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
CandidateMario Amoroso
Supervisor
Dott. Ing. Angelo Pasini
1
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.
2
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
3
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
4
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.
5
Nomenclature
A Cross-section area, m2.
a,a0 Unperturbed acoustic velocity, m/s. a Unsteady acoustic velocity, m/s.
T C Tank compliance, m4s2kg-1. H D Hydraulic diameter, m. e Unit vector. f Frequency of perturbation, Hz. u
f Maximum frequency of perturbation of interest, Hz.
g Steady gravity acceleration, m/s2.
g Unsteady gravity acceleration, m/s2.
h Tank height, m. ij
H 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. loss k Loss coefficient. loss
k Steady loss coefficient. loss
k 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. ˆ
m Unsteady mass flow rate, complex amplitude, kg/s.
P Pressure coefficient. p Static pressure, Pa. p Steady static pressure, Pa.
6 ˆp Unsteady static pressure, complex amplitude, Pa.
q Steady independent variable. q Unsteady independent variable.
R Resistance, m-1s-1.
r Vertical tank oscillation amplitude, m. T
r Pump tip radius, m.
r Steady inertial acceleration, m/s2.
r Unsteady inertial acceleration, m/s2.
t Time, s.
V Volume, m3.
0
w Unperturbed axial velocity, m/s.
Specific heat ratio. Density, kg/m3. ,0 Steady density, kg/m3. Cavitation number.
Flow coefficient.
Pump rotational speed, m/s. Radiant frequency of perturbation, rad/s.
7
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.
9 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 17thtest 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.
10 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.
11 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.
12 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.
13 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.
14 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.
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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.
16 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.
17 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: 𝜔1∗= [𝜌𝐿𝑆(𝐶𝑎+ 𝐶𝑏)]− 1 2 𝜔2∗ = [ 𝜌(𝐿𝑎+ 𝐿𝑏) 1 𝐶𝑎+ 1 𝐶𝑏 ] −12
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
18 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:
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Figure 1.17 Annular Accumulators (A.Swanson and T.Giel, 2009)
Defining the compliance and inertance of the devices as: 𝐶𝐴= 𝜌𝐿 𝑔𝑆 𝑉𝑜𝑙 𝛾𝑔 𝑃𝑔 𝑔𝑐 𝐼 =(𝑡𝑤+ 𝐷𝑝) (𝑔𝑆 𝐴𝑝) + ∫ 𝑑𝑠 𝑔𝑠 𝐴(𝑠) 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
𝑠 is the accumulator liquid flow path A(s) is the liquid flow path area at 𝑠
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:
d 11 12 u 21 22 d u
p
H
H
p
=
H
H
Q
Q
where
p
andQ
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:
( )
i tp t
p p e
( )
i tQ t
Q Q e
where p and Q (usually real) are the pressure and flow rate steady values,
p
andQ
