UNIVERSITA’ DEGLI STUDI DI PISA
Master Thesis in Space Engineering
DEVELOPMENT OF A MODELING FRAMEWORK OF THE FEEDING SYSTEM FOR THE CHARACTERIZATION OF POGO OSCILLATIONS
Candidate
Mario Amoroso
Supervisor
Dott. Ing. Angelo Pasini
Academic Year 2016-2017
Abstract
This thesis has been carried out within the framework of the MIT-UNIPI Project funded by MISTI Global Seed Funds and entitled “Dynamic Characterization of POGO Instabilities in Cavitating Turbopumps”, which provides a collaboration between the Massachusetts Institute of Technology and the University of Pisa, aiming to jointly develop a novel theoretical foundation capable of characterizing the dynamics of POGO oscillations and devising new design guidelines. The first section of this thesis deals with a literature review of the past POGO experiences of NASA human spaceflights and a collection of some procedures used to obtain a prediction of the dynamic performances of space rocket turbopumps. In the second part, a modeling framework defined in the time-domain has been developed to characterize the steady-state and dynamic behaviour of each component of a typical feeding system for liquid rocket engines. A typical water loop for experimental characterization of liquid rocket turbopumps has been modeled according to the modeling framework in order to understand the best way to perform forced experiments for the characterization of the transfer matrix of cavitating turbopumps necessary for understanding the POGO instability phenomena that affect rocket launchers. The best results in terms of capability of generating mass flow rate and pressure oscillations at the inlet of the inducer, have been obtained by means of a device that produces a volume oscillation located downstream of the pump.
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Contents
1 Detailed Review of the POGO Instability ...... 7 1.1 Introduction ...... 7 1.2 POGO Instabilities Episodes in NASA Human Spaceflight Vehicles ...... 8 1.2.1 Gemini – Titan II Experience ...... 8 1.2.2 Apollo – Saturn V Experience ...... 10 1.2.3 1970 POGO State of the Art ...... 15 1.2.4 Space Shuttle Experience ...... 17 1.2.5 Recent Challenges ...... 18 2 Characterization of the Dynamic Transfer Matrix of Space Rocket Turbopumps ...... 21 2.1 Introduction ...... 21 2.2 The Dynamic Transfer Matrix of a Cavitating Pump ...... 21 2.3 Analytical and Experimental Turbopump Matrix Characterization ...... 22 3 Mathematical Model of the System ...... 40 3.1 Introduction ...... 40 3.2 Development of the Dynamic Equations for the Subsystems ...... 41 3.2.1 Incompressible Duct: Straight (ID-S), Elbow (ID-E) and Tapered (ID-T) ...... 42 3.2.2 Compressible Duct Straight (CD-S)...... 42 3.2.3 Silent Throttle Valve (STV) – Exciter ...... 43 3.2.4 Volume Oscillator Valve (VOV) – Exciter ...... 44 3.2.5 Tank (T) – Exciter ...... 45 3.2.6 Pump (P) ...... 46 4 System Design Tools ...... 48 4.1 Introduction ...... 48 4.2 Incompressible VS Compressible Solution ...... 48 4.2.1 Downstream Mass Flow Rate and Pressure Signal Comparison ...... 48 4.2.2 Conclusions and Results ...... 73 4.2.3 Duct Transfer Matrix Comparison ...... 73 4.2.4 Conclusions and Results ...... 77 4.3 Hydraulic Loop System Design, Semi-Compressible Approach ...... 77 4.3.1 Conclusions and Results ...... 84 5 Conclusions and Future Developments ...... 85 6 Appendixes ...... 86 6.1 Appendix A, Mathematical Model Equations ...... 86 6.1.1 Introduction ...... 86
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6.1.2 Subsystems Mathematical Model ...... 86 6.1.3 Incompressible Duct (ID) ...... 87 6.1.4 Compressible Duct Straight (CD-S)...... 93 6.1.5 Silent Throttle Valve (STV) ...... 100 6.1.6 Volume Oscillator Valve (VOV) – Exciter ...... 104 6.1.7 Tank (T) – Exciter ...... 105 6.1.8 Pump (P) ...... 109 6.2 Appendix B, Matlab Code ...... 113 6.2.1 Introduction ...... 113 6.2.2 Steady-State System Parameters ...... 113 6.2.3 Dynamic System Parameters ...... 115 6.3 Appendix C, Simulink Modeling of the Hydraulic Loop ...... 118 6.3.1 Introduction ...... 118 6.3.2 Simulink Environment ...... 118 6.4 Appendix D, Simulink Modeling for the Comparison of the Incompressible Solution with the Compressible Solution ...... 127 6.4.1 Downstream Mass Flow Rate and Pressure Signal Comparison Circuit ...... 127 6.4.2 Duct Transfer Matrix Comparison Circuit ...... 128 6.5 Appendix E, Table of Figures ...... 130 6.6 Appendix F, List of Tables ...... 136
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Introduction
The subsequent work aims to:
1. Present a review of the POGO instability episodes occurred in the past and collect the analytical and experimental procedures exploited worldwide to study the behavior of the pump withstanding unsteady conditions and to prevent the occurring of unstable phenomena; 2. Develop a mathematical model able to predict the steady and unsteady-state of the subsystems operating within the hydraulic context of the pump; 3. Develop some tools useful to drive the design of an experimental apparatus, pointing out whether to exploit the simplified incompressible assumption of the working fluid and, in case, provide a solution able to take into account the compressibility effects.
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Nomenclature
A Cross-section area, m2. a , a0 Unperturbed acoustic velocity, m/s. a Unsteady acoustic velocity, m/s. 4 2 -1 CT Tank compliance, m s kg .
DH Hydraulic diameter, m. e Unit vector. f Frequency of perturbation, Hz. fu Maximum frequency of perturbation of interest, Hz. g Steady gravity acceleration, m/s2. g Unsteady gravity acceleration, m/s2. h Tank height, m.
Hij Transfer matrix component; ith row, jth column.
I Impedance, m-1s-1; Inertance, m-1s-1. i Complex unit. K Complex constant, kgm-2s-1. k Complex constant, m-1. kloss Loss coefficient. kloss Steady loss coefficient. kloss Unsteady loss coefficient. L Duct Length, m. M Mass flow rate coefficient. m Mass flow rate, kg/s. m Steady mass flow rate, kg/s. m Unsteady mass flow rate, kg/s. mˆ Unsteady mass flow rate, complex amplitude, kg/s. P Pressure coefficient. p Static pressure, Pa. p Steady static pressure, Pa. p Unsteady static pressure, Pa.
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pˆ Unsteady static pressure, complex amplitude, Pa.
q Steady independent variable.
q Unsteady independent variable.
-1 -1 R Resistance, m s . r Vertical tank oscillation amplitude, m.
rT Pump tip radius, m. r Steady inertial acceleration, m/s2. r Unsteady inertial acceleration, m/s2. t Time, s. V Volume, m3.
w0 Unperturbed axial velocity, m/s. Specific heat ratio.
Density, kg/m3.
3 , 0 Steady density, kg/m . Cavitation number. Flow coefficient. Pump rotational speed, m/s. Radiant frequency of perturbation, rad/s. Head coefficient.
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1 Detailed Review of the POGO Instability
1.1 Introduction
Propulsion stages can suffer from a dynamic coupling of the combustion process with structure and feed system dynamics, called POGO. As the name suggests, this causes rapid positive and negative accelerations along the thrust axis, as showed in Figure 1.1. POGO-type instabilities can result in severe vibration, interference with the guidance systems, and possible destruction of the stage. This instability is named after the children’s stick toy.
Figure 1.1 Typical occurrence of POGO vibration (NASA, 1970)
The so-called “POGO” instability consists of coupled vehicle structure/propulsion system oscillations, as schematically showed in Figure 1.2, resulting in a severe longitudinal unstable phenomenon.
Figure 1.2 Block diagram of POGO feedback process (NASA, 1970)
An overview of more than 45 years of NASA human spaceflight experience is presented with respect to the thrust axis vibration response of liquid fueled rockets known as POGO, inspired by the work of Larsen, 2008 and NASA, 1970.
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1.2 POGO Instabilities Episodes in NASA Human Spaceflight Vehicles 1.2.1 Gemini – Titan II Experience
The NASA history begins with the Gemini Program and adaptation of the USAF Titan II ballistic missile as a spacecraft launch vehicle. It continues with the pogo experienced on several Apollo-Saturn flights in both the first and second stages of flight. The defining moment for NASA’s subsequent treatment of pogo occurred with the near failure of the second stage on the ascent of the Apollo 13 mission. Since that time NASA has had a strict “no POGO” philosophy that was applied to the development of the Space Shuttle. These efforts lead to the first vehicle designed to be POGO-free from the beginning and the first development of an engine with an integral pogo suppression system. NASA first identified POGO as a threat to spaceflight vehicles and their crews in the early 1960’s during the tests of the Titan II launching vector, for the Gemini-Titan II program, showed in Figure 1.3.
Figure 1.3 Gemini-Titan
The USAF began test flights with the Titan II ballistic missile on March 16, 1962. Ninety seconds into the first-stage flight the missile began a longitudinal vibration going from 10-13 Hertz for roughly 30 seconds, reaching a maximum amplitude of 2.5 g’s at about 11 Hertz.
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Even for a military payload this design environment was excessive (the USAF considered 1.0 g as a tolerable design load for the structure of the Titan II and it’s payload), especially considering manned flights. NASA required that the vibrations be kept below 0.25 g’s. The first thought was that the POGO vibration might be caused by pressure oscillations in the propellant feedlines and after a mathematical modeling evaluation, a vertical surge-suppressor pipe was added to each of the oxidizer feedlines. Test results show a worsening of the vibrational condition, with a maximum value of 5 g’s reached. A partial explanation was given by sequent investigations: the accumulator on the oxidizer lines was prompted to the closeness between the frequency of vibration of the structure and the oxidizer one, while the fuel natural frequency was retained to be well above the structural ones; theoretically, then, having lowered the oxidizer frequency enough w.r.t. the structure and with a fuel response that shouldn’t have to be activated, the coupling action between propellant system and structure should have been weakened. This didn’t happen because of the presence of cavitating bubbles at the inlet of both the oxidizer and the fuel pumps in such a way that the fuel frequency were lowered resulting close to the structural one. Before the installation of the surge-suppressor, the two oscillating response of fuel and oxidizer competed through phasing, giving rise to a soft instability via thrust chamber pressure perturbation, then coupled with the structure, while with the presence of the accumulator in the oxidizer feedlines, the competing action of the oxidizer lines, missing, left free way to the action of the fuel lines perturbation which, coupling with the structure, gave rise to a worse instability. The addition of accumulators in each engine’s fuel line was shown to be essential to eliminate POGO on the Titan II. The next Titan II flew on December 19, 1962 with no standpipes, but increased fuel-tank pressure and aluminum oxidizer feedlines instead of steel. Surprisingly, the POGO amplitude was lessened but no reason for the effect was readily apparent. POGO on the tenth flight on January 10, 1963, was recorded at a new low of 0.6 g at the spacecraft interface. But the NASA requirement for the Titan II remained as 0.25 g at most due to the larger role astronauts were to play in piloting Gemini compared to Mercury. In a subsequent review with the commanding USAF general, the Titan II contractors argued POGO could be solved by increased fuel-tank pressure, and a combination of standpipes in the oxidizer lines and mechanical accumulators in the fuel lines. The 17th test flight on May 13, 1963, reached a new low amplitude record for Titan II POGO of 0.35g. Titan II launched again on September 23, 1963, and suffered a guidance malfunction unrelated to the Gemini booster configuration. Pogo on this launch was reached plus or minus 0.75g. An October meeting of the USAF management considered whether to follow through with plans to fly Missile N-25 with oxidizer standpipes and fuel-side piston accumulators. Engine tests begun in August had confirmed the fuel line resonance as the cause of the Missile N-11 failure and demonstrated that fuel accumulators would solve the problem. The extensive testing was used to generate test-verified equations describing the dynamics of structure, the propellant feed systems and the engines. Pump tests showed that as inlet pressures were reduced toward cavitation, the pump started acting as an amplifier and large oscillations resulted in the thrust chamber pressure. Aerospace and Space Technology Laboratories argued strongly for the planned flight and won the crucial decision to fly as planned. With both fuel and oxidizer suppressors installed, flight N-25 launched on November 1, 1963, recording the lowest vibration levels ever on Titan II of only 0.11 g's, well below the 0.25 g required by NASA as the upper limit for pilot safety. The first and second unmanned Gemini launches (April 8, 1964 and January 19, 1965) didn’t show significant levels of POGO, proving Gemini's spacecraft and launch. The first Gemini crew (March 23, 1965), also didn’t notice any remarkable level of vibration, while the Gemini V crew of Gordon Cooper and Pete Conrad reported experiencing POGO during launch at 126 seconds, although the booster systems engineer didn’t notice any evidence on telemetry. Unable to read the panel gauges to the desired degree of accuracy and finding speech difficult, the pilot estimated the magnitude at 0.5 g.
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Post-flight data analysis showed POGO onset after 92 seconds, lasting for 46 seconds, with maximum amplitude of 0.38g at the spacecraft-launch vehicle interface, as showed in the graph of Figure 1.4.
Figure 1.4 Comparison of Gemini-Titan POGO levels (NASA, 1965)
The subsequent mission of the Gemini program showed low values of vibrations. In retrospect to the Gemini-Titan experience, it was recognized that the longitudinal oscillations experienced on previous Mercury flights on the Redstone and Atlas launch vehicles were also POGO, with the astronauts withstanding about 0.45 g.
1.2.2 Apollo – Saturn V Experience
In parallel to the Gemini Program, NASA was developing the Saturn rockets to take the Apollo spacecraft to the moon. The Saturn I vehicles all flew with no occurrence of POGO. No sign of vibration were recognized on the first Saturn V launched on Nov. 9, 1967, showed in Figure 1.5, carrying the unmanned Apollo 4 spacecraft.
Figure 1.5 Saturn V First Stage S-IC
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Apollo 6, the second unmanned Saturn V launched on April 4, 1968, experienced pogo at 5 Hz between 105-140 seconds during first stage boost with 0.60 g maximum acceleration at the command module and 0.33 g at the aft of the vehicle. More subsequent analysis of the Apollo 4 and 6 flights demonstrated a previously unappreciated sensitivity of POGO to what may have been thought to be inconsequential changes to the Apollo spacecraft. In fact, Figure 1.6 shows that, though the Apollo 6 mission presented a rocket structure heavier w.r.t. the Saturn V staged for the Apollo 4 of just 45 kg’s and therefore, theoretically a slightly less first natural frequency, the coupling response of the system resulted to be much more amplified for the Apollo 6 as the oxidizer frequency line crosses the structural one during the flight. The POGO phenomenon, being so sensitive to small changes, confirms its unpredictable and therefore dangerous nature.
Figure 1.6 Illustration of Sensitivity to Small Changes: Comparison of AS-501 and AS-502 (Ryan, Robert S., 1985)
The POGO problem for the Saturn V launcher was underestimated. Several tests were conducted to decrease the susceptibility of the rocket to longitudinal vibrations, but the perturbation amplifying role of the cavitation phenomenon taking place at the inlet of the pump, was neglected in favor of a deeper investigation on how to modify the inlet line natural frequency in order to erase the coupling feedback with the structure vibrations. The solution proposed was to inject helium bubbles into the selected line to decrease enough its natural frequency w.r.t. to the structural response, but the test results showed that the frequency variation of the line could not be controlled under flight acceleration and tank pressure and therefore could not be implemented.
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The same concept, less extremist, was successfully suggested and applied by using helium gas from the tank pressurization system as trapped gas in the oxidizer pre-valve to create an accumulator, as in Figure 1.7.
Figure 1.7 Saturn I-C POGO Mitigation (Von Braun, Wernher, 1975)
Apollo 8, the third Saturn V and first manned flight, successfully demonstrated the effectiveness of this POGO mitigation for the first stage, while, unexpectedly, the second stage experienced POGO at about 50 seconds before engine cut-off. Data analysis revealed an 18 Hz vibration of the center engine of the five engine J-2 cluster, due to the LOX tank oscillation. The magnitude at the crew cabin wasn’t significant, but the local amplitude at the engine mount reached dangerous level for the supporting cross beam structure showed in Figure 1.8.
Figure 1.8 Saturn V Second Stage S-II Close up, J-2 Engine Cluster (Doiron, Harold H., 2003)
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A mitigation was attempted for the subsequent Apollo 9, increasing the LOX tank pressure and therefore the bulk modulus of the oxidizer line, but another POGO vibration was detected with a maximum amplitude of 12 g’s, as showed in Figure 1.9.
Figure 1.9 Apollo 8 and 9 POGO Episodes (Fenwick, J., 1992)
Since the structural vibrational load limit for the engine cluster structure was set at 15 g’s and thus there were substantially no margin with respect to the load experienced, it was applied the operational procedure to shut off about one minute earlier the central engine, keeping on burning a bit longer the others four. The most famous Apollo 11, experienced a small POGO vibration after 75 seconds into the second stage, while four different POGO episodes occurred during the Apollo 12 second stage burn reaching a maximum amplitude of 8 g’s. During the Apollo 13 second stage burn (April 11, 1970), two episodes of POGO occurred on the center J-2 engine as expected from previous missions, but the third occurrence diverged severely and acceleration at the engine attachment reached an estimated 34 g’s before the engine’s combustion chamber low-level pressure sensor commanded a shut down.
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Figure 1.10 shows a comparison between the Apollo 13 and previous Apollo mission center engine thrust pad acceleration of the second stage.
Figure 1.10 Comparison of Center Engine Thrust Pad Accelerations (Ryan, Robert S., 1985)
After the installation of a helium-bleed toroidal POGO suppressor for the oxidizer side of the J-2 engine for the subsequent Apollo missions, no further significant POGO episode occurred. Figure 1.11 shows a comparison of the vibration level reached during the flights of NASA space programs until the Apollo one and the French Diamant B space vehicle.
Figure 1.11 POGO Experienced in Flight (Rubin S., 1970)
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1.2.3 1970 POGO State of the Art
In October 1970 the previous experiences with POGO, were collected in a monograph entitled “Prevention of Coupled Structure-Propulsion Instability (POGO)”, authored by Sheldon Rubin of the Aerospace Corporation for NASA’s Langley Research Center. State of the art, criteria and recommended practices for mathematical modeling, preflight tests, stability analysis, corrective devices or modifications, and flight evaluation were provided, highlighting the concept that the POGO instability had to be eliminated rather than managed as a peculiar dynamic load condition, given the serious threat represented. In particular, a better understanding of the dynamic behavior of the pump and the structure, was indicated as a key point. The instability was addressed to a non-linear behaviour of the damping of the vehicle’s longitudinal modes and the compliance and the dynamic gain components of the pump matrix. Under the assumption of small perturbations, the linearization process led to a feasible analytical way to approach the oscillation phenomena with mathematical models that subsequently should have been validated. Moreover, since the rate of change of the system properties is relatively slow, a series of system parameters can be assumed constant for the stability analysis at successive time of flight. At that time, one of the major problems to tackle, together with the pump characterization, which is nowadays still only a partially solved component, was the modeling of the structural modal parameters. In fact, the most reliable source of structural-damping data was a carefully executed modal test of the full- scale vehicle. The approach to the pump characteristics was experimental; Figure 1.12 shows the cavitation compliance against the cavitation parameter response of the Titan I pumps.
Figure 1.12 Cavitation Compliance of Titan Stage 1 Pumps (NASA, 1970)
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Moreover, the dynamic gain of pumps does not increase as rapidly with reduction of cavitation index as steady-state characteristics would indicate. Figure 1.13 presents this behaviour with respect to oscillation data.
Figure 1.13 Pump Gain for Titan Stage I Pumps (NASA, 1970)
Figure 1.14 shows a schematic view of some accumulators solutions thought for the launch vehicles of the Gemini and Apollo program, close to the pump inlet, as recommended in the NASA monograph.
