure 7.7 Forces observed during eccentricity test. ... 110

Figure 7.8 Force measured by the dynamometer along y axis during eccentricity test of VAMPIRE. ... 111

Figure 7.9 Residual eccentricity vector. ... 112

Figure 7.10 Nondimensional normal force for a single set of discrete experiments for DAPROT3 with σ

_N

= 1.015, Φ = 0.065, and T = 20 °C. ... 113

Figure 7.11 Parabolic trend of the angular position as a function of the time. ... 115

Figure 7.12 Linear trend of the whirl rotation speed as a function of time. ... 116

Figure 7.13 Results for the nondimensional normal force in continuous experiments for DAPROT3 with σ

N

= 1.015, Φ = 0.065, and T = 20 °C. ... 117

Figure 7.14 Experimental curve without elimination of the low whirl speed results. ... 118

Figure 7.15 CPRTF configuration and pressure transducers position for DAPROT3 inducer. ... 120

Figure 8.1 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 1.015 and T = 20 °C. ... 126

Figure 8.2 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 1.015 and T = 20 °C. ... 127

Figure 8.3 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different flow coefficients when σ

_N

= 1.015 and T = 20 °C. ... 128

Figure 8.4 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.143 and T = 20 °C. ... 129

Figure 8.5 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.143 and T = 20 °C. ... 130

Figure 8.6 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view

(bottom) at different flow coefficients when σ

_N

= 0.143 and T = 20 °C. ... 131

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Figure 8.7 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.088 and T = 20 °C.

... 132 Figure 8.8 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.088 and T = 20 °C. ... 133 Figure 8.9 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different flow coefficients when σ

_N

= 0.088 and T = 20 °C. ... 134 Figure 8.10 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = Φ

_D

= 0.065 and T = 20

°C. ... 135 Figure 8.11 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ ₌ Φ

_D

= 0.065 and T = 20 °C. ... 136 Figure 8.12 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different cavitating conditions when Φ = 0.065 and T = 20 °C... 137 Figure 8.13 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.078 and T = 20 °C. 138 Figure 8.14 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.078 and T = 20 °C. ... 139 Figure 8.15 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different cavitating conditions when Φ = 0.078 and T = 20 °C... 140 Figure 8.16 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.052 and T = 20 °C. 141 Figure 8.17 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.052 and T = 20 °C. ... 142 Figure 8.18 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different cavitating conditions when Φ = 0.052 and T = 20 °C... 143 Figure 8.19 Color gradient for whirl frequency ratio in overall diagrams. ... 144 Figure 8.20 Overall diagrams of nonimensional rotordynamic force vector for positive (top) and negative (bottom) whirl frequency ratios for DAPROT3, at design flow coefficient and noncavitating condition (T = 20 °C). ... 145 Figure 8.21 Overall diagrams of nonimensional rotordynamic force vector for positive (top) and negative (bottom) whirl frequency ratios for DAPROT3, at low flow coefficient ( Φ =0.052) and noncavitating condition (T = 20 °C). ... 146 Figure 8.22 Overall diagrams of nonimensional rotordynamic force vector for positive (top) and negative (bottom) whirl frequency ratios for DAPROT3, at noncavitating condition and different flow coefficients (T = 20 °C). ... 147 Figure 8.23 Overall diagrams of nonimensional rotordynamic force vector for positive (top) and negative (bottom) whirl frequency ratios for DAPROT3, at various cavitation conditions and at Φ ₌ 0.078 (T = 20 °C). ... 148 Figure 8.24 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 1.015 and T = 70 °C.

... 150

Figure 8.25 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the

rotordynamic force with σ

_N

= 1.015 and T = 70 °C. ... 151

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Figure 8.26 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.143 and T = 70 °C.

... 152 Figure 8.27 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.143 and T = 70 °C. ... 153 Figure 8.28 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.088 and T = 70 °C.

... 154 Figure 8.29 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.088 and T = 70 °C. ... 155 Figure 8.30 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.065 and T = 70 °C. 156 Figure 8.31 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.065 and T = 70 °C. ... 157 Figure 8.32 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.078 and T = 20 °C. 158 Figure 8.33 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.078 and T = 70 °C. ... 159 Figure 8.34 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.052 and T = 70 °C. 160 Figure 8.35 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.052 and T = 70 °C. ... 161 Figure 8.36 Influence of the temperature on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.065 and σ

_N

= 1.015.

