Relative-edQt Induced SIip tn CenffigalImpellerslfor Engineering S tuden8

T.W. von Backstrdm*

The paper presents a new method for deriving therelative-eddy induced slip factor in centnfugalimpellers in an engineering teaching situation. Thesimple analytical method derives the slip velocity interms of a single relative eddy (SRE) centered on therotor axis instead of the usual multiple (one perblade passage) eddies. Features of the method are:the application of basic fluid dynamics to a consisf-ent control volume and logical determination of znt-pirical constants, The method shows corcect limitingbehaviour for zero blades and for 90" blade anglecombined with unity radius ratio, and excellentagreement with the accurate analytical method ofBusemann. The SRE method meets the main criteriafor presenting slip factor in an engineering teachingsituation. It is suggested as a replacement in theteaching situation for the commonly used methodsof Stodola, Stanitz and Wiesner.

NomenclatureRomanc blade lengthd rotor diametere eddy radius in Stodola derivationF blade angle functionRR radius ratior rotor radiuss blade spacing, distance along integration pathU rotor speedw circulation velocityZ number of rotor bladesGreekB blade anglef circulationIc circle circumference to diameter ratioo slip factorA rotor angular velocitySubscripts

i inletlim limitingvaluep pressure side of blade.r suction side of blade0, 1 ,2,3 blade angle function designations

Additional KeywordsEngineering education, slip factor, centrifugal impeller, eddy-

*Professor, MSA|MechE, Department of Mechanical andMechatronic Engineering, University of Stellenbosch, South AfricaE-mai: twvb@sun.ac.za

induced slip, pump.

1. lntroductionThe rate at which centrifugal compressors and pumps do flowwork is less than that calculated with the assumption that therelative flow at the exit of the rotor follows the blade trailingedges. The reduction in angular momentum imparted to the flowis determined by the slip factor. Engineering teachers andstudents need a reliable method for the calculation of slip factorin centrifugal impellers. It should be based on sound fluiddynamics, and be suitable for classroom derivation. The methodshould be widely applicable in terms of basic impeller geometrysuch as blade number and blade angle, and be relatively accuratefor typical values of impeller radius ratio.

The main mechanism usually considered when predictingslip factor in radial flow impellers is the so-called relative eddy.This is an inviscid flow effect. A fluid element entering a radialflow impeller does not rotate around its own axis with an angularvelocity equal to that of the rotor, but moves around the machineaxis while maintaining a constant orientation relative to themachine casing. Relative to the rotor, however, the fluid elementrotates at an angular velocity equal but opposite to the angularvelocity of the rotor. The relative vorticity of the flow in the rotorwill set up a recirculating flow pattern relative to the rotor. Incentrifugal impellers it results in a change in circumferentialvelocity component relative to the rotor at the rotor exit plane,causing the flow to deviate from the blade direction at the trailingedges.

Directly or by implication, textbooks have generally treatedthe relative eddy as the major factor causing slip in radial flow

b 06uoGlfrCL

604

0.2 0.4 0.6 0.8Blade radius ratio, RR

Figure 1: Busemann slip factor as dependent on RRlor 0 = 30o

exit

R & D Journal, 2007, 23 (l) of the South African Institution of Mechanical Engineering 21

Relative-eddy Induced Slip in Centrifugal lmpellers for Engineering Students

1

0.9

0.8

0.7

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0.5

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0.3

0.2

0.1

00 10 20 30 40 50 60 70 80 90

Blade angle

ollttrLolfrl-o+.()(E

rFCL

a-

-a

Figure 2: Busemann slip factor at RR -

0 as dependent on B

turbomachines, for example Stodolat, Eckert and Schnell2,Fergusoo3, Wislicenus4, Osbornes, Ecku, Watson and JanotaT,Cumpstys, Logarle, Dixon t o, Johnson I r, Wilson and Korakianiti s I 2,Aungiert3, and Saravanamuttoo et al.la. At least, they generallydo not attempt to model the other contributing factors. Dean andYoungtt, Whitfield and Bainesr6 and Japikse and Bainestt dohowever consider the effect of the wake region in the bladepassage, butjet-wake models still require a slip factorcorrelationin the jet flow region where viscous effects do not dominate.2. BackgroundB u semann t t propo sed a remarkable s lip factor prediction method.

