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90-GT-187

A Contribution to Designing a Radial FlowGas Turbine Stage

F. S. BHINDERProfessor, Engineering Research

Hatfield PolytechnicCollege Lane

HatfieldHerts, AL10 9AB

U.K.

ABSTRACT The performance of a radial inwardflow gas turbine stage depends on the designof the rotor as well as that of the casing.The paper describes a simple procedure fordesigning the complete stage comprising arotor and a single entry nozzle-less volutecasing. The rotor design follows aprescribed mean stream velocity distributionin a step by step manner and the casingdesign uses the radial equilibrium togetherwith the conservation of mass energy andmomentum equations.

NOTATION

A area normal to mean flow direction Cm 2)

b blade height Cm)C absolute flow velocity Cm/s)C specific heat at constant pressure

P

CkJ/kg KDd diameter Cm)f friction loss factorh specific enthalpy CkJ/kg)It meridional length Cm)m mass flow rate Ckg/s)M Mach numberP pressure Cbar or kN/m2)

R gas constant CkJ/kg K)T temperature (K)U rotor tip velocity Cm/s)V relative velocity of flow Cm/s)

Greek Symbols

a absolute flow angle Cdegrees)(3 relative flow angle Cdeg)s, incidence angle Cdeg)y ratio of specific heatsr isentropic efficiencyp gas density (kg/ma)0 azimuth angle Cdeg)

Subscripts

0 stagnation conditions12r etc. stations as shown on Fig. 23

INTRODUCTION

The overall performance of a radial flowturbine stage depends on the designs of boththe stator, generally known as thecasing, and the rotor. The casingaccelerates the flow and directs it into therotor at an optimum value of the absoluteflow angle. The rotor decelerates theabsolute flow and turns it from the radialto the axial plane, and in so doingproduces shaft work. The motion and thethermodynamic states ofthe fluid at variouslocations as it flows through the turbinestage are shown qualitatively in Figs.1(a)and 1(b) respectively.

The fluid motion in the casing as well asin the rotor is very complex, beingcompressible, viscous, three dimensional andinternally unsteady. The analysis methodswhich might take into account some or all ofthese real flow effects would be inevitablyvery involved, consequently they would notlend themselves to use as design tools. Fordesign purposes, simpler models have beenproposed by a number of researchers. Vonder Nuell Cl) and Balje C2,3) suggested thatthe choice of the principal dimensions mightbe based on specific speed and specificdiameter, the two well known similarityparameters widely used for classifyingpumps and hydraulic turbines. Rohlikdeveloped a relationship between specificspeed and a number of important designratios. He argued that for the samespecific speed it was possible to produce a

'Presented at the Gas Turbine and Aeroengine Congress and Exposition—June 11-14, 1990—Brussels, Belgium

Copyright © 1990 by ASME

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large number of rotor shapes by choosingdifferent combinations of these designratios. Therefore the problem was to findthe combination of these ratios at eachspecific speed that would result in maximumefficiency. Wallace et al C , 6) and Bensonand Fisher C7) used one dimensionalanalysis together with empirical loss modelsto determine the principal dimensions ofthe rotor. The hub to shroud and blade toblade profiles were obtained by joining theinlet and outlet sections by means ofmathematically definable smooth curves.The resulting geometry could then be usedfor quasi three dimensional flow studiessuch as those proposed by Katsanis CS).

The efficiency of a rotor is affected by themagnitude of the losses which would, inturn, be influenced by the shape of theflow channels in the blade to blade and hubto shroud planes. The axial length of therotor and the space rate of curvature of theflow passages do not appear to have beenused in the published literature as designvariables.

Published literature dealing with the designand performance of volute casings is scarce.Reference C9) shows a simple relationshipbetween the absolute flow angle at rotorentry, rotational speed, entry temperatureand the maximum value of the ratio Cvolutethroat area/radius of centroid of thisarea). This relationship was based on theassumption of the conservation of mass,energy and angular momentum. It was shownthat the absolute flow angle dependedlargely on the maximum value of CA /r 3,

the influence of the turbine ventrytemperature and speed was rather small. Theratio CA /r ) has since become the

v v max

industry standard for classifyingnozzle-less volute casings for inward flowradial turbines. It is a common practice tomake the ratio CA /r 3 and not the volute

v V

cross sectional area as a linear functionof the azimuth angle.

The aim of this paper is to describe asimple procedure for designing an IFRturbine stage. The design of the rotor usesa prescribed variation of the relativevelocity along the mean streamline. Hencethose losses which depend on the relativevelocity and curvature of the flow path canbe distributed. The casing design assumesthat the mass flow rate reduces as a linearfunction of the azimuth angle.

