SCIENTIFIC REVIEW (1996), Number 16 AIR INFLUENCE ON STABILITY OF REAL AND THEORETICAL HYDRAULIC SYSTE M1 Stanislav Pejovi} University of Belgrade, Faculty of Mechanical Engineering 27 marta 80, 11000 Beograd, Yugoslavia E-mails: earn_030@ubbg.etf.bg.ac.yu cane@fon.fon.bg.ac.yu Abstract: Hydraulic transients and vibrations can provoke very strong hydraulic loads in pumping systems, hydro power plants, or in any hydraulic system by setting in motion water masses enclosed in it. Natural frequencies and response of a system to the exciters are highly influenced by the content of air bubbles in the water flowing in it. Each operating point of a hydraulic machine is associated with matrix 2x2 and vector 1x2. All six elements are functions of frequency, cavitation, etc. They should be and could be measured when several meters of straight pipe exist at the inlet and outlet of the machine. Two-phase flow of air-water mixture that usually occurs in the pump discharge pipe and turbine draft tube must always be taken into account in the analysis of hydraulic transients and vibrations. This variation in wave speed changes the natural frequencies of the plants and systems in many different ways. The method that has been described for measuring of oscillatory behavior of hydromechanical components is applicable to the physiological measurements of cardiovascular systems. Numerical analyses, field tests and laboratory tests were usually carried out to investigate the natural frequencies, hydraulic vibrations and transients of hydraulic systems. The designer should be aware of various facts and should take adequate measures to prevent the worst. Keywords: Stability, Transfer Matrix, Hydraulic Machine, Pump, Turbine, Piping System, Measurement of Machine Matrix, Wave Speed Measurement, Air in Water, Cavitation, Natural Frequencies, Resonance, Pulsatile Human Blood Systems, Hemodynamics, Cardiovascular System. 1. INTRODUCTION 1 This paper was presented to the Scientific Society of Serbia on the 9 November 1995. Project 0402 Serbian Ministry of Science and Technology.
Abstract: Hydraulic transients and vibrations can provoke very stronghydraulic loads in pumping systems, hydro power plants, or in any hydraulicsystem by setting in motion water masses enclosed in it. Natural frequenciesand response of a system to the exciters are highly influenced by the contentof air bubbles in the water flowing in it. Each operating point of ahydraulic machine is associated with matrix 2x2 and vector 1x2. All sixelements are functions of frequency, cavitation, etc. They should be andcould be measured when several meters of straight pipe exist at the inlet andoutlet of the machine. Two-phase flow of air-water mixture that usuallyoccurs in the pump discharge pipe and turbine draft tube must always be takeninto account in the analysis of hydraulic transients and vibrations. This
variation in wave speed changes the natural frequencies of the plants andsystems in many different ways.
The method that has been described for measuring of oscillatorybehavior of hydromechanical components is applicable to the physiologicalmeasurements of cardiovascular systems.
Numerical analyses, field tests and laboratory tests were usuallycarried out to investigate the natural frequencies, hydraulic vibrations andtransients of hydraulic systems. The designer should be aware of variousfacts and should take adequate measures to prevent the worst.
Keywords: Stability, Transfer Matrix, Hydraulic Machine, Pump, Turbine,Piping System, Measurement of Machine Matrix, Wave SpeedMeasurement, Air in Water, Cavitation, Natural Frequencies,Resonance, Pulsatile Human Blood Systems, Hemodynamics,
Cardiovascular System.1. INTRODUCTION
1This paper was presented to the Scientific Society of Serbia on the 9 November 1995.
Project 0402 Serbian Ministry of Science and Technology.
Hydraulic vibrations are common problems in hydro power and pumpingsystems as well in any hydraulic structure. Masses of water (or any other
fluid) enclosed in pipelines, tunnels and penstocks, when set in oscillatorymotion by an exciter, can cause severe damage. The worst case is when thefrequency of forced oscillations fits one of natural frequencies of theplant. Then the resonance can destroy it. Such conditions should beprevented; the question is how to do it?