2m"lZ)cos

Figure 3: Geometry for slip derivation of Stodola

He analytically solved the inviscid flow field through a series oftwo-dimensional impellers with logarithmic spiral blades. Hegenerated maps of slip factor versus impeller radius ratio, withblade number as parameter, for various blade sweep angles forlogarithmic spiral blades. Blade radius ratio (RR) is the radialdistance of the blade leading edge from the axis divided by thatof the blade trailing edge. Wislicenuso and Wiesnertn repro-duced these maps (for example figure 1). The Busemann mapsindicated that slip factor depends on RR, but below a criticalvalue of RR it is relatively constant, especially for high bladenumbers. Figure2 shows the Busemann slip factors forRR - 0as dependent on blade angle, P.The Busemann values pre-sented here have recently been recalculated by Hassenpflug2o.

oIIttLolf-bou(J(tr

lf-

.g6

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.010 20 30 40 50 60 70 80 90

Blade angleFigure 4: Stanitz-Stodola slip factor compared to Busemann

Unfortunately the Busemann methodrs is mathematicallycomplex and not compact enough for inclusion in text books orderivation in the classroom, So various simplified approacheshave been tried. The most popular of these are the equations ofStodolar and Stanitz2r and the curve fit by Wiesnertn.

Stodolat presented a simplified and popular approximatederivation followed by many textbooks. He inserted a circular-shaped control volume between the blades, near the outer radiusof the rotor (figure 3). The circle touches the suction side trailingedge of one blade and is tangent to the pressure surface of itsneighbour. For a rotor with exit radius, r" and number of blades,Z,theblade spacing tsLnr" lZand the eddy diameter is2e = (2tcr,lZ) cos B, with pthe blade exit angle measured from the radialdirection. Stodola assumed the slip velocity caused by therelative eddy to be equal in magnitude to the speed of rotationof the circular eddy at its rim: Aw = Q e = {Zr"/t(cos P)lZ = U ,tT(cos P)IZ. A recent example of such an approach is the paper ofPaeng and Chung22.

The present study was started because the Stodolat assump-tion that the eddy rim velocity Aw may be applied along the rotorperimeter (the edge of another control volume) as the so-calledslip velocity was difficult to justify, especially in a teaching

22 R & D Journal, 2007, 23 (l) of the South African Institution of Mechanical Engineering

Relative-eddy Induced Slip in Centrifugal lmpellers lor Engineering Students

oltttl-othLo+,()(U

l|-.g6

1.0

0.9

0.8

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0.5

0.4

0.3

0.2

0.1

0.00 1020 3040 506070 8090

Blade angleFigure 5: Wiesner slip factor compared to Busemann

situation. The vaguely defined control volumes in the Stodolatapproach leads to the unrealistic conclusions that blade length(or radius ratio) does not matter, and that in the limit of zeroblades on the impeller, the slip factor is equal to minus infinity.Text books do not generally state the accuracy of the Stodolar,Stanitz2t and Wiesnertn approaches compared to the Busemannt8exact inviscid flow solution. As a consequence students end upwith a rather vague understanding of what causes slip factor,which method to apply where, and how well the popular ap-proaches agree with Busemanntt or with experimental data.

3. Definition of SIip FactorBefore defining slip factor, the normalised slip velocity shouldbe defined. In one common definition the slip is normalised bydividing the slip velocity by the rotor rim speed, and in anotherby the circumferential component of the ideal (slipless), absolutevelocity atthe rotor exit. The second one introduces the compli-cation that the ideal circumferential fluid velocity component isdependent on the flow through the impeller, except in the caseof radial blades (8"= 0) or zero flow through the impeller, whenthe two definitions are equivalent.

Slip factor is one minus the normalized slip velocity. For thesake of simplicity we shall follow Wiesnertn and use the firstdefinition of normalised slip velocity. It is known that in practiceslip factors are not independent of through-flow, but the presentinvestigation will focus on eddy-induced slip, which is indepen-dent of flow.

4. Traditional ApproachesStanitz2t presented the slip factor equation given below, basedon inviscid flow numerical modelling:

e --

1 - 0.63 TtlZ (1)Textbooks such as Dixonto recommend the Stanitzzt equation

(*f)t...i,

Figure 6: Geometry for SRE slip derivation

for use when B

Helative-eddy Induced Slip in Centrifugal lmpellers for Engineering Students

oltt,trL-o*-Lo+.o(E

larCL

.II

U'

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.010 20 30 40 50 60 70

Blade angle80 90

Figure 8: SRE slip factor for F1 = 5 (cosp)o's compared toBusemann

o- 1 /*r B lZoT (3)Figure 5 compares the Wiesner prediction (for RR ( RR,,,,)

with Busemann. The agreement is seen to be excellent for16 blades when B< 80". It is also good for p< 30" for 4or moreblades, but is difficult to find a contiguous region where theagreement is very good.

oilttL-olfrL-o+,o(E

tFCL

II

-a

1.0

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Blade angle70 80 90

None of the methods described above models the Busemanndata consistently well, and all predict the slip factor for thelimiting case of zero blades to be -oo, instead of 0.

5. The Single Relative Eddy ApproachVon Backstrcim23 has recently presented a new, approximatemethod for the derivation of relative-eddy induced slip in cen-trifugal impellers. The most important assumptions are listedbelow:tr Logarithmic spiral rotor blades.tr The two-dimensional control volume consists of a curvedsector bounded by five lines: two logarithmic spirals represent-ing adjacent blades, two radial lines between the blade leadingedges and the axis, and by the rotor perimeter between thetrailing edges (figure 6).tr The flow induced by the relative eddy causes no through-flow.tr There is only one relative eddy in the whole rotor: it revolvesaround the axis and protrudes into the blade passages, and whenit forms separate cells associated with each blade passage, thesecells are included in the main cell centered on the rotor axis (figure6).

With reference to its most distinguishing feature, it was calledthe Single Relative Eddy (SRE) method.

6. Derivation of SRE EquationsThe complete derivation is given by Von Backstrdm23, but itsfundamental principles are :tr Each fluid particle in the rotor has a vorticity equal inmagnitude to twice the rotor angular velocity, relative to therotor.tr There is a single average circulation velocity, Aw around theedges of the relative eddy.tr The integral of the circulation velocity around the controlvolume divided by the control volume area is equal to thevorticity.tr The integration path follows the suction surface from lead-ing to trailing edge, then the rotor exit rim from the blade trailingedge to the next blade pressure side trailing edge, then to itsleading edge, and then around its leading edge from pressure tosuction side (figure 6).

The slip velocity as a fraction of the rotor rim speed is thengiven by:

(4)

F is the sum of the average circulation velocities along theblade suction and pressure surfaces, divided by the averagecirculation velocity along the exit bounddt!, Aw,:

The function F must be found to give good agreement withBusemannt8 or with experimental data.

The norrnal definition of blade row solidity is the blade chorddivided by the spacing, but to keep things simple, we shallreplace the chord by the blade length (in a plane perpendicularto the rotor axis) and use the spacing at the radius, r, of the bladetrailing edges (rotor rim). The solidity is then:

(s)

Figure 9: SREBusemann

24

slip factor for F2 = 5 (cosp)o a5 compared to

R & D Joumal, 2007, 23 (l) of the South African Institution of Mechanical Engineering

Relative-eddy lnduced SIip in Centrifugal lmpellers lor Engineering Students

c I s" -

(r" -

,,) I cos p(2" ,") t z

_

(r -RR)z

2n cos pThe nonnalised slip velocity is then simply:

Ar"(J" l+F(tlr") (t)It is instructive to point out in class that the slip is fundamen-

tally determined by the ratio of the total blade length to theimpeller circumferential length. It is also worth noting in retro-spect that the derivation would have been possible without thelogarithmic spiralblade assumption. This probably explains whyBusemann's values have been successfully used for impellerblades with other shapes.

As defined above slip factor is one minus the normalizedslipvelocity:

e -l -(lw, lU)-

1- vQ+ F @tr")) (8)-l-

Since the magnitude of the other factors affecting slip, likeblade incidence angle, trailing edge pressure gradient relaxationand boundary layer blockage effect (including the existence ofwakes) are also primarily dependent on solidity, solidity shouldcorrelate measured slip factors well, at worst with a different Ffor each family of impellers.

7. The Dependence of F on Blade AngleThe next step in finding F is to determine its dependence on 2,,

ILF_c

{r,Co

II o4iEl+roOQo\)

10 20 30 40 50 60 70 80 90Blade angle

Figure 10: Required values of F tor agreement withBusemann

B and RR. Stodohr and Stanitz2r ignore RR as a parameter,Wiesnerle brings it in only as a correction, and Busemannshowed that at high blade numbers (Z> 8) slip factor is veryweakly dependent on RR, especially for RR < 0.5 (figure 1).Figure 7 shows the SRE prediction of slip factor with radiusratioRR, andF = 30'andforaprovisionalvalueofF = 4.6. Thegeneral trend of the lines of constant blade number is similarto that of figure 1. Both predict d =0 at RR= 1, with oincreasing as RR decreases, but the SRE prediction does notlevel off at low values of RR. Since most impellers have valuesof radius ratio between 0.4 and 0.6, a possible approach to fi ndinga suitable expression for F would be to choose a value that wouldensure good agreement between SRE with RR = 0.5, andBusemann with RR = 0, and then assuming that RR = 0.5 whenRR < 0.5 in the SRE prediction. These assumptions enable us tocalculate values of slip factorforall combinations ofZandBandcompare them, on the same basis as the Stodola, Stanitz andWiesner methods, to the corresponding Busemann values forRR=0.

8. Gomparison of SRE Predictions withBusemannBy finding the values of Fthat would give good agreement withthe data of Busemanntt measured from the graphs in Wiesnerre,vonBackstr0m23 showedthattheequation: F, =Fo (cos B)05, withFo= 5.0, represented the trend well enough. Figure 8 correlateswith the SRE slip factor for F, with accurate Busemann valuesfrom Hassenpflog'0. Also shown are lines of constant solidity.It is apparent that the equation correlates with the Busemanndata accurately when the solidity exceeds 0.5 and the blade angleis less than 70". The agreement can be slightly improved,however if the exponent in the equation is changed from 0.5 to0.45 (equation for Fr). The higher angle limit is then increased to80'(figure9).

When the required value of F for perfect agreement withB usemann is plotted against B for various blade numbers (figure10) it appears thatFr=)aJcos pshouldleadto good agreementfor 4 or more blades and blade angles up to 85o. The reason whyF = 2 when F=90 is that blades oriented at 90o do not deflectthe flow, so that the flow inside and outside the rotor remainsstationary in the absolute frame. The relative velocities over theblade suction and pressure sides are then equal, and equal to therelative velocity along the rotor rim (the slip velocity) and:

--a*o+aw, -aw"+aw" -n^ Aw. Aw" - (9)

Inserting F, into the SRE then leads to figure 1 I , which showsexcellent agreement (within 0.005) with Busemann for all bladeangles right up to 90o, for Z > 2 and c/s" > 0.5, and good agree-ment (within 0.03)forZ> I andc/s">0.25.9. DiscussionThe SRE method for the prediction of relative-eddy induced slipin inviscid flow in centrifugal impellers has been derived, basedon the following:tr Simple assumptions based on sound fluid dynamicsE Consistent control volumesE Logical determination of empirical constantsO Careful check against an accurate analytical method

The proposed equation for eddy-induced slip factor in

(6)

R & D Journal, 2007, 23 (1) of the South African Institution of Mechanical Engineering 25

Relative-eddy Induced Slip in Centrifugal lmpellers for Engineering Students

centrifugal impellers is then:

o:l-( 10)

where RR is taken as 0.5 when RR < 0.5. As pointed out, theSRE slip factor automatically approaches 0 as RR approaches 1 .0(figure 7), without the need for a separate correction as in theWiesnerre method. Application of the SRE equation above to the

ollttLorhLo+,ortrl+rCL

.I

-CN

1.0

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0.5

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Blade angleFigure 1 1: SRE slip factor forBusemann

experimental data presented by Wiesnerre shows excellent agree-ment between the SRE predictions and the Busemann valuesplotted against the modified solidity Fr( c/s"),forblade number,Z varyingfrom3 to 44, blade angle, B from 0" to 82" and radiusratio,RR from0.33 to 0.60 (figure 12).

The various SRE approaches represent the analytical methodof Busemann extremely well. Von Backstrtinf3 has shown thatthe SRE method also predicts a large range of experimental slipfactors as accurately as the method of Wiesnerre. Consequentlystudents can use the SRE method for the prediction of relative-eddy induced slip with confidence.

10. GonclusionsThe SRE method for the prediction of relative-eddy induced slipfactor in centrifugal impellers represents the mathematicallycomplex inviscid method of Busemannrs with sufficient accu-racy for class room use. It is derived by applying basic fluiddynamics to a consistent control volume. It exhibits the correctlimiting behaviour for zero blades and 9O'blade angle combinedwith unity radius ratio, and is more accurate than the methodsof Stodola, Stanitz and Wiesner. The SRE method meets the maincriteria for teaching slip factor, and can replace the commonlyused methods in an engineering teaching situation.

References1. Stodola A, Steam and Gas Turbines, McGraw-HiU, 1927.Reprinted by Peter Smith, New York, 1945.2. Eckert B and Schnell4 Axial- und Radialkompressore[Springer-Verlag, 1961, 345.3. Ferguson IB, The Centrifugal Compressor Stage, Butter-worths, Landon, 1963, 85

- 90.

4. Wislicenus G4 Fluid Mechanics of Turbomachinery, 2ded., in two volumes, Volume One. Dover Publications, Inc.,New York, 1965,269.5. Osbome I4lC Fans, Pergamon Press, Bell and Bain Ltd.,Glasgow, 1966, 129.6. Eck B 1973, Fans, Pergamon Press, Germany, 37.7. Watson N and Janota MS, Turbocharging the InternalCombustion Enging MacMillan Education Ltd. London,1986, 89.8. Cumpsty NA, Compressor Aerodynamics, Longman Scien-tific & Technical, England, 1989, 245

- 249.

9. Logan E Jnr, Turbomachinery Basic Theory and Applica-tions,2nd ed. Revised and expanded, Marcel Dekker, Inc.,New Yorh 1993, 167,248.10. Dixon Sl, Fluid Mechanics, Thermodynamics of Turbo-machinery. Pergamon Press, 1998, 222

- 227.

I I. Johnson RlV, The Handbook of Fluid Dynamics, CRCPress, Springer, U.S.A., 1998, 41-12

- 41-14.

I 2. Wils on DG and Ko rakianitis f,, The Design of High-Effi -ciency Turbomachinery and Gas Turbines,2d ed., PrenticeHaIl, New Jersey, 1998,240.13. Aungier RH, Centrifugal Compressors

-

A Strategy forAerodynamic Design and Analysis, ASME Press, New Yorh2000, 55.14. Saravanamuttoo HIH, Rogers GFC and Cohen II, GasTurbine Theory, 5'h ed. Prentice Hall, Cornwall,2001, 153,I 55.I5. Dean RC and Young IR, The fluid dynamic design ofadvanced centrifugal compressors, Creare TN-244, 1976,5-27.

r+(z+3cos p)((:-*):\''\,. 2ncosp )

2+Fr= 3 cosp compared to

Lo+,C)(E

l|rCL

rIlIo

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00.0 1.0 2.0 3.0 4.0

blade angleSolidity, modified forFigure 12: Busemann slip factors for Wiesner's test casesagainst Fr(c/s")

26 R & D Journal, 2007, 23 (1) of the South African Institution of Mechanical Engineering

Relative-eddy lnduced Slip in Centrifugal lmpellers for Engineering Studenfs

16. Whiffield A and Baines NC, Design of radial turbo-ma-chines, I-ongman Singapore Publishers (Pty) Ltd., Avon,uK, 1990,220

- 231.

17. Japikse D and Baines NC, Introduction to Turbo-machin-ery, Concepts ETI and Oxford University Press, 1997, 4-6

-

4-7.18. Busemann A, Das Fdrderhdhenverh[ltniss radialerKreiselpumpen mit logarithi sch- spiraligen S chaufeln, Ze it -schrift fur Angewandte Mathematik und Mechanik, 1928, E371

- 84.

19. Wiesner FJ, Areview of slip factors for centrifugal impel-lers, Trans. ASME Journal of Engineering for Power 1967,89, 558

- 72.

20. Hassenpflug WC, Personal communication, 2004.21. Stanitz JD, Some Theoretical Aerodynamic Investiga-tions of Impellers in Radial and Mixed-Flow Centrifugal Com-pressors, Cleveland, Ohio, Transactions of the ASME, 1952,74, 473

- 476.

22. Paeng KS and Chung MK, A new slip factor for centrifu-gal impellers, Proc. Instn. Mech. Engrs.,200l,2l5, Part A,64s

- 649.

23. Von Backstrdm TW, A unified correlation for slip factor incentrifugal impellers, ASME Journal of Turbomachinery,January 2006, 128, I

- 10.

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