... 162 Figure 8.37 Influence of the temperature on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.065 and σ

_N

= 1.015. ... 163 Figure 8.38 Influence of the rotordynamic effects on the hydraulic efficiency for different flow temperatures at Φ = 0.065 and σ

_N

= 1.015... 164 Figure 8.39 Influence of the rotordynamic effects on the hydraulic efficiency for different flow temperatures at Φ = 0.065 and σ

_N

= 0.143... 164 Figure 8.40 Influence of the rotordynamic effects on the hydraulic efficiency for different flow temperatures at Φ = 0.065 and σ

_N

= 0.088... 165 Figure 8.41 Influence of the rotordynamic effects on the hydraulic efficiency for different flow temperatures at Φ = 0.078 and σ

_N

= 1.015... 165 Figure 8.42 Influence of the rotordynamic effects on the hydraulic efficiency for different flow temperatures at Φ = 0.052 and σ

_N

= 1.015... 166 Figure 8.43 Influence of the rotordynamic effects on the hydraulic efficiency at different cavitating conditions when Φ = 0.065 and T = 70 °C. ... 166 Figure 9.1 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.604 and T = 20 °C.

... 170

Figure 9.2 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the

rotordynamic force with σ

_N

= 0.604 and T = 20 °C. ... 171

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Figure 9.3 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different flow coefficients when σ

_N

= 0.604 and T = 20 °C. ... 172 Figure 9.4 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.081 and T = 20 °C.

... 173 Figure 9.5 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.081 and T = 20 °C. ... 174 Figure 9.6 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different flow coefficients when σ

_N

= 0.081 and T = 20 °C. ... 175 Figure 9.7 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.053 and T = 20 °C.

... 176 Figure 9.8 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.053 and T = 20 °C. ... 177 Figure 9.9 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different flow coefficients when σ

_N

= 0.053 and T = 20 °C. ... 178 Figure 9.10 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ ₌ Φ

_D

= 0.092 and T = 20

°C. ... 179 Figure 9.11 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ ₌ Φ

_D

= 0.092 and T = 20 °C. ... 180 Figure 9.12 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different cavitating conditions when Φ = 0.092 and T = 20 °C... 181 Figure 9.13 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.111 and T = 20 °C. 182 Figure 9.14 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.111 and T = 20 °C. ... 183 Figure 9.15 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different cavitating conditions when Φ = 0.111 and T = 20 °C... 184 Figure 9.16 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.074 and T = 20 °C. 185 Figure 9.17 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.074 and T = 20 °C. ... 186 Figure 9.18 Influence of the rotordynamic effects on the hydraulic efficiency (top) with close-up view (bottom) at different cavitating conditions when Φ = 0.074 and T = 20 °C... 187 Figure 9.19 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.604 and T = 70 °C.

... 189 Figure 9.20 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.604 and T = 70 °C. ... 190 Figure 9.21 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.081 and T = 70 °C.

... 191

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Figure 9.22 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.081 and T = 70 °C. ... 192 Figure 9.23 Influence of the flow coefficient on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with σ

_N

= 0.053 and T = 70 °C.

... 193 Figure 9.24 Influence of the flow coefficient on the modulus (top) and phase(bottom) of the rotordynamic force with σ

_N

= 0.053 and T = 70 °C. ... 194 Figure 9.25 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = Φ

_D

= 0.092 and T = 70

°C. ... 195 Figure 9.26 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ ₌ Φ

_D

= 0.092 and T = 70 °C. ... 196 Figure 9.27 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.111 and T = 70 °C. 197 Figure 9.28 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.111 and T = 70 °C. ... 198 Figure 9.29 Influence of the cavitation number on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.074 and T = 70 °C. 199 Figure 9.30 Influence of the cavitation number on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.074 and T = 70 °C. ... 200 Figure 9.31 Influence of the temperature on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.093 and σ

_N

= 0.604.

... 201 Figure 9.32 Influence of the temperature on the modulus (top) and phase(bottom) of the rotordynamic force with Φ = 0.093 and σ

_N

= 0.604. ... 202 Figure 9.33 Influence of the temperature on the nondimensional normal force (top) and nondimensional tangential force (bottom) of the rotordynamic force with Φ = 0.111 and σ

_N

_{= 0.053.}

... 203

Figure 9.34 Influence of the temperature on the modulus (top) and phase(bottom) of the rotordynamic

force with Φ = 0.111 and σ

_N

= 0.053. ... 204

Figure 9.35 Influence of the rotordynamic effects on the hydraulic efficiency for different flow

temperatures at Φ = 0.111 and σ

_N

= 0.053... 205

Figure 9.36 Influence of the rotordynamic effects on the hydraulic efficiency for different flow

temperatures at Φ = 0.092 and σ

_N

= 0.081... 205

Figure 9.37 Influence of the rotordynamic effects on the hydraulic efficiency for different flow

temperatures at Φ = 0.074 and σ

_N

= 0.604... 206

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XV

List of tables

Table 5.1 Main characteristics of the vibrating table. ... 58

Table 5.2 Main characteristics of DILATOFLEX K. G. Pace [8] ... 61

Table 5.3 Main characteristics of the facility. A. Bonaguidi [4] ... 74

Table 5.4 Elements in CPRTF facility indicated in Figure 5.29. ... 74

Table 5.5 Main characteristics of CI

²

RTF configuration. G. Pace [8] ... 76

Table 6.1 Geometric dimensions and main characteristics of DAPROT3 inducer. ... 82

