fluid force moments open-type centrifugal compressor...

International Journal of Rotating Machinery2001, Vol. 7, No. 4, pp.237-251Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Rotordynamic Fluid Force Momentson an Open-type Centrifugal Compressor

Impeller in Precessing Motion

YOSHIKI YOSHIDAa’*, YOSHINOBU TSUJIMOTOa, DAIZO YOKOYAMAa,HIDEO OHASHIb and FUMITAKA KANO

aOsaka University, Engineering Science, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan;bKougakuin University, 1-24-2, Nishishijuku, Shinjuku, Tokyo 163-8677, Japan,"

CNara National College of Technology, 22, Yata, Yamatokoriyama, Nara 639-1058, Japan

(Received 15 May 2000; In finalform 23 May 2000)

In recent years, increasing interest has been given to the rotordynamic fluid forces onimpellers, from the view point of the shaft vibration analysis. Previous experimental andanalytical results have shown that the fluid-induced forces and moments on closed-type pump impellers contribute substantially to the potential destabilization ofsubsynchronous shaft vibrations. However, to date few papers are known of therotordynamic fluid forces on open-type centrifugal impellers. This paper reports aboutexperimental investigations of the rotordynamic fluid force moments on an open-typecentrifugal compressor impeller in precessing motion. For open-type impellers, thevariations of the tip clearance and the clearance between the back shroud and casing dueto the precessing motion contribute to the rotordynamic fluid force moments.Experiments were conducted to measure the rotordynamic fluid force moments directlyusing the 4-axis sensor, and the unsteady pressure on the front and back casing wall.

In this paper, following results are obtained: (1) The fluid force moment becomesdestabilizing in the region of negative precessing speed ratio (- 0.3 < f/c < 0), at thedesign flow rate; (2) At reduced flow rate, the destabilizing fluid force moments occurredat small positive precessing speed ratio (0.2 < f/co < 0.4); (3) From the comparison ofdirect measured fluid force moments with those estimated from the unsteady pressuremeasured on the front and back casing walls, it was found that the destabilizingmoments in the backward precession are mainly caused by the fluid forces on the frontsurface of the present impeller, where there is large clearance between the back shroudand casing.

Keywords: Rotordynamic fluid force moment; Centrifugal impeller; Precessing motion; Self-excited vibration; Unsteady pressure; Blade force

Corresponding author, e-mail: [email protected]

237

238 Y. YOSHIDA et al.

INTRODUCTION

For turbomachinery operating at supercriticalshaft speed it is important to understand thecharacters of rotordynamic fluid forces and mo-ments on impeller which occur in response to shaftvibration. Figure l(a) shows the typical vibration

of overhung impeller. The motion of impeller canbe divided into two fundamental modes (Ohashiet al., 1991). One is whirling motion, and the otheris precessing motion. The vibration of a rotatingimpeller (rotational speed ) can be dissolved intotwo modes; whirling with a constant eccentricity,c, as shown in Figure l(b), and precession with a

Bearing I

Bearing

Impeller(a) Vibrating motion of a overhung impeller

Impeller

Precession’orbit

apex angle

Impeller

Cemer f

(b) Whirling motion (c) Precessing motion

FIGURE Motion of a overhung impeller on a vibrating shaft, and two fundamental vibration modes of an impeller.

FLUID FORCE MOMENTS 239

constant shaft inclination, c, as shown inFigure l(c). In the precession, the center of theimpeller, "O", does not move but the direction ofthe shaft inclination rotates with constant speed f.For whirling motion, many experimental and

analytical data have been obtained on the rotor-dynamic fluid forces on pump impellers, mainly atCaltech and the University of Tokyo. It is nowwidely recognized that for closed-type centrifugalimpellers the fluid forces become destabilizing forthe forward whirl generally at whirl speed ratio (/a;) less than 0.5 (Jery et al., 1985, Bolleter et al.,1987, and Ohashi et al., 1988). The destabilizingforces are caused by the unsteady interactionbetween the impeller and volute casing (Adkinset al., 1988, and Tsujimoto et al., 1988A) or vaneddiffuser (Tsujimoto et al., 1988B), or by unsteadyleakage flow between impeller shroud and casing(Childs, 1989, and Guinzburg et al., 1994). Theresults are summarized in the recent textbooks ofChilds (1993) and Brennen (1994).For precessing motion, Ohashi et al. (1991) have