Figure 1.14 POGO Suppression Devices (NASA, 1970)
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The installations denoted by a), b), c) and e) successfully flight. In particular, the f) solution, thought to decrease the line acoustic velocity in such a way to provide the decoupling of the structure-propulsion feedback, didn’t work due to the modification of several line natural frequencies in the structural range. The idea behind these concepts is to lower the first resonance frequency of the line well under the first structural longitudinal mode and to increase the second frequency of the line well above the first structural one. The first and second line resonance frequencies, according to the article of Norquist L.W.S. et al., 1969, are:
1 ∗ − 휔1 = [휌퐿푆(퐶푎 + 퐶푏)] 2
1 − 2 휌(퐿 + 퐿 ) 휔∗ = [ 푎 푏 ] 2 1 1 + 퐶푎 퐶푏
Where the subscripts a, b and s respectively stands for accumulator, pump and line. It’s clear why the accumulators were designed to have a low internal inertance and a high compliance.
1.2.4 Space Shuttle Experience
The Space Shuttle Phase C/D development began in 1972. Great attention was paid to POGO, and for the first time, a POGO suppressor device was designed and installed as an integral component of the propulsion system rather than being added as a successive remedy. Studies showed that the best location was at the inlet of the LOX high-pressure turbopump. Figure 1.15 shows the actual Space Shuttle POGO Suppressor device.
Figure 1.15 Space Shuttle POGO Suppressor Device
It’s a spherical container charged with hot gaseous oxygen, acting as a Helmholtz resonator, to attenuate LOX flow oscillations in the 5-50 Hz frequency band for a smooth flow rate of LOX into the high pressure turbopump. Since it was clear the unpredictable nature of the POGO phenomenon, a much more conservative approach was used for the analysis of the possible instabilities, in fact, the safety margins to account for
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an instability behave were increased and uncertainty factors were applied to the components of the models exploited. The flight test program included instrumented flights on STS-1 through STS-5. The pogo instruments were basically pressure measurements installed in the liquid oxygen feedline and the SSMEs, and accelerometers on the Orbiter thrust structure. Examination of flight data indicated a POGO-free vehicle and the POGO flight measurement responses were as expected. The instrumentation verified that the suppressors were fully charged and operating throughout the boost phase of each flight. The flight data confirmed and validated the data developed in the ground test programs. The POGO effort for the Space Shuttle program taught that prevention is much more effective than providing an eventual remedy.
1.2.5 Recent Challenges
In the 2000’s NASA gave birth to the “Constellation Program”, which objectives, among the others, were the fabrication of two launch vectors: the Ares I and Ares V. We know that these objectives were never accomplished due to cost issues and subsequent cancellation of the program in 2011 in favor of the SLS, but nevertheless it’s interesting to highlight the studies performed for the design of a POGO suppressor device. L.A. Swanson and T.V. Giel, 2009, presented a trade-off between a branch and two annular accumulator concepts as schematically showed in Figure 1.16 and Figure 1.17:
Figure 1.16 LO2 Feed Line and Branch Accumulator (A.Swanson and T.Giel, 2009)
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Figure 1.17 Annular Accumulators (A.Swanson and T.Giel, 2009)
Defining the compliance and inertance of the devices as:
휌퐿 푔푆 푉표푙 퐶퐴 = 훾푔 푃푔 푔푐
(푡푤 + 퐷푝) 푑푠 퐼 = + ∫ (푔푆 퐴푝) 푔푠 퐴(푠)
Where: ρL is the liquid propellant (LO2) density gs is the gravitational acceleration at sea level Vol is the helium charge gas volume γg is the charge gas ratio of specific heats Pg is the absolute pressure of the charge gas volume gc is the conversion between mass and force tw is the feed line wall thickness Dp is the communication port diameter Ap is the communication port area 푠 is the accumulator liquid flow path A(s) is the liquid flow path area at 푠
Reminding that the two general guidelines of this kind of design are 1)High Compliance; 2)Low Inertance, with a safety check on the liquid height inside the accumulator, i.e. a margin of safety which ensure that the gas isn’t injected in the main feedline, Figure 1.18 and Figure 1.19 clarify why the shaped annular accumulator concept was the one selected:
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Figure 1.18 Accumulator Compliance versus Liquid Level (A.Swanson and T.Giel, 2009)
Figure 1.19 Accumulator Inertance versus Liquid Level (A.Swanson and T.Giel, 2009)
A passive control system for the gas parameters during the transient time of flight, was selected to complete the POGO mitigation design analysis. The whole POGO experience made clear that every family of launchers thought to be fabricate, has to go through a POGO suppressing design analysis, exploiting models of increasing precision adapted to the global design steps leading to the final configuration, to obtain a POGO-free launch vector.
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2 Characterization of the Dynamic Transfer Matrix of Space Rocket Turbopumps
2.1 Introduction
The dynamic transfer matrix of a component can be seen as the mathematical description of its physics, since, connecting the upstream values of fluctuating quantities to the downstream ones in a peculiar way, it unveils the nature of the component. The first steps in the analytical and experimental characterization of the dynamic matrix date back to the work of Brennen, Acosta and their collaborators in the 70s (Brennen & Acosta, 1976; Brennen, 1978; Ng & Brennen, 1978). However, more recent works have given important contributions by evaluating the previously obtained results through a careful analysis of the successive experimental and numerical data (Otsuka et al., 1996; Rubin, 2004). It is well known that many of the flow instabilities acting on axial inducers, including the above outlined POGO oscillations, are significantly influenced by the dynamic matrix of the propulsion system turbopumps (Kawata et al., 1988; Tsujimoto et al., 1993, 1998). This chapter will be consequently devoted to a collection of the past and recent efforts for the characterization of the dynamic matrix of cavitating/non cavitating pumps and its influence on the flow instabilities acting on the machine.
2.2 The Dynamic Transfer Matrix of a Cavitating Pump
Conventionally, the dynamics of hydraulic systems is treated in terms of “lumped parameter models”, which assume that the distributed physical effects between two measuring stations can be represented by lumped constants. This assumption is usually considered valid when the geometrical dimensions of the system are significantly shorter than the acoustic wavelength at the considered frequency. As a direct consequence of this assumption, the dynamic matrix of a generic system can be written as:
ppduHH11 12 = QQ HH 21 22 du where p and Q are, respectively, the pressure and flow rate oscillating components, and the subscripts u and d denote, respectively, the flow conditions upstream and downstream of the considered system. As a consequence of the well-known electrical analogy, the negative of the real part of H12 is usually denoted as the system “resistance”, the negative of the imaginary part of H12 is referred as the system “inertance”, the negative of the imaginary part of H21 is the system “compliance”, and the negative of the imaginary part of H22 is the system “mass flow gain factor” (Bhattacharyya, 1994; Kawata et al., 1988). If we assume unsteady, quasi 1-dimensional flow with small oscillations, pressure and flow rate can be written in complex form as follows:
it p(t) p pe it Q() t Q Q e where p and Q (usually real) are the pressure and flow rate steady values, p and Q (usually complex) are the pressure and flow rate oscillating components, is the frequency of the oscillations.
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Under the above assumptions, the dynamic matrix of a “passive” incompressible system (as simple duct lines filled with water or another liquid), as well as that of a non cavitating pump, typically has the following appearance:
ppdu1 -R -iωL = QQ 01 du or, in other words, only a resistance R and an inertance L are present. On the other hand, for a cavitating pump, the compressibility of the cavitating region leads to a more complicated form of the transfer matrix. Typical appearance of the matrix is as follows:
ppdu1-(S +iωX) -R -iωL = QQ -iωC 1-iωM du where S + iX is the pressure gain factor, R + iL is the pump impedance, C is the cavitation compliance and M is the mass flow gain factor. The last two parameters (C and M) are generally functions of the volume of the cavitating region inside the pump blades.
2.3 Analytical and Experimental Turbopump Matrix Characterization
C. Brennen and A. J. Acosta, 1975, presented a paper which shows an analytical approach to evaluate the components of the transfer matrix related to the discharge mass flow vibration. Linearizing the dynamics by confining attention to small oscillations about a particular steady operating point, the problem is therefore to determine the transfer function [Z] for the cavitating turbomachine where:
푝̃ − 푝̃ 푍 푍 푝̃ { 2 1 } = [ 11 12] { 1 } 푚̃2 − 푚̃1 푍21 푍22 푚̃1
Being the input and output quantities non-dimensional fluctuations of mass flow rate and pressure at the inlet and discharge sections of the turbopump. Mass flow rate and pressure are made dimensionless through the following expressions:
푝̃∗ 푚̃ ∗ 푝̃ = 1,2 ; 푚̃ = 1,2 1,2 1 1,2 휌푈2 휌퐴푖푈푡 2 푡
Where 푈푡 is the tip speed, 퐴푖 the inlet area, 휌 the fluid density and 휔 = 훺퐻/푈푡 a reduced frequency where 훺 is the actual frequency of the oscillations and H the distance between impeller blade tips. Being the reduced frequency small for the treated application, an expansion of the matrix elements through its power is performed:
2 푍21 = −푗휔퐾퐵 + 푂(휔 )
2 푍22 = −푗휔푀퐵 + 푂(휔 )
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The main concept behind this paper is to take into account the different conditions of pressure and velocity of the flow and shape of the blades through the radial dimension. Employing a linearized cavitating cascade theory, flow solutions are obtained in order to calculate the cavity area, function of the local shape of the blade and therefore of the radius coordinate, of the local cavitation number and of the operative conditions. This area, can be seen as a local cavity volume on unity of the radial coordinate and it’s exploited to calculate for example the local Compliance and Mass Flow Gain using the basic relations that follows:
푈 훥(푚̃ ∗ − 푚̃ ∗) 훺푈 휕푉 푍 = 푡 ( 1 2 ) = 푗 푡 ( ) 21 2퐴 훥푝̃∗ 2퐴 휕푝 푖 1 휑=푐표푛푠푡 푖 1 휑=푐표푛푠푡
∗ ∗ 훥(푚̃1 − 푚̃2) 훺 휕푉 푍22 = ( ∗ ) = 푗 ( ) 훥푚̃1 ∗ 퐴푖 휕푈푡 ∗ 푝̃1=0 푝̃1=0
The global parameters are then calculated by integration through the radial coordinate from the hub to the tip of the inducer and then the matrix components are obtained.
Sheung-Lip Ng, 1976, performed the dynamic characterization of the pump matrix, exploiting an experimental approach. Using the same notations and hypothesis of the previous paper, he exploited a closed-loop test facility where the main components were a tank, an inducer and two fluctuating valves placed upstream and downstream with respect to the pump as in Figure 2.1:
Figure 2.1 Schematic diagram of the mathematical pump loop (Ng, 1976)
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The set-up allowed direct measurements of the pressure perturbations through transducers and mass perturbation through a Laser Doppler Velocimeter system upstream and downstream of the pump. The steady state mass flow rate was controlled by a silent valve, while the mass oscillations were provided by two identical siren valves represented in Error! Reference source not found.:
Figure 2.2 Functional schematic of the fluctuator valve (Ng, 1976)
The siren valve design consists of two slotted concentric cylinders with the inside one made of bronze rotating within the outside stationary one, which was made of stainless steel, to avoid bonding with the bronze cylinder. A piece of sintered bronze cylinder was situated coaxially and next to the slotted cylinders providing a bypass to the flow when the slots were momentarily closed completely. The varying amplitude of fluctuation, proportional to the relative amount of area covered, was provided by a cylindrical sleeve which slid axially to cover the slotted cylinders and the sintered one. Since the matrix parameters are four and the perturbation equations for the mass and pressure are two, the relative amplitude and phases of the fluctuating valves where changed in order to obtain at least two sets of linearly independent test situation and therefore equations. The tests were conducted under different operative conditions (steady mass flow rate, cavitation number and frequency of perturbation). Then the data, through a square fit method, allowed the extrapolation of the pump matrix parameters. The tests were conducted for several combinations of relative amplitude and phases of the valves, obtaining several sets of independent condition. Though theoretically the extrapolated matrix components should have been coincident, the unavoidable errors led to difference from one set to the other, but what is to be highlighted is at least the consistence of the trend showed by the matrix component plotted versus the perturbation frequency as reported by the author.
S.L. Ng and C. Brennen, 1978, presented further results along the path of Ng’s PhD work. In fact, dealing with the same test set-up, three different set of matrix parameters plotted against the perturbation frequency are obtained in order to take into account respectively of absent, extensive and moderate cavitating conditions, as represented in Figure 2.3 and Figure 2.4:
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Figure 2.3 Left: the [ZP] transfer function for Impeller IV in the virtual absence of cavitation. The real and imaginary parts of the elements (solid and dashed lines, respectively) are plotted against both the actual and the nondimensional frequencies; Right: the [ZP] transfer function for Impeller IV under conditions of extensive cavitation (Ng and Brennen, 1978)
Figure 2.4 The [ZP] transfer function for Impeller IV under conditions of moderate cavitation (Ng and Brennen, 1978)
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During the Joint Symposium on Design and Operation of Fluid Machinery in 1978, C. Brennen presented the same experimental results obtained with Ng, plotted on the same graph for different cavitation numbers and relating them to the results of an analytical model predicting the pump matrix components, as showed in Figure 2.5:
Figure 2.5 Polynomial curve fitting to experimental pump transfer matrices, [ZP], obtained for Impeller IV at 흋=0.070 and a rotational speed of 9000 rpm. The real and imaginary parts of the matrix elements are presented as functions of frequency by solid and dash lines respectively. The letters A to E denote matrices taken at fives, progressively diminishing cavitation numbers, 흈, as follows: (A) 0.508 (B) 0.114 (C) 0.046 (D) 0.040 (E) 0.023 (Ng and Brennen, 1978)
According to the model, the flow through the impeller is divided into four parts and dynamic relations for each part are used to synthesize the dynamics of the pump: (i) the relations between the upstream inlet fluctuations and those at entrance to a blade passage; (ii) the bubbly flow region within a blade passage; (iii) the single phase liquid flow in the remainder of the blade passage following collapse of the cavities and (iv) the relations between the fluctuations at the end of a blade passage and the downstream conditions. Despite these complications, the overall transfer function for the pump resulted predominantly determined by the response of the bubbly region. The model, by C. Brennen, 1977, an article collected in the Journal of Fluid Mechanics, seems to catch the trends of the dynamic matrix components as showed in Figure 2.6:
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Figure 2.6 Theoretical pump transfer matrices, [ZP], obtained for Impeller IV at 흋=0.070 as functions of reduced frequency ퟂ. The lettered curves are for different fractional lengths, 휺, of the bubbly region and correspond to decreasing cavitation numbers, 흈: (A) 휺=0.2 (B) 휺=0.4 (C) 휺=0.6 (D) 휺=0.8. The curves are for one specific choice of the parameters K and M (See Brennen 1978) (Ng and Brennen, 1978)
C. Brennen, C.Meissner, E.Y. Lo and G.S. Hoffman, 1982, together with the previous experimental results, presented a similar test campaign, represented in Figure 2.7, using a bigger impeller (10.2 cm versus the 7.6 cm impeller in the 1978 experiments) and measuring mass flow perturbations with electromagnetic meters (EM), judged, in comparison with the LDVs, more accurate:
Figure 2.7 Polynomial curve fits to the 10.2 cm impeller transfer matrices at 흋=0.070, a rotational speed of 6000 rpm and various cavitation numbers as follows: (A) 0.37 (C) 0.10 (D) 0.069 (G) 0.052 (H) 0.044. The real and imaginary parts of the matrix elements are presented as functions of frequency by solid and dashed lines respectively. The quasistatic resistance from the slope is indicated by the arrow (Brennen et al., 1982)
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The trend, at decreasing cavitation parameter, for the 10.2 cm impeller, is quite similar to the one of the 7.6 cm one only for low values of the reduced frequency. On the other end, the bubbly flow model applied to the polynomial curve fits method with respect to the expressions built for the 10.2 cm impeller test results, shows that the curves of the matrix components, except for the real part of the compressibility component, are in accordance with the ones of the experimental results as can be noticed in Figure 2.8:
Figure 2.8 Transfer functions calculated from the complete bubbly flow model with 흋=0.07, 흲=9 deg, 흉=0.45, F=1.0, K=1.3 and M=0.8. Various cavitation numbers according to 휺=0.2/흈 are shown (Brennen et al., 1982)
A. Stirnemann, J. Eberl, U. Bolleter and S. Pace, 1987, described a semi-experimental approach to the characterization problem under non-cavitating and slightly cavitating conditions. The main aspect of their work was to measure directly only the pressure oscillations, by means of four couple of quartz transducers (stations n°1,2,5 and 6), placed symmetrically with respect to the pump as showed in Figure 2.9:
Figure 2.9 Schematic representation of the basic system and its nomenclature (A.Stirnemann et al., 1987)
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Stations n°2 and n°5 are installed at a certain distance from the inlet and discharge sections of the pump because of noise reasons. Then the four pipes are analytically characterized through their geometry and acoustic characteristics: the ones from station n°1 to 2, the one from n°1 to 3 and the symmetric w.r.t. the pump. The characterization is made via dynamic matrix as usual. Dealing with the pump subsystem, three special ratios are computed: the pressure ratios and the ratio of the two mass flow rate (inlet and discharge) with one of the pressure (the discharge one in this case):
푝3 푞3 푞4 퐻34 = ; 푦34 = ; 푦4 = 푝4 푝4 푝4
The next step is to express these three components as a function of three pressure ratios experimentally measured (stations n°1 and 2, 2 and 5, 5 and 6), together with the components of the four matrixes expressing the behavior of the four pipes previously stated:
퐻12(휂11훾12 − 휂12훾11) + 휂12 퐻25훿12 퐻34 = ∗ 퐻65(푘11훿12 − 푘12훿11) + 푘12 훾12
퐻12(휂11훾11 − 휂21훾12) − 휂11 퐻25훿12 푦34 = ∗ 퐻65(푘11훿12 − 푘12훿11) + 푘12 훾12
퐻65(푘21훿12 − 푘11훿11) + 푘11 푦4 = 퐻65(푘11훿12 − 푘12훿11) + 푘12
Being:
푝1 푝2 푝6 퐻12 = ; 퐻25 = ; 퐻65 = 푝2 푝5 푝5
These expressions are then incorporated in a system of linear equations to finally calculate the four components of the dynamic matrix of the pump as showed:
(퐻 ) 1 (푦 ) 34 1 4 1 ∗ ∗ ∗ 훼 ∗ = ∗ ∗ [ 11 ] ∗ ∗ ∗ 훼12 [(퐻34)푁] [ 1 (푦4)푁 ]
(푦 ) 1 (푦 ) 34 1 4 1 ∗ ∗ ∗ 훼 ∗ = ∗ ∗ [ 21 ] ∗ ∗ ∗ 훼22 [(푦34)푁] [ 1 (푦4)푁 ]
N is the number of independent conditions experimented. N=2 is the necessary and sufficient value to obtain the four matrix components, but to have a perspective of the uncertainty of the method and therefore an estimation of the errors, it was increased. To solve the linear system for each frequency value, a method extended to the complex numbers from R.I. Jennrich, 1977, was exploited.