The designers can carry out a sensitivity analysis of transients andpossible natural frequencies of the plant (Bikov, 1961; Krivchenko et al.,1975; Pejovi}, 1977; Pejovi}, Boldy 1992b; Pejovi} et al., 1987; Wylie etal., 1993; Chaudhry, 1979; Zielke, 1980). Most dangerous transient regimesand frequencies of exciters are known (Pejovi}, 1979; Brekke, 1984; Grein,1980; Vladislavlev, 1972) but data on air content in water and intensity ofexciters are very poor (Grein, 1980; Jemcov et al., 1980).
Changing some parameters, such as diameters of penstocks, machinecharacteristics (specific speed), or something else, and computing thecoefficient of attenuation for natural frequencies as well as responses tothe hydraulic transients and existing forcing exciters, it is possible toreduce amplitudes of oscillations and extreme waterhammer pressures.
The aim of transient and vibration analyses carried out during designand construction stage are to find out if the machines and associated systemscan operate safely avoiding any possible danger to the plant and personnel.Therefore, analyses of transients, resonance and stability are veryimportant.
In order to reduce the cost of structure, most turbines, and pumpsoperate under cavitation conditions followed by two-phase flow in thedownstream parts of the system. The wave speed, dependent mostly on thepressure and free gas content in the water, is much small (Lee, 1991; Perko,1984; Wylie, 1993). The mode shape, especially maximum and minimum pressurein the system, in the zone of characteristics excitations should be analyzed.The water hammer is also highly influenced by the wave speed in the system.
The transient flow in a pipeline could be divided into three cases:waterhammer one phase flow, two-phase flow caused by cavitation and/or airintroduction and water column separation. In the waterhammer phase therelease of dissolved air (gases) is small and the wave speed depends on voidfraction, which in turn depends on the local pressure. In the cavitationphase, gas bubbles are dispersed throughout the liquid owing to the reductionof the wave speed. At certain low pressure dissolved air very quickly formtwo-phase flow of air and water significantly reducing wave velocity; thisphenomenon is followed by incipient cavitation. The water boils at the vaporpressure partially forming three-phase flow of water-vapor-air mixture. Theexistence of column separation, trapped gas and/or vapor volume and entrainedfree gas bubbles greatly complicate vibration, stability, and transientanalysis.
The influence of the air bubbles in water on the wave speed is already
well known for a long time. In some pumping systems it is reduced from about1300 m/s to 500 m/s (Bikov, 1961). In the conduits of hydro power plants itis sometimes even more reduced (Perko, 1984; Wylie, 1993). In the turbinedraft tubes it could even be smaller than 50 m/s (Pejovi} et al., 1985a,1985b).
The oscillatory behavior of hydraulic turbomachines can be studiedperforming linearisations in the equation of motion and continuity equationin order to arrive to the transfer matrix method. Various authors deal withthe experimental determination of machine matrix. The general form of machineshould be completed with a vector 1x2 of excitation. Each operating point isassociated with matrix 2x2 and vector 1x2. All six elements are functions offrequency, cavitation, etc. They could be measured when several meters ofstraight pipe exist at the inlet and outlet of the machine.
Similarity of oscillatory (and transient) flow in a model and prototypeis highly influenced by the air content in the fluid. In the case ofcavitation and/or air entrained into the fluid, similarity is very poor. Very
often there is no similarity at all (Lee et al., 1995; Pejovi} et al., 1995).
2.2. OSCILLATORY FLOW
Unsteady flow is described by the equation of motion and the equationof continuity
g cdg
c c∂
∂
∂
∂
∂
∂
λ h
x
c
x
c
t + + + =
20 (1)
∂
∂
∂
∂
∂
∂
∂
∂
h
t
h
x
c
x
z
x+ + − =c
a
g c
2
0 (2)
where: h - piezometric head, c - flow velocity x - distance along pipe axis,
t - time, g - gravitational acceleration, λ - friction coefficient, d - pipe
diameter, α - pipe inclination, a - wave velocity. Equations (1) and (2) arepartial deferential equation of hyperbolic type.