shown that, from direct force measurements on a

closed-type pump impeller, the fluid force mo-ments become destabilizing for the forward pre-cessing speed ratio (f/) less than 0.5. This rangeof destabilizing speed ratio is quite similar to thatof the whirling cases as mentioned above. Yoshidaet al. (1997) and Tsujimoto et al. (1997) investi-gated the unsteady flow in the back shroud/casingclearance of a precessing pump impeller. They alsoobserved that the fluid force moment on the backshroud is destabilizing at small positive precessingspeed ratio. In addition, the frequency range ofdestabilizing moment increases as the increase inthe leakage flow rate, caused by the increase in thesteady circumferential velocity, Vo, in the backshroud/casing clearance.Open-type impellers have been widely used for

high-speed and high-pressure centrifugal compres-sors. For open-type impellers, the variation of thetip clearance due to the shaft vibration contributesto the rotordynamic fluid forces. Nevertheless, asfar as the authors are aware, no rotordynamic datafor these machines are available except the recentreport of Yoshida et al. (1999). Yoshida et al.,

reported experimentally that for an open-typecentrifugal compressor impeller in whirling motionthe fluid force becomes destabilizing at smallpositive whirl speed ratio throughout all the flowrange without the interaction of a volute or vaneddiffuser. In addition, the fluid forces changedramatically into destabilizing near the whirlingspeed ratio f/=0.8 at the low flow rate. Thispeak of destabilizing fluid force is caused by thestrong interaction of the whirling motion with therotating flow instability, similar to "rotating stall",with propagating speed ratio fl/;=0.79.

This paper presents the results from an investi-gation of the rotordynamic fluid force moments onan open-type centrifugal compressor impeller inprecessing motion. For the precessing motion, thevariations of the tip clearance and the clearancebetween the back shroud and casing contribute tothe rotordynamic fluid force moments. Fluid forcemoments were measured directly with a forcebalance device. Discussions on the unsteady pres-sure on the front and back casing walls, and bladeto blade pressure distribution are also presented.

EXPERIMENTAL FACILITY

Description of Test Facility

Figure 2 shows the sketch of the mechanism usedto generate the precessing motion. The inner sleevesupports the main shaft through two eccentricinner bearings set to produce a pure precessingmotion. The main shaft is driven by an AC motorwith the rotational speed (w) through a universaljoint, and the outer sleeve is driven by a DC motorcontrolled to run at a prescribed speed (f). This interm is the precessing speed. The precessing ang-ular velocity, 9t, is defined as positive when it is inthe same direction as the impeller rotation, w. Themain shaft speed was maintained at 400 + rpm,and the precessing speed ratio (f/) was varied inthe range from -1.4 to / 1.4. Uncertainty in theprecessing speed ratio, 2/, is + 0.002.

Figure 3 shows the details around the impeller.The test impeller is a model of an industrial

Shaft center. Casing center

I’ a: apex angle(1) Shaft

(4) Outer bearin

(2) Inner bearin

(3) Inner sleeve

(5) Outer sleeve

(6) 4-Axis force sensor (7) Impeller

O" Center of precession

FIGURE 2 Mechanism to produce the impeller precessing motion.

r4=240

Vaneless Back casing

,

[’ Impeller’?2

Collector

Units

c:0. 19 deg.Back shroud

’:..., :.- I_,:!...r=160rz=149

t, 4-|s free

o

sensor

O Center of precession

FIGURE 3 Cross section of the test rig (impeller, casing and shaft).


centrifugal compressor with 12 blades (Zi), inletblade angle 32 deg. and outlet 45 deg. at the tip(i.e., back swept blade), outer radius (r2) 149mm,exit width (b2) 23 mm; its non-dimensional typenumber is 1.3. The impeller is equipped with avaneless diffuser with radius ratio r4/r2-- 1.61 anda symmetrical collector to minimize the nonuni-formity of exiting flow. Although the test impellerwas designed for gas, water was used as theworking fluid to facilitate the measurement of thefluid forces. The Reynolds number (Re uzrz/u,uz=r2co) is 3.04.0 x 106 for actual condition(air, u2=300400m/s), and 0.92 x 106 for thislaboratory test condition (water, u2 6.2 m/s). Theeffect of compressibility of actual working fluid(gas) is neglected in the test condition (water).Under the condition without a shaft inclination(a 0 deg.), the normal blade tip clearance (’) isconstant (- mm) from the inlet to outlet. In thepresent tests, the center of precessing motion wasplaced at the impeller center "O", (see Fig. 3,placed on the center line of the impeller outletheight), and the apex angle a 0.19 deg. was usedfor the precessing motion. In this condition, thetip clearance varies in the angular direction in therange of 0.7mm 1.3mm at the impeller inlet,and 0.6mml.4mm at the outlet due to theprecessing motion. The clearance between the backshroud and casing varies in the range of19.5 mm 20.5 mm at the impeller outlet (r r2)with the precessing motion.

In present study, we focus on the fluid forcemoment caused by the forces on the front sur-face of the impeller. For fluid force moment onthe impeller back shroud, Tsujimoto et al. (1997)reported analytically and Yoshida et al. (1997) ex-

perimentally that the destabilizing moment causedby the pressure on the back shroud increasesas the increase in the leakage flow through theback shroud/casing clearance due to the in-crease in the steady circumferential velocity, Vo.The magnitude of the moment increases as thedecreases of the axial clearance between backshroud and casing, H, and of the radial clearancebetween impeller side plate and casing, Gap-A (see

Fig. 3), caused by the increase in unsteady pressurefluctuation on the back shroud. From theseresults, the back casing of the present modelcompressor was designed with a wider clearance H(= 20mm), and a wider Gap-A (= 11 mm) thanactual ones, and with no leakage flow in theclearance to minimize the fluid force moment onthe back shroud.

Instrumentation and Data Acquisition System

The impeller is supported by the main shaftthrough a rotating force balance with a 4-axisforce sensor, as shown in Figure 2. The forcebalance is composed of two couples of parallelplates and 4 strain gauges per plate to measure the4-axis forces (2 forces and 2 force moments). Thestrain signals are taken out through a slipring. Theoutput signals of the strain gauges are converted to

two forces and two moments components usinga transfer matrix determined from a dynamiccalibration test. Output signals are ensemble-averaged over 32 precessing orbits based on atrigger signal that indicates the instance whenboth the direction of the shaft inclination and theimpeller rotating angle come to a prescribed orien-tation. Force and force moment are measuredtwice, that is, in air and in water at the samerotational and precessing speed. The former meas-urement values merely inertia force and gyro-scopic moment of the impeller itself due to theprecessing motion, therefore, the fluid-inducedforce and force moment can be obtained by sub-tracting the former from the latter. In thepresent paper we focus only on the fluid forcemoments, which determines the stability for theprecessing motion. Figure 4 shows the coordi-nate system. The r-axis is set in the direction ofmaximum blade tip clearance and the t-axis per-pendicular to it, directed by 90 deg. from ther-axis in the direction of the impeller rotation. Thefluid force moment M is represented with its radial(Mr) and tangential (Mr) component. Measuredfluid force moments are normalized as (mr, mr)=(Mr, Mt)/Mo, where Mo Ica;, I prcr22bzi2,


Precesion’-orbit -"-]11111

.’[_] Back casingBack shroud.(’"’)

Impeller r

Front casing

t

FIGURE 4 Scheme showing the rotordynamic fluid forcemoments. Mr and Mt are the radial and tangential componentto the precessing motion (a: apex angle, : shaft rotationalspeed, 2: precessing speed, (r, t): radial and tangential to theprecessing motion).

v/r/4 + b/12. In the present experiment, un-i=

certainty in the dimensionless fluid force mo-ments rn and mt is -t-2.0. We should note herethat the radial component of fluid force moment,mr, is destabilizing for the precessing motion when

rnr x (f/) > O. (i.e., rnr and f/co are both positiveor both negative.) In this case, rn x (f/) > 0, theradial moment feeds kinetic energy to the vibratingshaft and thus excitatory.P1 P6 on the front casing, and R1 R4 on the

back casing in Figure 3 show the locations ofpressure taps to measure the steady and unsteadypressure. P1 P3, P6, and R1 R4 were used tomeasure the steady pressure with a manometer. Inaddition to this, P1 P5 and R1 R4 were used tomeasure the unsteady pressure using pressuretransducers to estimate the pressure force mo-ments on the impeller (described later). P1 P3 onthe casing, wall are facing the impeller tip, andR1 R4 on the back casing wall are facing theimpeller back shroud. The diameter of the pressuretaps, P1 P3, is mm and silicone oil is filled in

the cavity in front of the pressure transducers. Thepressure transducers at P4, P5 and R1 R4 wereinstalled flush with the casing wall. P4 and P5are located at different circumferencial positions(separation angle 60 deg.) to facilitate the exam-

ination of the circumferential propagation of arotating flow instability. The resonance frequencyof the measurement system is 2.2 kHz, while theblade passing frequency is 80 Hz (= Z x /27r.).

RESULTS AND DISCUSSIONS

Compressor Pressure Performance

Figure 5 shows the static pressure coefficient (b) atthe collector and the steady pressure at variouslocations on (a) front casing, and (b) back casingwall plotted against the flow coefficient (b), underthe condition without the shaft inclination. Thedesign flow coefficient is ba= 0.424. From the flowvisualization through the transparent casing withair bubble, backflow onset at the impeller inletwas observed at b=0.32, where the pressure risereaches a local peak. The performance curve has apositive slope in a range of b 0.30 0.32. For themeasurements of fluid force moments, the flowrate was varied from b=0.285 to b=0.508. Thesteady circumferential velocity on the back shroud,Vo, was estimated from the pressure distribu-tions in Figure 5(b). Assuming that the pressuregradient is balanced by the centrifugal force (i.e.,dp/dr- pV/r- pK22r), it was found that K=Vo/r is in the range of 0.460.51 at the lo-cations R1 R4 throughout all the flow range.This result shows that the swirl on the backshroud has nearly forced vortex pattern, sincethere is no leakage flow in the back shroud/casingclearance.

Fluid Force Moments on Impeller Measuredwith Force Balance

Figure 6 shows the dimensionless radial, m andtangential, mr, fluid force moments on the impeller


0.6

0.2

0

-0.2

Total pressure

’O..o 1,!- Static’Cllectrpressure

0.2 0.4 0.6

Flow coefficient 0

(a) Front casing (PI---P3, P6)

0.6

0 0.2 0.4 0.6

Flow coefficient

(b) Back casing (RI---R4)

Total pressureCollector

Static pressureOR1A R2!1 R3AR4

P6

FIGURE 5 Pressure performance of the test impeller andsteady pressure on (a) front casing wall, and (b) back casingwall without a shaft inclination. Pressure coefficient versusflow coefficient 05 (uncertainty in b + 0.005, in 05 + 0.01).

measured directly by the force balance versus theprecessing speed ratio, f2/co, for various flow rates.At the design flow rate, q5 qSd=0.424, it can beseen that the radial fluid force moment, mr, on theopen-type centrifugal impeller are destabilizing(i.e., mr < 0) for the backward precession in therange -0.3 < f/w < 0. This destabilizing range ofthe fluid force moment extends to the negativedirection of the precessing speed ratio as thedecrease in flow rate. In addition, at lower flowrate, the radial fluid force moments become

destabilizing (i.e., mr > 0) at positive precessingspeed ratio in the range 0.2 < f/a < 0.4. Figure 7shows the comparison of the dimensionless fluidforce moments of the present open-type com-pressor impeller with those of the closed-typepump impeller from Ohashi et al. (1991), at thedesign flow rates respectively. The present open-type impeller has the destabilizing range (mr < O)with negative small precessing speed ratio, whilethe closed-type impeller has the destabilizingrange (mr > 0) with positive small speed ratio.Although the fluid force moments on the open-type impeller are smaller than those of the closed-type impeller, the radial component is roughlylinear and the tangential component is quadraticwith precessing speed ratio for both cases. Asmentioned in the previous section, the clearancebetween the back shroud and casing of the presentimpeller, H- 20 mm, is wider than that of theclosed-type impeller, q 7 ram. This wider clear-ance H and wider Gap-A may be one of the reasonsof the smaller fluid force moments on the presentopen-type impeller than those on the closed-typeimpeller.The peak of the force moment occurred near [2/

co=0.20.3 at low flow rate 05=0.330. For thewhirling motion of the present impeller, Yoshidaet al. (1999) observed that the fluid force increasesdramatically near the whirling speed ratio fUco0.8 at the low flow rate (q5 < 0.32), caused bythe interaction of the whirling motion with therotating flow instability at the impeller inlet,similar to "rotating stall", with propagating speedratio fZ’/w=0.79. Childs (1989) predicted in hiscalculation that the peak of the force on thefront shroud occurred near fUco=0.3 0.5 due tothe fluid structure interaction under the whirlingmotion. On the other hand, Tsujimoto et al.(1987) calculated the fluid forces on a whirlingimpeller in a vaneless diffuser using 2-dimensionalvortical flow analysis. They reported that, at lowflow rate, the tangential fluid force becomes de-stabilizing at the whirling speed ratio close tothe propagating speed ratio f’/co=0.157 of thediffuser rotating stall. In the present experiment,


0 mr, mt15

10

-15-1.5 -1 -0.5 0 0.5 1 1.5

(a)

15O mr, m

-1.5 -1 -0.5 0 0.5 I 1.5

(c)

-10 0

-15 ..............15-1.5 -1 -0.5 0 0.5 1 1.5 -1,5 -1 -0,5 0 0.5 1 1.5

(b) ,i=0.330 (d) ,I)=0.508

FIGURE 6 Dimensionless fluid force moments on the impeller, radial mr and tangential mt components versus precessing speedratio, 2/w, for various flow coefficients , =0.285, 0.330, 0.424(=a), 0.508 (uncertainty in mr, mt +/- 2.0, in /w +/- 0.002, in+/-O.Ol).

Present impeller3-D closed-type pump impeller from Ohashi et al. [1991]

-20-1.5 -1 -0,5 0 0.5 1 1.5

3O

2O

o".2o ..

-1,5 -1 -0.5 0 0.5 1.5

FIGURE 7 Comparison of the dimensionless fluid force moments on the impeller, radial mr and tangential mt components, withthose on the closed-type pump impeller, from Ohashi et al. (1991), at design flow rates respectively.


we couldn’t observe the rotating stall in the vane-less diffuser nor the rotating stall in the impellerinlet at this flow rate q5=0.330 from the meas-urements of the pressure fluctuation at the dif-fuser inlet, P6, and the impeller inlet, P4 and P5.Unfortunately, full explanation of this peak forcemoment near f/c 0.2 0.3 at the low flow ratehas not been obtained.

Fluid Force Moments on the Back Shroudand Front Surface of Impeller

The fluid force moments on the back shroud, andthe front surface of the impeller are calculatedusing the direct measurements of the fluid forcemoment and the unsteady pressure measure-ments on the back casing wall to obtain a betterunderstanding of their origin.

Unsteady pressure on the back casing wall (Apat locations R1, R2, R3, and R4) consists mainly ofthe precessing frequency component. Figure 8(a)shows a typical example of unsteady pressure, Ap,measured on the back casing wall at the locationof R1, for qS=qd=0.424, and f/c=0.61. In this

figure, the horizontal axis represents the phase ofthe precessing during a period, in which theclearance gap is widest at r, and smallest at 0 and27r. The pressure fluctuation with precessing fre-quency, denoted as Apa, was obtained from theFourier analysis of Ap, as shown in the right ofFigure 8(a). The fluid force moments on the backshroud were obtained from the integration of thepressure distribution between the radii r 149 mmand 60mm using Apa at locations R1, 2, 3, andR4.

Figure 9 shows the comparison of the dimen-sionless fluid force moments on the back shroud,m and mr,’ obtained by integrating the pressuredistribution, Apa, with the total moments on theimpeller, mr and mr, measured directly by theforce balance, at =qSd=0.424. In this figure,theoretical moments on the back shroud (fromTsujimoto et al., 1997) are also shown. It is

calculated under the conditions of no leakageflow, K= 0.5, and no resistance at Gap-A. Underthese conditions, the calculated radial com-ponent is 0 for all values of f/c, and the tan-gential component is 0 at f/a=-0.5 and + 1.5with the maximum at f/co= +0.5. It can befound that the tangential fluid force moment,

shows almost the same character as the cal-mt,culated moment with precessing speed ratio.The radial fluid force moment, mr, on the backshroud is nearly 0 in the range -0.3 < f/a < 0,however the radial fluid force moment, mr, on theimpeller directly measured by the force balancebecomes negative (i.e., destabilizing). Further-more, the influence of the flow rate upon the fluidforce moment on the back shroud was very small.To obtain the understanding of the fluid force

moments on the "front surface" of the impeller," and " the fluid force moments on the backmr mt

andshroud, m me, were subtracted from the directmeasured moments on the impeller, mr and mr, atthe same test conditions. Here, we call the sur-face of the impeller except the ""back shroud" asthe front surface". Figure 10 shows the dimen-

" andsionless fluid force moments, m mr, on thefront surface of the impeller estimated by theabove method, versus the precessing speed ratiofor various flow rates. A solid line, mtgyro inFigure 10(c) shows the gyro moment of the fluid inthe impeller due to the precession. This mtgyro isagree fairly well with the tangential moment, mso that for the design flow rate some fraction ofthe fluid force moment on the front surface of theimpeller is caused by the inertia of the fluid in theimpeller. The radial component, m on the frontsurface of the impeller becomes destabilizing(m < 0) for the backward precession with a de-crease in flow rate. This tendency is similar tothe radial component, mr, on the impeller. Fromthese results it can be concluded that the destabi-lizing moment (mr < 0) in backward precession ismainly caused by the forces on the front surfaceof the impeller due to the variation of the tipclearance, or the blade channel height.


V=d=0.424, /O----0.61

0.02Location R1

Precessing phase 2

0.02

0 Precessing phase 2

(a)/Cp on back casing wall

Blade passing

r ]] Location Pl0.1

-0.10 Precessing phase 2

(b) Cp on front casing wall

0.02r

0 Precesslng phase 2

Smax0.I’

0 Precessing phase 2

FIGURE 8 Typical wave from of unsteady pressure ACp, (a) on back casing wall at R1 and (b) on front casing wall at P1, forb ba 0.424, Ct/w 0.61. AC shows the component of frequency f, and AC shows the plot of the amplitude of the blade passingfluctuation (uncertainty in ACp + 0.005, in b + 0.01, in f/w= -t- 0.002).

m. m Total moment on the impellerO mr’, O mt Moment on the back shroud

Theoretical moment on the back shroudfrom Tsujimoto et al. [1997]

15

lO o,

-lO .[1

-15-1.5 -t -0.5 0 0.5 1.5 1.5 -1 -0.5 0 0.5 1.5

[=a=o.424

and tangential components,FIGURE 9 Comparison of the dimensionless fluid force moments on the back shroud, radial m mwith moments on the impeller, mr and mr, for design flow rate b bd= 0.424 (uncertainty in m, mr-- 4- 2.0, in f/w= + 0.002).


15

10

"10

"15

0 mr", mr"

-1.5 -1 -0.5 0 0.5 1.5/o

(a) ([r0.285

mtgyroGyro moment on the fluid

in the impeller

0 mr",

-1.5-1-0.5 0 0.5 1.5

(c) Wd=0.424

0 mr"15

.-" 0 :--.,,.

-15-1.5 -1 -0.5 0 0.5 1.5

(b) =0.330

0 HIr’V mt

-1.5 -1 -0.5 0 0.5 1..5

(d) (I)=0.508

and tangential components,FIGURE 10 Dimensionless fluid force moments on the front surface of the impeller, radial m mversus precessing speed, Ft/co, for various flow coefficients , =0.285, 0.330, 0.424 (=d), 0.508 (uncertainty in 4-0.01, in/o 4- 0.002).

Force Moments on Front Surface of ImpellerEstimated from Unsteady Pressure

Two simple estimations of the force moment on

the front surface of the impeller are employed. Thefirst is the integration of the pressure distributionon the front casing wall. In this case the forcesresulted from the pressure distribution and themomentum transfer at the impeller inlet andoutlet, and the rate of change of fluid momentumin the impeller are neglected. The second is theintegration of the blade forces evaluated from thepressure difference across the blade measured on

the front casing wall. This corresponds to the non-uniform blade loading model in axial flowturbine proposed by Thomas (1958) and Alford(1965).

During one period of the precession, the numberof blades passing by the pressure transducer on thefront casing is Zi ]/1. For the condition shownin Figure 8(b), twenty waves (Z; ]/] 19.7)due to the blade passing are clearly observed. Theamplitudes of the component with blade pass-ing frequency, denoted by Ap and used for theestimation of the blade loading, were obtainedfrom the reading of peak-to-peak values for eachblade passing, as shown in the lower right ofFigure 8(b). On the other hand, the pressurefluctuation at the precessing frequency, denotedby Ap and used for the evaluation of the casingpressure force moment, was obtained from theFourier analysis of Ap shown in the upper right ofFigure 8(b).

248 Y, YOSHIDA et al.

For the casing pressure force moments evalu-ation, the fluid force moments (mp, m")tp areobtained by integrating the pressure distributionsApa from the blade leading edge to trailing edgeon the casing wall. The pressure distributions areinterpolated and extrapolated from Lkpa at P1,P2 and P3. For the blade load evaluation, it isassumed that Ap represents the pressure differ-ence across the blade at the tip, and the pressuredifference is proportional to the square of itsradius from the tip to hub. The force moments

(m’r m")tb are estimated by integrating the assumedpressure differences on the three segments of theblade using/kp at P1, P2 and P3 taking accountof the back swept blade.

Figure 11 shows the comparison of the esti-mated force moments (a) (m, " "mtp), and (b) (mrb,

m’t) with the moments on the front surface ofthe impeller (rn m"), for the design flow rate

(0 4d- 0.424). The estimated moments (mp,m) and (rn, rn) are not in good agreementwith (m, m). Thomas (1958) and Alford (1965)explained the destabilizing mechanism in axialflow turbines from the blade loading nonunifor-mity due to the change in tip clearance. Yoshidaet al. (1999) reported that for the whirling open-type centrifugal impeller some fraction of thefluid force on the impeller is caused by thenonuniform pressure distribution on the casingwall. However, the present results suggest thatthe fluid force moment on the front surface of thetested impeller cannot be approximated by thenonuniform blade loading nor nonuniform pres-sure distribution on the casing wall. Therefore, a

15

10

"10

"151.5 -1 -0.5 0 0.5

0 m", mr" 0 mtp’, lilt"

, o

-0

51.5 -1.5 -I -0.5 0 0.5 1.5

(a) mrp" mtp" 01=d=0.424)

5

0

5

(b) mrb’, mt" (l=d---0;424)

and tangentialFIGURE 11 Comparison of the dimensionless fluid force moments on the front surface of the impeller, radial m mcomponents with (a) moments, mp and mtp, estimated from component of unsteady pressure on the front casing, and (b) moments,mrb" and mtb", estimated from the pressure tifference across the blades, for the design flow =0.424 (uncertainty in -+- 0.01, in/ + o.oo2).


different flow model is needed to explain therotordynamic fluid force moment on open-typecentrifugal impellers. More detailed experimentsand sophisticated analyses are necessary to esti-mate the fluid force moment on open-type centri-fugal impellers in the future.

CONCLUSIONS

From the experimental results and discussions, thefollowing conclusions can be drawn:

(1) For an open- type centrifugal compressorimpeller, the radial fluid force moment be-comes destabilizing for the backward preces-sing motion in the range -0.3 < f/cv < 0 at thedesign flow rate (b=0.424). This destabiliz-ing range of the fluid force moment extends tothe negative direction of the precessing speedratio as the decrease in flow rate (q5 0.2850.330).

(2) At lower flow rate (b=0.2850.330), theradial fluid force moment becomes destabiliz-ing also for the forward precessing motion inthe range 0.2 < f2/a < 0.4.

(3) From the direct measured moments andthe unsteady pressure measurements on theback casing, it was shown that the destabiliz-ing fluid force moments on the impeller atnegative precessing speed ratio (-0.3 < f/c < 0) are mainly caused by the forces onthe front surface of the present impeller, wherethere is a large clearance in the back casingwall.

(4) The force moments on the front surface of theimpeller estimated from the unsteady pressureon the front casing wall, and the pressure dif-ference across the blade don’t agree with thedirect measured force moments.

(5) More detailed experiments and sophisticatedanalyses are needed to explain the fluid forcemoments on open-type centrifugal impel-lers in precessing motion, and get more generalconclusion.

Acknowledgments

The authors wish to express their gratitude for theeffort of Mr. Isamu Fukushima in support of thisprogram as an undergraduate project at OsakaUniversity. The authors would like to thank GasTurbine Society of Japan for permission to publishthis paper that was presented in the 7th interna-tional Gas Turbine Congress 1999, Kobe. Thepresent study was partly supported by the Ministryof Education, Science, Sports and Culture throughthe Grant-in Aid for Developmental ScientificResearch.