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The independent condition of oscillating flow and pressure was achieved changing the impedance by means of a series of five accumulators placed either on the suction or on the discharge line, while the oscillation was provided by an Electro-Dynamic Exciter placed at a different line w.r.t. to the series of accumulators as schematically represented in Figure 2.10:
Figure 2.10 Schematic representation of the experimental facility (A.Stirnemann et al., 1987)
The following Figure 2.11 shows the results for the Mass Gain component of the pump matrix plotted against the perturbation frequency, under slightly cavitating condition (0.2). The dotted lines represent an estimation of the standard deviance:
Figure 2.11 Mass Gain obtained from excitation on the suction side (A.Stirnemann et al., 1987)
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Y. Kawata et al., 1988, presented an experimental way to obtain the matrix components. A direct measurement of the mass flow rate and pressure oscillation were performed at two measuring stations, 6.5 m before the pump inlet and 8.2 m after the pump discharge section, as represented in Figure 2.12:
Figure 2.12 Test Loop for measuring the dynamic behaviour of the prototype multi-stage pump (Kawata et al., 1988)
If two independent conditions lead to one result for the matrix component, three independent experimental loops get to three comparable results (theoretically coincident), to catch an estimation of the robustness of the method; the 70 m long pipe and the accumulator are activated through three valves to change the characteristics of the hydraulic loop leading to different experimental conditions. The last step of the test campaign is represented to the analytical adjustment of the values of the matrix components represented in Figure 2.13, since the measuring station are distant enough from the sections of interest to deal with a likely modifying effect:
Figure 2.13 Corrected transfer matrix of the prototype multi-stage pump (Kawata et al., 1988)
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S. Rubin, 2004, dealt with the mathematical modeling and interpretation of the frequency response of a pump at a fixed flow coefficient and different cavitating conditions. The pump characteristics data are extrapolated from tests conducted on a scaled model of the Space Shuttle Main Engine low pressure oxidizer turbopump using room temperature water. Pressure transducers and electromagnetic flowmeters were exploited at two measuring stations upstream and downstream of the pump location to acquire data, while a siren-valve and a throttle valve exciter were respectively placed upstream and downstream to provide the mass flow perturbation; they were activated individually to achieve the two sets of data needed. Since the distance between the measuring stations and the inlet and discharge sections of the pump wasn’t negligible, the actual pump transfer matrix was obtained after the analytical step accounting for the impedance of the pipelines connecting the downstream measuring station to the inlet pump section and the pump discharge section to the upstream measuring station. The new interpretation of the data, emerges from the intuitively assumption that the cavitation bubbles, having an inertia, respond to inlet mass flow and pressure perturbations with a delay. This phenomenon is taken into account assuming a complex value of compliance and mass flow gain, through a phase lag. So, being the cavitation volume:
휕푉푠 휕푉푠 푉 = 푝1 + 푚̇ ̇1 휕푝1 휕푚̇ 1
The non-dimensional continuity equation is expressed as:
푚1 − 푚2 = −휌푉 = 퐶̅푝1 + 푀̅푚̇ 1
Using the Laplace variable 푠 = 푗휔, the complex compliance and mass flow gain are given by:
휕푉 퐶 휕푉 푀 −휌 푠 = ; 푠 = 휕푝1 1 + 휏푐푠 휕푚̇ 1 1 + 휏푀푠
With the non-dimensional transfer equations of the pump:
푝2 = 푇11푝1 + 푇12푚̇ 1
푚̇ 2 = 푇21푝1 + 푇22푚̇ 1
The transfer matrix non-dimensional components can be expressed as:
−푗휔퐶 −푗휔푀 푇21(휔) = ; 푇22(휔) = 1 − 1 + 푗휔휏푐 1 + 푗휔휏푀
Having plotted the transfer matrix components against the frequencies of perturbation of 4, 7, 14, 21, 28, 35 and 42 Hz, the frequency dependent compliance and mass flow gain values and respective time delays, where extracted as:
1 푅푒[푇21(휔)] 휏푐(휔) = 휔 퐼푚[푇21(휔)]
1 (푅푒[푇22(휔)] − 1) 휏푀(휔) = − 휔 퐼푚[푇22(휔)]
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1 퐶(휔) = − 퐼푚[푇 (휔)](1 + 휔2휏2) 휔 21 푐
1 푀(휔) = − 퐼푚[푇 (휔)](1 + 휔2휏2 ) 휔 22 푀
The non-dimensional pressure equation is expressed by:
푝2 = 퐺̅푝1 − (푅 + 푗휔퐿)푚̇ 2
It’s worth to notice that the pump impedance is applied to the discharge mass flow perturbation. This choice is dictated by two reason: 1)The impedance experimental value is weak dependent form the cavitation number, in particular the resistance can be assumed constant while the inertance face only one change of value at increasing cavitation conditions; 2)The pump gain results to be real. The pump gain and impedance can be extracted from the transfer matrix results as:
푇 푍 = 푅 + 푗휔퐿 = 12 푇22
푇12푇21 퐺̅ = 푇11 + 푇22
S. Rubin obtained, except for the mass flow gain which data were inconsistent, simple functional forms of the real component of the pump parameters that fit the data, as showed in the following Figure 2.14, Figure 2.15 and Figure 2.16:
Figure 2.14 Left: Compliance Lag data and fits; Right: Compliance data and fits (Rubin S., 2004)
퐶0 퐶(휔) = 2 1 + 9퐶0휔
휏푐0 휏푐(휔) = 2 1 + 5휏푐0휔
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Figure 2.15 Left: Resistance data and fits; Right: Inertance data and fits (Rubin S., 2004)
2 푅0(1 + 2.3훼푅휔 ) 푅 = 2 1 + 훼푅휔
2 퐿0(1 + 0.5훼퐿휔 ) 퐿 = 2 1 + 훼퐿휔
Figure 2.16 Pump Gain data and fits (Rubin S., 2004)
2 퐺 = 퐺0(1 + 훼퐺휔 )
The reference value, subscripted with o, at the different cavitation numbers used for the test campaign, are shown in Table 1:
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Table 1 Values of fit parameters and references to equations and figures. Dual values for flow gain indicate uncertainty (Rubin S., 2004)
To maintain a second order form in the Laplace variable s, taking into account the previous frequency relations expressing the pump parameters, the continuity and pressure equations can be expressed as a system of equations through the exploitation of auxiliary variables which have no true physical meaning; The continuity equation is expanded as:
푚푐 + 휏푐0푠푚푐푐 + 퐶0푝1푐 = 0
2 푚푐 − (1 − 9휏푐0푠 )푚푐푐 = 0
2 푝1 − (1 − 5퐶0푠 )푝1푐 = 0
(1 + 휏푀푠)푚푀 + 푀푠푚1 = 0
푚푐 − 푚2 + 푚푐 + 푚푀 = 0
While the pressure equation becomes:
2 2 2 푝2 = 퐺0(1 − 훼퐺푠 )푝1 − 푅0(1 − 35푠 )푚2푅 − 퐿0(1 − 36푠 )푚2퐿
2 (1 − 15푠 )푚2푅 − 푠푚2 = 0
2 2 (1 − 72푠 )푚2퐿 − 푠 푚2 = 0
This formulation is heavier, but can be useful for a stability analysis through eigenvalues method.
A. Cervone et al., 2009, presented an interesting way to compute the calculation of the matrix components. The main concept underneath this study is to analytically characterized the facility under the assumption that the compressibility has to be addressed only to the pump (under cavitating condition) and the tank that behaves as an ideal capacitor, while the pipelines are treated as impedance as usual; each of the four components of the pump matrix are expressed as function of the operative condition (steady flow rate and
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cavitation number) and geometry of the inducer. In particular, the cavitating volume is expressed by means of a polynomial relationship through the difference between the actual cavitation number and the so called “choked cavitation number” (breakdown value), suggested by the data of a 10.2 cm impeller from C. Brennen, 1994. The test facility of the Osaka laboratory showed in Figure 2.17 clarifies the schematization underneath the model concept:
Figure 2.17 Schematic of the experimental facility (Cervone et al., 2009)
The exciter highlighted above provides a known flow rate perturbation to the loop. Pressures and flow rates at station n°1 and 2 are then calculated. The whole procedure is repeated on an independent system. As has been already noticed in some of the previous papers, the linearly independent condition is achieved varying the impedance of a pipeline. At this point two independent sets of data of pressure and flow rate at station n°1 and 2 are available and the four components of the pump matrix can now be obtained with what the authors logically call “backward calculation” as showed:
̂ 푝1̂ 푎 푄1푎 0 0 퐻푀 푝̂2푎 11 푝̂ ̂ 0 0 푝̂ 1푏 푄1푏 퐻푀12 2푏 [푇][퐻푀] = [ ] = ̂ 0 0 푝̂ 푄̂ 퐻푀21 푄2푎 1푎 1푎 퐻푀22 ̂ [ 0 0 푝1̂ 푏 푄̂1푏 ] [푄2푏]
Obviously the HM matrix encloses, together with the pump components, the contribution of the two pipelines that links the two measuring station to the pump. If the two sets of data were linearly dependent, the determinant of the T matrix would be zero; that’s why the authors suggest that an intuitively way to check the degree of independence of the two conditions selected is to calculate the value of the determinant of the data matrix, judging the correctness of the selection through the magnitude of its value. The author state that the best way to achieve the two linearly independent condition is to change the impedance of the line which is not subjected to direct perturbation, therefore the suction line in this case.
A. Cervone et al., 2010, applied the same main concept to the 2009 paper to the Alta Space facility, in Ospedaletto, Pisa. The assumptions of the model are the same of the previous works (1-D flux, small perturbations, quasi- steady response of the system components, incompressible flow in the pipelines); the analytical
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characterization of the facility is performed by means of non-dimensional variables. The main difference with the test campaign carried out in Osaka is the more precise characterization of the tank, by means of continuity and momentum equations, used as a generator of perturbation (on the suction and discharge line contemporary), by means of a controlled mechanical vertical oscillation. Eight linear equations are obtained in order to compute the mass and flow oscillations at each strategic section (begin and end section of the suction and discharge pipelines) as showed in Figure 2.18:
Figure 2.18 Top View of the Cavitating Pump Rotordynamic Test Facility (Cervone et al., 2010)
The procedure is repeated to obtain the same results for a different operative condition and again an analytical model of this kind shows its power into the ability to predict which way is the best to proceed with a setup modification in order to obtain the second set of data necessary for an experimental characterization of the pump transfer matrix. Modification of the impedance of the suction line is confirmed to be the most effective way; Figure 2.19 shows how the hydraulic circuit is modified for the achievement of the independent operative condition:
Figure 2.19 Suggested modification to the original facility setup for obtaining the second linearly independent test configuration (added pipe lines are coloured in red) (Cervone et al., 2010)
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Then, dealing with the actual measurement of the oscillating variables, it is showed that the mass and pressure perturbation measurement can be approached in an indirect way, through the direct measurement of the pressure in two section of each pipeline, obtaining the mass flow oscillation analytically through the knowledge of the impedance of the pipelines. The necessity to by-pass the direct measurement of the mass is due to the quite inaccurateness of the frequency response of the electromagnetic flowmeters. The results shows that it works for frequency above a certain value for a small oscillation of the tank, while an increase of the amplitude of its mechanical oscillation is needed to overcame the low frequency constraint.
G. Pace, L. Torre, A. Pasini, D. Valentini and L. d’Agostino, 2013, presented the results of the experimental procedure described in the previous paper. The tests were carried out on a high-head three-bladed inducer in the Cavitating Pump Rotordynamic Test Facility (CPRTF) at Alta, Pisa. The two configurations exploited are showed in the following Figure 2.20:
Figure 2.20 The “short” (top) and “long” (bottom) configurations of the test loop used for the experimental characterization of the dynamic transfer matrix of cavitating inducers (Pace G. et al., 2013)
Figure 2.21 shows a comparison between the experimental and analytical results, plotting the non- dimensional inducer matrix components against the frequency of perturbation. The bold lines represent the results of the model:
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Figure 2.21 Dynamic matrix for DAPAMITOR3 inducer: experimental points are in squares and the points obtained by using the model are in star (Pace G. et al., 2013)
It’s evident that the results are quite different; the author suggest that the experimental setup lacks in accuracy because of the ill-conditioned nature of the measurements and due to an uncertainty in the dynamic modeling of the pipeline interposed between the inducer and the downstream measurement section, not considered and likely home of high-compliant gas bubbles trapped able to modify the compliance of the flow.
K. Yamamoto, A. Müller, T. Ashida, K. Yonezawa, F. Avellan and Yoshinobu, 2015, applied the same experimental concept to the characterization of a resistance (orifice) and a compliance (accumulator). The measurements, in fact, are carried out with four pressure transducers placed downstream and upstream w.r.t. the component to characterize. The independent conditions are achieved exploiting two mass flow exciters, upstream (piston) and downstream (rotary valve) of the component of interest. The results are in good agreement with the analytical calculation, validating in a certain way the concept of by-passing the mass flow direct measurement.
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3 Mathematical Model of the System
3.1 Introduction
To characterize the dynamic behavior of the pump, a good modeling of all the subsystems interacting with it, is essential. In our case we want to develop a reduced-order model for the characterization of the pressure and flow rate oscillations in a given experimental facility. The model is based on the following initial assumptions:
Unsteady, one-dimensional flow. Small perturbations of the steady state flow (linearized equations under unsteady condition). The response of all the components of the system is assumed quasi-steady
Even if it has been shown that the last assumption is not valid in real pumps under cavitating condition (Rubin, 2004; Tsujimoto et al., 1996, 2008), it could represent a good starting point for the simplified reduced order analysis presented in this thesis. Under the made assumptions, pressure and mass flow rate can be written in complex form as follows:
p t p p t pp Re e ˆ it m t m m t m Re mˆ eit where p and m are the pressure and mass flow rate steady values, p and m are the pressure and flow rate oscillating components, is the frequency of the oscillations. A system of equations characterizing the steady and unsteady (oscillating) condition of the working fluid is obtained for each component of the hydraulic loop, exploiting the continuity and momentum equations. The definition of the steady condition is essential to the definition of the unsteady regime condition. The following subsystems can be identified:
Incompressible Duct Straight (ID-S) Incompressible Duct Elbow (ID-E) Incompressible Duct Tapered (ID-T) Compressible Duct Straight (CD-S) Silent Throttle Valve (STV) Volume Oscillator Valve (VOV) Tank (T) Pump (P)
The exciter will be represented by three different devices:
Silent Throttle Valve (pressure drop oscillation) Volume Oscillator Valve (volume oscillation) Tank (vertical oscillation of the tank).
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Figure 3.1 Schematic view of the hydraulic loop system (Torre et al., 2011)
Figure 3.1 shows an example of hydraulic loop that can be represented by the mathematical model developed in the following sections.
3.2 Development of the Dynamic Equations for the Subsystems
As a general approach, the steady and unsteady pressure and mass flow rate downstream (subscript d) of each component are expressed as functions of their independent variables (respectively qn and qn ) qn qn qn qn through suitable coefficients (such as Pname and Pname for the pressure or M name and M name for the mass flow rate) as reported in the following expressions:
name qn name qn pd P name q n pd P name q n n n name qn name qn md M name q n md M name q n n n
In particular, the model includes also the effects of body forces (such as the gravitational force expressed through the gravitational acceleration g) and inertia related to the rigid body acceleration of the component r . In the one-dimensional approximation, the work done by these forces can be obtained by integrating the force along the streamline that connect the inlet and the outlet of each components. The following expressions report the corresponding coefficient related to gravitational and inertial forces in both steady and unsteady conditions (both g and r are supposed positive while g and r are oscillating around zero): x x g d g d Pdname exg Pdname exg xu xu
x x r d r d Pdname exr Pdname exr xu xu
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3.2.1 Incompressible Duct: Straight (ID-S), Elbow (ID-E) and Tapered (ID-T)
A duct can be considered incompressible if its length is so small that the compressibility effects can be neglected. Further consideration in this regard will be provided in the next sections. The application of the continuity and momentum equations for this component in steady state regime yields to the following coefficients for the expression of the steady state pressure and mass flow rate downstream of the incompressible duct element:
pu PID 1 22 kmloss du 1 AAmx x du Pmdu ud ID 22AA22 xxuu M mu 1 ID
mmdu where mdu 2 is the averaged mass flow rate of the upstream and downstream sections. In unsteady condition, the oscillations of the downstream pressure and mass flow rate can be obtained through the sum of the following contributions:
pu PID 1 22 kmloss du 1 AAmx x du Pmdu ud ID AA22 xxuu xd mdu dx PID x Ax u M mu 1 ID
Whenever the assumptions of incompressible duct are satisfied, this model can be applied either to straight duct (ID-S) or Elbow (ID-E) or tapered duct (ID-T) or even ducts with more complex geometry. To fully characterize a 1-D incompressible duct the needed parameters are:
x d dx g r g r ;;;;;;;;;mdu Ax A x k loss P ID P ID P ID P ID ud x Ax u
3.2.2 Compressible Duct Straight (CD-S)
In steady state condition, the compressible duct behaves as an incompressible duct:
P pu 1 CD S kmloss du Pmdu CD S 2A2 xu M mu 1 CD S
While the unsteady behaviour of a compressible duct substantially defers from the incompressible case according to the following coefficients:
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K eik12 L K e ik L pu 12 PCD S KKKK2 1 2 1 K K eik12 L K K e ik L mu 1 2 1 2 1 PCD S KKKK A 2 1 2 1 xu ik12 L ik L a K1 K 2 e K 1 K 2 e PCD S KKKK2 1 2 1 ik12 L ik L pu ee MACD Sx u KKKK2 1 2 1 K eik12 L K e ik L mu 21 M CD S KK21 KK21
ik12 L ik L a K21 e K e MACD Sx 1 u KKKK2 1 2 1
where the constants K1 , K2 , k1 and k2 are defined as follows:
2 0ak 0 1 ww00 2 K1 wk w0 2 i fi f 01 DD kikk2 kHH 0, 2 a2 w 22 2 a w 12 0ak 0 2 0 00 0 K2 wk02
where 0,,,w 0 a 0 L are, respectively, the unperturbed density of the fluid, the unperturbed axial velocity, the speed of sound and the length of the duct. The independent variable a can be obtained from the unsteady gravitational and inertial forces g and ()r according to the following equation:
da w fa0 g r e dt D x H
To fully characterize a 1-D straight compressible duct the needed parameters are:
aa ;;;;;;;mdu D L f a P M H0 CD S CD S since:
2 DH mdu Axx Aw 4 ;; 00 A du xu
3.2.3 Silent Throttle Valve (STV) – Exciter
The Silent Throttle Valve is a device used in the water loop to regulate the load of the pump without introducing cavitation inside the rig. In steady state condition, the relevant parameters for the continuity and momentum equations are:
pu PSTV 1
22 kmloss du 1 AAmx x du Pmdu ud STV 22AA22 xxuu M mu 1 STV
The unsteady behaviour of the Silent Throttle Valve can be obtained according to the following coefficients:
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pu PSTV 1
22 kmloss du 1 AAmx x du Pmdu ud STV AA22 xxuu
xd mdu dx PSTV x Ax u mm Pkloss du du STV 2A2 xu M mu 1 STV
where kloss is the pressure loss coefficient in steady state regime and kloss is the unsteady pressure loss coefficient used to excite the flow oscillations inside the rig during the forced experiments. To fully characterize a Silent Throttle Valve the needed parameters are:
x d dx g r g r ;;;;;;;;;;mdu Ax A x k loss kP loss P P STV P STV STV STV ud x Ax u
3.2.4 Volume Oscillator Valve (VOV) – Exciter
The Volume Oscillator Valve is a variable volume device capable of changing its cross-section area in such a way to introduce volume flow rate oscillation. In steady state condition, the relevant parameters of the Volume Oscillator Valve for the continuity and momentum equations are:
pu PVOV 1
22 kmloss du 1 AAmx x du Pmdu ud VOV 22AA22 xxuu M mu 1 VOV
The unsteady behaviour of the Volume Oscillator Valve can be obtained according to the following coefficients:
pu PVOV 1 m Pmu u VOV A2 xu m md P d VOV A2 xd kmloss du Pmdu VOV A2 xu xd Pmdu dx VOV Ax xu
mu MVOV 1
mOV MVOV 1
VOV where mOV t is the mass flow rate oscillation produced by this device to excite the flow oscillations inside the rig during the forced experiments. To fully characterize a Volume Oscillator Valve the needed parameters are:
x d dx g r g r ;;;;;;;;;mdu Ax A x k loss P VOV P VOV P VOV P VOV ud x Ax u
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3.2.5 Tank (T) – Exciter
Following the direction of the flow in the water loop, the tank presents an upstream inlet section and a downstream outlet section. The tank has either an air-bag on the top of the tank or an equivalent free- surface air volume in which the pressure of the air ( pG ) can be controlled. In presence of the gravitational force, the pressure at the inlet and outlet sections differs from the pressure of the gas also for the hydrostatic head. In steady state condition, the general expression for the downstream pressure and mass flow rate can be obtained from the following expressions:
pu PTANK 1
22 kmloss du 1 AAmx x du Pmdu ud TANK 22AA22 xxuu M mu 1 TANK
The unsteady behaviour of the tank that combines both the properties of a pressure accumulator and a gravity tank is described through the following coefficients:
pu PTANK 1 m Pmu u TANK A2 xu m md P d TANK A2 xd km Pmdu loss du TANK A2 xu xd mdu dx PTANK x Ax u MCpu TANK Teq M mu 1 TANK Cm mu Teq u MTANK 2 Au MCrin FSh 2 TANK Teq r
Under the assumption of imposing the external forced oscillation as a vertical vibration of the tank, the resulting compliance of the tank becomes:
C ACTT Teq 2 in FS AT C T g r h u t x
where AT is the cross-section area of the tank and the compliance of the air-bag is
VVGG CT ppGG obtained by supposing an isentropic transformation in the air-bag. The pressure oscillation at the upstream section of the tank can be related to the unsteady acceleration of the tank through the following equations expressed in the time domain
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CT mu in FS p p 2 m h r uCTeq G A u r xu dpG mmud C dt T
Or in the frequency domain:
ˆˆ1 md1 i C Teq m u m u A2 xu in FS ˆ phˆu r r iC Teq
To fully characterize a vertical oscillating Tank the needed parameters are:
x d dx in FS in FS ;;;;;;;;;mdu Ax A x kC lossT A Tu h t h r u dx x Ax u
3.2.6 Pump (P)
The pump is usually characterized in terms of non-dimensional number such as the head coefficient, the flow coefficient and the cavitation number defined as follows:
ppdu 22 rT 1 mm 2 ud r3 T ppdv 1 22r 2 T
The unperturbed characteristic curve of the pump , is typically obtained from experimental data. In steady state condition, the momentum and continuity equations yield to the following coefficients:
pu PPUMP 1 22 PrPUMPT M mu 1 PUMP
The dynamic behaviour of the pump is obtained by the unsteady momentum and continuity equation in terms of the following coefficients:
pu PPUMP 1 22 PrPUMP T Pr 22 PUMP T xd Pmdu dx PUMP x A' u mu M PUMP 1 M mC 1 PUMP
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VC where mC t is the mass flow rate oscillation associated to the evolution of the cavity volume around the impeller in cavitating regime. To fully characterize a cavitating pump needed parameters are:
x d dx g r g r ;;;;;;;;;;;;mdu rm P P P P TCx PUMPA' PUMP PUMP PUMP u
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4 System Design Tools
4.1 Introduction
This chapter aims to show some useful tools to achieve fundamental informations about the quality of the mathematical model exploited to characterize the system and to judge the effectiveness of different way to generate mass flow rate and pressure perturbations inside the hydraulic loop, by means of four experimental set-ups.