(usually complex) are the pressure and flow rate oscillating components, is the frequency of the oscillations.22 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: d u d u
p
1 -R - iωL
p
=
0
1
Q
Q
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:
d u d u
p
1- (S +iωX) -R - iωL
p
=
-iωC
1- iωM
Q
Q
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: {𝑚̃𝑝̃2− 𝑝̃1 2− 𝑚̃1} = [ 𝑍11 𝑍12 𝑍21 𝑍22] { 𝑝̃1 𝑚̃1}
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:
𝑝̃1,2= 𝑝̃1,2∗ 1 2 𝜌𝑈𝑡2 ; 𝑚̃1,2= 𝑚̃1,2∗ 𝜌𝐴𝑖𝑈𝑡
Where 𝑈𝑡 is the tip speed, 𝐴𝑖 the inlet area, 𝜌 the fluid density and 𝜔 = 𝛺𝐻/𝑈𝑡 a reduced frequency
where 𝛺 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:
𝑍21= −𝑗𝜔𝐾𝐵+ 𝑂(𝜔2)
23 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: 𝑍21= 𝑈𝑡 2𝐴𝑖 (𝛥(𝑚̃1 ∗− 𝑚̃ 2∗) 𝛥𝑝̃1∗ )𝜑=𝑐𝑜𝑛𝑠𝑡= 𝑗 𝛺𝑈𝑡 2𝐴𝑖 (𝜕𝑉 𝜕𝑝1 ) 𝜑=𝑐𝑜𝑛𝑠𝑡 𝑍22 = ( 𝛥(𝑚̃1∗− 𝑚̃2∗) 𝛥𝑚̃1∗ ) 𝑝̃1∗=0 = 𝑗𝛺 𝐴𝑖 (𝜕𝑉 𝜕𝑈𝑡 ) 𝑝̃1∗=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:
24 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:
25
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)
26 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 𝝋=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, 𝝈, 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:
27
Figure 2.6 Theoretical pump transfer matrices, [ZP], obtained for Impeller IV at 𝝋=0.070 as functions of reduced
frequency 𝟂. The lettered curves are for different fractional lengths, 𝜺, of the bubbly region and correspond to decreasing
cavitation numbers, 𝝈: (A) 𝜺=0.2 (B) 𝜺=0.4 (C) 𝜺=0.6 (D) 𝜺=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 𝝋=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)
28 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 𝝋=0.07, 𝝲=9 deg, 𝝉=0.45, F=1.0, K=1.3
and M=0.8. Various cavitation numbers according to 𝜺=0.2/𝝈 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:
29 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):
𝐻34= 𝑝3 𝑝4 ; 𝑦34= 𝑞3 𝑝4 ; 𝑦4= 𝑞4 𝑝4
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:
𝐻34= 𝐻12(𝜂11𝛾12− 𝜂12𝛾11) + 𝜂12 𝐻65(𝑘11𝛿12− 𝑘12𝛿11) + 𝑘12 ∗𝐻25𝛿12 𝛾12 𝑦34= 𝐻12(𝜂11𝛾11− 𝜂21𝛾12) − 𝜂11 𝐻65(𝑘11𝛿12− 𝑘12𝛿11) + 𝑘12 ∗𝐻25𝛿12 𝛾12 𝑦4 = 𝐻65(𝑘21𝛿12− 𝑘11𝛿11) + 𝑘11 𝐻65(𝑘11𝛿12− 𝑘12𝛿11) + 𝑘12 Being: 𝐻12= 𝑝1 𝑝2 ; 𝐻25= 𝑝2 𝑝5 ; 𝐻65= 𝑝6 𝑝5
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:
[ (𝐻34)1 ∗ ∗ ∗ (𝐻34)𝑁] = [ 1 (𝑦4)1 ∗ ∗ ∗ ∗ ∗ ∗ 1 (𝑦4)𝑁 ] [ 𝛼 𝛼11 12 ] [ (𝑦34)1 ∗ ∗ ∗ (𝑦34)𝑁] = [ 1 (𝑦4)1 ∗ ∗ ∗ ∗ ∗ ∗ 1 (𝑦4)𝑁 ] [ 𝛼 𝛼21 22 ]
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.
30 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:
31 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:
32 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:
𝑉 = 𝜕𝑉𝑠 𝜕𝑝1 𝑝1+ 𝜕𝑉𝑠 𝜕𝑚̇1 𝑚̇̇ 1
The non-dimensional continuity equation is expressed as:
𝑚1− 𝑚2= −𝜌𝑉 = 𝐶̅𝑝1+ 𝑀̅𝑚̇1
Using the Laplace variable 𝑠 = 𝑗𝜔, the complex compliance and mass flow gain are given by:
−𝜌𝜕𝑉𝑠 𝜕𝑝1 = 𝐶 1 + 𝜏𝑐𝑠 ; 𝜕𝑉𝑠 𝜕𝑚̇1 = 𝑀 1 + 𝜏𝑀𝑠
With the non-dimensional transfer equations of the pump: 𝑝2= 𝑇11𝑝1+ 𝑇12𝑚̇1
𝑚̇2= 𝑇21𝑝1+ 𝑇22𝑚̇1
The transfer matrix non-dimensional components can be expressed as: 𝑇21(𝜔) = −𝑗𝜔𝐶 1 + 𝑗𝜔𝜏𝑐 ; 𝑇22(𝜔) = 1 − −𝑗𝜔𝑀 1 + 𝑗𝜔𝜏𝑀