THE PROPOSED DESIGN METHOD

The fluid dynamic design of a radial flowturbine may be conveniently divided intotwo stages, namely:

ci) the preliminary design dealing withthe choice of the principal dimensions;

cii) the detailed design describing the

cross sectional shapes of the flowpassages at convenient stations along theflow path.

FIG. 1(a) AN IFR TURBINE STAGE

0.

• !r toss due to friction the coshp

1 IIQ^ Qp

^ I

loss due to Incidence

V • Vi(2 ,hteh _

• ^ Qt

Ha toss due totrlctbnD a 2 In the rota.

F4 '^ D^ 2

C'^2I

l H,EL .3 U 2

Entropy

FIG. 1(b) THERMODYNAMIC STATES OF THE WORKINGFLUID IN A IFR TURBINE STAGE

The design specification would normallyinclude: Ci) stagnation pressure andtemperature of gas at entry; Cii) eithermass flow rate of gas or power output. Thedesign may also be subject to two importantconstraints, namely: the rotational speedand the overall size.

Rotor Design For given values of the massflow rate, pressure ratio and the entrytemperature, the concept of specific speedmay be used for selecting the rotationalspeed of turbine wheel. The procedure isshown below:

Specific speed n = I a Cl)H9'4

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Where w = angular velocity Crad/s)V =volume flow rate at rotor exit3

C m3/S)

H = Isentropic specific enthalpydrop CkJ/kg)

Kofsky and Nusbaum CIO) found that theoptimum value of specific speed at thedesign point for small turbines wasapproximately 0.75.

The test turbine was designed by using thefollowing design data

mass flow rate in = 0.15 kg/stotal to static pressure ratio

P iP = 1.5501 3

gas temperature at entry Tai

= 873 K

ambient pressure = 101379 N/m2gas constant R = 0.2866 kJ/kgratio of specific heats

= 1.33

For angular velocity calculations a highdegree of accuracy is not necessary,therefore the flow can be assumed to beisentropic. Hence the gas desity at rotorexit and the specific work output can becalculated from the following relationships:

P3 = PO1CP3/POt)1/y and V. = m/p C2a)

Cy-j)/?

H = R Y1- TOt f 1- C P3/PO, ) IC 2b)

The rotational speed can be found bysubstituting first the design datainequations C2a) and C2b) and then the resultsin equation Cl):

the angular velocity w = 7532 rad/s[71926 rev/min]

For small turbocharger turbines the rotortip velocity usually lies between 300 and350 m/s. The limiting value of the tipvelocity isdictatedby the properties of thedisk material at elevated temperatures andthe permissible level of the stresses.Taking an average value of 320 m/s:

rotor tip diameter d = 320 x 60 x 1000i rr x 71925

= 84.97 mm

For this design the rotor entry diameter wasfixed at 85mm. The valuesof otherdimensionless parameters are given below:

lC1'mass flow parameter m I P O1 = 0.1328

d2 P1 01

d Nspeed parameter 2 - = 0.1014

7 prat

Uvelocity ratio s = 0.702

Cs

In order to calculate the rotor tip widthb, the following relationships can be

written from the rotor entry velocitytriangle, Fig. 2:

tan a tan (3

Cma - Ui tan a + tan (3 ( 3)s s

in lip t_ y C.AIPoiy - 1 y^ 1

Pi

2^

1 2 lC

C

Tmi 1 ? -Il 1 - JC4J

2 sinaI r P 0i

A = B .77.d .b C5)1 1 s i

where B is the blockage factor to takes

account of the blades at rotor entry and theboundary layer effects.

Experience has shown that the design valuesof at and (3 are known to lie within a

narrow range. Jamieson Cli) demonstratedthat optimum efficiency would be obtainedwhen a = 17° . In order to minimise the

i

incidence loss, (3 should not be much1

greater than -10° from the radial direction(12). It is of course necessary to ensurethat at the chosen incidence angle the swirlat rotor outlet is zero. By substitutingthe assumed values of a and (3 in

equations (3) and C5) the rotor tip widthcould be calculated by solving theseequations iteratively.

In order to calculate the principaldimensions of the exducer, it is necessaryto know the gas density and the axialcomponent of the relative velocity at rotoroutlet. However, the density would dependon the loss of stagnation pressure fromrotor inlet to rotor outlet.

AP =AP + AP0 0.orotor tncidenco curvature

+ AP (5)friction

The incidence loss at rotor entry can beestimated by using flat plate model (13),and the other two losses can be determinedfrom the intially prescribed values of themean stream relative velocity. Theprocedure, which was originally developedfor designing centrifugal compressorimpellers (14), is described in thefollowing.