4.2 Incompressible VS Compressible Solution
As we could have noticed in the previous studies about the characterization of the dynamic matrix of the pump, and, more in general, for the characterization of an experimental facility, the assumption of concentrating the compressibility of the working fluid to the pump subsystem is one of the pillars of the exploited models. An analytical solution for the perturbation of a compressible fluid moving into a straight duct has been obtained, leading to the possibility to relate the behaviour of the oscillating pressure and mass flow rate into two geometrical and physical identical ducts, exploiting the incompressible duct straight solution and the compressible one.
4.2.1 Downstream Mass Flow Rate and Pressure Signal Comparison
This section shows the results about the oscillation of the pressure and mass flow rate at the end of a straight duct for the two cases, varying the frequency of perturbation, the acoustic velocity and the duct length at different phase shift between the upstream pressure and mass flow rate perturbation, whose oscillation amplitude is set as follows:
푘푔 Upstream mass flow rate perturbation amplitude = 1 푠 Upstream pressure perturbation amplitude = 103 푃푎
The simulation will show the oscillating behaviour of the downstream variables considered. We expect that a combination of length of the duct, low acoustic velocity and high frequency of perturbation, will lead to substantial differences between the two solutions. Figure 4.1, Figure 4.2, Figure 4.3 and Figure 4.4 show the comparison of the results of the two mathematical models against the frequency, with the phase between the upstream mass flow rate and pressure perturbation equal to zero.
Case 1) 5 퐻푧 < 푓 < 140 퐻푧 푚 푎 = 1400 푠 퐿 = 1 푚
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Figure 4.1 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.2 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.3 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.4 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.5, Figure 4.6, Figure 4.7 and Figure 4.8 show the comparison of the results of the two mathematical models against the frequency, with the phase between the upstream mass flow rate and 휋 pressure perturbation equal to 2
Figure 4.5 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.6 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.7 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.8 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.9, Figure 4.10, Figure 4.11 and Figure 4.12 show the comparison of the results of the two mathematical models against the frequency, with the phase between the upstream mass flow rate and pressure perturbation equal to 휋
Figure 4.9 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.10 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.11 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.12 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.13, Figure 4.14, Figure 4.15 and Figure 4.16 show the comparison of the results of the two mathematical models against the frequency, with the phase between the upstream mass flow rate and 3휋 pressure perturbation equal to 2
Figure 4.13 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.14 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.15 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.16 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Case 2) 푚 푚 50 < 푎 < 1400 푠 푠 푓 = 5 퐻푧 퐿 = 1 푚
Figure 4.17, Figure 4.18, Figure 4.19 and Figure 4.20 show the comparison of the results of the two mathematical models against the acoustic velocity, with the phase between the upstream mass flow rate and pressure perturbation equal to zero.
Figure 4.17 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.18 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.19 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.20 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.21, Figure 4.22, Figure 4.23 and Figure 4.24 show the comparison of the results of the two mathematical models against the acoustic velocity, with the phase between the upstream mass flow rate 휋 and pressure perturbation equal to 2
Figure 4.21 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.22 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.23 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.24 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.25, Figure 4.26, Figure 4.27 and Figure 4.28 show the comparison of the results of the two mathematical models against the acoustic velocity, with the phase between the upstream mass flow rate and pressure perturbation equal to 휋
Figure 4.25 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.26 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.27 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.28 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.29, Figure 4.30, Figure 4.31 and Figure 4.32 show the comparison of the results of the two mathematical models against the acoustic velocity, with the phase between the upstream mass flow rate 3휋 and pressure perturbation equal to 2
Figure 4.29 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.30 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.31 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.32 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Case 3) 1 푚 < 퐿 < 28 푚 푓 = 5 퐻푧 푚 푎 = 1400 푠
Figure 4.33, Figure 4.34, Figure 4.35 and Figure 4.36 show the comparison of the results of the two mathematical models against the duct length, with the phase between the upstream mass flow rate and pressure perturbation equal to zero.
Figure 4.33 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.34 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.35 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.36 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.37, Figure 4.38, Figure 4.39 and Figure 4.40 show the comparison of the results of the two mathematical models against the duct length, with the phase between the upstream mass flow rate and 휋 pressure perturbation equal to 2
Figure 4.37 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.38 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.39 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.40 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.41, Figure 4.42, Figure 4.43 and Figure 4.44 show the comparison of the results of the two mathematical models against the duct length, with the phase between the upstream mass flow rate and pressure perturbation equal to 휋
Figure 4.41 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.42 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.43 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.44 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.45, Figure 4.46, Figure 4.47 and Figure 4.48 show the comparison of the results of the two mathematical models against the duct length, with the phase between the upstream mass flow rate and 3휋 pressure perturbation equal to 2
Figure 4.45 Pressure amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.46 Mass flow rate amplitude comparison between the compressible and the incompressible solution vs frequency of perturbation at fixed acoustic velocity and duct length
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Figure 4.47 Phase shift between the compressible and the incompressible pressure solution vs frequency of perturbation at fixed acoustic velocity and duct length
Figure 4.48 Phase shift between the compressible and the incompressible mass flow rate solution vs frequency of perturbation at fixed acoustic velocity and duct length
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4.2.2 Conclusions and Results
These results confirm the criteria which state that the use of the incompressible model must be restricted to duct lengths that do not exceed one-tenth of wavelength over the frequency range of interest, mathematically:
a L 10 f u where fu is the highest frequency of interest in Hz, and a is the effective acoustic speed (Oppenheim and Rubin, 1993). In fact, the two solutions are in great accordance when the duct length is small w.r.t. the maximum length provided by the criteria. The phase shift between the two solutions is negligible, in particular the phase difference of the two mass flow rate never reaches the degree, while the maximum phase shift for the pressures is higher, but doesn’t overcome the value of 10 degrees. It can be noticed that for the limit cases i.e. when the length of the duct is equal to one, the two pressure solutions remain in good agreement with a maximum percentage error of about 10%, while the two mass flow rate solutions can present an error of 20% in the limit cases, overcoming 30% when the acoustic velocity drops to its minimum, with respect to the incompressible upstream/downstream value. In conclusion, as expected, getting close to the limit condition imposed by the Rubin-Oppenheimer criteria, the two solutions tends to diverge and, in particular, the two mass flow rate present a more developed tendency to depart moving towards the duct length criteria boundary under the point of view of the amplitude of oscillations, while for the phase shift it can be noticed the opposite behaviour which is not so appreciable as the amplitude phenomenon.
4.2.3 Duct Transfer Matrix Comparison
This section, conceptually, presents the same comparison of the previous section, exploiting the more classical matrix notation to compare the oscillating behaviour of a 3 meter long straight duct using the ID- S and the CD-S solutions. Being the transfer matrix of an incompressible duct (using the complex domain):
ˆˆ ppdu 1 Id ˆˆ01 mmdu where Id is the so called “Impedance” (complex) of the duct, next plots show that, at decreasing acoustic velocity of the line (water used as working fluid), is not correct to exploit the incompressible approach for an appreciable range of frequency of perturbation (and length of the duct, which is taken constant in this case), to model the unsteady-state of the oscillating system, by comparing the matrix components of the above ID-S representation, with a general one which take into account the compressibility effects of the fluid, exploiting our wave solution.
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Figure 4.49 Comparison between the real components of the principal diagonal of the matrix of a 3 meters long ID-S (black line) with a geometrically identical CD-S at different acoustic velocity: a=1400 m/s (red); a=1000 m/s (blue); a=700 m/s (cyan); a=500 m/s (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Figure 4.50 Comparison between the imaginary components of the principal diagonal of the matrix of a 3 meters long ID- S (black line) with a geometrically identical CD-S at different acoustic velocity: a=1400 m/s (red); a=1000 m/s (blue); a=700 m/s (cyan); a=500 m/s (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Figure 4.49 and Figure 4.50 represent the results of the diagonal of the CD-S matrix, which is composed from two equal components, for a range of frequency from 0.1 Hz to 100 Hz, as for all the others simulations of this section. The imaginary contribution of the compressibility is negligible w.r.t. the real one, which can lead to an appreciable error even for high acoustic velocity and low frequencies.
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Figure 4.51 and Figure 4.52 show that the Resistance component tends to depart from the ID-S value at lower frequencies w.r.t. the imaginary component (product between the frequency in rad/s and the Inertance) and that, for the value of the steady-state mass flow rate selected for these simulations (the same resulted from the evaluation of the steady-state condition of the hydraulic loops studied in the next section), for an appreciable range of frequency of perturbation, the imaginary part of the impedance is predominant.
Figure 4.51 Comparison between the Re[H12] of a 3 meters long ID-S (black line) with a geometrically identical CD-S at different acoustic velocity: a=1400 m/s (red); a=1000 m/s (blue); a=700 m/s (cyan); a=500 m/s (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Figure 4.52 Comparison between the Im[H12] of a 3 meters long ID-S (black line) with a geometrically identical CD-S at different acoustic velocity: a=1400 m/s (red); a=1000 m/s (blue); a=700 m/s (cyan); a=500 m/s (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
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Figure 4.53 Comparison between the Re[H21] of a 3 meters long ID-S (black line) with a geometrically identical CD-S at different acoustic velocity: a=1400 m/s (red); a=1000 m/s (blue); a=700 m/s (cyan); a=500 m/s (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Figure 4.54 Comparison between the Im[H21] of a 3 meters long ID-S (black line) with a geometrically identical CD-S at different acoustic velocity: a=1400 m/s (red); a=1000 m/s (blue); a=700 m/s (cyan); a=500 m/s (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Figure 4.53 and Figure 4.54 show that the imaginary part of H21 tends to depart from zero at lower frequencies w.r.t. the real one, being also dominant for an appreciable range of frequency of perturbation. The algorithm behind the previous results can be exploited to check, knowing the range of frequencies of our interest, an estimation of the acoustic velocity and the geometrical characteristics of the facility, which error lies behind the choice to use the incompressible model for each straight duct component and, in case this wasn’t negligible, to replace the ID-S with a CD-S to obtain more accurate results.
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4.2.4 Conclusions and Results
These results, again, confirm the Oppenheim and Rubin criteria, in fact, the CD-S matrix components tends to defer from the ID-S ones for an appreciable range of frequency of perturbation, at decreasing acoustic velocity. The results show that the real component of the principal diagonal of the matrix and the impedance of the CD-S present the higher tendency to depart from the ID-S ones for an appreciable range of frequency of perturbation and acoustic velocity. In conclusion, as expected, the incompressible model fails to achieve a good prediction of the oscillating behaviour of a straight duct for an appreciable range of combination of duct length, frequency of perturbation and acoustic velocity of interest.
4.3 Hydraulic Loop System Design, Semi-Compressible Approach
This section presents the results of four different experimental set-ups, discriminating the capability to create a detectable regime oscillating condition, aiming to the experimental characterization of the pump transfer matrix. The longitudinal acceleration of the components has been neglected to simplify the qualitative assessment of the set-ups. For each one of the four set-ups, has been developed a model on Matlab/Simulink, to calculate the steady- state condition of the hydraulic loop and then to perform the unsteady-state analysis of the system, measuring the oscillating condition at the inlet section of the pump to have an idea of the magnitudes in play. After having selected the elements composing the loop and the order in which they are placed with respect to each other, the hydraulic loop can be built by connecting them through their two input/output. The STV, the VOV and the tank, exploited as generators of perturbation, can be commanded in frequency, phase and amplitude to observe, after having properly characterized all the parameters of interest of all the subsystems of the loop, the time history of the two variables considered on whichever signal line. The upper line represents the pressure, while the lower, the mass flow rate signal. It can be noticed that, while there are as many pressure sensors as the elements, we need less mass flow rate sensors, since they have to be placed only at the downstream section of the elements taking into account the compressibility effects. The results have been obtained assuming a non-cavitating pump and replacing the suction line ID-S model with the CD-S one due to its length, since it approached the maximum length given by the Oppenheim and Rubin criteria, at increasing frequency of perturbation. Figure 4.56 and Figure 4.57 show the results for the hydraulic loop which exploits the longitudinal oscillation of the STV as source of perturbation. As it can be noticed by the equations of the model of the STV, the mathematical source of oscillation is the oscillating loss coefficient of the valve which has been varied as a fraction of the steady-state one. The mass flow rate has been divided by the steady-state mass flow rate, while the pressure by the dynamic pressure at the tip radius of the inducer, to achieve the non- dimensional condition. The results have been obtained for a range of frequency of perturbation from 0.1 Hz to 30 Hz. The behaviour of the pressure and the mass flow rate of the STV set-up, with the frequency of perturbation, is the opposite. We can notice that at low frequencies, the STV is able to create appreciable mass flow rate oscillation, which decrease quite heavily at increasing frequency, while the system cannot create appreciable pressure oscillation at very low frequencies and about beyond 5 Hz the pressure amplitude is weakly affected by the frequency.
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Case 1) STV; the hydraulic loop is represented in Figure 4.55.
Figure 4.55 Hydraulic loop logic by Simulink
Figure 4.56 Comparison between the pressure amplitudes of oscillation at different values of the STV unsteady-state loss coefficient: K=Kv/100 (red); K=Kv/25 (blue); K=Kv/5 (cyan); K=Kv/2.5 (yellow) where Kv is the steady-state STV loss coefficient. The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
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Figure 4.57 Comparison between the mass flow rate amplitudes of oscillation at different values of the STV unsteady-state loss coefficient: K=Kv/100 (red); K=Kv/25 (blue); K=Kv/5 (cyan); K=Kv/2.5 (yellow) where Kv is the steady-state STV loss coefficient. The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Case 2) TVO; the 1st level Simulink logic of the hydraulic loop is represented in Figure 4.55. The tank subsystem would act as the exciter instead of the STV, as it will be showed in detail into the corresponding appendixes.
Figure 4.58 and Figure 4.59 show the results for the hydraulic loop which exploits the vertical oscillation of the tank as source of perturbation. As it can be noticed by the equations of the model of the tank with vertical oscillation, the mathematical source of oscillation is the vertical acceleration of the tank which has been related to the vertical displacement. The qualitative behaviour of the mass flow rate and the pressure oscillations created is similar, while quantitative a huge problem seems to verify, since, not only at low frequencies this set-up is not capable to create an appreciable mass flow rate oscillation, but when this start becoming detectable at high frequency, the pressure amplitude grows too fast towards such high values that the model assumption of small perturbations is violated.
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Figure 4.58 Comparison between the pressure amplitudes of oscillation at different values of maximum vertical displacement of the tank: r=0.2 mm (red); r=1 mm (blue); r=5 mm (cyan); r=1 cm (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Figure 4.59 Comparison between the mass flow rate amplitudes of oscillation at different values of maximum vertical displacement of the tank: r=0.2 mm (red); r=1 mm (blue); r=5 mm (cyan); r=1 cm (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
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Case 3) VOV placed downstream of the tank (far upstream of the pump) as showed in Figure 4.60.
Figure 4.60 Hydraulic loop logic by Simulink with the VOV (red) placed downstream of the tank (blue)
Figure 4.61 and Figure 4.62 show the results for the hydraulic loop which exploits the longitudinal oscillation of the VOV, placed right after the tank, as source of perturbation. As it can be noticed by the equations of the model of the VOV, the mathematical source of oscillation is the VOV perturbation mass flow rate, related to the oscillating volume occupied by the valve. We can notice that the system fails to create an appreciable mass flow rate oscillation, which, beyond 5 Hz is weakly dependent by the frequency, while this set-up, except for a narrow range of very low frequencies, seems to be able to create an appreciable pressure oscillation, which is highly frequency dependent. The VOV set-up was very slow to achieve an oscillating regime condition at very low frequencies.
Figure 4.61 Comparison between the pressure amplitudes of oscillation at different values of the VOV mass flow rate amplitude of oscillation (in percentage of the steady-state mass flow rate): mvov=0.25% (red); mvov=1% (blue); mvov=5% (cyan); mvov=10% (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
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Figure 4.62 Comparison between the mass flow rate amplitudes of oscillation at different values of the VOV mass flow rate amplitude of oscillation (in percentage of the steady-state mass flow rate): mvov=0.25% (red); mvov=1% (blue); mvov=5% (cyan); mvov=10% (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Case 4) VOV placed downstream of the pump, as represented in Figure 4.63.
Figure 4.63 Hydraulic loop logic by Simulink with the VOV (red) placed downstream of the pump (orange)
Figure 4.64 and Figure 4.65 present the results for the hydraulic loop which exploits the longitudinal oscillation of VOV, placed after the pump, as source of perturbation. We can notice that the system succeeds to create an appreciable mass flow rate oscillation, which, beyond 5 Hz is very weakly dependent by the frequency, while, dealing with pressure oscillation, this set-up shows difficulties in creating an appreciable oscillating regime condition at very low frequencies, presenting a quite relevant dependence from the frequency of perturbation. The regime condition at very low frequencies, again, is achieved much more slowly w.r.t. the other solutions.