The transfer matrix method is based upon the assumption thatoscillations of pressure and flow around their mean values are rather small.By introducing hydraulic inertance L = 1/(gA), hydraulic resistance R
= λ Q/(gdA 2) and hydraulic capacitance C = gA/a2 as well as the complex
frequency s i= +σ (σ - coefficient of attenuation, ω - angular frequency),the equation for oscillatory motion may be written as
and the amplitudes of pressure, H and flow, Q oscillations at the ends of a single pipe, seeFig. 1, are related as
( )
H
Q
Z
Z
H
Q D
C
C U
=−
−
l l
l cosh l
cosh sinh
sinh
γ γ
γ γ (5)
Here the characteristic impedance of the
pipe is Z CsC = γ with γ = +Cs Ls R( ) -
propagation constant (a complex number), A -
cross-section, and l - pipe length. Subscripts"D" denote the downstream, and "U" the upstreamend (Wylie, 1993; Chaudhry, 1979; Pejoci},1979). In matrix notation, this reads
{ } [ ]{ }U D
V F V = (6)
Other components of the plant, such as turbines, pumps, valves, airvessels, surge tanks, bifurcations, branches, dead-end pipes, loops, storagebasins, etc., are represented by similar equations.
A hydraulic machine, see Fig. 1 is usually represented by the simpleform of transfer matrix:
[ ] P M =
10
-1
(7)
This is not enough, especially in the case of hydraulic turbine, since manyexcitations are concentrated in it. One of the most dangerous is a vortexcore downstream of the runner. The matrix (7) must bee completed with anexcitation vector
{ } E H
Q
E
E
=
(8)
and a transfer matrix presentation of hydraulic machine reads
where: tg = - slope of head-discharge curve and H E and Q E are
excitations of head and flow, respectively. The value of is usually
supposed a real constant for pumps and for turbines if the opening of wicketgates remains constant during the transient regime; the elasticity ofstructure and cavitation are neglected too; otherwise, if the governorcontrols wicket gates, is a function of several variables: governor'scharacteristics, frequency of excitation, electric machine behavior etc. Inthe case of valve M = 2H/Q (H - means head losses, Q - means discharge).
The system consisting of several nodes and pipes could be described ina similar fashion, by the matrix equation
{ } [ ] { } { }V U V C D S U = + (10)
The matrix [ ]U S is the transfer matrix for the whole system determined by
multiplying matrices of unit elements in the proper way. The vector { }C is the
excitation of the system. Details are given in (Brekke, 1981; Chaudhry, 1979;Pejovi}, 1979; Wylie, 1993).
At any position x in the system the oscillations about the mean valuesare the real parts of equations
′ = =h H x e H x e e st t i t ( ) ( ) σ ω , q Q x e Q x e e st t i t ' ( ) ( )= = σ ω
(11)
By substituting the boundary conditions at the upstream and downstream
ends as well { }C = 0 two equations in two unknowns are obtained. A nontrivialsolution of these equations determines the resonant frequencies s i k k k k = +σ ω , = 1, 2, 3,..., of the system (Chaudhry, 1979; Wylie at al., 1993). An
infinite number of eigenvalues, or roots of the characteristic equation, maybe found since the fluid system under consideration is a continuous system
with an infinite number of degrees of freedom. Each of the eigenvaluescorresponds to a solution
′ = =h H x e H x e e s t t i t k k k ( ) ( )
σ ω , ′ = =q Q x e Q x e e
s t t i t k k k ( ) ( )σ ω
(12)
where H(x) and Q(x) are functions of position only. The solutions show thatthe free vibration in the system consists of only harmonics at particular
frequencies, ωk , k = 1, 2,.3, ... , which are the natural frequencies of the
entire fluid system. The sinusoidal fluid motion may either attenuate or
amplify depending on the sign of σ k . In stable hydraulic systems σ k is
negative and the oscillation decays exponentially in time. For this reason,
σ k , k = 1, 2, 3,... is defined as decaying factor. The value of σ k is a
property of the entire system, not just an individual pipeline, and isindependent of position and time.
