research article dynamic response in transient operation

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2013 Article ID 735321 11 pageshttpdxdoiorg1011552013735321

Research ArticleDynamic Response in Transient Operation ofa Variable Geometry Turbine Stage Influence ofthe Aerodynamic Performance

Nicolas Binder Jaime Garcia Benitez and Xavier Carbonneau

Departement drsquoAerodynamique Energetique et Propulsion Universite de Toulouse ISAE BP 5403231055 Toulouse Cedex 4 France

Correspondence should be addressed to Nicolas Binder nicolasbinderisaefr

Received 8 April 2013 Accepted 4 July 2013

Academic Editor Dimitrios T Hountalas

Copyright copy 2013 Nicolas Binder et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The transient response of a radial turbine stage with a variable geometry system is evaluated Mainly the consequences of thevariations of the aerodynamic performance of the stage on the response time are checked A simple quasi-steadymodel is derived inorder to formalize the expected dependencesThen an experimental campaign is conducted a brutal step in the feeding conditionsof the stage is imposed and the response time in terms of rotational speed is measured This has been reproduced on differentdeclinations of the same stage through the variation of the stator geometry and correlated to the steady-state performance of theinitial and final operating points of the transient phase The matching between theoretical expectation and results is surprisinglygood for some configurations less for othersThemost important factor identified is themass-flow level during the transient phaseIt increases the reactivity even far above the theoretical expectation for some configurations For those cases it is demonstratedthat the quasi-steady approach may not be sufficient to explain how the transient response is set

1 Introduction

Many applications make use of the interesting propertiesof radial inflow turbines namely the compactness andthe ability of reaching high pressure ratios for acceptableefficiencies The description of the radial turbines operatingin steady flows has been widely presented in the literature[1 2] together with an extensive description of the flowpattern even for off-design conditions (eg in [3]) Somedesign processes have also been presented to ensure the goodperformance of the energy-recovering process of the stage(see [4 5]) The favourable pressure gradient due to theexpansion of the flow through the turbine stage stabilizes theboundary layer behaviour and makes separations unlikely ifthe guidance of the flow is appropriate As a consequence aone-dimensional description of the flow is adequate to reachan accurate description of the radial turbine functioning [4]This is transposable to operational and geometric off-designoperation [6] But the problem gets more complex when the

inlet conditions are time dependent Most of the contribu-tions analysing radial turbines operating in unsteady flows arefound in the literature dedicated to possible turbochargingapplications The turbine stage experiences some severepulsating inlet conditions due to its position in the exhaustmanifold The research activity in that field is currently veryactive A review of the state-of-the-art and of the maincontributors of the field can be found in [7 8] The fact thatthe rotor part of the stage operates in a quasi-steady manneris one of the most significant conclusions of the differentstudies Even if this assessment needs to be generalized(far from the design point the rotor is likely to be moresensitive to some unsteady processes) it points out thatthe performances of the static part of the stage (volute andstator) are strongly affected by the time-variation of the inletconditions

Another kind of time-dependent operation is definedwhen dealing with transient phases of the turbine stage

2 International Journal of Rotating Machinery

There again most of the research is motivated by automotivepropulsion since the turbocharger often experiences such akind of request on a typical run The literature depicts howthe response time of the turbocharger impacts the transientresponse of the engine This is presented through the mod-elling of the engine dynamic response when coupled to a tur-bocharger compared with experimentations (in [9 10]) Theturbocharger is thenmodelled as a systemhaving a first-ordertransfer function which is a good representation accordingto [11] The general assessment is that the mass-moment ofinertia of the rotating assembly is the most dominant factorinfluencing the response time of the turbocharger (eg in[12]) A direct consequence of this variation of the mass-moment of inertia of the turbocharger shaft on the engineoverall response is presented in [10] The issue of reducingthe mass-moment of inertia is that the turbine stage mustsustain high temperatures and thus requires high-densitymaterials of construction Some solutions such as scallopingare recommended to decrease as much as possible thismoment of inertia [12] even if it deteriorates the aerodynamicefficiency Anyway the presumed importance of the momentof inertia of the turbocharger shaft is still in discussionA great part of this discussion is focused on the overallreactivity of the engine for which the mechanical chemicaland thermodynamic inertia of the engine itself should beconsidered (according to [9 13 14]) A marginal part of thediscussion deals with the isolated turbocharger to understandhow its own transient response is affected by other factorsOne solution to increase the reactivity is the use of variablegeometry devices either on the compressor or on the turbinestage This is demonstrated in different studies [13 15 16]and highlights once again the importance of the static partof the stage here on the dynamic response of the rotor Thecontrol strategy of the variable geometry device is then crucialas demonstrated by [13] In [14] is mentioned the fact thatthe efficiency of the stage is of the same order of importanceas the mass-moment of inertia in the establishment of theresponse time It is indeed surprising that such an extensiveknowledge of flow pattern in radial turbines as describedabove is that poorly correlated to the dynamic responsein transient operation No clear correlation between thedynamic performance and the efficiency of the turbine ispresented Since the definition of efficiency in unsteady flowsis a complex matter [17] the only indicator available is theefficiency measured in steady flows The comparison of timescales of the stage (rotor rotation propagation of the pressureormasswave in the stator) to that of a typical transient requesttends to validate this ldquosteadyrdquo indicator as representative ofthe performance But this is questionable since a pressurewave will not be instantaneously distributed all around therotor and will definitely affect the boundary layers in thestage The present work is a first contribution to understandhow the efficiency of a radial turbine stage interferes with itsdynamic response for different geometric configurationsThefirst part of the paper focuses on the theoretical backgroundin order to identify the most relevant factors of the problemThe second part of the paper presents the experimentalapproach and the facility The last part of the paper presentsthe results which are discussed in the conclusion

2 Theoretical Background

The time evolution of the rotational speed during transientoperation is based on a rather simplemechanical equilibriumIf the turbine stage is connected to another device througha solid crank arm (compressor electric generator etc) forwhich the torque is quoted 120591

119861 it can be written as follows

119869 sdot

119889120596 (119905)

119889119905

= 120591119879minus 120591119861 (1)

Since

= 120591119879sdot 120596 (2)

if there is no gearbox plugged between the two systems insteady operation both the torque and the rotational speedof the turbine and of the braking device are equal (thecontribution of the bearings is taken into account as partof the ldquobrakingrdquo device) For the turbine in steady-stateoperation the power extracted from the flow is given by thefirst law of thermodynamics

= 1205781198791198621199011198791198791199051[1 minus (

1

120587119879

)

(120574minus1)120574

] (3)

This statement forbids the direct integration of (1) becausethe dependence of the quantities (mass flow pressure ratioand efficiency) to the rotational speed is not explicit Insteady-state operation a complete map is required to identifythis dependence it does not have a simple analytical expres-sion And in transient operation the validity of this map isnot demonstrated Equally the braking work (

119861) depends

on the rotational speed and also on the characteristics of thedevice to which the turbine stage is connected

The exact modelling of the transient behaviour is not thepurpose of this work But a theoretical reference is needed tomeet the two main objectives listed here

(1) the identification of the most influential factors con-trolling the setting of the response time (in additionto the obvious contribution of the mass-moment ofinertia)

(2) the verification of the validity of a quasi-steadyapproach in transient processes

In that purpose the most simple quasi-steady theoreticalexpression of the response time for transient operation isbuilt The transient phase is defined as follows from aninitial steady-state operating point (denoted with ldquoirdquo indices)a sudden rise (characteristic time of the step lt 10ms) ofpressure and mass-flow at inlet of the stage leads to a secondsteady-state operating point (denoted by ldquof rdquo indices) Onehas to drastically simplify the problem to integrate (1) Thosesimplifications are as follows

(i) an intermediate point is defined (denoted by ldquoi1015840rdquoindices) The transient phase is then decomposedin two subphases [indash1198941015840] and [1198941015840ndashf ] as illustrated inFigure 1

International Journal of Rotating Machinery 3

120587t

i

120596i

120596f

i998400

f

mrt

(a) i-i1015840-f phases in the turbine map

120591

120591i998400

120591f

120591i

i998400

i

120596i 120596f

120596

f

Turbine

Braking device

(b) i-i1015840-f torque-to-rotational speed characteristics

Figure 1 Decomposition of the transient phase from initial (i) to final (f ) operating point through intermediate (1198941015840) operating point

(ii) the first subphase ([indash1198941015840]) considers the instantaneousvariation of the inlet pressure and the mass flowto their final values at a fixed rotational speedThis is supported by the characteristic velocity ofpropagation of the pressure perturbation inside theturbine stage which is near the speed of sound (aspresented in [7] and discussed in [8]) the time scaleis thus shorter than the one of the shaft velocityvariation which keeps its initial value Then 120596

1198941015840 = 120596119894

1198941015840 asymp 119891 and 120587

1198791198941015840 asymp 120587119879119891

The torque 1205911198941015840 is discussed

later

(iii) the second subphase ([i1015840ndashf ]) consists in the variationof the rotational-speed due to the difference of torquebetween turbine and braking device

(iv) a linear evolution of the torque-to-rotational-speedcharacteristic for both the turbine and the brakingdevice is assumed since it is difficult to evaluate it apriori (see Figure 1)

In that quasi-steady situation a simple expression for both theturbine and braking torques can be obtained On the brakingdevice side the functioning characteristic is linear and mustbe compatible with the final operating point of the turbine(to reach the power equilibrium in steady state operation) Itmeans that

120591119861 (119905) =

120591119879119891

120596119891

120596 (119905) (4)

The same treatment is applied on the turbine side namelylinear variation and compatibility with the final operatingpoint

120591119879 (119905) = 120591119879119891

+

120591119879119891minus 1205911198791198941015840

120596119891minus 120596119894

(120596 (119905) minus 120596119894) (5)

Equation (1) then gives

119869

119889120596 (119905)

119889119905

+ (

120591119879119891

120596119891

minus

120591119879119891minus 1205911198791198941015840

120596119891minus 120596119894

)120596 (119905)

=

120596119894

120596119891minus 120596119894

(1205911198791198941015840 minus 120591119879119891) + 1205911198791198941015840

(6)

this equation reduces to a first-order differential equation asfollows

119889120596 (119905)

119889119905

+ [

1205961198911205911198791198941015840 minus 120596119894120591119879119891

119869120596119891(120596119891minus 120596119894)

]120596 (119905) = 119870 (7)

where 119870 is a constant For such an equation it is possibleto identify a time parameter 120581 which is representative of theresponse time of the shaft Its expression is

120581 = 119869

120596119891(120596119891minus 120596119894)

1205961198911205911198791198941015840 minus 120596119894120591119879119891

(8)

An estimation of the torque 1205911198791198941015840 generated by the turbine at

point i1015840 is difficult to give Such a transient operating pointis hardly accessible by measurement The correspondingsteady-state operating point if it exists is generally out of therange of conventional test benches since a large overlappingzone between the speed-lines of the map is required tocapture it This is hardly possible when the braking deviceis a compressor limited by blockage and surgestall regionsA last estimation is then proposed for the torque at thisintermediate step of the transient in order to close the modelas simply as possible

(v) The variations of mass-flow efficiency and pressureratio are supposed to be small between the points 1198941015840and f Then

1198941015840 asymp

119891 which leads to the following

approximation

1205911198941015840 asymp

120596119891

120596119894

120591119891 (9)


The time parameter is written as follows

120581 = 119869

120596119891

120591119879119891

120590 (10)

where 120590 is a parameter only depending on the initial and finalrotational speeds It is written as follows

120590 = (

120596119891minus 120596119894

1205962

119891120596119894minus 120596119894

) (11)

Finally according to (2) and (3) and to the definition ofisentropic speed one gets

120581 =

119869

119891120578119891

8

1198632

4

(

1198804

119862119904

)

2

119891

120590 (12)

The response time to a brutal modification of the operatingpoint from the steady conditions ldquoirdquo to the steady conditionsldquof rdquo obviously depends on moment of inertia of the shaftBut it also depends on the initial rotational speed (through120590) and on the final operating point performance It meansthat a given jump of rotational speed will not have the sameduration depending on the region of the map considered(illustrated in Figure 2) According to the simplified modelproposed in (12) decreasing the moment of inertia willimprove the reactivity of the stage at every position on themap but a general improvement in the stage efficiency willhave the same effect Otherwise if a general improvementcannot be achieved this simple relation points out the factthat the low mass-flow configuration should be favouredin efficiency since the high mass-flow regions are naturallyreactive

This result has been derived with the restrictive hypothe-ses above mentioned The validity of these hypotheses is alsodependent on the position of the operating in the map andon the geometric configuration of the stator For exampleconsidering the mass flow constant during the subphase [1198941015840ndashf ] in a transient which is near the blockage region seemsmorevalid than in other regions of the map or the modification ofthe stator geometry may shift the reduced tip speed from itsoptimum value of 07 The real dependence of the transientresponse on the factors identified still needs to be checked

3 Experimental Approach

The approach is based on a single geometric configuration ofthe rotating assembly radial inflow turbine and centrifugalcompressor (see Figure 3(a)) The stator of the turbine stageis modified to create different flow configurations and checkthe dependence of the factors identified in (12) Threemodifications are imposed and presented in Figure 3(b) (i)opening angle of the stator blades (ii) height of the statorblades and (iii) presence of the spacers and clearances inthe stator The first two modifications aim at the alterationof the mass-flow level inside the stage without changingthe final pressure ratio The last modification interferes withthe efficiency of the stage for the presence of the spacerscreates some clearances at the tip and at the hub of the stator

120587t

Δ120596

Δ120596

Δ120596

Fast resp

Fast resp

Slow resp

mrt

Figure 2 Transient response of the stage depending on the positionof ldquoirdquo and ldquof rdquo in the map for the same rotational speed jump Δ120596

blades and generates some losses It is an artificial methodto change the efficiency of a given operating point Whendealing with variable geometry stators the importance of thereduced section (119878lowast

3) is demonstrated in [6] This parameter

is a nondimensional expression of the throat section of thestator its expression is given in (13) The setting of thisreduced section defines the operating point of the stage Thestator opening angle and height combinations are chosen sothat some different geometric configurations generate similarvalues of reduced section For most of the configurationssome tests with andwithout spacerclearances are conductedThe different geometric configurations are labelled accordingto the opening position of the stator (1 closed 2 intermediateclosedreference 3 referencesdot sdot sdot 5 opened) and subscriptedwith the height of the stator (s small r reference h high)The configurations tested and the associated values of theparameters are presented in Table 1

119878lowast

3=

1198783

1198632

4(11986331198634) sin (120572

3)

(13)

31 Steady-State Measurements The ldquosteady-staterdquo identifi-cation of the initial and final operating points is obtainedon our PTM (Petite TurboMachine) test rig dedicated toturbocharger applications This steady flow test rig allowseither global or local instrumentation of small stages fora wide range of inlet temperatures (from 20∘C to 620∘C)up to 250 gs The air is supplied by a pressure source (6bars) stabilized by two regulation stagesThe compressor andturbine flows are independent They are thermally isolatedfrom the ambient air and from each other A specific circuitsupplies hot and pressurized oil to the bearings with a properregulation Inletoutlet pressure and temperature are mea-sured in order to get the performance of both stages Mass-flow measurement is operated through a Coriolis flowmeterRotational speed is measured using a proximity sensor on thecompressor stage For more details about the test facility see[18]The repeatability quality of the experimental device is thegood indicator for the experimental uncertainties since the


47mm52mm

J = 124 times 10minus5 kgmiddotm2

(a) Shaft characteristics

SpacerBlade angle

(120572)

Blade height(H)

Clearances

(b) Modifiable parameters of the turbine stage

Figure 3 Illustration of the geometric characteristics

Table 1 Characteristics of the stators

Stator Δ120572 119867119867ref 119878lowast

3

1r minus15∘ 1 00743r 0∘ 1 02155r +22∘ 1 05201s minus15∘ 071 00472s minus10∘ 071 00864s +10∘ 071 02301h minus15∘ 113 00863h 0∘ 113 0257

experimental study deals with comparative results The con-fidence ranges are given in Table 2 for the nondimensionalquantities presented in the results

For machining convenience the prototype stators werebuilt with aluminum alloy (AU4G) which does not sustainhigh temperatures The air-flow temperature supplied to theturbine stage is thus left at ambient value The differentpressure ratio lines are then described (120587

119905119904from 135 to 16) by

modifying the loading of the compressor which is controlledby a discharge valve

32 Unsteady Measurements An improvement of the PTMtest rig is developed to meet unsteady measurement require-ments A rapid solenoid valve generates a sharp pressurevariation at the inlet of the turbine stage A flow derivationsupplies air for the stage when the valve is closed so that therotor never stops This flow supply defines the initial oper-ating point Then the valve is opened A rapid modificationof the inlet conditions (the typical time of variation of theinlet pressure is 10 ms) forces the turbine stage to reach thefinal operating point A typical transient phase imposed tothe stage by the valve opening in terms of initial and finalpressure ratios is from 120587

119905119894= 115 to 120587

119905119891= 160 Some

unsteady measurement devices were implemented two fast-response pressure probes at the inlet of the stage and atthe outlet of the compressor stage a fast-response proximity

Table 2 Repeatability uncertainty of the bench

Quantity 95 confidence range of ref conditions119903119905

plusmn05 gs 05120596 plusmn50 rads 05120587119905

plusmn0003 02120578 plusmn05 pts 08

sensor near the compressor impeller to detect the bladepassage

The pressure sensors are conditioned by in-house ampli-fiers designed to set the cut-off frequency 10 times greaterthan the maximum frequency expected in the pressure exci-tation signal The sampling frequency of the data acquisitionsystem is set at 200KHz to ensure an accurate resolution inthe observation of the blade passage by the proximity sensorThe raw signal of the blade passage recorded is postprocessedto extract the temporal evolution of the rotational speed

The initial and final operating points are identified bythe use of the steady-state metrology The extraction of theresponse time of the shaft is performed through the postpro-cess of the unsteady measurements For safety reasons someregulation of the pressure upstream of the solenoid valvehas been implemented This regulation process is less rapidthan the solenoid valve and suffers some lag This makes thepressure step at the inlet of the stage unperfect (see Figure 4)Thus the response time is not accessible through directmeasurements of the raw signal The shaft should behave likea first-order system according to (7) since the initial and finaloperating points are fixed It has thus been decided to usean identification function to express the impulse responseof our system The entry of the system is the pressure riseat inlet and the response is the rotational speed evolutionThe mean square method is applied to match the coefficientsto the experimental evolution of the two quantities Theproduction of the final model is then compared with the dataThe matching between the prediction and the experimentaldata is in the range of 98 for all of the configurationsThe characteristic response time of the first-order model istaken as the estimation of the actual response time of the


0 1 2 3 5 6Time (s)

06

08

1

12

14

16

120587T

4k

6k

8k 120596(r

ads

)

4

10k

12k

14k

Figure 4 Time evolution of the pressure ratio in the stage (uppercurve) and of the rotational speed (lower curve)

stage Ten repetitions of each experiment give access to thestandard deviation of the response time and to the associatedconfidence range at 95 This range is expressed in thefollowing results as error bars around the response time data(typical value plusmn25ms)

33 Numerical Simulations The intermediate torque (1205911198941015840)

is only estimated in the model and never measured Asstated above the validity of the estimation depends on thelocation of the transient phase in the map and cannot beexperimentally confirmed Some numerical simulations werepunctually conducted in order to reach estimations of thetorque for some operating points out of reach of the testbench Those simulations were performed with the Euranussolver of the FineTurbo software suite of Numeca Int Thestator-rotor configuration is meshed with Autogrid 5 (15million cells 119884+ lt 5) The steady simulations (mixing planecondition at the rotorstator interface) use the Spalart-All-maras turbulence model The inlet total conditions outletstatic pressure and rotational speed are imposed as boundaryconditions

4 Results

In this section the analysis of the results is divided in twoparts First one focusses on the factors highlighted by thetheoretical development for their supposed importance in theestablishment of the response time (mainly the final mass-flow and the final efficiency) Second the discrepancy of thetheoretical model is analysed to feed the discussion about therelevance of the quasi-static approach for transient operation

41 Identification of the Influential Factors Figure 5 presentsthe general influence of both the mass-flow and the efficien-cies of the final operating point on the transient responseof the stage The mass-flow and efficiency plotted valuesare relative to that of the final point for the referenceconfiguration (3119903 with spacers) The steady-state part of the

results (Figure 5(a)) shows the performance of the stage forthree opening configurations (closed 1119903 nominal 3119903 andfully opened 5119903) The value of the final pressure ratio is thesame for the three configurations (120587

119905119891= 160) The results

confirm that it is possible to downgrade the efficiency bythe use of the spacers and the clearances without drasticmodifications of the operating point The penalty inducedby the spacers and the clearances is far more important inthe closed configuration than in the fully opened one Thishas already been explained long ago by [19] since the relativeimportance of the clearances increases as the throat sectionof the nozzle decreases In response to the alteration of boththe mass-flow and the efficiency levels the reactivity of thestage is modified as presented in Figure 5(b) The generaltrend is in agreement with the theoretical expectation whenthe mass-flow level in the stage increases at the end of thetransient phase the response time decreases It also decreaseswhen the efficiency is increased (for the same mass-flowlevel across the stage) even if the tendency is less clear Forthe closed configurations the theoretical prediction basedon the steady-state measurements of the initial and finaloperating points is surprisingly accurateThis accuracy is lostfor the two other configurations for which the predictionstrongly overestimates the response time the shaft is almosttwice more reactive than expected by the model The strongdependence of the response time on the final operatingpoint definition is examined in Figure 6 The steady-stateresults show the position of the initial point and that of thethree final operating points in the turbine map for the stageconfiguration 3ℎ The gap in rotational speed between initialand final operating points is successively increased (from Δ120596= 3000 rad sdot sminus1 for transient 1 to Δ120596 = 6000 rad sdot sminus1 fortransient 3) together with the final mass-flow rate and finalpressure ratio as shown in Figure 6(a) Figure 6(b) presentsthe response time of the three imposed transient phases Itdecreases despite the apparent increase of the distance inthe map between the initial and final points The model alsopredicts this trend Equation (12) shows how the ldquogeographicrdquodistance between the initial and final operating points in themap is irrelevant in the prediction of the response time theproximity obviously does not ensure a short transient phaseThe good reactivity is better ensured by a large mass-flowthrough the stage at the end of the transient and to a lesserextent by a good efficiency

The results examined in Figure 6 also reinforce theconclusion that in certain conditions the experimental valueof the response time is strongly overestimated by the quasi-static approach The accuracy of the theoretical prediction isthus investigated

42 Validity of the Quasi-Static Approach A general evalu-ation of the theoretical model is proposed in Figure 7 Allthe configurations are classified in terms of reduced sectionas presented in Figure 7(a) where the corrected mass-flowof the final operating point is plotted This diagram showsthe importance of the value of the reduced section (119878lowast

3)

for variable geometry stages for a given pressure ratio themass-flow values of the operating point collapse on a singletrend This occurs despite the fact that the different stators


04 06 08 1 12 14mf

05

06

07

08

09

1

11

120578 f

Stator 1r-no spacerStator 1r-spacerStator 3r-no spacer

Stator 3r-spacerStator 5r-no spacerStator 5r-spacer

(a) Efficiency as a function of the mass-flow for the final operating point

04 06 08 1 12 14mf



0

005

01

015

02

025

03

120581(s

)

Model

(b) Response time as a function of the mass-flow of the final operatingpoint comparison of the theoretical prediction (12) and experimentalresults

Figure 5 Final operating point and reactivity of the stage for three opening configurations of the stator

1 12 14 16 18 2120587t

0

04

08

12

16

2

m

120596 = 5100 radmiddotsminus1

120596 = 8200 radmiddotsminus1120596 = 10500 radmiddotsminus1


Transient 1

Transient 2

Transient 3

Initial pointFinal point-1

Final point-2Final point-3

(a) Position of the initial and three final operating points in the turbinemap

04 06 08 1 12 14mf

0

005

01

015

02

025

03

120581(s

)

Model

Transient 1Transient 2

Transient 3

(b) Response time as a function of the final mass-flow

Figure 6 Modification of the gap in rotational speed during the transient for the stage 3ℎ


0 02 04 06Slowast3

0

05

1

15m

rtf

Stator 1rStator 3rStator 5rStator 1s

Stator 2sStator 4sStator 1hStator 3h

(a) Corrected mass-flow rate as a function of the reduced section

0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590

Stator 1rStator 3rStator 5rStator 1sStator 2s

Stator 4sStator 1hStator 3hModel

(b) Transient mass as a function of the square of the reduced tip speed

Figure 7 Agreement between theoretical prediction and experimental results for all the configurations for a final operating point of120587119905= 16

differ geometrically (see [6] for more details) The reducedsection actually defines the permeability of the stage aslong as the minimum section of the stage is in the statorThis classification in terms of reduced section is importantbecause the validity of the theoretical prediction of 120581 ispartially related to it A reversal formulation of (12) is

120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)

This specific formulation is convenient because there isa separation between the response time and the aerodynamicperformance on the left-hand side (which are results) and therequest in terms of final rotational speed with the characteris-tics of the shaft on the right-hand side (which are imposed bythe transient operation) Both terms of the equation have thedimension of a mass On the left-hand side this ldquotransientmassrdquo is roughly the mass of fluid crossing the stage duringthe transient phase and is supposed to evolve linearly withthe square of the final reduced tip speed of the rotor Theslope of the line (81198691198632

4) is a characteristic mass of the

shaft which is common to all configurations This supposedlinearity is checked in Figure 7(b) where the ldquotransient massrdquois plotted as a function of the square of the reduced tipspeed What was observed in Figure 5(b) is here generalizedto every configuration tested The experimental results andthe theoretical predictions are in a good agreement for someconfigurations those having a small value of the reducedsection A strong discrepancy appears for the others whenthe reduced section of the geometry is increased The actual

response time is then much shorter than the prediction ofthe quasi-static approach almost halved For the theoreticalmodel it means that

(1) either the quasi-static approach is correct and all theother approximations (such as the decomposition ofthe transient phase in two subphases the linearity ofthe torques and the estimation of the intermediatetorque (120591

1198791198941015840)) lead to the deviation of the model

(2) or those approximations are acceptable but withthe quasi-static approach some phenomena in thetransient phase are out of reach

Whatever its origin is the deviation is inactive for theconfigurations having a small value of the reduced sectionand it is very intense for the others This duality is surprisingsince the different stages tested are not extremely differentfrom one another in terms of geometry and the hypothesesof the model are always the same

Some additional tests have thus been carried out toincriminate one of the two previous propositions For thefirst one the possible decomposition in two subphases issupported by the literature and by recent unsteady mea-surements in the intermediate locations of the stage itseems reasonable enough The linearity of the torque duringthe phase [1198941015840ndashf ] has been checked for the braking torqueby reproducing the expected transient phase through asuccession of steady points In that quasi-static approachthe linearity has been observed For the turbine the sameapproach has not been possible on the complete segment [i1015840ndashf ] because of the limitations imposed by the blockage and


0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)


Stator 4sStator 1hStator 3hy = x

(a) Comparison for the different stator geometries

0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm

)Stator 3rStator 5rStator 3r-CFD

Stator 5r-CFDy = x

(b) Comparison of the CFD prediction of 1205911198941015840 with 120591

11989410158401015840 for two stator

geometries

Figure 8 Comparison of the initial estimation of the torque at the point 1198941015840 given by (9) with the back-deduction of the torque through theresponse time (120591

11989410158401015840 )

surge regions of the compressor used as ldquobraking devicerdquo butthe portion observed was also linear

Only two possible explanations remain a strong underes-timation of 120591

1198791198941015840 for some configurations and the occurrence

of an unsteady phenomenon activating the response timeNo measurement of 120591

1198791198941015840 can be reached experimentally (the

steady representation of point 1198941015840 corresponds to a high mass-flow (

1198941015840 asymp

119891) and pressure-ratio (120587

1199051198941015840 asymp 120587

119905119891) and a

small rotational speed (1205961198941015840 = 120596

119894) it is far over the possible

absorption of energy of our compressor at this rotationalspeed for steady conditions) But though (8) it is possible toback-deduce a value of 120591

1198791198941015840 from the experimental results of

the response time This back-deduction quoted 12059111987911989410158401015840 is thus

the value that the instantaneous torque should take at point1198941015840 to match the measured response time It is compared inFigure 8(a) to the initial estimation of 120591

1198791198941015840 given by (9) and

implemented in the quasi-static model For the configurationpresenting a small reduced-section value (white symbols inFigure 8(a)) as another expression of the good matchingbetween the prediction and the results we find that 120591

11987911989410158401015840 asymp

1205911198791198941015840 For intermediate and high values of the reduced section

(grey and black symbols) the torque back deduced from theresponse timemeasurement is far more important up to 30more than the approximation Indeed the approximationproposed in (9) is quite simplistic and does not have an equalvalidity for the different stator configurations or the differentregions of the map This validity is checked through the

simulation of the operating point 1198941015840 with CFD calculationssince it is out of the possible range of the test bench Somesimulations of the operating points 119894 and 1198941015840 were performedfor the stators 3119903 and 5119903 (see Section 3 for the details ofmethodology)The simulation of the point 119894 aims at validatingthe torque predicted by CFD compared with the steadyresults Then the same methodology and postprocess areapplied for the prediction of the torque for 1198941015840 This predictionis hopefully more accurate than (9) for the point 1198941015840 andcompared with the results of the quasi-steady approach (120591

1198791198941015840

simulated comparedwith 12059111987911989410158401015840 back-deduced)The results are

presented in Figure 8(b) and show that the approximationof (9) is acceptable More important even with the properestimation of 120591

1198791198941015840 the quasi-static prediction still fails to

match the measurement even if it gets closer to it For thenominal configuration a gap is still present For the full-opened configuration the torque at point 1198941015840 is still morethan 20 lower than the value compatible with the measuredresponse time

Summarizing the quasi-steady inaccuracy increases asthe characteristic time of the transient decreases Since theshort response-time cases are also the ones for which themass-flow is important this inaccuracy is observed for theconfigurations presenting the less restrictive stator geome-tries (full-opened geometries)The fact that the response timeis significantly shorter than the one predicted is noteworthyThus the possibility of a phenomenon out of the scope of


the quasi-steady approach rises but still needs to be verifiedThe creation of an unsteady ldquoovertorquerdquo due to the passageof the pressure front across the stage during the brutaltransient could be considered Since the pressure front isprobably damaged by the very restrictive closed nozzles itis not active for the small reduced-section configurationsWhen the front keeps some coherence through the statorthe transient phenomenon appears and the torque 120591

1198941015840 is

reinforced to an instantaneous value 12059111989410158401015840 This has not been

demonstrated and is still speculative It has not been observedbut it is not surprising since most of the unsteady resultsfound in the literature concern periodic pulsating flows fornominal geometries of the stator (for those configurations thetransient phenomenon is not strongly active)

5 Conclusion

Some measurements of the transient response of a variablegeometry turbine have been conducted and compared tosome theoretical predictions The matching is good forsome opening configurations of the stator less for othersMore specifically when the reduced section of the stator isimportant some very strong reactivity has been observedThe different conclusions can be summarized as follows

(1) It is possible to change the transient response of ashaft without changing its polar moment of inertiaA substantial gain of efficiency will produce a gain ofreactivity

(2) The main factor is the mass flow rate To increasethe reactivity of a given shaft through a transientoperation the most efficient way is to increase themass flow level The geometry of the stator should beadapted to the increase of mass-flow rate

(3) For configurations presenting an important valueof the reduced section the actual response time issignificantly smaller than the quasi-steady prediction

(4) An unsteady creation of torque is suspected if notdemonstrated The origin of this ldquoovertorquerdquo is con-jecturally related to the passage of the pressure front

(5) If the geometry of the stator is too restrictivethe ldquoovertorquerdquo is ineffective and the quasi-steadyapproach is accurate enough to predict correctly theresponse time

This process needs to be verified in detail But should itbe demonstrated this could explain some of the unsteadymeasurements of efficiency presented in the literature whichare above 1 even if it is also agreed that the inappropriatedefinition of the efficiency in an unsteady context is also partof such amazing results

Nomenclature

Quantities

119862119901 Specific heat at constant pressure (J(kg sdot K))

119862119904 Isentropic speed (ms)

1198634 Rotor inlet diameter (m)

H Stator-blade opening height (m)J Mass-moment of inertia (kg sdotm2)1198783 Stator throat section

119878lowast

3 Stator reduced section

T Temperature (K)t Time (s)1198804 Rotor inlet tip speed (ms)

Power (W) Mass-flow rate (kgs)120572 Stator-blade opening angle120596 Rotational speed (rads)120578 Efficiency120574 Specific heat ratio120581 Response time (s)120587119879 Turbine pressure ratio

120591 Torque (N sdotm)120590 Transient parameter

Subscripts

B Relative to the braking devicef Final point of the transienti Initial point of the transient1198941015840 Intermediate point of the transientrt Reduced quantityt Total stateT Relative to the turbine

References

[1] H Moustapha M Zelesky N C Baines and D Japiske Axialand Radial Turbines Concepts ETI 2003

[2] B Lakshminarayana Fluid Dynamics and Heat Transfer ofTurbomachinery John Wiley and Sons New York NY USA1996

[3] G Cox A Roberts and M Casey ldquoThe development of adeviation model for radial and mixed-flow turbines for use inthroughflow calculationsrdquo in Proceedings of the ASME TurboExpo Conference vol 7 pp 1361ndash1374 Orlando Fla USA June2009

[4] R H Aungier Turbine Aerodynamics Axial-Flow and Radial-inFlow Turbine Design and Analysis ASME press 2006

[5] A Whitfield and N C BainesDesign of Radial TurbomachinesLongman Scientific and Technical New York NY USA 1990

[6] N Binder S le Guyader and X Carbonneau ldquoAnalysis ofthe variable geometry effect in radial turbinesrdquo Journal ofTurbomachinery vol 134 no 4 Article ID 041017 9 pages 2011

[7] D E Winterbone and R J Pearson ldquoTurbocharger turbineperformance under unsteady flowmdasha review of experimentalresults and proposed modelsrdquo In IMechE C554031 1998

[8] N C Baines Turbocharger Turbine Pulse Flow Performance andModellingmdash25 Years on Concepts NREC 2010

[9] D A Ehrlich Characterization of unsteady on-engine tur-bocharger turbine performance [Purdue University Thesis] 1998

[10] C D Rakopoulos and E G Giakoumis ldquoAvailability analysisof a turbocharged diesel engine operating under transient loadconditionsrdquo Energy vol 29 no 8 pp 1085ndash1104 2004


[11] G G Venson and J E M Barros ldquoTurbocharger dynamicanalysis using first order sytem step responserdquo In GT2009-59822 2009

[12] NWatson and S JanotaTurbocharging the Internal CombustionEngine MacMillan New York NY USA 1982

[13] R Barkhage Evaluation of a variable Nozzle turbine tur-bocharger on a diesel engine under steady and transient con-ditions [PhD thesis] Chalmers University of TechnologyGAuteborg Sweden 2002

[14] C Brustle J Wagner K T Van and K Burk ldquoTurbochargingtechniques for sports car enginesrdquo IMechE C405055 1990

[15] Z Filipi Y Wang and D Assanis ldquoEffect of variable geometryturbine (vgt) on diesel engine and vehicle system transientresponserdquo SAE Paper 2001-01-1247 2001

[16] H Uchida ldquoTransient performance prediction for turbocharg-ing systems incorporating variable-geometry turbochargersrdquoRampD Review of Toyota CRDL vol 41 no 3 2006

[17] A Suresh D C Hofer and V E Tangirala ldquoTurbine efficiencyfor unsteady periodic flowsrdquo Journal of Turbomachinery vol134 no 3 Article ID 034501 6 pages 2011

[18] N Binder X Carbonneau and P Chassaing ldquoInfluence of avariable guide vane nozzle on the design parameters of a radialturbine stagerdquo in Proceedings of the 6th European Conference onTurbomachinery Fluid Dynamics and Thermodynamics LilleFrance 2005

[19] P L Meitner and A J Glassman ldquoOff-design performance lossmodel for radial turbine with pivoting variable-area statorsrdquoNASA-TP- 1708 1980

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



There again most of the research is motivated by automotivepropulsion since the turbocharger often experiences such akind of request on a typical run The literature depicts howthe response time of the turbocharger impacts the transientresponse of the engine This is presented through the mod-elling of the engine dynamic response when coupled to a tur-bocharger compared with experimentations (in [9 10]) Theturbocharger is thenmodelled as a systemhaving a first-ordertransfer function which is a good representation accordingto [11] The general assessment is that the mass-moment ofinertia of the rotating assembly is the most dominant factorinfluencing the response time of the turbocharger (eg in[12]) A direct consequence of this variation of the mass-moment of inertia of the turbocharger shaft on the engineoverall response is presented in [10] The issue of reducingthe mass-moment of inertia is that the turbine stage mustsustain high temperatures and thus requires high-densitymaterials of construction Some solutions such as scallopingare recommended to decrease as much as possible thismoment of inertia [12] even if it deteriorates the aerodynamicefficiency Anyway the presumed importance of the momentof inertia of the turbocharger shaft is still in discussionA great part of this discussion is focused on the overallreactivity of the engine for which the mechanical chemicaland thermodynamic inertia of the engine itself should beconsidered (according to [9 13 14]) A marginal part of thediscussion deals with the isolated turbocharger to understandhow its own transient response is affected by other factorsOne solution to increase the reactivity is the use of variablegeometry devices either on the compressor or on the turbinestage This is demonstrated in different studies [13 15 16]and highlights once again the importance of the static partof the stage here on the dynamic response of the rotor Thecontrol strategy of the variable geometry device is then crucialas demonstrated by [13] In [14] is mentioned the fact thatthe efficiency of the stage is of the same order of importanceas the mass-moment of inertia in the establishment of theresponse time It is indeed surprising that such an extensiveknowledge of flow pattern in radial turbines as describedabove is that poorly correlated to the dynamic responsein transient operation No clear correlation between thedynamic performance and the efficiency of the turbine ispresented Since the definition of efficiency in unsteady flowsis a complex matter [17] the only indicator available is theefficiency measured in steady flows The comparison of timescales of the stage (rotor rotation propagation of the pressureormasswave in the stator) to that of a typical transient requesttends to validate this ldquosteadyrdquo indicator as representative ofthe performance But this is questionable since a pressurewave will not be instantaneously distributed all around therotor and will definitely affect the boundary layers in thestage The present work is a first contribution to understandhow the efficiency of a radial turbine stage interferes with itsdynamic response for different geometric configurationsThefirst part of the paper focuses on the theoretical backgroundin order to identify the most relevant factors of the problemThe second part of the paper presents the experimentalapproach and the facility The last part of the paper presentsthe results which are discussed in the conclusion

2 Theoretical Background

The time evolution of the rotational speed during transientoperation is based on a rather simplemechanical equilibriumIf the turbine stage is connected to another device througha solid crank arm (compressor electric generator etc) forwhich the torque is quoted 120591

119861 it can be written as follows

119869 sdot

119889120596 (119905)

119889119905

= 120591119879minus 120591119861 (1)

Since

= 120591119879sdot 120596 (2)

if there is no gearbox plugged between the two systems insteady operation both the torque and the rotational speedof the turbine and of the braking device are equal (thecontribution of the bearings is taken into account as partof the ldquobrakingrdquo device) For the turbine in steady-stateoperation the power extracted from the flow is given by thefirst law of thermodynamics

= 1205781198791198621199011198791198791199051[1 minus (

1

120587119879

)

(120574minus1)120574

] (3)

This statement forbids the direct integration of (1) becausethe dependence of the quantities (mass flow pressure ratioand efficiency) to the rotational speed is not explicit Insteady-state operation a complete map is required to identifythis dependence it does not have a simple analytical expres-sion And in transient operation the validity of this map isnot demonstrated Equally the braking work (

119861) depends

on the rotational speed and also on the characteristics of thedevice to which the turbine stage is connected

The exact modelling of the transient behaviour is not thepurpose of this work But a theoretical reference is needed tomeet the two main objectives listed here

(1) the identification of the most influential factors con-trolling the setting of the response time (in additionto the obvious contribution of the mass-moment ofinertia)

(2) the verification of the validity of a quasi-steadyapproach in transient processes

In that purpose the most simple quasi-steady theoreticalexpression of the response time for transient operation isbuilt The transient phase is defined as follows from aninitial steady-state operating point (denoted with ldquoirdquo indices)a sudden rise (characteristic time of the step lt 10ms) ofpressure and mass-flow at inlet of the stage leads to a secondsteady-state operating point (denoted by ldquof rdquo indices) Onehas to drastically simplify the problem to integrate (1) Thosesimplifications are as follows

(i) an intermediate point is defined (denoted by ldquoi1015840rdquoindices) The transient phase is then decomposedin two subphases [indash1198941015840] and [1198941015840ndashf ] as illustrated inFigure 1


120587t

i

120596i

120596f

i998400

f

mrt


120591

120591i998400

120591f

120591i

i998400

i

120596i 120596f

120596

f

Turbine

Braking device




1198941015840 = 120596119894

1198941015840 asymp 119891 and 120587

1198791198941015840 asymp 120587119879119891


later




120591119861 (119905) =

120591119879119891

120596119891

120596 (119905) (4)


120591119879 (119905) = 120591119879119891

+

120591119879119891minus 1205911198791198941015840

120596119891minus 120596119894

(120596 (119905) minus 120596119894) (5)


119869

119889120596 (119905)

119889119905

+ (

120591119879119891

120596119891

minus

120591119879119891minus 1205911198791198941015840

120596119891minus 120596119894

)120596 (119905)

=

120596119894

120596119891minus 120596119894

(1205911198791198941015840 minus 120591119879119891) + 1205911198791198941015840

(6)


119889120596 (119905)

119889119905

+ [

1205961198911205911198791198941015840 minus 120596119894120591119879119891

119869120596119891(120596119891minus 120596119894)

]120596 (119905) = 119870 (7)


120581 = 119869

120596119891(120596119891minus 120596119894)

1205961198911205911198791198941015840 minus 120596119894120591119879119891

(8)




1198941015840 asymp


approximation

1205911198941015840 asymp

120596119891

120596119894

120591119891 (9)



120581 = 119869

120596119891

120591119879119891

120590 (10)


120590 = (

120596119891minus 120596119894

1205962

119891120596119894minus 120596119894

) (11)


120581 =

119869

119891120578119891

8

1198632

4

(

1198804

119862119904

)

2

119891

120590 (12)





120587t

Δ120596

Δ120596

Δ120596

Fast resp

Fast resp

Slow resp

mrt





119878lowast

3=

1198783

1198632

4(11986331198634) sin (120572

3)

(13)



47mm52mm



SpacerBlade angle

(120572)

Blade height(H)

Clearances





3




119905119904from 135 to 16) by



119905119894= 115 to 120587

119905119891= 160 Some










0 1 2 3 5 6Time (s)

06

08

1

12

14

16

120587T

4k

6k

8k 120596(r

ads

)

4

10k

12k

14k





4 Results




119905119891= 160) The results




3)



04 06 08 1 12 14mf

05

06

07

08

09

1

11

120578 f




04 06 08 1 12 14mf



0

005

01

015

02

025

03

120581(s

)

Model



1 12 14 16 18 2120587t

0

04

08

12

16

2

m




Transient 1

Transient 2

Transient 3




04 06 08 1 12 14mf

0

005

01

015

02

025

03

120581(s

)

Model


Transient 3




0 02 04 06Slowast3

0

05

1

15m

rtf




0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590






120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)











0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120587t

i

120596i

120596f

i998400

f

mrt


120591

120591i998400

120591f

120591i

i998400

i

120596i 120596f

120596

f

Turbine

Braking device




1198941015840 = 120596119894

1198941015840 asymp 119891 and 120587

1198791198941015840 asymp 120587119879119891


later




120591119861 (119905) =

120591119879119891

120596119891

120596 (119905) (4)


120591119879 (119905) = 120591119879119891

+

120591119879119891minus 1205911198791198941015840

120596119891minus 120596119894

(120596 (119905) minus 120596119894) (5)


119869

119889120596 (119905)

119889119905

+ (

120591119879119891

120596119891

minus

120591119879119891minus 1205911198791198941015840

120596119891minus 120596119894

)120596 (119905)

=

120596119894

120596119891minus 120596119894

(1205911198791198941015840 minus 120591119879119891) + 1205911198791198941015840

(6)


119889120596 (119905)

119889119905

+ [

1205961198911205911198791198941015840 minus 120596119894120591119879119891

119869120596119891(120596119891minus 120596119894)

]120596 (119905) = 119870 (7)


120581 = 119869

120596119891(120596119891minus 120596119894)

1205961198911205911198791198941015840 minus 120596119894120591119879119891

(8)




1198941015840 asymp


approximation

1205911198941015840 asymp

120596119891

120596119894

120591119891 (9)



120581 = 119869

120596119891

120591119879119891

120590 (10)


120590 = (

120596119891minus 120596119894

1205962

119891120596119894minus 120596119894

) (11)


120581 =

119869

119891120578119891

8

1198632

4

(

1198804

119862119904

)

2

119891

120590 (12)





120587t

Δ120596

Δ120596

Δ120596

Fast resp

Fast resp

Slow resp

mrt





119878lowast

3=

1198783

1198632

4(11986331198634) sin (120572

3)

(13)



47mm52mm



SpacerBlade angle

(120572)

Blade height(H)

Clearances





3




119905119904from 135 to 16) by



119905119894= 115 to 120587

119905119891= 160 Some










0 1 2 3 5 6Time (s)

06

08

1

12

14

16

120587T

4k

6k

8k 120596(r

ads

)

4

10k

12k

14k





4 Results




119905119891= 160) The results




3)



04 06 08 1 12 14mf

05

06

07

08

09

1

11

120578 f




04 06 08 1 12 14mf



0

005

01

015

02

025

03

120581(s

)

Model



1 12 14 16 18 2120587t

0

04

08

12

16

2

m




Transient 1

Transient 2

Transient 3




04 06 08 1 12 14mf

0

005

01

015

02

025

03

120581(s

)

Model


Transient 3




0 02 04 06Slowast3

0

05

1

15m

rtf




0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590






120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)











0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120581 = 119869

120596119891

120591119879119891

120590 (10)


120590 = (

120596119891minus 120596119894

1205962

119891120596119894minus 120596119894

) (11)


120581 =

119869

119891120578119891

8

1198632

4

(

1198804

119862119904

)

2

119891

120590 (12)





120587t

Δ120596

Δ120596

Δ120596

Fast resp

Fast resp

Slow resp

mrt





119878lowast

3=

1198783

1198632

4(11986331198634) sin (120572

3)

(13)



47mm52mm



SpacerBlade angle

(120572)

Blade height(H)

Clearances





3




119905119904from 135 to 16) by



119905119894= 115 to 120587

119905119891= 160 Some










0 1 2 3 5 6Time (s)

06

08

1

12

14

16

120587T

4k

6k

8k 120596(r

ads

)

4

10k

12k

14k





4 Results




119905119891= 160) The results




3)



04 06 08 1 12 14mf

05

06

07

08

09

1

11

120578 f




04 06 08 1 12 14mf



0

005

01

015

02

025

03

120581(s

)

Model



1 12 14 16 18 2120587t

0

04

08

12

16

2

m




Transient 1

Transient 2

Transient 3




04 06 08 1 12 14mf

0

005

01

015

02

025

03

120581(s

)

Model


Transient 3




0 02 04 06Slowast3

0

05

1

15m

rtf




0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590






120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)











0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










47mm52mm



SpacerBlade angle

(120572)

Blade height(H)

Clearances





3




119905119904from 135 to 16) by



119905119894= 115 to 120587

119905119891= 160 Some










0 1 2 3 5 6Time (s)

06

08

1

12

14

16

120587T

4k

6k

8k 120596(r

ads

)

4

10k

12k

14k





4 Results




119905119891= 160) The results




3)



04 06 08 1 12 14mf

05

06

07

08

09

1

11

120578 f




04 06 08 1 12 14mf



0

005

01

015

02

025

03

120581(s

)

Model



1 12 14 16 18 2120587t

0

04

08

12

16

2

m




Transient 1

Transient 2

Transient 3




04 06 08 1 12 14mf

0

005

01

015

02

025

03

120581(s

)

Model


Transient 3




0 02 04 06Slowast3

0

05

1

15m

rtf




0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590






120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)











0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 5 6Time (s)

06

08

1

12

14

16

120587T

4k

6k

8k 120596(r

ads

)

4

10k

12k

14k





4 Results




119905119891= 160) The results




3)



04 06 08 1 12 14mf

05

06

07

08

09

1

11

120578 f




04 06 08 1 12 14mf



0

005

01

015

02

025

03

120581(s

)

Model



1 12 14 16 18 2120587t

0

04

08

12

16

2

m




Transient 1

Transient 2

Transient 3




04 06 08 1 12 14mf

0

005

01

015

02

025

03

120581(s

)

Model


Transient 3




0 02 04 06Slowast3

0

05

1

15m

rtf




0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590






120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)











0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










04 06 08 1 12 14mf

05

06

07

08

09

1

11

120578 f




04 06 08 1 12 14mf



0

005

01

015

02

025

03

120581(s

)

Model



1 12 14 16 18 2120587t

0

04

08

12

16

2

m




Transient 1

Transient 2

Transient 3




04 06 08 1 12 14mf

0

005

01

015

02

025

03

120581(s

)

Model


Transient 3




0 02 04 06Slowast3

0

05

1

15m

rtf




0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590






120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)











0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06Slowast3

0

05

1

15m

rtf




0 02 04 06 08 1(UCs)

2f

0

001

002

003

004

005

120581m

f120578 f

120590






120581119891120578119891

120590

= 8

119869

1198632

4

(

1198804

119862119904

)

2

119891

(14)











0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

06

08

1

0 02 04 06 08 1120591i998400 (Nmiddotm)

120591 i998400998400

(Nmiddotm

)




0 02 04 06 08 1120591i998400 (Nmiddotm)

0

02

04

06

08

1

120591 i998400998400

(Nmiddotm


Stator 5r-CFDy = x


11989410158401015840 for two stator

geometries


11989410158401015840 )







1198941015840 asymp


1199051198941015840 asymp 120587

119905119891) and a







1198791198941015840 given by (9) and


11987911989410158401015840 asymp




1198791198941015840








1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198941015840 is



5 Conclusion








Nomenclature

Quantities





119878lowast





Subscripts


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article dynamic response in transient operation

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