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Figure 4.64 Comparison between the pressure amplitudes of oscillation at different values of the VOV mass flow rate amplitude of oscillation (in percentage of the steady-state mass flow rate): mvov=0.25% (red); mvov=1% (blue); mvov=5% (cyan); mvov=10% (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
Figure 4.65 Comparison between the mass flow rate amplitudes of oscillation at different values of the VOV mass flow rate amplitude of oscillation (in percentage of the steady-state mass flow rate): mvov=0.25% (red); mvov=1% (blue); mvov=5% (cyan); mvov=10% (yellow). The asterisks represent the data points resulting from the simulation process, the lines are the corresponding polynomial fitting curves.
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4.3.1 Conclusions and Results
The previous results seem to point to the VOV set-up as the most suitable to create an appreciable oscillating regime condition, by placing such a device in the correct position w.r.t. the other components of the hydraulic loop. Exploiting an electrical analogy, as the current tends to flow more towards the low- resistance path, the oscillation of mass flow rate tends to be directed to a high-compliance path and, as it can be noticed from the first of the two VOV set-up, having the tank very close and placed to the opposite flow direction w.r.t. the pump, can result in “stealing” the oscillation from it. Since our objective is to create an appreciable oscillation regime that “wraps” the pump, the VOV must be placed in a smart way.
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5 Conclusions and Future Developments
A modeling framework defined in the time-domain has been developed to characterize the steady state and dynamic behavior of each component of a typical feeding system for liquid rocket engine. A typical water loop for experimental characterization of liquid rocket turbopumps (composed by a tank, a suction line, a pump and a discharge line equipped with a silent throttle valve) has been modeled according to the modeling framework. In order to characterize the transfer matrix of cavitating pumps, the oscillations of pressure and mass flow rate upstream and downstream of the pump must be measured. The modeling framework has been employed to understand the best way to generate the pressure and flow oscillations in a forced experiment in view of the evaluation of the transfer matrix. In particular, three different main set- ups of the forcer have been considered:
Oscillation of the load of the silent throttle valve; Vertical oscillation of the tank; Oscillation of the volume inside a suitable device.
The best results in terms of capability of generating mass flow rate and pressure oscillations at the inlet of the inducer have been obtained by means of a Volume Oscillator Valve located downstream of the pump. Moreover, in view of the experimental measurement of the mass flow rate oscillations upstream and downstream of the pump the goodness of the approach that uses pressure transducers at the inlet and outlet of a duct to obtain the indirect measurement of the flow rate oscillation has been assessed by comparing the dynamic behavior of incompressible ducts and compressible ducts. The main outcome of this analysis is that the incompressible assumptions and the corresponding modeling framework can introduce significant errors in the estimation of the mass flow rate oscillations especially at high frequency. Therefore, a compressible model that relies on a reliable measurement of the speed of sound in the duct should be employed to enhance the quality of the measurement of the mass flow rate oscillations. The procedure exploited throughout the simulation processes, can be used to predict the oscillating behaviour of any hydraulic loop and therefore, every kind of experimental set-up which aims to perform a forced experiment, under the same assumptions, can be assessed. In the end, the dynamic model created in the Simulink environment, can be easily exploited to assess the goodness of the experimental set-ups designed to obtain (at least) two sets of linearly independent data for the experimental characterization of the pump matrix.
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6 Appendixes
6.1 Appendix A, Mathematical Model Equations
6.1.1 Introduction
This section aim to show the analytical process that led to the equations characterizing the subsystems in Chapter 3.
6.1.2 Subsystems Mathematical Model
The pressure and mass flow rate in a given point of the test facility can be written in complex form, as functions of time, as follows:
p() tp p e it
Q() t Q Q eit p() t p p eit m() t m m eit
()te it
it gx () t gxx g e pRe p eit mRe m eit Re eit ggRe e it xx
where p and Q (usually real) are the pressure and flow rate steady values, p and Q (usually complex) are the pressure and flow rate oscillating components, is the frequency of the oscillations. As pointed out before, the relevant components for the dynamic matrix are the oscillating ones; the set of equations used for the evaluation of the matrices of the components of the facility are presented in the following sections. Aiming to describe the behavior of the system, an unsteady characterization, dividing the steady and the assumed perturbation condition, exploiting dimensional and non-dimensional pressures and mass flow rates linked to the following relations:
ˆ pp 2 2 2 2 rrTT Q m m ˆ rr33 r 3 TTT
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where is the water density, is the pump rotational speed and rT is the pump outlet tip radius, is performed for each one of the subsystems.
6.1.3 Incompressible Duct (ID)
General Case: ppp uuu ggg mmm
Governing Equation: D uu Dt t DPuu 1 1 4 1 uu u up g u u Cff p g u u C p g f Dt t2 A 2 DHH 2 D
D u Dt Du 1 uu pf g DtD 2 H u 1 uu u u pf g tD2 H
Along the streamline in the 1-D approximation:
xdu x d x d x d x d 1 uu dx u u d x p d x g d x f d x x x x x x u u u u u tD2 H
xxddu 1 1 1 L x dx u22 u p p g dx u u f xxd u d u x u u uutD2 2 2 eq
mxxdd dx1 1 1 L u22 u p p g dx u u f xxd u d u x u u uu t Ax 2 2 2 D eq
mxxdd dx 1 1 1 u22 u p p g dx u u k xxd u d u x u u loss uu tAx 2 2 2
mdxxddx1 1 1 m m u22 u p p g dx k xxd u d u x loss t uuAAA2 2 2 x xuu x 22 mxxdd dx1 m 1 m 1 m m p p g dx k xxd u x loss tuu A2 A 2 A 2 A A x xd x u x u x u
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Under the assumption of incompressible flow: m m m mud m 22 111 m mm m xd p p g dxk d ux x loss 222AAA A u xxxduu x u
Taking into account the perturbation component of the pressure and the mass flow rate:
22 mm xd dx11 m m m m xu tAx 22AA xxdu
xd 1 m m m m p p p p g g dxk d d u u x x x loss u 2 AA xxuu
2222 mm m m 22 mm m m mm xd dx 11 x 22 u tAx 22AA xxdu m m mm2 m m xd 1 p p p p g g dx k d d u u x x x 2 loss u 2 A xu Subtracting the steady-state solution from the total one:
m 22mm mm 2 mm xxdddx 1 11 p p g dx k xx2 22 d u x loss uu tAx 2A 22 AA xd xx uu
Leading to the steady-state momentum and continuity equations: 22 1m 1 m 1 m m xd p p k g dx d ux loss x 2AAAA 2 2 u xd x u x u x u mmdu x 22 d 2 2 2 2 g dx 1r 1 r 1 r r x x TTTT k u d u2A 2 A 2 A A loss 22 r xd x u x u x u T du
Time-domain and frequency-domain unsteady-state momentum and continuity equations:
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22mm2 m m 1 1 1 xxdddx p p k m g dx d u2 2 2 loss xx x uu 2AAA 2 2 tAx xd x u x u mmdu x 2 2 2 d 2 2 2 gx dx r r r r xd dx x TTTT u d u k loss 22 A A A txu A r xd x u x d x T du 22mm2 m x i t i t1 1 1 d dx i t p e p e k i mu e du 2 2 2 loss x u 2AAA 2 2 Ax xd x u x u x d g dx eit x x u i t i t mdu e m e 222 2 2 2 r r r r xd dx ˆˆeei t i t TTT k i T ˆ eit du loss x AA AA u xxdu xxd x d it gx dx e xu 22 rT ˆˆi t i t du ee
The inputs needed to evaluate pressure and mass flow rate from the previous dimensional relations are: ;m xd dx A;;; A k xud x loss x u Ax
xxdd g dx; g dx xxxx uu
While the corresponding known data for the non-dimensional relations are: ;;rT xd dx A;;; A k xud x loss x u Ax
xxdd g dx; g dx xxxx uu
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6.1.3.1 Incompressible Duct Straight (ID-S)
Steady-state momentum and continuity equations: 1 mm xd p pkg dx d ux loss x u 2 AAxx uu mmdu
xd 22 g dx 1 rr x x TT u d uloss k 22 2 Ax Ar xT uu du
Time-domain unsteady-state momentum and continuity equations: 2 m m 1 xxdddx p p k m g dx d ux 2 loss xx uu 2 A tAx xu mmdu
x 2 d 2 gx dx rr xd dx x TTu d u k loss 22 A txu A r xxd T du
Frequency-domain unsteady-state momentum and continuity equations: 2 m 1 xd dx p p k i mu x x g d ux 2 lossx d u u 2 A Ax xu mmdu x 2 d 2 gx dx rr xd dx x ˆˆ TT ki ˆ u d u loss x 22 Au A r xd x T ˆˆ du
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6.1.3.2 Incompressible Duct Elbow (ID-E)
Steady-state momentum and continuity equations: 1 mm xd p pkg dx d ux loss x u 2 AAxx uu mmdu
xd 22 g dx 1 rr x x TT u d uloss k 22 2 Ax Ar xT uu du
Time-domain unsteady-state momentum and continuity equations: 2 m m 1 xxdddx p p k m g dx d ux 2 loss xx uu 2 A tAx xu mmdu
x 2 d 2 gx dx rr xd dx x TTu d u k loss 22 A txu A r xxd T du
Frequency-domain unsteady-state momentum and continuity equations: 2 m 1 xd dx p p k i mu x x g d ux 2 lossx d u u 2 A Ax xu mmdu x 2 d 2 gx dx rr xd dx x ˆˆ TT ki ˆ u d u loss x 22 Au A r xd x T ˆˆ du
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6.1.3.3 Incompressible Duct Tapered (ID-T)
Steady-state momentum and continuity equations: 22 111 m m m m xd p pk g dx d ux loss x 222AA A A u xxduu u x x mmdu x 22 d 2 2 2 2 g dx 1r 1 r 1 r r x x TTTT k u d uloss 2A 2 A 2 A Ar 22 xd x u x u x u T du
Time-domain unsteady-state momentum and continuity equations: 22mm2 m m 1 1 1 xxdddx p pk m g dx d ux 2 2 2 loss xx uu 2AAA 2 2 tAx xd x u x u mmdu
x 2 2 2 d 2 2 2 gx dx r r r r xd dx x TTTT u d uloss k 22 A A A txu A r xd x u x d x T du
Frequency-domain unsteady-state momentum and continuity equations: 22mm2 m 1 1 1 xd dx p p k i mu du 2 2 2 loss x u 2AAA 2 2 Ax xd x u x u x d g dx x x u mmdu 2 2 2 2 2 2 r r r r xd dx ˆˆTTTT ki ˆ d u loss x AAAA u xd x u x d x
xd g dx x x u 22 rT ˆˆ du
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6.1.4 Compressible Duct Straight (CD-S)
Governing Equations (Continuity and Momentum): D uu Dt t DPuu 11 4 1 uu u up g u u C p g u u C p g f ff Dt tADD 222 HH D u Dt Du 1 uu pf g DtD 2 H
Linearization:
0 ' p p0 p ' u u0 u' g g g ' 0
Continuity equation: D uu Dt t ' 0 u u'''' u u t 0 0 0 0 0 uu t 0 0 0 0 ' uu '' t 00 ' uu '' t 00 D u Dt t 0 D ' 0 u' Dt
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Momentum equation:
uu0 ' 1 u00 u'' u u u0 u'''' u 0 ug g p 0 pf 0 tD2 H
u0 1 uu00 u0 ug 0 pf 0 0 tD2 H
u'12'uu0 u0 u''' pf g tD2 H
u'12 2'uu0 u00 u''' af g tD2 H
Du'12 2'uu0 af0 '' g DtD 2 H
Du' uu0 ' af2 '' g 0 Dt 0 0 0 D H
Linearized Governing Equations (Continuity and Momentum): D ' 0 u' Dt D u0 Du' uu0 ' Dt t af2 '' g 0DtD 0 0 0 H
Cross Differentiation:
D ' 2 00 uu'' Dt 2 DDDDu' uu0 ' af2 '' g 0Dt 2 0Dt 0 Dt 0 Dt D H
General wave equation: 2 DDDu' uu0 ' af22 ug'' 0Dt 2 0 0 0Dt 0 Dt D H
2 DDDu' uu0 ' af22 ug'' Dt 2 0 Dt Dt D H
2 DDDu' uu0 ' af22 ug'' Dt 2 0 Dt Dt D H 2 2 2 D u'''''''' u u u u u22 u u 2 2 uuu0' 0 uu 0 ' 2 u 0 u 0 uuu 0 0 ' 2 2 u 0 uu 0 ' Dtt t t t t t t t
2 D ug''u00 u'' u u af22 u'' u g u 2 0 0 0 Dt t t DHH D
2 D u'''22 gu0 u u 0 u 0 u 0 2 af0u''''' u 0 g u u 0 u u 0 u Dt t D t t D D D HHHH
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General wave equation: 2 D u'22 gu '' u0 u 0 u 0 u 0 2 af0u' u 00 gu ''' u 0 ' u u u Dt tD t t D D D HHHH
Wave equation, particular case assumptions:
u0 0 tDH
u0 0 DH g'0
Hence: 2 u' ugu ''' 2 2 2 2 u0 u 0 u 0 u 0 2 2uuuuug0 0 'af 0 ' 00 ' 0 u '' u ' uuu t tt D t t D D D HHHH 2 u'''' u2 guu u 2u u2 u 'af 2 2 u '' 00 u u 2 0 0 00 t t t DHH t D u u0 u', w w z w z t it g g0 g ' g g g t g geˆ uu0 0z 0 0 w 0 z 0 0 w z
it uu' 'z 0 0 w ' z 0 0 Re w z , t 0 0 Re wˆ z e
DH constant w constant 0
1-D wave equation: 2 2 2 2 2 w w22 w w gww00 w w 2w0 w 0 a 0 f t2t z z 2 z 2 t D t D z HH
Tentative solution: w z, t wˆ z eit
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wˆ z eikz a wzˆ ikz ike z 2 wzˆ ke2 ikz z 2 wˆ z Aeik12 z Be ik z a wzˆ ik12 zik z ik12 Ae ik Be z 2 wzˆ 22ik12 zik z 2 k12 Ae k Be z
2 2 2 2 2 w w22 w w gww00 w w 22w0 w 0 2 a 0 2 f tt z z z t DHH t D z wˆ z 22 w ˆ z w ˆˆ zw z ww2 2ˆˆit ˆ it 2 it 2 itit 00 it it w z e 2 i w0 e w 022 e a 0 e i ge f i w z e e zDzz D z HH wˆ z 22 w ˆ z w ˆ z ww2 w ˆ z 2 ˆ 22 ˆˆ00 w z 2 i w0 w0022 a i g f i w z zzz DHH D z 2 wˆˆ z ww2 w z 2 200 2 ˆˆ a0 w 0 2 2 i w 0 f if w z i g z DHH z D 2 w0 2 w0 2i w0 f if 2 wˆˆ z D w z D igˆ H H wzˆ z2 2 2z 2 2 22 a0 w 0 a 0 w 0 aw00
ww00 w0 2 i f i i f 2 wˆˆ z D w z D ig ˆ HH wzˆ z2 2 2z 2 2 2 2 a0 w 0 a 0 w 0 a 0 w 0
ww00 w0 2 i f i i f 2 wˆˆ z D w z D ig ˆ HH wzˆ z2 2 2z 2 2 2 2 a0 w 0 a 0 w 0 a 0 w 0
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ww00 w0 2 i f i i f DDig ˆ k2 eikeeikzikzikz a HH 2 22 22 2 a0 wa 00 wa 00 w 0
www000 w0 2 i f i i f i i f DDD ig ˆ k2 eikeeaikzikzikz HHH 2 22 22 2 2 2 a0 wa 00 w 00 0 a 0 w 0 a w gˆ a w fi0 DH
ww00 w0 2 i f i i f DD k2 eikeeikzikzikz HH 2 22 2 a0 wa 00 w 0
ww00 2 w0 2 i fi f DD kikk2 k HH 0, 2 2 2 2 12 a0 w 0 a 0 w 0
Final solution:
ww00 2 w0 2 i f i f DD k,0 k k2 HH ik 12 2 2 2 2 ik z ik z a0 w 0 a 0 w 0 wˆ z Ae12 Be a gˆ a w fi0 D H
1-D continuity equation: D ' u' Dt 0 Dw Dtz 0 pˆ Ceik12 z De ik z pˆ ik Ceik12 z ik De ik z z 12 1 p p w 2 w00 a0 t z z 1 pwˆˆ ˆ 2 i p w00 a zz 0 1 pˆ ˆ ik12 z ik z 2 i p w0 0 ik 1 Ae ik 2 Be a0 z
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1 i Ceik1 z De ik 21212 zik z w ik Ce ik zik z ik Deik ik z Ae ik Be 2 0 120 12 a0
2 ik2 z 2 1 0a 0 ik 2 e ak ik222 zik zik z 0 0 2 2 i e w 0 ik 20 e 2 D ik e B DB DB a ik22 zik z wk 0 i e w02 ik e 02
2 ik1 z 2 1 0a01ik e ak ik111 z ik zik z 0 0 1 2 i e w 0 ik 10 e 1 C ik e A C A CA a ik11 z ik z wk 0 i e w01 ik e 01 22 a ka k ik zik z pAeBeˆ 0 0 10 0 2 12 w0 kw 10 k 2
Boundary Conditions: w z, t wˆ z eit it p z, t pˆ z e 1 z, tz e ˆ it a2 0
wˆ z Aeik12 z Be ik z a 22 0a 0 k 1 ik zik z 0 a 0 k 2 pˆ z Ae12 Be w k w k 0 1 0 2
wˆˆ z 00 A B a A B w z a 2 2 2 2 001a k 002 a k 001 a k 002 a k pˆˆ z00 A B A B p z w0 k 1 w 0 k 2 w 0 k 1 w 0 k 2
A B wˆ z 0 a 2 2 2 0a 0 k 2 0 a 0 k 1 0 a 0 k1 B pˆ z 0 wˆ z0 a w k w k wk 0 2 0 1 01
ak2 pˆˆ z00 0 0 1 w z a wk01 Aw z a ˆ 0 22 0a 0 k 2 0 a 0 k 1 w k w k 0 2 0 1 ak2 pˆˆ z00 0 0 1 w z a wk B 01 a22 k a k 0 0 2 0 0 1 w k w k 0 2 0 1
Leading to the steady-state momentum and continuity equations:
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1 mm xd p pkg dx d ux loss x u 2 AAxx uu mmdu
xd 22 g dx 1 rr x x TT u d uloss k 22 2 Ax Ar xT uu du
Frequency-domain unsteady-state momentum and continuity equations: a22 k a k 0 0 1ik12 L 0 0 2 ik L pd p z L Ae Be w0 k 1 w 0 k 2
ik12 L ik L md AwzL A Ae Be a xxdd 1 a22 k a k 0 0 1 ik12 L 0 0 2 ik L d z LAe Be r 2rr 3 3 0 T TTkk 12 AAxx dd A xd ik12 L ik L d z L 3 Ae Be a rT
p z0 p u p p z 0 u mu wz 0 mu A w z 0 A xu xu 2 2 rT u p z 0 p z0 rT u 3 3 r u A w z 0 r u Tx u wz 0 T A xu
Time-domain unsteady-state momentum and continuity equations:
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ppRe e it dd it mmddRe e it ppuuRe e it mmuuRe e it dd Re e it dd Re e it uu Re e it uu Re e i pRe p epd eit p cos t p cos t d d d pp d dd i md it mdRe m d e e m d cos t m d cos t mmdd i p it pRe p eu e p cos t p cos t u u u pp u uu i mu it muRe m u e e m u cos t m u cos t mmuu i d it dd Re ee ddcos tt cos dd i d it dRe de e d cos t d cos t dd i u it uRe ue e u cos t u cos t uu i u it uRe ue e u cos t u cos t uu
6.1.5 Silent Throttle Valve (STV)
General Case: p p p u u u g g g m m m
Governing equations:
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D uu Dt t DPuu 11 4 1 uu u up g u u Cff p g u u C p g f Dt tADD 222 HH
D u Dt Du 1 uu pf g DtD 2 H u 1 uu u ug pf tD2 H
Along the streamline in the 1-D approximation:
xdu xx dd x d x d 1 uu ddx u pu d x d x f g xx d x xx x x u uu u u tD2 H
xxddu 1 1 1 L x dx u22 u p p g dx u u f xxd u d u x u u uutD2 2 2 eq
xxdd mL1 1 1 u22 u p p g dx u u f xx d u d u x u u uu t ADx 2 2 2 eq
mxxdd dx 1 1 1 xd 1 L m dx u22 u p p g dx u u f xx d u d u x x u u uu u t Axx t A 22 2 D eq
mxd dx xx dd 1 1 1 1 m dx u22 u p p g dx u u k x xx d u d u x u u loss u uu t Axx t A 2 2 2
mxd dxm xx mdd 1 1 1 1 m dx u22 u p p g dx k x xx d u d u x loss tu A uu t AA A2 2 2 x xx u xu 22 mxd dx xx dd 1 1 m 1 m 1 m m m dx p p g dx k x xx d u x loss tu A uu t A2 A 2 A 2 A A x x xd x u x u x u
Under the assumption of incompressible flow: m m m mud m kloss kloss k loss
AAAx xx 11 0 t Ax t AAxx
11 Ax 2 0 t Axx A t
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22 1m 1 mxd 1 m m p p g dx k loss d ux x 2AAAA 2 u 2 xd x u x u x u
Leading to the steady-state momentum and continuity equations: 22 111 m m m m xd p pk g dx d ux loss x 222AA A A u xxduu u x x mmdu x 22 d 2 2 2 2 g dx 1r 1 r 1 r r x x TTTT k u d uloss 2A 2 A 2 A Ar 22 xd x u x u x u T du
Taking into account the perturbation component of the pressure and the mass flow rate:
22 mm xd dx11 m m m m xu tAx 22AA xxdu
xd 1 m m m m p p p p g g dx kloss k loss d d u u x x x u 2 AA xxuu
2222 mm m m 22 mm m m mm xd dx 11 x 22 u tAx 22AA xxdu mmmm 22 mm mmmm mm xd 11 p p p p g g dx kloss k loss d d u u x x x 22 u 22AA xxuu Subtracting the steady-state solution form the total one:
m 22mm mm 2 m m m m xxdddx 1 1 1 1 p p g dx kloss k loss xx2 2 d u x 2 2 uu tAx 2AAAA 2 2 2 xd x u x u x u
Leading to the time-domain unsteady-state momentum and continuity equations:
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22mm2 mm m m 1 1 11 xd dx p pk mk du 2 2 22 lossloss x u 2A 2 A 22 AA tAx xd x u xx uu x d g dx x x u mmdu 2 2 2 2 2 2 2 2 r r r r xd dx 1 r TTT k TT k d uloss x loss AAA tu A2 A xd x u x d xxu
xd g dx x x u 22 rT du
Frequency-domain unsteady-state momentum and continuity equations: 22mm2 m x i t i ti t 1 1 1 d dx p e p ek i m e loss u du 2 2 2 x u 2AAA 2 2 Ax xd x u x u mm x 1 i td i t kloss e g dx e 2 x x 2 A u xu i t i t mdu e m e 2 22 2 22 r r r r xd dx ˆˆ eei t i t T TTT k i ˆ eit du loss x A AAA u xd xud x x x 2 d it 2 g dx e 1 r x x T keˆ it u 2 Arloss 22 xTu ˆˆi t i t du ee
22mm2 m m m 1 1 1 xxdddx 1 p p kloss i m u k loss g dx d u2 2 2xx 2 x uu 2AAAA 2 2 Ax 2 xd x u x u x u mmdu 2 2 2 2 2 2 r r r r xd dx ˆˆTTTT ki ˆ d u loss x AAAA u xd x u x d x x 2 d 2 g dx 1 r x x T kˆ u 2 Arloss 22 xTu ˆˆ du
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6.1.6 Volume Oscillator Valve (VOV) – Exciter
Following the same procedure exploited for the characterization of the IDs, with the presence of a forced mass flow rate contribution given by the exciting purpose of the device into the unsteady-state equations: Steady-state momentum and continuity equations: 22 111 m m m m xd p pk g dx d ux loss x 222AAAA u xxduu u x x mmdu
Time-domain unsteady-state momentum and continuity equations: 22mmdu mm 2 mm m 1 1 1 xxdddx pd pk ux 2 g 2 dx 2 loss xxuu 2AAA 2 2 tAx xd x u x u VD md m u m u m D t mmud 11VD m mu m u m D 2 2t 2 22mm2 m mu 1 1 1 1 xd dx p pk mu du 2 2 2 loss x u 2AAA 2 2 mu tAx xd x u x u 2m 2 m 2 1 1 1 1 1 VVxd dx DD 22 kloss 2 V xu 2AA 2 2 2 D t Ax t xxdu t x d g dx x x u
Frequency-domain unsteady-state momentum and continuity equations: 22mm2 m x i t i ti t 1 1 1 d dx p e p e k i mu e du 2 2 2 loss x u 2AAA 2 2 Ax xd x u x u 2m 2 m x 1 1 1 1 d dx it k i mD e 22loss x u 2AA 2 2 2 Ax xxdu xd g dx eit x x u it i t i t med muD e m e
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6.1.7 Tank (T) – Exciter
It is assumed that in steady-state condition, the tank acts as an ideal incompressible duct, without any contributions of inertial and viscous losses. General momentum equation:
xdu x d x d x d x d x d 1 uu dx u u d x p d x g d x r d x f d x x x x x x x utD u u u u u 2 H
Taking into account the 1-D streamline connecting the free surface of the tank and its downstream duct section which communicates with the hydraulic loop, and defining some useful parameters, we obtain:
xxoutoutu 1 x out 11 L x dx u22 u p p g dx r dx u u f xx x out FS out FS x x out out FSFStD2 FS 22 eq
1 xout hFS out g dx g x x g FS
1 xout hFS out r dx r x x r FS
1 xout u hFS out x dx utx x FS ux t t
xout u 1 11 L x dx u22 u p p gh rh u u f x out FS out FSgr out out FS t 2 22 D eq
122 1FS out FS out 1 L ux FS out pout p FS u FS u out ghgr rh u out u out f h u t 2 2 2 Dt x eq
The compliance of the tank is taken into account by means of the dynamics of the interface of its upper part (free surface): dV mm T out in dt dV dz T A FS dtT dt dz u FS FS dt
Under the assumption of adiabatic transformation:
11dpTTTT dV dV dp pT VC TT constant 0 pTT dt V dt dt dt
VT CT pT ppT FS dpA dz mm FST FS out in dt C dt C TT
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Taking into account the 1-D streamline connecting the upstream duct section of the tank which communicates with the hydraulic loop and its free surface, and defining some useful parameters, we obtain:
xxFSFSu 1 x FS 11 L x dx u22 u p p g dx r dx u u f xx x FS in FS in x x in in inintD2 in 22 eq
1 xFS hin FS g dx g x x g in
1 xFS hin FS r dx r x x r in
1 xFS u hdxin FS x utx x in ux t t
1122in FS in FSin FS1 L ux pFS u FS p in u in ghgr rh uin u inu f t h 22 2 Dt x eq
Having obtained the previous relations, we can now eliminate the free surface pressure contribution: 1 1 1 L u p p u22 u ghFS out rh FS out u u f x h FS out out FS FS outgr out out ux t 2 2 2 Dteq 1 1 1 L u p u22 p u ghin FS rh in FS u u f x h in FS FS FS in ingr in in ux t 2 2 2 Dteq
1 221 in FS FS out in FS FS out pout p in u inu out g hg h g r h r h r 2 2 11LL uin u in f u out u out f 22DD eq eq u x hhin FS FS out uxx t u t t
Dealing again with the free surface of the tank and the connected compliance, we aim to relate the remaining parameters related to the free surface, to the time derivative of the vertical displacement of the free surface, which will be related to the frequency of perturbation.
dV mm T out in dt dV dz T A FS dtT dt 23 dzFSCT dp FS d u FS d z FS uFS 23 dt AT dt dt dt dz m m A FS out in T dt
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dp m m C FS out in T dt CAdp dz dp dz TTFS FS FS FS ATT dt dt dt C dt
dpA dz FS T FS dt CT dt 1 11 L u p up22 u ghin FS rh in FSin FS uufh FS FS FS in inin inugr t x 2 22 Dteq Taking into account the assumptions of negligible viscous losses and small perturbations: 1 L uin u in f 0 2 D eq dp dp du dud udhin FS du dhin FS dr 2 dhin FS FS in u in u FSFS gg FS rr hin FS h in FS utx dt dtin dt FSu dt t dt dt dtr dt2 x dt dt dp dp du dhin FS dhin FS dr du2 FS in u in gg r r hhin FSFS in FS dt dtin dt dt dt dtr dt 2 utx dp dp dud z dhin FS dhin FS dr 3 FS in u inFS gg rr hin FS h in FS dt dtinu dt t dt dt dtr dt3 x dhin FS dhin FS dz C dp g r FS T FS dt dt dt AT dt
Aiming to obtain, as usual, unsteady-state momentum and continuity equations that relate the upstream and downstream sections pressure and mass flow rate of the component, to allow the implementation of the tank into our dynamic model: eecos ggzz eerrzzcos 11dhin FS dhin FS dz C dp g r FS T FS cosgrzdt cos z dt dt A T dt 3 dzFS A dz dp du dz dzdr 3 dz T FS in u in gcos FS r cos FS hin FS dt FS h in FS ing z r zu t r dz x CT dt dt dt dt dt dtFS dt dt 3 dzFS A 3 dz dp du dr T gcos r cos dt hin FSFS in u in h in FS gz r zdz ux t in r CT FS dt dt dt dt dt dzFS11 dp in du in 1 dr in FS uin hr dt33 dt dt 3 dt d zFS d z FS dzFS AA33 A 3 TTgcos r cos dt hin FS g cos r cos dt h in FS T gcos r cos dt hin FS gz r zdz uxx t g z r z dz u t grz zdz ux t CCTTFS FS CT FS dt dt dt
ACTT dpinACACTTTT du in dr in FS mmout in uin hr 3 dt 33 dt dt dzFS d zFS d z FS 3 33 A C gcos r cos dt hin FS A C gcos r cos dt hin FS A C g cos r cos dt h in FS T Tgr z z ux t TTzzg r utxx TTzz g r ut dzFS dzFS dz FS dt dt dt
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dzFS mmAoutinT dt
dpdu dr m m CC inin u C h in FS out in Teqdtdtdt Teq in Teq r
ACTT CTeq 3 dzFS 3 A C gcos rh cos dt in FS T Tgr z zu t x dzFS dt AC C TT Teq 2 in FS AT C T gcos gr z rh cos z u t x
Particular case:
Ain and Aout placed at the same height; Vertical tank vibration (coordinate z);
gg z
Obtain: cosgz 1 cosrz 1 AC C TT Teq 2 in FS AT C T g r h u t x
Obtaining the unsteady-state momentum and continuity equations in time and frequency domain:
dpin1 dm in dr in FS mout m in C Teq C Teq2 m in C Teq hr dt Ain dt dt ACTT CTeq A C g r 2 hin FS T T ux t 22mm2 m m 1 1 1 xd dx p p k min out in2 2 2 loss x u 2AAA 2 2 tAx xout x in x in 22mm2 m x i t i t1 1 1 d dx i t p e p e k i min e out in 2 2 2 loss x u 2AAA 2 2 Ax xout x in x in p p R i L min out in 1 ˆˆ ˆ2 in FS ˆ miCpout Teq in 1 iC Teq2 mmiChr in in Teq r Ain
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6.1.8 Pump (P)
Under the 1-D approximation inside the channel (only fully guided velocity): =0 xx22 uP'1 x 2 u''22 11 u p p dxr2 r 1 2g dx 1 C2 u 2 dx 2 22 '0 tA 2 2 221 22 xf xx11 x 1 p p u''22 u 1 1'1 x2 uP x 2 x 2 2 1 2 1 2r 2 2 r 22 dxgdxCu ' dx 2 2 221 22 tA xf x1 x 1 x 1 p p u22 u x2 uP'1 x 2 x 2 2 1 2 1 rvrv dx gdx C u'2 dx 2 22 2 2 1 1 tA xf x1 x 1 x 1 pp xx221'Pu x 2 tt21rvrv C u'2 dx dx gdx 2 2 1 1 fx2 At xx11 x 1 pp r v rv xxx222 1u '2 Pu 1 ' 1 tt212 2 1 1 C dxdx g dx 2 2 2 22 2 fx 2 22 2 rT rr TT A 2 r TT t r xxx111
In steady-state conditions: pp xx221 P tt21rv rv C u'2 dx gdx 2 2 1 1 fx2 A xx11 pp r v rv xx221uP '2 1 tt212 2 1 1 C dx g dx 2 2 2 2fx 2 2 2 2 rTTTT r2 r A r xx11
Transforming in non-dimensional form: pp r v rv xx221uP '2 1 tt21 2 2 1 1 C dx g dx 2r 2 2 r 2fx2 2 r 2 A 2 r 2 TTTTxx11 ,,,, 0 0 0 0 0 0 0 0
0
0 ,,,, 0 0 0 0 0 0
In unsteady-state conditions:
pt2 p t 2 p t 1 p t 1 r2 r 2 v 2 v 2 r 1 r 1 v 1 v 1
x2 x 2 x 2 1'2 Pu C uu dx dx ggdx f2 At x x x1 x 1 x 1
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p p p p t2 t 1 t 2 t 1 rv rv rv rv rv rv rv rv 22 11 22 11 22 11 22 11
x2 x 2 x 2 x 2 x 2 122P 1 P u ' Cf u dx C f2 uu u dx dx g x dx g x dx 22A A t x1 x 1 x 1 x 1 x 1 pp xx221 P tt21r v rv C u2 dx g dx 2 2 1 1 fx xx2 A 11 xx22 x 2 pptt21 1'2 Pu r22 v rv 11 r 22 v rv 11 r 22 v rv 11 Cfx2 uu u dx dx g dx 2 At xx11 x 1
pp r v rv xxx222 1u '2 Pu 1 ' 1 tt212 2 1 1 C dxdx g dx 2r 2 2 rr 22 2A fx 2 r 2 22 2 t r T TTxxx111 TT
22 xx222 x 2 pp 11uu '21 r v 1 rv ' 1 u P u 21 2 2 1 1 C dx dx g dx 22r 2222 rr22 r 22 A fx r 22 t r 22 T TT Txx11 T x 1 T
pp21 22 rT
22 xx222 11uu '21 r v 1 rv uP 2 2 1 1 C dx g dx 2r 222 2 r 2fx 2 r 2 A 2 r 2 TTTTxx11 1xx22 u ' 1 dx g dx 2 2 2 2 x rTT t r xx11
Expansion in Taylor’s series: ,,,, 0 0 0 0 0 0 0 0 0 0 ,,,, 0 0 0 0 0 0 ,,, 0 0 0 0 0 0
Summarizing:
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11 mxx22 dx ,,,, g dx 0 00 00 0 2r 22 2 t A' r x TTxx11 r 3 xx22dx 1 ,,,, T g dx 0 00 00 0 2r 22 2 t A' r x TTxx11
r xx22dx 1 ,,,, T g dx 0 00 00 0 t Ar' 22 x xx11T r xx22dx 1 ,,, T g dx 0 00 0 22 x tA r' T xx11 ,, r xx22dx 1 , 0 0 0 0 T g dx 22 x t A' rT xx11
1 V 1 VC C du 3 u 3 rt 2 rtT T 1 1 V C du d 3 2 2 rtT
x ˆ ˆ ˆ d dx ˆ d u 2 u ir T xu A L Average oscillations of the average mass flow rate calculation: ˆ ˆ2 ˆ 12 ˆ d u u du nˆ ˆ ˆ xxdddxdx d i r i r TTxx uu LL AA
Leading to the steady-state characteristic pressure rise equation: 22 xx222 1uu21 r v rv 1uP ' 1 2 2 1 1 C dx g dx 2 2 2 2fx 2 2 2 2 rTTTT22 rxx r A r 11 ,,,, 0 0 0 0 0 0 0 0 0 0 ,,,, 0 0 0 0 0 0
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Time-domain unsteady-state characteristic pressure rise and continuity equations: xx r 22dx 1 , ,, T g dx 0 0 0 0 22 x t A' rT xx11
1 V 1 VC C du 3 u 3 rt 2 rtT T 1 1 V C du d 2 rt3 2 T
Frequency-domain unsteady-state pressure rise and continuity equations:
x ˆ ˆˆ d dx ˆ d uuT 2 ir x u A L ˆ Vˆ VC ˆ ˆ C ˆˆ u i du i 3 3 r 2 rT T 1 Vˆ ˆ ˆˆ ˆ ˆ i C du d 3 2 2 rT
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6.2 Appendix B, Matlab Code
6.2.1 Introduction
The subsequent section aim to unveil the code used to calculate all the parameters to proceed with the unsteady analysis of a wide-world typology of hydraulic loop system exploited to perform experiments concerning a pump. Knowing the geometry properties of the different subsystems identified in 3.2, composing the desired loop, the code is able to calculate the steady-state operative condition and the unsteady parameters exploited for the small-perturbation dynamic analysis, which is provided by a parallel cooperation between a Matlab live script and a Simulink model described in 6.3. The code is linked to the model equations derived previously and therefore, coherently with the mathematical characterization of the subsystems, it will be presented the steady and unsteady-state parameters required by the program for each one of the system elements.
6.2.2 Steady-State System Parameters
The following subsections show the parameters and the functions needed to obtain the operative condition.
6.2.2.1 Incompressible Duct Straight (ID-S)
Experimental and geometrical parameters to input for the calculation of the steady-state operative condition:
Loss coefficient: k_sd Upstream duct diameter: Du_sd Unperturbed density of the working fluid: d_sd
6.2.2.2 Incompressible Duct Elbow (ID-E)
Experimental and geometrical parameters to input for the calculation of the steady-state operative condition:
Loss coefficient: k_ed Upstream duct diameter: Du_ed Unperturbed density of the working fluid: d_ed
6.2.2.3 Incompressible Duct Tapered (ID-T)
Experimental and geometrical parameters to input for the calculation of the steady-state operative condition:
Loss coefficient: k_td Upstream duct diameter: Du_td Downstream duct diameter: Dd_td Unperturbed density of the working fluid: d_td
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6.2.2.4 Silent Throttle Valve (STV)
Experimental and geometrical parameters to input for the calculation of the steady-state operative condition:
Loss coefficient: k_v Upstream duct diameter: Du_v Downstream duct diameter: Dd_v Unperturbed density of the working fluid: d_v
6.2.2.5 Volume Oscillator Valve (VOV)
Experimental and geometrical parameters to input for the calculation of the steady-state operative condition:
Loss coefficient: k_vov Upstream duct area: Au_vov Downstream duct area: Ad_vov Unperturbed density of the working fluid: d_vov
6.2.2.6 Pump (P)
Experimental and geometrical parameters to input for the calculation of the steady-state operative condition:
Unperturbed density of the working fluid: d_p Rotational speed velocity of the volute: rpm Pump tip radius: rt_p Flow coefficient vector (non-dimensional): phi Pressure rise coefficient vector (non-dimensional): psi
Since we expect a parabolic behavior of the pump characteristic curve, the next script shows the interpolation procedure exploited to obtain the dimensional pressure rise of the pump as a second order polynomial of the mass flow rate: c_i = polyfit(phi,psi,2) a = c_i(1)/(d_p*pi^2*rt_p^4) b = c_i(2)*(rpm*pi/30)/(pi*rt_p) c = c_i(3)*d_p*(rpm*pi/30)^2*rt_p^2
Since in a closed loop system the pressure rise of the pump has to be equal to the pressure drop obtained through the others components, and, having all quadratic relations through the mass flow rate, for all the pressure variation across the system elements, the solution of the operative condition i.e. the steady-state mass flow rate, is given by the positive value of the mass flow rate which provides the intersection between the concave pump parabolic characteristic curve and the summation of the elements convex parabolic curves which results in an equivalent convex parabolic curve. The operative point is calculated by the so called ‘OperativeCondition” function; the script presents the steady-state solution for an example of a hydraulic loop composed by a tank, three ID-Es, four ID-Ss, a pump and a STV.