NOMENCLATURE

b2

fGap-A

H

mr, mt

impeller axial width at outlet 23 mm(see Fig. 3)coefficient of unsteady pressure Ap,normalized by p(r2)2

coefficient of unsteady pressure Apa,normalized by p(rza)2

coefficient of unsteady pressurenormalized by p(r2cv)frequency (Hz)clearance between impeller side plateand casing 11 mm (see Fig. 3)clearance between back shroudand casing (normal clearance20mm)moment of inertia of a disk with ra-dius r2, thickness, b2 pTrrb2 i2, where

i= r/4 + b22/12velocity ratio (= Vo/rcv)components of fluid force moment;radial (r) and tangential (t) component(see Fig. 4)reference value of moment =Iccv

p:rr22bi, i= r22/4 + blwhere I 12dimensionless fluid force momenton impeller, radial (r) and tangen-tial (t) components, normalized byMo


mr, m

II II

Itmrp, mtp

II IImrb, mtb

mtgyro

PPtl

(r,t)’2

Zi

dimensionless fluid force moment onback shroud, radial (r) and tangential(t) components, normalized by Modimensionless fluid force moment onfront surface of impeller, radial (r) andtangential (t) components, normalizedby Modimensionless fluid force moment dueto pressure distribution on casing wall:Ap, radial (r) and tangential (t)components, normalized by Modimensionless fluid force moment dueto pressure difference across the blades:Ap, radial (r) and tangential (t) com-ponents, normalized by Mogyro moment of the fluid in the impellerdue to the precessionpressuretotal pressure at inletunsteady pressureamplitude of unsteady pressure on

casing wall at precessing frequencypeak-to-peak amplitude of unsteadypressure on casing wall at blade passingfrequencyradiusradial and tangential axis (see Fig. 4)impeller outlet radius 149mm (seeFig. 3)blade tip clearance (normal clearanceS- lmm)timemean circumferential velocity in theclearance between back shroud andcasingnumber of impeller blades 12apex angle of precessing motion 0.19deg.radius of circular whirl orbitfluid densityflow coefficient-flow rate/(27rrb2)pressure coefficient (p-Ptl)/p(rzco)2

precessing angular velocityangular velocity of rotating stall

angular velocity of impellerprecessing speed ratio or whirling speedratiopropagating speed ratio of rotating stall

References

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Brennen, C. E. (1994) Hydrodynamics of Pumps, Concept ETI,and Oxford University Press.

Childs, D. W. (1989) "Fluid Structure Interaction Forces atPump-Impeller-Shroud Surfaces for Rotordynamic Cal-culation", ASME Journal of Vibration, Acoustics, Stress andReliability in Design, 109, 144-151.

Childs, D. W. (1993) Turbomachinery Rotordynamics, Wiley,New York.

Guinzburg, A., Brennen, C. E., Acosta, A. J. and Caughy, T. K.(1994) "Experimental Results for the Rotordynamic Char-acteristics of Leakage Flow in Centrifugal Pump", ASMEJournal of Fluids Engineering, 116, 11O- 115.

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Tsujimoto, Y. and Acosta, A. J. (1987) "Theoretical Studyof Impeller and/or Vaneless Diffuser Attributed RotatingStall and Their Effects on Whirling Instability of Centri-fugal Impeller", Work Group on the Behavior of HydraulicMachinery under Steady Oscillatory Conditions, Lille,France.

Tsujimoto, Y., Acosta, A. J. and Brennen, C. E. (1988A) "Theo-retical Study of Fluid Forces on Centrifugal Pump ImpellerRotating and Whirling in a Volute", ASME Journal ofVibration, Acoustics, Stress and Reliability in Design, 110,263-269.


Tsujimoto, Y., Acosta, A. J. and Yoshida, Y. (1988B) "ATheoretical Study of Fluid Forces on Centrifugal PumpImpeller Rotating and Whirling in a Vaned Diffuser", NASACP. 3026, pp. 307-322.

Tsujimoto, Y., Yoshida, Y., Ohashi, H., Teramoto, N. andIshizaki, S. (1997) "Fluid Moment on a Centrifugal ImpellerShroud in Precessing Motion", ASME Journal of FluidsEngineering, 119, 366-371.

Yoshida, Y., Tsujimoto, Y., Ohashi, H., Saito, A. and Ishizaki, S.(1997) "Measurements of the flow in Backshroud/CasingClearance of Precessing Centrifugal Impeller", InternationalJournal ofRotating Machinery, 3(4), 259-568.

Yoshida, Y., Tsujimoto, Y., Ishi, N., Ohashi, H. and Kano, F.(1999) "The Rotordynamic Forces on an Open-typeCentrifugal Compressor Impeller in Whirling Motion",ASME Journal of Fluids Engineering, 121, 259-265.

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