Since linearized systems are being studied σ k is also independent of
the amplitude of oscillation. It is, however, generally a function offrequency. In the case of resonance the system is not linear any more and
measured and calculated results of H(x) and Q(x) usually cannot match eachother. The mathematical model was linearized and a real system is not linear.
A digital program makes the necessary computations easy, regardless ofthe complexity or size of the system in question (Pejovi}, 1979). Thisprogram and model were used for investigating hydraulic vibrations in thepumped storage power plants, hydro-power plants, pumping stations, highpressure oil hydraulic systems, .... Some results were published in thepapers (Gaji} et al., 1993; Obradovi} et al., 1986; Pejovi}, 1979, 1986,1989a, 1990a, 1990b, 1990c, 1991a, 1991b; Pejovi} et al., 1982, 1983, 1984a,1984b, 1885a, 1985b, 1986a, 1986b, 1986c, 1988, 1991, 1992a, 1993, 1994a,1994b, 1995; Sekulic et al., 1990).
2.3. TRADITIONAL MACHINE MATRIX
The dynamic instability of power plants and pumping systems whichcreated various difficulties from acoustic noise to huge pressure fluctuationand machine and pipe vibrations, making system sometimes inoperable, couldnot be explained by such a simple model of machines (Eq. 7). Most of systemselements are easy understood and described in relatively simple mathematicalform, but hydraulic machines usually represent a more complex boundarycondition. In the last two decades several papers, were presented on thismatter (Anderson et al., 1971; Black et., 1975; Fanelli, 1972; Fanelli etal., 1983, 1984; Guarga, 1990; Goto, 1990; Jacob, 1990; Jacob et al., 1990;Kawata et al., 1987a, 1987b; Ng, 1978; Stirnemann et al., 1987) analyzing theform of machine matrix. Finally it had been presented by a 2x2 matrix
[ ]T T
T =
11 12
21 22
T
T (13)
with its four complex functions of frequency which can be measured in thelaboratory or even in field tests by introducing four pressure transducers,two on suction and two on pressure side if several meters of straight line
exist (Bolleter, 1981).The pump matrix measured in the laboratory, for the operating point: Q
= 1.3 l/s and head 10 m with rotational pump speed 900 rpm under steadyoscillatory conditions, is presented in Fig. 2. The measured wave speed inthe piping system during the tests was 370 m/s since cavitation had happened.So the air bubbles (two-phase flow) reduced the wave speed in the metal pipesdown to this value which is less then a half comparing to the one-phase flowof water. The influence of the machine matrix on stability, response, modeshape, and on the machine operation is very high. It is necessary to have ameasured full data on behavior under oscillatory conditions (Pejovi} et al.,1993; 1994a) to correctly predict system response.
For the analyses of steady one-phase and some undercritical cavitatingtwo-phase flows in turbomachines system matrix (13) reduces to matrix (7) forvery small frequencies only. For elastic machines as well as the cavitating
and two-phase flow the real matrix is the measured one only (Dörfler 1979;Kawata et al., 1987b; Pejovi} et al., 1993, 1994a).
The influence of machine matrix on stability, response, mode shape andgenerally on machine operation is very high and sensitive, so in the case ofimportant analysis, it is necessary to have a measured full data on behaviorunder oscillatory conditions.
2.4. REAL PRESENTATION OF THE MACHINE
Numerous field and model measurements as well as theoretical analysesrevealed that the hydraulic excitation forces developed by the interferenceof fluid and mechanical parts of hydraulic machines induce vibrationproblems. Draft tube outlet vortex core appears at partial-load and over-loadexciting the system with very high amplitudes of pressure and flowfluctuations. As conclusion, to the matrix (13) must be added excitationvector (8). The transfer matrix equation that relates downstream state vector
in terms of upstream state vector has a general form
There are a lot of theoretical analyses of matrix [ ]T and excitation
vector { } E . Both of them are still unknown functions of: machine operatingpoint, cavitation, frequency of excitation, quantity of dissolved gas in theliquid, content of gas bubbles, mechanical elasticity of machine components,etc. Most important and most difficult to determine is the influence ofcavitation since it depend on air content in water and amplitude of pressurehead fluctuation. The quantity of dissolved gas in the liquid is usuallysmall and the diffusion of gas toward gas bubbles is slow too. Very smallquantities of evolve during the steady and transient flows. In the case ofcavitation the quantities of gas bubbles are much higher, but after theircollapse it is still very high since dissolving of gas bubbles is very slow.However, even a small quantity of free gas has a great influence on the wavespeed (Perko, 1984; Wylie, 1993), reducing it down to 20 or 50 m/s in theturbine draft tube, depending on how small is the pressure.
Cavitation is related to NPSH (net positive suction head) or cavitation
coefficient σ (= NPSH/H ). In the case of inlet and outlet pressureoscillation, see Fig 3, head H, NPSH, σ , and other machine characteristicsoscillate too. What is a real cavitation characteristic? It is to bedetermined. How? The pressure and pressure distribution in the turbine orpump influences the intensity of cavitation and air release of dissolved air
in water. Which H, NPSH, and σ correlate to the size of air-vapor cavities inthe machine and content of air in water downstream of the cavitation?
Similarity law of flow in hydraulic machine is valid only in the caseof one-phase flow. If the cavitation coefficient is higher than incipientone, there is no air in the liquid. Guidelines and standards accept thesimilarity of steady flow in hydraulic machines until the critical cavitationcoefficients are reached. Within the zone of cavitation between the incipient
and critical cavitation, see Fig. 4 (Karelin, 1970), it was assumed similarenough for internal flow analyses in machines. However, a small quantity ofgas bubbles will highly reduce the wave speed downstream of cavitation sothat similarity of transients and hydraulic vibrations do not exist. (Lee,Pejovi}, 1995; Pejovi}, 1989b). What are the lines of incipient and criticalcavitation in oscillatory operating condition?
Which a corresponds to the measured wave velocity should determine someadditional correlation. This problem is solved if two or more values of
( ) H k ω , k = 1, 2, ...,K, (K ≥ 2) could be determined from measured data.
Greater number of values of ( ) H k ω increase accuracy of measurement.
4. MEASUREMENT OF MACHINE BEHAVIOR.
Four complex functions of matrix [ ]T and two complex functions of excitation
vector { } E , see Eq. 14, can be measured in the laboratory or in the fieldtests by introducing four pressure transducers, two on suction and two onpressure side, if a several meters of straight pipe exist. For calculation ofsix unknown oscillatory characteristics six equations are necessary for eachoperating point of a machine, see Fig. 5. Three different boundary conditionsis needed for each operating point of the machine. To achieve this, thepiping system should have an additional dead end piping to change the modeshape not influencing operating point of the hydraulic machin.
The same method can be applied to measurement of the matrix andexcitation vector of any part and component of a hydraulic system. The
systems could be: high pressure oil hydraulic systems, aircraft, satellite,and space shuttle systems, human blood artificial circulating systems,cardiovascular physiological pulsation, ... Application in hemodynamics couldbe of great importance (Milnor, 1989).
5. CONCLUSIONS
The aim o hydraulic vibration analysis carried out during design andconstruction is to find if the machines and associated systems can operatesafely avoiding any possible danger to the plant and personnel. Therefore,analysis of resonance and stability is very important.
Stability and behavior of the hydraulic system under steady oscillatingconditions is dependent upon the characteristics of all its parts. Changingsome parameters, such as diameters of penstocks, machines characteristics, or
something else, and computing coefficient of attenuation for naturalfrequencies, as well as responses to the existing forcing exciters, it ispossible to reduce amplitudes of oscillations making stability to be better.
Real, measured characteristics of machines and other components must beavailable for an accurate analysis. Hydraulic systems of airplanes, spaceshuttles, high pressure oil hydraulic systems as well as internal andexternal artificial systems of pulsatile human blood systems should becarefully analyzed too, to discover instabilities.
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