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function m0 = =OperativeCondition(a,b,c,k_ed1,d_ed1,Au_ed1,k_ed2,d_ed2,Au_ed2,... k_ed3,d_ed3,Au_ed3,k_sd1,Au_sd1,d_sd1,k_sd2,Au_sd2,d_sd2,... k_sd3,Au_sd3,d_sd3,Au_v,Ad_v,k_v,d_v)
m = roots([-a+0.5*k_ed1/(d_ed1*Au_ed1^2)+0.5*k_ed2/(d_ed2*Au_ed2^2)+... 0.5*k_ed3/(d_ed3*Au_ed3^2)+0.5*k_sd1/(Au_sd1^2*d_sd1)+... 0.5*k_sd2/(Au_sd2^2*d_sd2)+0.5*k_sd3/(Au_sd3^2*d_sd3)+... +k_v/(d_v*Au_v^2)+1/(d_v*Ad_v^2)-1/(d_v*Au_v^2) -b -c]);
if m(1) > 0 m0 = m(1); else m0 = m(2); end end
The design flow coefficient can be calculated from the steady-state mass flow rate with the following expression: phi_op = m0/(d_sd1*pi^2*rt_p^3*rpm/30)
With the characteristic curve gradient at the operative point parameter named: dpsi_dphi_op
6.2.3 Dynamic System Parameters
The following sections show the parameters and the functions needed to obtain the unsteady-state parameters, that will be consequently exploited into the Simulink workspace to simulate a small perturbation experiment.
6.2.3.1 Incompressible Duct Straight (ID-S)
Geometrical and steady-state derived parameters to input for the calculation of the unsteady-state oscillating condition:
Duct length: L_sd Inertance calculation: I_sd = L_sd/Au_sd Resistance calculation: R_sd = k_sd*m0/(d_sd*Au_sd^2)
6.2.3.2 Incompressible Duct Elbow (ID-E)
Geometrical and steady-state derived parameters to input for the calculation of the unsteady-state oscillating condition:
Duct length: L_ed Duct area as a function of the 1-D space variable (linearly decreasing in the example): A_ed = pi*(Du_ed^2)/4 - z*0.001 Inertance calculation: I_ed = vpa(int(1/A_ed,z,0,L_ed)) Resistance calculation: R_ed = k_ed*m0/(d_ed*Au_ed^2)
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6.2.3.3 Incompressible Duct Tapered (ID-T)
Geometrical and steady-state derived parameters to input for the calculation of the unsteady-state oscillating condition:
Duct length: L_td Taper coefficient: tap_coeff Duct area as a function of the 1-D space variable: A_td = Au_td - z*tap_coeff Inertance calculation: I_td = vpa(int(1/A_td,z,0,L_td)) Resistance calculation: R_td = m0*(k_td/(d_td*Au_td^2)+1/(d_td*Ad_td^2)-1/(d_td*Au_td^2))
6.2.3.4 Compressible Duct Straight (CD-S)
Under steady-state conditions, the CD-S behaves as an ID-S, therefore, for the steady-state operative point calculation, it can be replaced by a geometrical identical ID-S, for what concerns the steady-state parameters. Under unsteady-state conditions, as previously stated, the situation is different and a dedicated subsection to describe the unsteady-state parameters and functions of the CD-S has to be considered. Geometrical and steady-state derived parameters to input for the calculation of the unsteady-state oscillating condition:
Unperturbed density of the working fluid: d_cd Friction loss coefficient: fr Duct length: L_cd Duct section diameter: D_cd Duct section area calculation: A_cd = 0.25*pi*D_cd^2 Frequency of perturbation: f Acoustic velocity: a0 Steady-state axial velocity calculation: u0_cd = m0/(d_cd*A_cd) Exponential coefficient calculation: k_cd = (k1_cd,k2_cd) = k_function(a0,D_cd,fr,2*pi*f,u0_cd);
Where the k_function is represented by: function k_cd = k_function(a0,D_cd,fr,2*pi*f,u0_cd)
k_cd = roots([1 (u0_cd*2*2*pi*f +sqrt(-1)*fr*u0_cd/D_cd)/(a0^2-u0_cd ^2)-... ((2*pi*f)^2+sqrt(-1)*2*pi*f*fr*(u0_cd/D_cd))/(a0^2-u0_cd ^2)]); end
6.2.3.5 Silent Throttle Valve (STV) - Exciter
Experimental and steady-state derived parameters to input for the calculation of the unsteady-state oscillating condition:
Experimentally characterized valve inertance: I_v Resistance calculation: R_v = m0*(k_v/(d_v*Au_v^2)+1/(d_v*Ad_v^2)-1/(d_v*Au_v^2)) Forcing coefficient calculation: F_v = 0.5*m0^2/(d_v*Au_v^2)
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6.2.3.6 Volume Oscillator Valve (VOV) - Exciter
Experimental and steady-state derived parameters to input for the calculation of the unsteady-state oscillating condition:
Experimentally characterized valve inertance: I_vov Resistance calculation: R_vov = m0*(k_vov/(d_vov*Au_vov^2)+1/(d_vov*Ad_vov^2)+1/(d_vov*Au_vov^2)) Forcing coefficient calculation: F_vov = m0*(0.5*k_vov/(d_vov*Au_vov^2)+1/(d_vov*Ad_vov^2))+0.5*I_vov
6.2.3.7 Tank (T) - Exciter
Geometrical and steady-state derived parameters to input for the calculation of the unsteady-state oscillating condition taking into account the exciting action:
Unperturbed density of the working fluid: d_t Unperturbed gas volume: V_g_t Heat specific ratio of the gas: hsr Unperturbed gas/working fluid pressure: p_t Upstream duct area: Au_t Downstream duct area: Ad_t Tank loss coefficient: k_t Experimentally characterized tank inertance: I_t Tank section area: A_t Gravity acceleration: g Tank water height: h_t Tank water height component w.r.t. the gravity acceleration direction: h_t_r Resistance calculation: R_t = m0*(k_t/(d_t*Au_t^2)+1/(d_t*Ad_t^2)-1/(d_t*Au_t^2)) Compliance calculation: C_t = V_g_t/(hsr*p_t) Equivalent compliance calculation: C_t_eq = A_t*C_t/(A_t+d_t*C_t*(g-4*pi^2*f^2*h_t))
6.2.3.8 Pump (P)
To evaluate the momentum and continuity equation parameters and functions of the pump, a non- cavitating configuration, as a first approximation of its dynamic behavior, have been exploited:
Resistance calculation: R_p = -dpsi_dphi_op*rpm/(30*rt_p) Semi-Experimental Inertance calculation: I_p = 29/(pi*rt_p)
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6.3 Appendix C, Simulink Modeling of the Hydraulic Loop
6.3.1 Introduction
This section will show how the parameters and functions developed in the previous section, have been used into the dynamic modeling environment of Simulink, to simulate the oscillating behavior of the hydraulic loop.
6.3.2 Simulink Environment
Simulink software models algorithms and physical systems using block diagrams which can be connected by way of signal lines to establish mathematical relationships between system components. We designed our models to be hierarchical by organizing groups of blocks into subsystems, enabling to build discrete components that reflect our real-life system and simulate the interaction of those components. For each one of the previous components, an appropriate blocks logic able to satisfy the continuity and momentum equations has been built. Each subsystem is provided by two inputs and two outputs lines that respectively represent the upstream pressure and mass flow rate signal and the downstream pressure and mass flow rate one. In this way we can build our experimental facility by connecting the selected components taking into account their relative positions and simulate the experiment.
6.3.2.1 Incompressible Duct Straight (ID-S)
A detailed view of the ID-S subsystem is showed in Figure 6.1:
Figure 6.1 Blocks logic of the ID-S
The input n°1 represents an incoming oscillating pressure while the n°2 the mass flow rate one. The circuit embraces the unsteady-state momentum and continuity equations. The triangular blocks are multipliers, the small rectangular one performs the time derivative and the big rectangular one is an adder. We can notice how the perturbation of the mass flow rate goes straight to the output, given the
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incompressibility of the duct, while the pressure one (output n°1) is obtained taking into account the resistance and the inertance of the duct calculated in the workspace of Matlab.
6.3.2.2 Incompressible Duct Elbow (ID-E)
A detailed view of the ID-E subsystem is showed in Figure 6.2:
Figure 6.2 Blocks logic of the ID-E
The input n°1 represents an incoming oscillating pressure while the n°2 the mass flow rate one. The circuit embraces the unsteady-state momentum and continuity equations. The triangular blocks are multipliers, the small rectangular one performs the time derivative and the big rectangular one is an adder. We can notice how the perturbation of the mass flow rate goes straight to the output, given the incompressibility of the duct, while the pressure one (output n°1) is obtained taking into account the resistance and the inertance of the duct calculated in the workspace of Matlab.
6.3.2.3 Incompressible Duct Tapered (ID-T)
A detailed view of the ID-T subsystem is showed in Figure 6.3: :
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Figure 6.3 Blocks logic of the ID-T
The input n°1 represents an incoming oscillating pressure while the n°2 the mass flow rate one. The circuit embraces the unsteady-state momentum and continuity equations. The triangular blocks are multipliers, the small rectangular one performs the time derivative and the big rectangular one is an adder. We can notice how the perturbation of the mass flow rate goes straight to the output, given the incompressibility of the duct, while the pressure one (output n°1) is obtained taking into account the resistance and the inertance of the duct calculated in the workspace of Matlab.
6.3.2.4 Compressible Duct Straight (CD-S)
A general view of the CD-S subsystem is showed in Figure 6.4:
Figure 6.4 Blocks logic of the CD-S
The compressible case, as can be noticed, is of a much more complex implementation than the incompressible one, and deserves a deeper insight to be appreciated in Figure 6.5.
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Figure 6.5 Blocks logic of the ID-S, left subsystem
The previous scheme represents the two subsystems on the left of the global compressible duct system. Since the wave solution in closed form has been obtained in the complex/frequency field and the input signal is a real/time function, the input has to be manipulated in order to assume the appropriate complex suit. Mathematically we have an input signal of this type:
푅푒푎푙(|퐴|푒−푖휔푡푒푖휙) = |퐴|cos (−휔푡 + 휙)
The Subsystem purpose is this:
푅푒푎푙(|퐴|푒−푖휔푡푒푖휙) |퐴|푒푖휙
The lower sub-subsystem, aim to extrapolate the magnitude |퐴| of the input signal and, since it works for complex quantity only, it manipulates the input signal adding it the same signal with the appropriate time delay and multiplying it with the unity complex number, to obtain a complex quantity of the same magnitude of the initial time dependent signal. Mathematically we have:
|퐴|(푐표푠(−휔푡 + 휙) + 푖푠푖푛(−휔푡 + 휙)) = |퐴|푒−푖휔푡푒푖휙 sin(−휔푡 + 휙) = 푐표푠(−휔푡 − 휋/2 + 휙)
Solving for the time delay 푇0:
휋 흅 ퟑ흅 ퟑ 푇 > 0 : 푐표푠(−휔푡 − + 휙) = 푐표푠(−휔(푡 − 푇표) + 휙) 푇 = − => = 0 2 0 ퟐퟂ ퟐퟂ ퟒ풇
The upper sub-subsystem, aim to extrapolate the phase of the input signal and again, since it works for complex quantity only, it manipulates the input signal in the same way of the lower circuit, but ending as
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the input of the Complex Phase Difference block, which, extrapolates the desired result comparing it with the basic complex exponential generated from a sine wave generator as showed:
휋 푒−푖휔푡 = cos(휔푡) − 푖 sin(휔푡) = 푠푖 푛 (휔푡 + ) − 푖 sin(휔푡) 2
The phase is then multiplied to the appropriate complex exponential and then to the previously extrapolated magnitude |퐴| to achieve the desired solution |퐴|푒푖휙.
Figure 6.6 Blocks logic of the ID-S, central subsystem
The central subsystem represented in Figure 6.6 provides the complex solution of the wave equation for pressure and velocity perturbation of the working fluid. Without dealing with too many programming details, since each one of the blocks, with the exception of the big rectangular ones that represent sums or products, performs a function created in the Matlab workspace coherently with the CD-S solution, it can be highlighted that the upper part work on the complex pressure perturbation signal, while the lower part manipulate the complex velocity perturbation signal.
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Figure 6.7 Blocks logic of the ID-S, right subsystem
The right subsystem presented in Figure 6.7 manipulates the complex downstream signal obtained from the wave solution procedure, back-transforming it in an oscillating real signal by performing (pressure as example, of course the same stands for the mass flow rate) the following operation: −푖휔푡 −푖휔푡 푝̂푑푒 and subsequently 푅푒푎푙(푝̂푑푒 ).
6.3.2.5 Silent Throttle Valve (STV) - Exciter
A detailed view of the STV subsystem is showed in Figure 6.8:
Figure 6.8 Blocks logic of the STV
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The input n°1 represents an incoming oscillating pressure while the n°2 the mass flow rate one. The circuit embraces the unsteady-state momentum and continuity equations. The triangular blocks act as multipliers, the small rectangular one performs the time derivative and the big rectangular one is an adder. The Silent Throttle Valve acts as a generator of perturbation, through the oscillation, at a given frequency, of its loss coefficient. Starting from the upper part of the subsystem it can be noticed that the second line is the only one of all the subsystems previously considered, to start with an imposed input. It represents our 푘̃퐿푂푆푆 met during the characterization of the unsteady-state behavior of the valve. When the STV will act as a non-oscillating valve, i.e. when the other two exciters will be exploited, 푘̃퐿푂푆푆 = 0, which in Simulink language means that the generator will be set to the ground option.
6.3.2.6 Volume Oscillator Valve (VOV) - Exciter
A detailed view of the VOV subsystem is showed in Figure 6.9:
Figure 6.9 Blocks logic of the VOV
The input n°1 represents an incoming oscillating pressure while the n°2 the mass flow rate one. The circuit embraces the unsteady-state momentum and continuity equations. The triangular blocks are multipliers, the small left rectangular ones perform the time derivative and the right rectangular ones are adders. We can notice how the perturbation of the downstream pressure is composed by four adding components, with one of them taking into account the exciting contribution, as described by the corresponding momentum equation (top adder), while the mass flow rate one (output n°2) is obtained directly taking into account the contribution of the forcing action of the device when exploited as the exciter of the system (down adder), embraced by a sinusoidal wave generator of mass flow rate.
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6.3.2.7 Tank (T) - Exciter
A detailed view of the tank subsystem is showed in Figure 6.10:
Figure 6.10 Blocks logic of the tank
The input n°1 represents an incoming oscillating pressure while the n°2 the mass flow rate one. The circuit embraces the unsteady-state momentum and continuity equations. The triangular blocks are multipliers, the small left rectangular ones perform the time derivative and the big right rectangular ones are adders. We can notice how the perturbation of the downstream pressure is composed by three adding components as described by the corresponding momentum equation (top adder), while the mass flow rate one (output n°2) is obtained taking into account the contribution of the forcing action of the device when exploited as the exciter of the system (down adder), embraced by a sinusoidal wave generator of vertical acceleration of the tank. When the tank won’t oscillate, i.e. when the other two exciters will be exploited, 푉푒푟푡푖푐푎푙 푎푐푐푒푙푒푟푎푡푖표푛 = 0, which in Simulink language means that the generator will be set to the ground option.
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6.3.2.8 Pump (P)
A detailed view of the pump subsystem is showed in Figure 6.11. In our simulations has been used a non- cavitating configuration, which can be represented for a 1st level configuration, as a particular incompressible duct, with its own impedance (lower part), set in the Matlab code and dependent on the steady-state characteristic curve of the inducer.
Figure 6.11 Blocks logic of the pump
The input n°1 represents an incoming oscillating pressure while the n°2 the mass flow rate one. The triangular blocks are multipliers, the small rectangular one performs the time derivative and the big rectangular one is an adder. The pump circuit embraces the unsteady-state momentum and continuity equations transformed in a dimensional form.
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6.4 Appendix D, Simulink Modeling for the Comparison of the Incompressible Solution with the Compressible Solution
Chapter 4.2 shows the results and conclusions of the comparison between two geometrical identical ducts, perturbed in the same way, exploiting the wave solution for the compressible case and the incompressible solution for the incompressible one. The logic behind the CD-S and the ID-S subsystems is showed in the previous appendix, this section aim to show the interaction between them to achieve the desired results.
6.4.1 Downstream Mass Flow Rate and Pressure Signal Comparison Circuit
A 1st level view of the Simulink logic behind the results obtained in 4.2.1 is showed in Figure 6.12:
Figure 6.12 Circuit for the comparison between the amplitude and the phase of the downstream mass flow rate and pressure signal of the ID-S and CD-S models
The two green blocks on the left embrace the already unveiled CD-S and ID-S logic; the four small blue rectangles are sensors which show the time dependent behavior of the variable of interest clearly depicted in the scheme; the two gray blocks perform the phase shift of the two input signals, which is visualized on the screen in the rectangular display placed on the top left corner of the two right yellow rectangular subsystems acting as sensors. Figure 6.13 unveils the logic exploited by the Phase Shift block.
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Figure 6.13 Circuit for the calculation of the phase shift between the CD-S and ID-S signal
It can be noticed that the subsystem presents the same component of the left section of the Compressible Duct scheme, concerning the extrapolation of the phase from a sinusoidal signal, while the right sector performs the subtraction of the phase obtained from the two input signals. The phase shift is calculated subtracting the phase of the Incompressible oscillating variable of interest from the Compressible one.
6.4.2 Duct Transfer Matrix Comparison Circuit
A 1st level view of the Simulink logic behind the results obtained in 4.2.3 is showed in Figure 6.14:
Figure 6.14 Circuit for the comparison between the components of the dynamic transfer matrix of an ID-S and a CD-S
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Exploiting the steady-state mass flow rate resulted from the first of the loop analysis showed in Error! Reference source not found., the ID-S matrix is known, while for the CD-S, as we already know, we need two sets of linearly independent data obtained from the two green blocks on the left embracing the already unveiled CD-S logic, with linearly independent upstream pressure and mass flow rate perturbation. The four grey rectangular blocks extract the complex amplitude of the corresponding signals to obtain the well-known matrix equation of oscillating variables in complex notation. The right part of the circuit represents the four matrix components (generally complex) of the CD-S, with the four big yellow rectangular blocks that calculate the corresponding components and the eight other ones that show the corresponding value of the real and imaginary parts. The yellow top-left subsystem performs the following calculation to get the H11 component:
pˆu mˆ pˆˆ p 2 u1 d 2 d 1 pˆ pˆ u1 d1 pˆu mmˆˆ 2 uu21pˆ H u1 11 pˆ u1
The yellow top-right subsystem performs the following calculation to get the H12 component:
pˆu ppˆˆ 2 dd21pˆ H u1 12 pˆu mmˆˆ 2 uu21 pˆu 1
The yellow down-left subsystem performs the following calculation to get the H21 component:
pˆu mˆ m ˆ m ˆ 2 u1 d 2 d 1 pˆ mˆ u1 d1 pˆu mmˆˆ 2 uu21pˆ H u1 21 pˆ u1
The yellow down-right subsystem performs the following calculation to get the H22 component:
pˆu mmˆˆ 2 dd21pˆ H u1 22 pˆu mmˆˆ 2 uu21 pˆu 1
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6.5 Appendix E, Table of Figures
FIGURE 1.1 TYPICAL OCCURRENCE OF POGO VIBRATION (NASA, 1970) ...... 7 FIGURE 1.2 BLOCK DIAGRAM OF POGO FEEDBACK PROCESS (NASA, 1970) ...... 7 FIGURE 1.3 GEMINI-TITAN ...... 8 FIGURE 1.4 COMPARISON OF GEMINI-TITAN POGO LEVELS (NASA, 1965) ...... 10 FIGURE 1.5 SATURN V FIRST STAGE S-IC ...... 10 FIGURE 1.6 ILLUSTRATION OF SENSITIVITY TO SMALL CHANGES: COMPARISON OF AS-501 AND AS-502 (RYAN, ROBERT S., 1985) ...... 11 FIGURE 1.7 SATURN I-C POGO MITIGATION (VON BRAUN, WERNHER, 1975) ...... 12 FIGURE 1.8 SATURN V SECOND STAGE S-II CLOSE UP, J-2 ENGINE CLUSTER (DOIRON, HAROLD H., 2003) ...... 12 FIGURE 1.9 APOLLO 8 AND 9 POGO EPISODES (FENWICK, J., 1992) ...... 13 FIGURE 1.10 COMPARISON OF CENTER ENGINE THRUST PAD ACCELERATIONS (RYAN, ROBERT S., 1985) ...... 14 FIGURE 1.11 POGO EXPERIENCED IN FLIGHT (RUBIN S., 1970) ...... 14 FIGURE 1.12 CAVITATION COMPLIANCE OF TITAN STAGE 1 PUMPS (NASA, 1970) ...... 15 FIGURE 1.13 PUMP GAIN FOR TITAN STAGE I PUMPS (NASA, 1970) ...... 16 FIGURE 1.14 POGO SUPPRESSION DEVICES (NASA, 1970) ...... 16 FIGURE 1.15 SPACE SHUTTLE POGO SUPPRESSOR DEVICE ...... 17 FIGURE 1.16 LO2 FEED LINE AND BRANCH ACCUMULATOR (A.SWANSON AND T.GIEL, 2009) ...... 18 FIGURE 1.17 ANNULAR ACCUMULATORS (A.SWANSON AND T.GIEL, 2009) ...... 19 FIGURE 1.18 ACCUMULATOR COMPLIANCE VERSUS LIQUID LEVEL (A.SWANSON AND T.GIEL, 2009) .... 20 FIGURE 1.19 ACCUMULATOR INERTANCE VERSUS LIQUID LEVEL (A.SWANSON AND T.GIEL, 2009) ...... 20 FIGURE 2.1 SCHEMATIC DIAGRAM OF THE MATHEMATICAL PUMP LOOP (NG, 1976) ...... 23 FIGURE 2.2 FUNCTIONAL SCHEMATIC OF THE FLUCTUATOR VALVE (NG, 1976) ...... 24 FIGURE 2.3 LEFT: THE [ZP] TRANSFER FUNCTION FOR IMPELLER IV IN THE VIRTUAL ABSENCE OF CAVITATION. THE REAL AND IMAGINARY PARTS OF THE ELEMENTS (SOLID AND DASHED LINES, RESPECTIVELY) ARE PLOTTED AGAINST BOTH THE ACTUAL AND THE NONDIMENSIONAL FREQUENCIES; RIGHT: THE [ZP] TRANSFER FUNCTION FOR IMPELLER IV UNDER CONDITIONS OF EXTENSIVE CAVITATION (NG AND BRENNEN, 1978) ...... 25 FIGURE 2.4 THE [ZP] TRANSFER FUNCTION FOR IMPELLER IV UNDER CONDITIONS OF MODERATE CAVITATION (NG AND BRENNEN, 1978) ...... 25 FIGURE 2.5 POLYNOMIAL CURVE FITTING TO EXPERIMENTAL PUMP TRANSFER MATRICES, [ZP], OBTAINED FOR IMPELLER IV AT 흋=0.070 AND A ROTATIONAL SPEED OF 9000 RPM. THE REAL AND IMAGINARY PARTS OF THE MATRIX ELEMENTS ARE PRESENTED AS FUNCTIONS OF FREQUENCY BY SOLID AND DASH LINES RESPECTIVELY. THE LETTERS A TO E DENOTE MATRICES TAKEN AT FIVES, PROGRESSIVELY DIMINISHING CAVITATION NUMBERS, 흈, AS FOLLOWS: (A) 0.508 (B) 0.114 (C) 0.046 (D) 0.040 (E) 0.023 (NG AND BRENNEN, 1978) ...... 26 FIGURE 2.6 THEORETICAL PUMP TRANSFER MATRICES, [ZP], OBTAINED FOR IMPELLER IV AT 흋=0.070 AS FUNCTIONS OF REDUCED FREQUENCY ퟂ. THE LETTERED CURVES ARE FOR DIFFERENT FRACTIONAL LENGTHS, 휺, OF THE BUBBLY REGION AND CORRESPOND TO DECREASING CAVITATION NUMBERS, 흈: (A) 휺=0.2 (B) 휺=0.4 (C) 휺=0.6 (D) 휺=0.8. THE CURVES ARE FOR ONE SPECIFIC CHOICE OF THE PARAMETERS K AND M (SEE BRENNEN 1978) (NG AND BRENNEN, 1978) ...... 27 FIGURE 2.7 POLYNOMIAL CURVE FITS TO THE 10.2 CM IMPELLER TRANSFER MATRICES AT 흋=0.070, A ROTATIONAL SPEED OF 6000 RPM AND VARIOUS CAVITATION NUMBERS AS FOLLOWS: (A) 0.37 (C) 0.10 (D) 0.069 (G) 0.052 (H) 0.044. THE REAL AND IMAGINARY PARTS OF THE MATRIX ELEMENTS ARE PRESENTED AS FUNCTIONS OF FREQUENCY BY SOLID AND DASHED LINES RESPECTIVELY. THE QUASISTATIC RESISTANCE FROM THE SLOPE IS INDICATED BY THE ARROW (BRENNEN ET AL., 1982) ...... 27
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FIGURE 2.8 TRANSFER FUNCTIONS CALCULATED FROM THE COMPLETE BUBBLY FLOW MODEL WITH 흋=0.07, 흲=9 DEG, 흉=0.45, F=1.0, K=1.3 AND M=0.8. VARIOUS CAVITATION NUMBERS ACCORDING TO 휺=0.2/흈 ARE SHOWN (BRENNEN ET AL., 1982) ...... 28 FIGURE 2.9 SCHEMATIC REPRESENTATION OF THE BASIC SYSTEM AND ITS NOMENCLATURE (A.STIRNEMANN ET AL., 1987) ...... 28 FIGURE 2.10 SCHEMATIC REPRESENTATION OF THE EXPERIMENTAL FACILITY (A.STIRNEMANN ET AL., 1987) ...... 30 FIGURE 2.11 MASS GAIN OBTAINED FROM EXCITATION ON THE SUCTION SIDE (A.STIRNEMANN ET AL., 1987) ...... 30 FIGURE 2.12 TEST LOOP FOR MEASURING THE DYNAMIC BEHAVIOUR OF THE PROTOTYPE MULTI-STAGE PUMP (KAWATA ET AL., 1988) ...... 31 FIGURE 2.13 CORRECTED TRANSFER MATRIX OF THE PROTOTYPE MULTI-STAGE PUMP (KAWATA ET AL., 1988) ...... 31 FIGURE 2.14 LEFT: COMPLIANCE LAG DATA AND FITS; RIGHT: COMPLIANCE DATA AND FITS (RUBIN S., 2004) ...... 33 FIGURE 2.15 LEFT: RESISTANCE DATA AND FITS; RIGHT: INERTANCE DATA AND FITS (RUBIN S., 2004) .. 34 FIGURE 2.16 PUMP GAIN DATA AND FITS (RUBIN S., 2004) ...... 34 FIGURE 2.17 SCHEMATIC OF THE EXPERIMENTAL FACILITY (CERVONE ET AL., 2009) ...... 36 FIGURE 2.18 TOP VIEW OF THE CAVITATING PUMP ROTORDYNAMIC TEST FACILITY (CERVONE ET AL., 2010) ...... 37 FIGURE 2.19 SUGGESTED MODIFICATION TO THE ORIGINAL FACILITY SETUP FOR OBTAINING THE SECOND LINEARLY INDEPENDENT TEST CONFIGURATION (ADDED PIPE LINES ARE COLOURED IN RED) (CERVONE ET AL., 2010) ...... 37 FIGURE 2.20 THE “SHORT” (TOP) AND “LONG” (BOTTOM) CONFIGURATIONS OF THE TEST LOOP USED FOR THE EXPERIMENTAL CHARACTERIZATION OF THE DYNAMIC TRANSFER MATRIX OF CAVITATING INDUCERS (PACE G. ET AL., 2013)...... 38 FIGURE 2.21 DYNAMIC MATRIX FOR DAPAMITOR3 INDUCER: EXPERIMENTAL POINTS ARE IN SQUARES AND THE POINTS OBTAINED BY USING THE MODEL ARE IN STAR (PACE G. ET AL., 2013)...... 39 FIGURE 3.1 SCHEMATIC VIEW OF THE HYDRAULIC LOOP SYSTEM (TORRE ET AL., (12)) ...... 41 FIGURE 4.1 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 49 FIGURE 4.2 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 49 FIGURE 4.3 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 50 FIGURE 4.4 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 50 FIGURE 4.5 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 51 FIGURE 4.6 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 51 FIGURE 4.7 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 52 FIGURE 4.8 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 52 FIGURE 4.9 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 53
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FIGURE 4.10 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 53 FIGURE 4.11 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 54 FIGURE 4.12 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 54 FIGURE 4.13 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 55 FIGURE 4.14 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 55 FIGURE 4.15 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 56 FIGURE 4.16 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 56 FIGURE 4.17 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 57 FIGURE 4.18 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 57 FIGURE 4.19 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 58 FIGURE 4.20 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 58 FIGURE 4.21 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 59 FIGURE 4.22 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 59 FIGURE 4.23 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 60 FIGURE 4.24 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 60 FIGURE 4.25 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 61 FIGURE 4.26 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 61 FIGURE 4.27 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 62 FIGURE 4.28 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 62 FIGURE 4.29 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 63
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FIGURE 4.30 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 63 FIGURE 4.31 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 64 FIGURE 4.32 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 64 FIGURE 4.33 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 65 FIGURE 4.34 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 65 FIGURE 4.35 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 66 FIGURE 4.36 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 66 FIGURE 4.37 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 67 FIGURE 4.38 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 67 FIGURE 4.39 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 68 FIGURE 4.40 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 68 FIGURE 4.41 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 69 FIGURE 4.42 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 69 FIGURE 4.43 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 70 FIGURE 4.44 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 70 FIGURE 4.45 PRESSURE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 71 FIGURE 4.46 MASS FLOW RATE AMPLITUDE COMPARISON BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 71 FIGURE 4.47 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE PRESSURE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ...... 72 FIGURE 4.48 PHASE SHIFT BETWEEN THE COMPRESSIBLE AND THE INCOMPRESSIBLE MASS FLOW RATE SOLUTION VS FREQUENCY OF PERTURBATION AT FIXED ACOUSTIC VELOCITY AND DUCT LENGTH ... 72 FIGURE 4.49 COMPARISON BETWEEN THE REAL COMPONENTS OF THE PRINCIPAL DIAGONAL OF THE MATRIX OF A 3 METERS LONG ID-S (BLACK LINE) WITH A GEOMETRICALLY IDENTICAL CD-S AT DIFFERENT ACOUSTIC VELOCITY: A=1400 M/S (RED); A=1000 M/S (BLUE); A=700 M/S (CYAN); A=500
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M/S (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 74 FIGURE 4.50 COMPARISON BETWEEN THE IMAGINARY COMPONENTS OF THE PRINCIPAL DIAGONAL OF THE MATRIX OF A 3 METERS LONG ID-S (BLACK LINE) WITH A GEOMETRICALLY IDENTICAL CD-S AT DIFFERENT ACOUSTIC VELOCITY: A=1400 M/S (RED); A=1000 M/S (BLUE); A=700 M/S (CYAN); A=500 M/S (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 74 FIGURE 4.51 COMPARISON BETWEEN THE RE[H12] OF A 3 METERS LONG ID-S (BLACK LINE) WITH A GEOMETRICALLY IDENTICAL CD-S AT DIFFERENT ACOUSTIC VELOCITY: A=1400 M/S (RED); A=1000 M/S (BLUE); A=700 M/S (CYAN); A=500 M/S (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 75 FIGURE 4.52 COMPARISON BETWEEN THE IM[H12] OF A 3 METERS LONG ID-S (BLACK LINE) WITH A GEOMETRICALLY IDENTICAL CD-S AT DIFFERENT ACOUSTIC VELOCITY: A=1400 M/S (RED); A=1000 M/S (BLUE); A=700 M/S (CYAN); A=500 M/S (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 75 FIGURE 4.53 COMPARISON BETWEEN THE RE[H21] OF A 3 METERS LONG ID-S (BLACK LINE) WITH A GEOMETRICALLY IDENTICAL CD-S AT DIFFERENT ACOUSTIC VELOCITY: A=1400 M/S (RED); A=1000 M/S (BLUE); A=700 M/S (CYAN); A=500 M/S (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 76 FIGURE 4.54 COMPARISON BETWEEN THE IM[H21] OF A 3 METERS LONG ID-S (BLACK LINE) WITH A GEOMETRICALLY IDENTICAL CD-S AT DIFFERENT ACOUSTIC VELOCITY: A=1400 M/S (RED); A=1000 M/S (BLUE); A=700 M/S (CYAN); A=500 M/S (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 76 FIGURE 4.55 HYDRAULIC LOOP LOGIC BY SIMULINK ...... 78 FIGURE 4.56 COMPARISON BETWEEN THE PRESSURE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF THE STV UNSTEADY-STATE LOSS COEFFICIENT: K=KV/100 (RED); K=KV/25 (BLUE); K=KV/5 (CYAN); K=KV/2.5 (YELLOW) WHERE KV IS THE STEADY-STATE STV LOSS COEFFICIENT. THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 78 FIGURE 4.57 COMPARISON BETWEEN THE MASS FLOW RATE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF THE STV UNSTEADY-STATE LOSS COEFFICIENT: K=KV/100 (RED); K=KV/25 (BLUE); K=KV/5 (CYAN); K=KV/2.5 (YELLOW) WHERE KV IS THE STEADY-STATE STV LOSS COEFFICIENT. THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 79 FIGURE 4.58 COMPARISON BETWEEN THE PRESSURE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF MAXIMUM VERTICAL DISPLACEMENT OF THE TANK: R=0.2 MM (RED); R=1 MM (BLUE); R=5 MM (CYAN); R=1 CM (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 80 FIGURE 4.59 COMPARISON BETWEEN THE MASS FLOW RATE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF MAXIMUM VERTICAL DISPLACEMENT OF THE TANK: R=0.2 MM (RED); R=1 MM (BLUE); R=5 MM (CYAN); R=1 CM (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES. ... 80 FIGURE 4.60 HYDRAULIC LOOP LOGIC BY SIMULINK WITH THE VOV (RED) PLACED DOWNSTREAM OF THE TANK (BLUE) ...... 81 FIGURE 4.61 COMPARISON BETWEEN THE PRESSURE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF THE VOV MASS FLOW RATE AMPLITUDE OF OSCILLATION (IN PERCENTAGE OF THE STEADY-
STATE MASS FLOW RATE): MVOV=0.25% (RED); MVOV=1% (BLUE); MVOV=5% (CYAN); MVOV=10%
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(YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 81 FIGURE 4.62 COMPARISON BETWEEN THE MASS FLOW RATE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF THE VOV MASS FLOW RATE AMPLITUDE OF OSCILLATION (IN PERCENTAGE OF THE
STEADY-STATE MASS FLOW RATE): MVOV=0.25% (RED); MVOV=1% (BLUE); MVOV=5% (CYAN);
MVOV=10% (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 82 FIGURE 4.63 HYDRAULIC LOOP LOGIC BY SIMULINK WITH THE VOV (RED) PLACED DOWNSTREAM OF THE PUMP (ORANGE)...... 82 FIGURE 4.64 COMPARISON BETWEEN THE PRESSURE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF THE VOV MASS FLOW RATE AMPLITUDE OF OSCILLATION (IN PERCENTAGE OF THE STEADY- STATE MASS FLOW RATE): MVOV=0.25% (RED); MVOV=1% (BLUE); MVOV=5% (CYAN); MVOV=10% (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 83 FIGURE 4.65 COMPARISON BETWEEN THE MASS FLOW RATE AMPLITUDES OF OSCILLATION AT DIFFERENT VALUES OF THE VOV MASS FLOW RATE AMPLITUDE OF OSCILLATION (IN PERCENTAGE OF THE STEADY-STATE MASS FLOW RATE): MVOV=0.25% (RED); MVOV=1% (BLUE); MVOV=5% (CYAN); MVOV=10% (YELLOW). THE ASTERISKS REPRESENT THE DATA POINTS RESULTING FROM THE SIMULATION PROCESS, THE LINES ARE THE CORRESPONDING POLYNOMIAL FITTING CURVES...... 83 FIGURE 6.1 BLOCKS LOGIC OF THE ID-S ...... 118 FIGURE 6.2 BLOCKS LOGIC OF THE ID-E ...... 119 FIGURE 6.3 BLOCKS LOGIC OF THE ID-T ...... 120 FIGURE 6.4 BLOCKS LOGIC OF THE CD-S ...... 120 FIGURE 6.5 BLOCKS LOGIC OF THE ID-S, LEFT SUBSYSTEM ...... 121 FIGURE 6.6 BLOCKS LOGIC OF THE ID-S, CENTRAL SUBSYSTEM ...... 122 FIGURE 6.7 BLOCKS LOGIC OF THE ID-S, RIGHT SUBSYSTEM ...... 123 FIGURE 6.8 BLOCKS LOGIC OF THE STV ...... 123 FIGURE 6.9 BLOCKS LOGIC OF THE VOV ...... 124 FIGURE 6.10 BLOCKS LOGIC OF THE TANK...... 125 FIGURE 6.11 BLOCKS LOGIC OF THE PUMP ...... 126 FIGURE 6.12 CIRCUIT FOR THE COMPARISON BETWEEN THE AMPLITUDE AND THE PHASE OF THE DOWNSTREAM MASS FLOW RATE AND PRESSURE SIGNAL OF THE ID-S AND CD-S MODELS ...... 127 FIGURE 6.13 CIRCUIT FOR THE CALCULATION OF THE PHASE SHIFT BETWEEN THE CD-S AND ID-S SIGNAL ...... 128 FIGURE 6.14 CIRCUIT FOR THE COMPARISON BETWEEN THE COMPONENTS OF THE DYNAMIC TRANSFER MATRIX OF AN ID-S AND A CD-S ...... 128
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6.6 Appendix F, List of Tables
TABLE 1 VALUES OF FIT PARAMETERS AND REFERENCES TO EQUATIONS AND FIGURES. DUAL VALUES FOR FLOW GAIN INDICATE UNCERTAINTY (RUBIN S., 2004) ...... 35
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Reference Documents
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Acknowledgements (Ringraziamenti)
I’d like to express my gratitude to prof. Zoltan Spakovszky for his support and his welcome during the project meetings occurred in Cambridge, Boston. Special thanks go to prof. Angelo Pasini for his constant support and patience throughout the period of research. Il ringraziamento più importante, però, non può che essere riservato ai miei genitori, Anna e Vincenzo, mio zio Antonio e mia sorella Alessia, per il loro costante e incondizionato supporto, senza il quale nulla di quanto fatto sarebbe stato possibile.
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