The relative velocity V at any point withinthe flow channel is the vector sum of its

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I

C

components in the radial, axial andtangential direction, i.e.,

y = y + V + V (7)r a v

From rotor entry to rotor exit, V reducesr

from a maximum value to zero. Va increases

from zero to a maximum and V increasesv

tan a1from V

C 8)v1 tana + tan (1 31

to V = U (because for designvz z

C must be zero) C9)vz

By assuming an initial value for themeridional length L, the components of therelative velocity can be drawn as shown inFig. 3. Hence, the relative velocity andthe curvature of the mean streamline can becalculated. Losses due to friction andcurvature along the mean streamline may nowbe calculated in a step by step manner byusing expressions developed by D'Arcy forfriction and Ito C15: for curvature.

The above procedure, together withanumerical optimisation routine, was used tofinalise the passage geometry. The finaldimensions of the rotor are shown in Fig. 4.

1^ f

FIG. 3 RELATIVE VELOCITY ALONG THEMEAN STREAM-LINE

n(_ ---a

SECTION A -A(schematic)

nvavr rntei UutIet velocityvelocity triangle triangle

v VCi E V^ C^u ^

u2Ut

FIG. 2 THE INLET AND OUTLET VELOCITY TRIANGLESIN AN IFR TURBINE STAGE

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4.

FIG. 4 DIMENSIONS OF THE TEST ROTOR

Casing Design The rotor was tested with asingle entry nozzle-less volute casing whichwas designed on the assumption of freevortex flow. The design objectives for thecasing were:

Ca) to accelerate the absolute flowand to direct it in tothe rotor at the prescribedabsolute flow angle with theraliius vector;

Cb) to ensure congrence of the rotorinlet velocitytriangleat every point around the rotorperiphery;

Cc) to achieve Ca) and Cb) with aminimum loss of stagnationpressure.

For the sake of simpilicity it was decidedto keep the cross-section circular. Thedesign data is given in Table 1.

It would be pertinent to mention thatinvestgations on a similar casing (16)revealed that the circular section did notachieve the stated design objectives. Thevariation of a with 0 was large rangingibetween 4° to 21.

TABLE 1

r /r ; A /r ...;

360 : 1.200 ; 1.9676330 : 1.233 3.8443300 : 1.268 ; 5.7492270 : 1.305 ; 7.6759240 : 1.344 ; 9.6117210 : 1.385 11.5391180 : 1.428 13.5089150 : 1.475 ; 15.4583120 : 1.562 17.419690 : 1.580 ; 19.356960 : 1.673 ; 21.302430 ; 1.700 ; 23.7519

PERFORMANCE CHARACTERISTICS

The performance of the turbine was measuredby means of a specially designed vortex airdynamometer. The cross section of thedynamometer is shown in Fig. 5. Theprinciple of operation is that powerdeveloped by the turbine is absorbed by animpeller which draws ambient air and givesangular momentum to it. Flow straighteningvanes which are connected to the statorconvert the angular momentum into torque andtransfer it to the stator. This torque canbe measured directly by using a load cell.

The entry pressure and temperature, exitpressure, gas mass flow rate, rotationalspeed and torque were measured at a range ofpressure ratios and speeds. Theexperimental data was processed to producedimensionless performance parameters. Theresults are shown in Fig. 6.

It can be seen from Fig.6Ca3 that at thedesign pressure ratio of 1.65:1, the valueof the mass flow parameter changes from0.212 to 0.234 as the speed parameter isincreased from 0.0425 to 0.1136. Thisrather narrow flow range suggests that theturbine is highly sensitive to speedchanges. The curve representing the designspeed, however, is fairly flat over a widerange of pressure ratios.

The isentropic efficency vs velocity ratiocurves shown in Fig. 6(b) indicate that atthe optimum value of U /C , i changes fromz •0.755 to 0.83 as turbine speed is increased,the maximum efficiency reached is 83%. Itshould be remembered that the shape of thecross-section of the single entrynozzle-less volute casing used in this studywas circular. A similar casing producedlarge variations of the absolute flow anglea aroundthe rotor perphery. As was

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mentioned earlier, for optimum efficiency a

should be constant close to 17 degrees.

j I ^.20 I

1 I

.la :2L{ .

o.osa'- ' o

- 0.oes^ : yAAA / I 4.0995 ! 4 4

1 ^ i.I ,_ ^' a P P4.1176

^.._ 11_., 1 ' . ` 3.. q .. .1 i ,_ 3j5. : _ .: ll a 1 7

. 1 Ia. 19Ir ._

.^ _2.0 2,1 2.2 2.3

Preisur9 Artjo (P /P

FIG. 6(a) MASS FLOW VS PRESSURE RATIO CHARACTERISTICS

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AL.p 1''

-rte — ---

_____________________-

1/ -.

_;_0.J _--.?- - - - - 0.0568 r ca

0.7 0. 0 831_i--- - ---:-----'---.—-------- 0.0995 AA m.

. _ - _. ._.-._ _ _- _.. _. —._._ -_______ .._.._ _ ____ _ _ _ ._ _ . -0:"1736 ` . C^'L•^ C^

0.2

0.2 0.3 0.6 0.5 0.6 0.7 0.8 0.9

Velocity Ratio (U 2 /C 0 )

FIG. 6(b) THE ISENTROPIC EFFICIENCY VSVELOCITY RATIO CHARACTERISTICS

It may be pertinent to mention that thefirst rotor was machined from wroughtaluminium bar using a simple copy millingmachine. Therefore, the blade thickness fromhub to shroud was constant. This rotorproduced an efficiency of 86%, however, theblades suffered from vibrations. The nextrotor had uniformly tapered blades in thehub to shroud direction. As a result ofthis modification the effiency reduced by 3percentage points.

CONCLUSIONS

C13A procedure for designing radial inwardflow gas turbine rotors has been described.The procedure uses a prescribed mean streamvelocity distribution in a step by stepmanner to calculate the hub to shroudprofile.

C2DLosses due to incidence, friction andturning of the flow channels can becalculated by introducing user defined lossmodels. Since friction and turning lossesare calculated along the mean streamline,therefore the axial length and the spacerate of turning of the flow channels can beoptimised.

C3)The measured performance characteristicsof the turbine are given. They indicatethat the mass flow parameter is verysensitive to speed changes, but at thedesign speed the mass flow rate vs pressureratio curve is fairly flat over a wide rangeof pressure ratios.

C4DThe measured isentropic efficiency isgood, ri = 83%, but further improvement inefficiency may be achieved by optimising thedesign of the casing. The circular crosssection of the casing as used inthese tests,although simple to construct, would not meetthe design criteria

CSDThe procedure described in the paperallows the designer to optimise the axiallength and the meridional profile of the IFRturbine rotor.

REFERENCES

Cl) Von der Nuell W.T. Single stage radial

turbines for gaseous substanceswith high rotative and low specificspeed. ASME Paper 51-F-16 FallMeeting, Minneapolis, Sept. 1951.

C2) Balje O.E. A contribution to theproblem of designing radialturbomachines. ASME Paper 51-F-12 FallMeeting, Minneapolis, Sept.1951 .

(3) BAl je 0. E. A study of design criteriaand matching of turbomachines:Part A - Similarity relations anddesign criteria of turbines. ASMEpaper No. 60-WA-230. Winter AnnualMeeting, New York Nov-Dec 1960.

C43 Rohlik H.E. Analytical determinationof radial inflow turbine designgeometry for maximum efficiency. NASATN D 4384, Oct. 1967

C 53 Wallace F.J. , Baines N.C. andWhitfield A. A unified approach to theone dimensional analysis and design ofradial and mixed flow turbines.ASME Paper No. 76-GT-100, March 1976.

CS) Baines N.C. , Wallace F.J. andWhitfield A. Computer aided design ofmixed flow turbines forturbochargers. ASME Paper No.78-GT-191, April 1978.

C 7a Benson R. S. and Fisher U. A proposedscheme for computer aided designand manufacture of radial turbinerotors. ASME Paper 78-GT-156, April1978.

(8) Katsanis T. Computer program forcalculating flow distribution in aradial-inflow turbine. NASA TM X-52161,1965.

C9) Bhinder F.S. Investigation of flow inthe nozzle-less spiral casing ofa radial inward-flow gas turbine. PaperNo. 21, IMechE Proc. Vol. 184,Part 3C,1969-70

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C10: Kofskey M.G. and Nusbaum W.J. Effectof specific speed on the experimentalperformance of a radial inflowturbine. NASA TN-D 6182, October 1970.

C11:) Jamieson A.W.H. The Radial Turbine.Gas Turbine Principles and

Practice edited by Sir H. Roxbee Co.George Newes, London, 1956.

C12) Knoernschils E.M. The radial Turbinefor low specific speeds and lowvelocity factors. ASME Paper 59-A-294,Journal of Eng. for Power, Jan.,1961.

( 13) Kastner L. J. and Bhinder F. S. A methodfor predicting the performance of acentripetal gas turbine fitted with anozzle-less volute casing. ASME Paper75-GT-65, March, 1975.

C143 Jawad S.N. and Bhinder F.S. The designof a high pressure ratio centrifugalcompressor impeller for small gasturbines. Proc.CIMAC, Paris, 1983.

C15: Ito H. Friction factors for turbulentflows in curved pipes. ASME Trans. D.Journal of Basic Eng. June 1959, Vol810 No.2 p 123

(16) Hussain M, I l yas M and Bhinder F S. Acontribution to designing a nozzle-lessvolute sasing for the inward flowradial gas turbine. IMechE Proc.Turbochargers and Turbocharging VolC35/82, 1982.

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