Table 6.2 Operating conditions for cavitating performance tests on DAPROT3 inducer. ... 86

Table 6.3 Geometric dimensions and main characteristics of VAMPIRE. ... 90

Table 6.4 Operating conditions for cavitating performance tests on VAMPDAP pump. ... 94

Table 7.1 Experimental procedure with tests performed with CPRTF facility. ... 108

Table 7.2 Test matrices of DAPROT3 (left), VAMPIRE (center), and VAMPDAP (right) configurations... 119

Table 7.3 Correspondence between nominal flow coefficient and experimental volumetric flow rate in liters per second measured by the flowmeter for the three pump configurations. ... 119

Abstract ... I Contents ... III List of figures ... VII List of tables ... XV

III

Contents

Abstract ... I Contents ... III List of figures ... VII List of tables ... XV

1. INTRODUCTION ... 1

1.1. CAVITATION ... 2

1.2. TYPES OF CAVITATION ... 4

1.3. TURBOMACHINERY INSTABILITIES ... 6

1.4. ROTORDYNAMIC FORCES ... 8

1.5. THESIS OBJECTIVES ... 9

1.6. REFERENCES ... 10

2. PERFORMANCE ANALISYS ... 11

2.1. TURBOMACHINES GEOMETRY ... 12

2.2. NONDIMENSIONAL PARAMETERS AND PUMP PERFORMANCE ... 14

2.3. FLOW FEATURES ... 17

2.4. PREROTATION AND DISCHARGE FLOW ... 21

2.5. REFERENCES ... 23

3. CAVITATION ... 25

3.1. FUNDAMENTAL PARAMETERS... 25

3.2. NUCLEI AND CAVITATION INCEPTION ... 27

3.3. BUBBLE DYNAMICS ... 28

3.4. CAVITATING PUMP PERFORMANCE ... 34

3.5. THERMAL EFFECTS ON CAVITATION ... 36

3.6. CAVITATION EROSION ... 39

3.7. REFERENCES ... 41

4. ROTORDYNAMICS ... 43

4.1. ROTORDYNAMIC INSTABILITY ... 43

4.2. ROTORDYNAMIC THEORY ... 45

IV

4.3. NONDIMENSIONAL PARAMETERS ... 50

4.4. FURTHER CONSIDERATIONS ON RADIAL FORCES ... 52

4.5. REFERENCES ... 54

5. CAVITATING PUMP ROTORDYNAMIC TEST FACILITY (CPRTF) ... 55

5.1. MAIN TANK ... 57

5.2. VIBRATING TABLE ... 57

5.3. SUPPORTS AND CONCRETE COLUMNS ... 58

5.4. TEMPERATURE EXCHANGER ... 59

5.5. SACRIFICIAL ANODE ... 59

5.6. PRESSURIZATION SYSTEM ... 59

5.7. PIPES ... 60

5.8. FLOW STRAIGHTENER ... 60

5.9. EXPANSION JOINTS ... 61

5.10. FLOWMETERS ... 62

5.11. AUXILIARY PUMP ... 63

5.12. SILENT THROTTLE VALVE ... 63

5.13. MAIN MOTOR ... 65

5.14. SECONDARY MOTOR ... 65

5.15. CV JOINT ... 66

5.16. TEST CHAMBER ... 66

5.17. ROTATING DYNAMOMETER ... 68

5.18. SHAFT, BEARINGS AND SEALS ... 71

5.19. CIRCUIT CONFIGURATIONS ... 71

5.19.1. CPRTF... 72

5.19.2. CI

TF ... 75

5.19.3. CI

RTF ... 75

5.20. DATA ACQUISITION SYSTEM ... 76

5.21. LOSSES ACROSS A TEST CHAMBER ... 78

5.22. REFERENCES ... 79

6. TURBOPUMP CONFIGURATIONS ... 81

6.1. DAPROT3 ... 81

6.1.1. DAPROT3 NONCAVITATING PERFORMANCE ... 83

6.1.2. DAPROT3 CAVITATING PERFORMANCE ... 85

6.2. VAMPDAP ... 89

6.2.1. VAMPDAP AND VAMPIRE NONCAVITATING PERFORMANCE ... 92

V

6.2.2. VAMPDAP CAVITATING PERFORMANCE ... 94

6.3. REFERENCES ... 97

7. EXPERIMENTAL PROCEDURE ... 99

7.1. BACKGROUND THEORY ... 99

7.2. INITIAL ANGLES... 103

7.3. SINUSOIDAL SIGNALS ... 104

7.4. EXPERIMENTAL PROCEDURE ... 106

7.5. OFFSET TEST ... 108

7.6. ECCENTRICITY TEST ... 110

7.7. RESIDUAL ECCENTRICITY TEST ... 111

7.8. DISCRETE PROCEDURE ... 112

7.9. CONTINUOUS PROCEDURE... 113

7.10. TEST MATRICES ... 118