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:
𝜏𝑐(𝜔) = 1 𝜔 𝑅𝑒[𝑇21(𝜔)] 𝐼𝑚[𝑇21(𝜔)] 𝜏𝑀(𝜔) = − 1 𝜔 (𝑅𝑒[𝑇22(𝜔)] − 1) 𝐼𝑚[𝑇22(𝜔)]
33 𝐶(𝜔) = −1 𝜔 𝐼𝑚[𝑇21(𝜔)](1 + 𝜔 2𝜏 𝑐2) 𝑀(𝜔) = −1 𝜔 𝐼𝑚[𝑇22(𝜔)](1 + 𝜔 2𝜏 𝑀2)
The non-dimensional pressure equation is expressed by:
𝑝2= 𝐺̅𝑝1− (𝑅 + 𝑗𝜔𝐿)𝑚̇2
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: 𝑍 = 𝑅 + 𝑗𝜔𝐿 =𝑇12
𝑇22
𝐺̅ = 𝑇11+
𝑇12𝑇21
𝑇22
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)
𝐶(𝜔) = 𝐶0
1 + 9𝐶0𝜔2
𝜏𝑐(𝜔) =
𝜏𝑐0
34
Figure 2.15 Left: Resistance data and fits; Right: Inertance data and fits (Rubin S., 2004)
𝑅 =𝑅0(1 + 2.3𝛼𝑅𝜔 2) 1 + 𝛼𝑅𝜔2 𝐿 =𝐿0(1 + 0.5𝛼𝐿𝜔 2) 1 + 𝛼𝐿𝜔2
Figure 2.16 Pump Gain data and fits (Rubin S., 2004)
𝐺 = 𝐺0(1 + 𝛼𝐺𝜔2)
The reference value, subscripted with o, at the different cavitation numbers used for the test campaign, are shown in Table 1:
35
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:
𝑚𝑐+ 𝜏𝑐0𝑠𝑚𝑐𝑐+ 𝐶0𝑝1𝑐= 0
𝑚𝑐− (1 − 9𝜏𝑐0𝑠2)𝑚𝑐𝑐 = 0
𝑝1− (1 − 5𝐶0𝑠2)𝑝1𝑐= 0
(1 + 𝜏𝑀𝑠)𝑚𝑀+ 𝑀𝑠𝑚1= 0
𝑚𝑐− 𝑚2+ 𝑚𝑐+ 𝑚𝑀= 0
While the pressure equation becomes:
𝑝2= 𝐺0(1 − 𝛼𝐺𝑠2)𝑝1− 𝑅0(1 − 35𝑠2)𝑚2𝑅− 𝐿0(1 − 36𝑠2)𝑚2𝐿
(1 − 15𝑠2)𝑚
2𝑅− 𝑠𝑚2= 0
(1 − 72𝑠2)𝑚
2𝐿− 𝑠2𝑚2= 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
36 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:
[𝑇][𝐻𝑀] = [ 𝑝̂1𝑎 𝑝̂1𝑏 𝑄̂1𝑎 𝑄̂1𝑏 0 0 0 0 0 0 0 0 𝑝̂1𝑎 𝑝̂1𝑏 𝑄̂1𝑎 𝑄̂1𝑏] [ 𝐻𝑀11 𝐻𝑀12 𝐻𝑀21 𝐻𝑀22 ] = [ 𝑝̂2𝑎 𝑝̂2𝑏 𝑄̂2𝑎 𝑄̂2𝑏]
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
37 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)
38 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:
39
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.
40
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:
Re
ˆ i t
p t p p t p p e
Re
ˆ i t
m t m m t m m e 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).
41
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) through suitable coefficients (such as qn
name
P and qn
name
P for the pressure or qn
name
M and qn
name
M for the mass flow rate) as reported in the following expressions:
n n q name d name n n q name d name n n p P q m M q
n n q name d name n n q name d name n n p P q m M q
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 gand r are supposed positive while g and r are oscillating around zero): d u d u x g name x x r name x P d P d
g r e x e x d u d u x g name x x r name x P d P d
g r e x e x42
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: 2 2 2 2 1 2 2 1 1 u x x du loss du u d du xu xu u p ID A A m k m m ID A A m ID P P M where 2 d u m m du
m 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:
2 2 2 2 1 1 1 u x x du loss du u d du xu xu d du x u u p ID A A m k m m ID A A x m dx ID x A m ID P P P M
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:
; ; ; ; ; d ; ; ; ;
u d x u x g r g r dx du x x loss A ID ID ID ID x m A A k P P P P
3.2.2 Compressible Duct Straight (CD-S)
In steady state condition, the compressible duct behaves as an incompressible duct:
2 2 1 1 u loss du du xu u p CD S k m m CD S A m CD S P P M
While the unsteady behaviour of a compressible duct substantially defers from the incompressible case according to the following coefficients:
