aerodynamic optimization design of a multistage...

Research ArticleAerodynamic Optimization Design of a Multistage CentrifugalSteam Turbine and Its Off-Design Performance Analysis

Hui Li12 and Dian-Gui Huang12

1School of Energy and Power Engineering University of Shanghai for Science and TechnologyShanghai 200093 China2Shanghai Key Laboratory of Power Energy in Multiphase Flow and Heat Transfer Shanghai 200093 China

Correspondence should be addressed to Dian-Gui Huang dghuangussteducn

Received 8 April 2017 Revised 28 June 2017 Accepted 7 August 2017 Published 27 September 2017

Academic Editor Ryoichi Samuel Amano

Copyright copy 2017 Hui Li and Dian-Gui HuangThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Centrifugal turbine which has less land occupation simple structure and high aerodynamic efficiency is suitable to be used assmall to medium size steam turbines or waste heat recovery plant In this paper one-dimensional design of a multistage centrifugalsteam turbine was performed by using in-house one-dimensional aerodynamic design program In addition three-dimensionalnumerical simulation was also performed in order to analyze design and off-design aerodynamic performance of the proposedcentrifugal steam turbine The results exhibit reasonable flow field and smooth streamline the aerodynamic performance of thedesigned turbine meets our initial expectations These results indicate that the one-dimensional aerodynamic design program isreliable and effective The off-design aerodynamic performance of centrifugal steam turbine was analyzed and the results showthat the mass flow increases with the decrease of the pressure ratio at a constant speed until the critical mass flow is reached Theefficiency curve with the pressure ratio has an optimum efficiency point And the pressure ratio of the optimum efficiency agreeswell with that of the one-dimensional designThe shaft power decreases as the pressure ratio increases at a constant speed Overallthe centrifugal turbine has a wide range and good off-design aerodynamic performance

1 Introduction

With the continuous reduction of fossil energy and theenhancement of peoplersquos environment awareness there hasbeen increasing attention to the high efficiency utilization ofenergy where the turbine is energy conversion component ifthe turbine is improved effectively it can help to improve theefficiency utilization of energy

Centrifugal turbine is a new type of turbine engine whichhas many advantages In the centrifugal turbine the gas flowsoutward the center and the channel area of the flow pathincrease naturally as the fluid volume flow increases duringthe expansion process That meets the principle of aerody-namic and geometric matching In addition centrifugal tur-bine can be used to achieve multistage design [1] more easilythan centripetal turbine Thus it can avoid the limitation ofsupersonic flow This is more useful for design conditionsespecially for off-design performance conditions

For turbine as the key component many scholars in thisarea have done some relevant researches but there is notmuch research on centrifugal turbine Domestically mainlyJing and Peng had studied the pneumatic analysis of a rocketcentrifugal turbine prototype starter and designed the modi-fication turbine [2] Yin-Ge et al [3] and Xin et al [4 5]had researched the single-stage centrifugal turbine designand its off-design performance Abroad the preliminaryhydrodynamic design of a small centrifugal turbine for theORC was studied by Casati and others of the Delft Universityof Technology They introduced the optimization based onthe intermediate streamline method for evaluating turbinedesign and performance [6ndash10]

In this paper one-dimensional design method of cen-trifugal steam turbine is proposed by drawing on theconventional turbine [11] And numerical simulation andoptimization of multistage centrifugal turbine were studied

HindawiInternational Journal of Rotating MachineryVolume 2017 Article ID 4690590 11 pageshttpsdoiorg10115520174690590

2 International Journal of Rotating Machinery

by referring to the inlet and outlet thermal parameters of asmall axial steam turbine

2 Aerodynamic Design of a SteamCentrifugal Turbine

21 One-Dimensional Design Method A one-dimensionaldesign of centrifugal turbine has been developed which isbased on the one-dimensional design method of conven-tional turbineTheone-dimensional thermodynamic calcula-tion programwas developed by FORTRANThemain designprinciples of centrifugal turbine one-dimensional designprogram are as follows

(1) The expansion flow is assumed to be adiabatic steadyand one-dimensional in the cascade channel

(2) The properties of working fluids are obtained bycalling Refpro 90 which is applicable to a variety ofworking fluid

(3) In order to simplify the design of the blade the bladeis designed to be of straight and constant height

(4) Stator and rotor velocity coefficients120593 and120595 are basedon previous experience [12]

(5) Each stage of the rotor absolute flow angle is 90degrees

(6) The program is mainly composed of continuity andenergy equations to achieve one-dimensional design

Figure 1 presents the design process of the centrifugal one-dimensional aerodynamic program The thermodynamicparameters of the centrifugal turbine inlet stagnation tem-perature 119879lowast0 inlet stagnation pressure 119875lowast0 outlet pressure119875119873 mass flow rate 119866 and rotation speed 119899 are based onoriginal conventional turbineThe other parameters impellerdiameter ratio 119887 (the influence of diameter ratio on thecentrifugal turbine wheel efficiency is referred to in [3] theoptimum wheel efficiency calculated at the optimum degreeof reaction and speed ratio increases as the diameter ratiodecreases 119887119873 = 119863119873out119863119873in) each stage outlet flow angle ofstator 1205721 the radial gap 120575 (120575 = 119877119903inminus119877119904out or 120575 = 119877119904inminus119877119903outbecause the passage area along the flowpath remains constantin axial flow turbine or the flow area reduces in radial flowturbine so the radial gap can be large in traditional turbinewhile in centrifugal turbine the flow area increases withthe working fluid expansion If the radial gap is too large itwill make the working fluid compressive in the gap whichis bad for turbine working So the radial gap for centrifugalturbine should be small) and stage number119873 are previouslyestimated Iterative and screeningmethods are used to searchthe maximum wheel efficiency of the centrifugal turbine

Figure 2 shows the overall stages119867-119878 diagramThe super-script lowast represents the stagnant state The second numbersof subscript are as follows 1 represents the stator and 2represents rotor Taking the first stage as an example theintroduction of design process is as follows

(1) In the stator the steam expands from state 0 tostate 1 Lines 0-1 represent the actual expansion and

lines 0-1119904 is the ideal expansion In this process thepressure energies are transferred to kinetic energiesThen the thermal parameters velocity and geometryparameters can be calculated by using

isentropic expansion 1199040 = 11990411119904 = 119891 (119875lowast0 119879lowast0 )ℎ11119904 = 119891 (11990411119904 11987511)

energy equation ℎlowast0 = ℎ11119904 +1198882111199042 = ℎ11 +

1198882112

11988811 = 1205931198881111990411990411 12058811 = 119891 (ℎ11 11987511)

continuity equation 119866 = 120588011988801198600 = 12058811119888111198601 sin1205721

1205941 =1199061111988811119904

= 12058711989911986311out6011988811119904

(1)

(2) Then in the rotor the steam continues to expandfrom state 1 to state 2 Lines 1-2 represent the actualexpansion and lines 1-2119904 is the ideal expansion Thefluid kinetic energies are transferred to mechanicalenergies whichmake the turbine outputs shaft powerThe thermal parameters and the velocity triangles canbe calculated by using

11990811 = radic119888211 + 119906211 minus 21198881111990611 cos1205721

sin1205731 =11988811 sin120572111990811

continuity equation 12058811119888111198601 sin1205721

= 120588121198602 sin1205722radic1205952 (11990821 + 2Δℎ2119904 + 119906212 minus 119906211) minus 119906212

energy equation ℎ11 +1199082112 minus 119906

211

2

= ℎ12119904 +1199082121199042 minus 119906

212

211990812 = 12059511990812119904isentropic expansion 11990412119904 = 11990411 = 119891 (119875lowast11 119879lowast11)

11987512 = 119891 (11990412119904 ℎ12119904)

11990412 120588101584012 = 119891 (11987512 ℎ12)

1205722 = 90∘

11988812 = radic119908212 minus 119906212

1199051198921205732 =1198881211990612

(2)

(3) The rotor actual outlet density 120588101584012 is compared withestimated density 12058812 If |120588101584012 minus 12058812| lt 120576 (120576 is

International Journal of Rotating Machinery 3

No

Yes

No

Yes

Calculate the wheel efficiency at different speed ratioGeometry and thermodynamic parameters are determined at the optimum speed ratio

One-dimensional analysis of other stages

Output the one-dimensional results of centrifugal turbine

Initial parameters N

Geometry parameters b 1

Thermodynamic parameters Tlowast0 P

lowast0 PN n G

Parameters at section 2 w12s w12 c12 P12 s12 ℎ12 12

and first-stage rotor outlet density 12

Estimated speed ratio 1 (01sim09 Δ = 001)

Estimated initial pressure at the first-stage stator outlet P11

12 minus 12

lt 1 times 10minus6

12 = 12

Parameter at outlet of the last stage cN2 wN2 PN2 sN2 ℎN2

PN2 minus PN lt 1 times 10minus6

Parameters at section 1 c11s c11 w11 ℎ11s s11 ℎ11 11

Figure 1 One-dimensional design process of centrifugal turbine

a minimum) the assumption density 12058812 is validThen the rotor outlet point is determined If it isnot satisfied then reassume 12058812 and repeat the abovesteps until the conditions are met

(4) Similar to the axial flow turbine the relevant adiabaticefficiency is given by (3) which is referred to in [13]

120578 = ℎ0 minus ℎ12ℎ0 minus ℎ101584012119904

(3)

The wheel efficiency is calculated at different speed ratioto select the optimum speed ratio and other correspondingparameters atmaximumefficiency Geometry and thermody-namic parameters are identified at the optimum speed ratio

The above-mentioned steps are also adopted to design otherstages

22 One-Dimensional Design Results The one-dimensionaldesign program of the centrifugal turbine is used to designthe multistage centrifugal turbine The initial design thermalparameters are derived from a small axial turbine andsummarized in Table 1

Because there is no stage number information aboutthe original axial turbine from the product information thedifferent stage number was tested to design the centrifugalturbine by using the one-dimensional design program Theresults show that if the stage number is 1 and 2 there issupersonic in the centrifugal turbine As the stage number


h

s

1

2

0

N

0lowast

ℎ0 minus ℎ12

1s

2s2s

Δℎ2s

ℎ0 minus ℎ12s

Figure 2 Turbine expansion process

Table 1 The initial design parameters of the centrifugal turbine

Total pressure 1198750 (Pa) 1275000Total temperature 1198790 (K) 613Back pressure 119875119873 (Pa) 294000Rotational speed 119899 (rpm) 6500Mass flow 119866 (kgs) 167Rated power119882 (kW) 750

increased there is no supersonic in the centrifugal turbinebut the size of the centrifugal turbine becomesmore andmorelarge When the stage number is 3 and the diameter ratiois 112 the centrifugal turbine is subsonic and it can satisfythe enthalpy drop as well as high efficiency So the centrifugalturbine is designed with three stages

Then the same blade height and the absolute airflowangle of each stage are 90 degrees as the criteria impellerdiameter ratio is estimated to be 112 for each stage eachstage outlet flow angle of stator is assumed to be 12 degreesand radial clearance between stator and rotor is set tobe 2mm The optimal efficiency and reasonable structureare the objectives The design parameters are calculatedby using the above-mentioned one-dimensional calculationprogram with iterative and screeningmethods In this paperthe designed three-stage centrifugal turbine just meets therequired enthalpy drop and has a very high wheel efficiencyThemain aerodynamic parameters of each stage are shown inTable 2 and themain aerodynamic parameters of overall cen-trifugal turbine are shown in Table 3 The geometry parame-ters are shown in Table 4 and the speed triangle data is shownin Table 5 The speed triangle schematic diagram is shown asFigure 3

1st

2nd

3rd

c1 c21 2

u1 u2

1

2

w1

w2

c1 c2

1 2

u1 u2

12

w1w2

c1c2

1 2

u1 u2

12

w1 w2

Figure 3 The speed triangle of the centrifugal turbine

3 Airfoil Design and Optimization

The working fluid flows are outward the center in the cen-trifugal turbine The passage area along the flow path in-creases naturally with the expansion of the working fluidwhich makes the variety of specific volume matches thechange of flow passage area It can be seen that the centrifugalturbine is much superior to the conventional turbine fromthe structure In the centrifugal turbine the blade planeflow channel is expanding outward with the increase of theradius It is obviously inappropriate to use the airfoil of con-ventional turbine at this time So designing a suitable airfoilfor centrifugal turbine is needed

31 Parametric Expression of Airfoil Angle and thicknessdesign method is adopted to design airfoil by using the inletand outlet geometry angle blade height and leading andtrailing edge diameters which are based on one-dimensionaldesign results in Tables 3 and 4 On the BladeGen platformtwo-dimensional structure of the stator and rotor is iden-tified Stator and rotor of the first stage are an example asshown in Figures 4(a) and 4(b) The blade surface includesfour patches namely the leading edge and the trailing edgeand the suction side and the pressure side The mean camberline is controlled by cubic Bezier curve The leading andtrailing edges are both ellipse Blade number is determinedinitially by referring to the relative pitch and expellingcoefficient and the data is provided in [12]The final numbersof stator and rotor for each stage are both 65

32 Blade Optimization for Three-Stage Centrifugal SteamTurbine In the optimization process of blade the bladenumber the blade inlet and outlet geometry angles and theleading and trailing edge thicknesses are the fixed parametersand two control points of the tangent angle and two controlpoints of the blade thickness are taken as the optimizationparameters Workbench is as the optimization software andNLPQL algorithm is used as optimization method

In the stator optimization process the minimum losscoefficient is the object and stator back pressure is the con-straint condition Stators of three-stage centrifugal turbine


Table 2 The main aerodynamic parameters of each stage

Stage number 1st 2nd 3rdSpeed ratio119883119894 069 086 087Reaction degreeΩ 0322 0453 0468Flow coefficient 120593 097 097 097Flow coefficient 120595 095 095 095Mach number of stator Ma 055 058 076Shaft power119882 (kW) 1205 1551 2436Wheel efficiency 120578 9120 9097 9098

Mean camber line

Suction side

Pressure side

(a) Initial profile of the first-stage stator

Leading edge

Trailing edge

(b) Initial profile of the first-stage rotor

Figure 4

Table 3 The main aerodynamic parameters of overall centrifugalturbine

Overall wheel efficiency 120578 9050Overall shaft power119882 (kW) 5192Mass flow 119866 (kgs) 167Back pressure 119875119873 (Pa) 294000

are optimized respectively The optimization mathematicalmodel of stator is expressed as

Object Min (lossCoefficient)Constraint 119875te1 le 1198751198731

= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (4)

The design variables 119883120579[12] 119884120579[12] respectively are thehorizontal and vertical coordinates of control points of thetangent angle119883119905[12] and119884119905[12] are the horizontal and verticalcoordinates of control points of the blade thickness A totalof eight variables are involved in optimization (the stator

of the first stage is an example as shown in Figure 5) 119875te1represents the pressure at the trailing edge of the stator and1198751198731 represents the stator back pressure of one-dimensionaldesign (119873 represents stage number)

Then the rotor is added behind the optimized stator Themaximum shaft power is the object and rotor back pressureis the constraint condition The optimization mathematicalmodel of rotor is expressed as

Object Max (ShaftPower)Constraint 119875te2 le 1198751198732

= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (5)

The same as the stator the design variables 119883120579[12] and119884120579[12] are the horizontal and vertical coordinate points forcontrolling the tangent angle of the rotor mean camber linerespectively 119883119905[12] and 119884119905[12] are the horizontal and verticalcoordinate points for controlling the thickness of the blade Atotal of eight variables are involved in optimization (the rotorof the first stage is an example as shown in Figure 6) 119875te2represents the pressure at the trailing edge of the rotor and


Table 4 The main geometrical parameters of the centrifugal turbine

Stage number 1st 2nd 3rdStator inlet diameter119863119904in (m) 0597 0758 0959Rotor outlet diameter119863119903in (m) 0754 0955 1208Radial clearance 120575119873 (m) 0002 0002 0002Blade height119867 (m) 003 003 003Symmetrical cone angle 120574119873 (degree) 0 0 0Diameter ratio of stator119863119904out119863119904in 112 112 112Diameter ratio of rotor119863119903out119863119903in 112 112 112

M-prime (LE to TE)0000 0025 0050 0075 0113

(X1 Y1)

(X2 Y2)

minus55

125

375

625

835

Ang

les (

∘ )

(a) Tangential angular distribution of medial camber line and controlpoints selection of rotor blade

01

25

50

75

100

125

154

Nom

al la

yer t

hick

ness

(mm

)150 30020010000 35825050M (LE to TE)

(Xt2 Yt2)

(Xt1 Yt1)

(b) The thickness distribution and control points selection of rotorblade

Figure 5

1198751198732 represents the rotor back pressure of one-dimensionaldesign

The optimization strategy and process are shown inFigures 7 and 8 respectively

Figure 8 shows the blade optimization process All ofthosewere simulated automatically on theworkbench Firstlythe value range of optimization parameters (119883120579[12] 119884120579[12]119883119905[12] and 119884119905[12]) constraint condition and object are setartificially Then Design Exploration selects a set of data inthe value range of optimization parameters automaticallyThe blade geometry and parameterization are obtained bygeometry software After that the blade mesh is generatedby TurboGrid and centrifugal turbine is simulated by CFXFinally if the results satisfy the constraint and object theparameter values of blade control points are the optimizationvalues If not theDesign Explorationwill select another set ofdata in the value range of optimization parameters automati-callyThen repeat the above steps until the conditions aremetSince the optimized stages deviate from the design conditionwhen they are calculated together the leading and trailingedge thicknesses of stator and rotor are needed to be changedslightly for local adjustment then the centrifugal turbine canobtain a better performance

Since the SST model is used in turbomachinery in mostcases it requires 119884+ to be very small (119884+ lt 2) So the layermesh quality requirement of SST model is higher than 119896-epsilonmodel If the SST uses the samemesh generated for 119896-epsilon the layermesh quality for SST is not good and cannotsatisfy the requirement The global size factors were changedto increase the mesh number and the mesh quality was

improved So the grid number of SST is more than 119896-epsilonIt is about 8627000 while the grid number of 119896-epsilonmodel is 4010000 If the multistage centrifugal turbinesimulation turbulence model was SST a lot of computingmemory will be needed meanwhile a lot of calculating timewill be cost We found that the calculation results of SST and119896-epsilon models are almost the same as shown in Table 6so there is no difference between SST model and 119896-epsilonmodel to simulate themultistage centrifugal turbine In orderto relax the demand on computer memory and to raisethe efficiency when the computational grid number is verylarge the 119896-epsilon model is used as the turbulence model tosimulate the centrifugal turbine

The optimized centrifugal turbine is simulated by CFXbased on Navier-Stokes equations Stators and rotors usestructured grid generated in TurboGrid and the total gridnumber is about 4010000 the blade global size factors ofmesh are 12 12 125 125 125 and 12 for each stator androtor respectively The computational model mesh used forsimulation is shown in Figure 9 The boundary conditionsare inlet total pressure total temperature back pressure andadiabatic wall The turbine rotation speed is 6500 rpm Ano-slip boundary condition is applied at all the solid wallsThe stator-rotor interface is Frozen Rotor Separate periodicconditions are used for rotor and stator regions 119870-epsilon isused as the turbulencemodel and the fluid isWater Ideal Gas

The optimal parameters of stator and rotor are almostconsistent with one-dimensional aerodynamic design Theoptimized stator and rotor of whole stages are both straightblades and the blade diagram is referred to in Figure 10


minus848

minus500

minus250

00

250

610A

ngle

s (∘ )

01130050M-prime (LE to TE)

00750000 0025

(X1 Y1) (X2 Y2)

(a) Tangential angular distribution of medial camber line and control pointsselection of stator blade

200 40430010000M (LE to TE)

03

50

100

176

Nom

al la

yer t

hick

ness

(mm

)

(Xt2 Yt2)(Xt1 Yt1)

(b) The thickness distribution and control points selection of statorblade

Figure 6

Table 5 The speed triangle data of the centrifugal turbine

Stage number 1st 2nd 3rd1205721 (∘) 12 12 121205731 (∘) 3793 6447 68441198621 (ms) 3221 3287 40811198821 (ms) 1089 7574 91221198801 (ms) 2291 2903 36701205722 (∘) 90 90 901205732 (∘) 1418 1232 13841198622 (ms) 6481 7102 10131198822 (ms) 2646 3328 42331198802 (ms) 2566 3251 4110

Table 6 The contrast of SST and 119896-epsilon models

Variable 119870-epsilon SST ΔWheel efficiency 120578 9329 9301 029Shaft power119882 (kW) 5387 52568 242Mass flow 119866 (kgs) 16708 16443 159Back pressure 119875119873 (Pa) 291170 293999 097

4 Results and Discussion

41 DesignConditionAerodynamic Performance Thenumer-ical simulation results of the optimized centrifugal turbine arebasically consistent with the one-dimensional aerodynamicdesign which shows that the one-dimensional design pro-gram is reliable and effective The overall performance dataof centrifugal turbine is shown in Table 7 Comparison ofperformance data at each stage is shown in Table 8 Seenfrom Table 8 the performance data has some deviations ateach stage This is because the optimized stages influenceeach other during the matching process So it results in somedeviations in the performance data And these deviations aregradually accumulated in the process of fluid flow As thestage increases the deviations are more But from the overallperformance data seen in Table 7 the overall performance

data is within 4 deviation This is due to the fact thatthere are positive and negative deviations at each stagewhich can offset each other So the overall performancedata of one-dimensional design and simulation results isnot quite different From the total shaft power and wheelefficiency the simulation results of overall centrifugal turbineare better than one-dimensional design values The three-stage centrifugal turbine meets the design requirements andhas good performance

Figures 11ndash13 show the distribution of pressure Machnumber and velocity streamline at the middle plane of thethree-stage centrifugal turbine respectively From the dia-grams the pressure distribution is uniform the main flow inthe impeller is the pressure flowThe flow field in the cascadechannel is reasonable and the streamline is smooth and theaerodynamic performance is in line with the expectation As


Stator optimization rotor optimization



Whole stage optimizationand local adjustment

1st

2nd

3rd

Figure 7 Optimization strategy

Table 7 The contrast of design value and CFD value of the overall performance of the centrifugal turbine

Variable Design value Simulation value ΔWheel efficiency 120578 9050 9329 +308Shaft power119882 (kW) 5192 5387 +375Mass flow 119866 (kgs) 167 16708 +005Back pressure 119875119873 (Pa) 294000 291170 minus096

Ansys-BladeGen blade design

Ansys-Geometry blade parameterization

Ansys-TurboGrid mesh generation

Ansys-CFX numerical simulation

Output optimized blade parameters

Ansys-Design Exploration optimization

Satisfying the requirements

Not satisfyingthe requirements

Figure 8 Optimization process

can be seen from Table 8 the stator and rotor outflow anglesof the first and second stage have deviation from the one-dimensional design value From the analysis of the flow fielddiagram there is a flow separation in the small area nearthe trailing edge the fluid expands at the chamfered part ofthe stator outlet and the gap between the rotor and statorwhich results in deflection of the flow angle The flow losscaused by the separation and the deviation of the inflow anglecaused by the flow deflection are not yet considered in theone-dimensional design procedure This is also the reasonwhy the simulation results deviate from the one-dimensionaldesign values

42 Off-DesignConditionAerodynamic Performance Figures14 15 and 16 show the relationship between mass flow 119866efficiency 120578 and shaft power 119882 at different rotation speedswith variable pressure ratio 120587 = 119875119873119875lowast0 The results indicate

that as 120587 decreases the mass flow increases until the criticalflow is reached and then the mass flow remains constantwhen the speed is constant This is because the flow processappeared supersonic and the flow reached its maximumAnd the flow passage had the blocking phenomenon but theturbine was still able to work There is an optimal efficiencypoint on the efficiencywith pressure ratio curve At both sidesof the optimal efficiency point the efficiency decreases withthe increase or decrease of 120587 This is due to the change of theback pressure and the enthalpy drop is changed the speedratio is deviated from the optimum value the flow angel isdeflected and the blade surface occurs the flow separationThose lead to the flow loss increase and efficiency decreaseFor shaft power the power decreases as 120587 increases Becauseof increase in back pressure the enthalpy drop of the wholestage is reduced Then the work capacity of working fluid isreduced So the shaft power decreases with the increasing120587 When the rotation speed changes the optimum efficiencypoint moves in the direction of pressure ratio decreasing andthe optimum efficiency value decreases with the increase ofthe rotation speed But the efficiency trend with the pressureratio is consistent at different speeds For the mass flow theflow characteristics at different rotation speeds are basicallythe same It can be seen that the mass flow variation atdifferent rotation speeds is the same and the change has littleeffect on mass flow At different rotation speeds the trendsof shaft power curves are basically the same But the curve ofshaft powerwith the pressure ratio is steeperwith the increaseof rotation speed

5 Conclusions

The three-stage centrifugal turbine is simulated by CFXwith Water Ideal Gas as working fluid drawing on theconventional turbine aerodynamic design method and off-design condition performance researchmethod Conclusionsare as follows

The numerical simulation results of the whole stages arebasically consistent with the one-dimensional design resultsand the aerodynamic performance meets the expectedrequirements which indicates the reliability and effectiveness


Periodic boundary

Inlet

Interface of rotor and stator Outlet

(a) Computational domain for the simulation of stages (b) Mesh distribution along the height and tangential direction ofblade

Figure 9

Stator Rotor

1st2nd

3rd

(a) Optimized stators and rotors

0minus002

06

05

04

03

H(m

)

R (m)

(b) 3D model of stator and rotor blades

Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)

Figure 11 Static pressure distribution

of the centrifugal turbine design Because each stage of cen-trifugal turbine is optimized the simulated overall efficiencyis 308 higher than the one-dimensional design efficiencyand the shaft power is 375 more than the one-dimensionaldesign shaft power At the design condition the streamlineof three-stage centrifugal steam turbine is smooth at cascadeflow channel the pressure distribution is uniform and theflow field is reasonable

At the off-design condition when the speed is constantthe pressure ratio 120587 reaches the critical ratio and the massflow reaches the maximum If 120587 continues to decrease themaximum flow value remains the same and the flow passage

11

08

09

07

06

05

04

02

01

00

Figure 12 Mach number distribution

has the blocking phenomenon But the turbine is still able towork At the same 120587 the speed decreases then the mass flowincreases but the impact is quite small

At a constant speed the efficiency is highest at optimumpressure ratio 120587 The pressure ratio of the optimum efficiencyagrees well with that of the one-dimensional design Whenthe speed decreases the efficiency curve moves to theplace where 120587 increases and the corresponding maximumefficiency increases

The shaft power curve trend with pressure ratio is sim-ilar at different rotation speeds But as the rotation speed


Table 8 The contrast of design value and CFD value of each stage performance of the centrifugal turbine

VariableStage number

1st 2nd 3rdDesign value Simulation value Δ () Design value Simulation value Δ () Design value Simulation value Δ ()

Ma 055 055 0 058 052 minus1034 076 082 +789Ω 032 030 minus625 045 051 +1333 047 042 minus1064120594 069 063 minus869 086 073 minus1512 087 067 minus2299119882 (kW) 1205 1231 +216 1551 1384 minus1077 2436 2771 +1375120578 () 9120 9334 +235 9097 8736 minus397 9098 8959 minus1531205721 (∘) 120 123 +25 120 142 +183 120 120 01205722 (∘) 900 789 minus123 900 803 minus108 900 900 0

5435

4529

3623

2718

1812

906

01

(ＧＭminus1)

Figure 13 Streamline distribution

0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point

Figure 14The variation of mass flowwith pressure ratio at differentspeeds

increases the power curve drops faster when the pressureratio increases

Overall it can be seen that the centrifugal turbine hasa wide range and good off-design performance from the

0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point

Figure 15The variation of efficiency with pressure ratio at differentspeeds

numerical simulation results of the three-stage centrifugalturbine and the analysis of the off-design conditions

Nomenclature

119879 Temperature K119875 Pressure Paℎ Enthalpy kJkg119904 Entropy kJ(Ksdotkg)119899 Rotational speed rmin119866 Mass flow kgs119887 Diameter ratio119863 Diameter m119877 Radius m119888 Absolute velocity ms119908 Relative velocity ms119906 Circumferential velocity ms119867 Height m119873 Stage number119882 Shaft power kWMa Mach number


0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point

Figure 16 The variation of shaft power with pressure ratio atdifferent speeds

Greek Letters

120588 Density kgm3120574 Radial symmetry cone angle ∘120575 Radial clearance between stator and rotor m120572 Absolute angle ∘120573 Relative angle ∘120593 Stator flow coefficient120595 Rotor flow coefficient120594 Speed ratio120578 Wheel efficiencyΩ Reaction degree120587 Pressure ratio

Subscripts

0 First stagein Inletout Outlet119904 Stator119896 Stage number

Superscripts

lowast Stagnation condition

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by National Natural science Foun-dation of China no 51536006 and Shanghai Science and

Technology Committee with Grant no 17060502300 andfunded by Opening Project of Shanghai Key Laboratory ofMultiphase Flow andHeat Transfer in Power Engineering no13DZ2260900

References

[1] E Coomes R Dodge and D Wilson ldquoDesign of a highpower-density ljungstrom turbine using potassium as aworkingfluidrdquo in Proceedings of the 21st Intersociety Energy ConversionEngineering Conference vol 3 San Diego CA USA 1986

[2] L Jing and S Peng ldquoModification design and flowfield analysisof a centrifugal turbine stage with discrete nozzlesrdquo Journal ofAerospace Power vol 23 no 6 pp 1047ndash1053 2008

[3] L Yin-Ge T Xin and L Xian-Qiao ldquoThe thermal design andanalysis of centrifugal turbinerdquo Journal of Engineering Thermo-physics vol 37 no 10 pp 2103-2010 2016

[4] T Xin L Yin-Ge L Xian-Qiao et al ldquoResearch on off-designcharacteristics of centrifugal turbinerdquo Journal of EngineeringThermophysics vol 37 no 6 pp 1201ndash1208 2016

[5] T Xin L Hui S Yan-Ping et al ldquoDesign of the auxiliary com-ponents of the centrifugal turbinerdquo Journal of EngineeringThermophysics vol 0253-231X pp 04ndash0000 2017

[6] M Pini G Persico E Casati et al ldquoPreliminary design of a cen-trifugal turbine for organic rankine cycle applicationsrdquo Engi-neering for Gas Turbines amp Power vol 135 no 4 2013

[7] E Casati S Vitale M Pini G Persico et al ldquoCentrifugal tur-bines for mini-organic rankine cycle power systemsrdquo Journal ofEngineering for Gas Turbines amp Power vol 136 no 136 ArticleID 122607 11 pages 2014

[8] G Persico M Pini V Dossena and P Gaetani ldquoAerodynamicdesign and analysis of centrifugal turbine cascadesrdquo in Proceed-ings of the ASME Turbo Expo Turbine Technical Conference andExposition GT rsquo13 vol 6C 12 pages San Antonio Texas TexUSA 2013

[9] G Persico M Pini V Dossena et al ldquoAerodynamics of centri-fugal turbine cascadesrdquo Journal of Engineering for Gas Turbinesamp Power vol 137 no 11 Article ID 112602 2015

[10] M Pini A Spinelli G Persico and S Rebay ldquoConsistent look-up table interpolation method for real-gas flow simulationsrdquoComputers amp Fluids vol 107 pp 178ndash188 2015

[11] L Yan-Sheng and L Gui-Lin Radial Turbine and CentrifugalCompressor Machinery Industry Press Bei Jing China 1987

[12] W Xin-Jun and L Liang Principle of Steam Turbine XIrsquoanJiaotong University Press Xirsquoan Shaanxi China 2014

[13] S L Dixon and C A Hall Fluid Mechanics andThermodynam-ics of Turbomachinery Elsevier Inc Press 2010

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

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



by referring to the inlet and outlet thermal parameters of asmall axial steam turbine

2 Aerodynamic Design of a SteamCentrifugal Turbine

21 One-Dimensional Design Method A one-dimensionaldesign of centrifugal turbine has been developed which isbased on the one-dimensional design method of conven-tional turbineTheone-dimensional thermodynamic calcula-tion programwas developed by FORTRANThemain designprinciples of centrifugal turbine one-dimensional designprogram are as follows

(1) The expansion flow is assumed to be adiabatic steadyand one-dimensional in the cascade channel

(2) The properties of working fluids are obtained bycalling Refpro 90 which is applicable to a variety ofworking fluid

(3) In order to simplify the design of the blade the bladeis designed to be of straight and constant height

(4) Stator and rotor velocity coefficients120593 and120595 are basedon previous experience [12]

(5) Each stage of the rotor absolute flow angle is 90degrees

(6) The program is mainly composed of continuity andenergy equations to achieve one-dimensional design

Figure 1 presents the design process of the centrifugal one-dimensional aerodynamic program The thermodynamicparameters of the centrifugal turbine inlet stagnation tem-perature 119879lowast0 inlet stagnation pressure 119875lowast0 outlet pressure119875119873 mass flow rate 119866 and rotation speed 119899 are based onoriginal conventional turbineThe other parameters impellerdiameter ratio 119887 (the influence of diameter ratio on thecentrifugal turbine wheel efficiency is referred to in [3] theoptimum wheel efficiency calculated at the optimum degreeof reaction and speed ratio increases as the diameter ratiodecreases 119887119873 = 119863119873out119863119873in) each stage outlet flow angle ofstator 1205721 the radial gap 120575 (120575 = 119877119903inminus119877119904out or 120575 = 119877119904inminus119877119903outbecause the passage area along the flowpath remains constantin axial flow turbine or the flow area reduces in radial flowturbine so the radial gap can be large in traditional turbinewhile in centrifugal turbine the flow area increases withthe working fluid expansion If the radial gap is too large itwill make the working fluid compressive in the gap whichis bad for turbine working So the radial gap for centrifugalturbine should be small) and stage number119873 are previouslyestimated Iterative and screeningmethods are used to searchthe maximum wheel efficiency of the centrifugal turbine

Figure 2 shows the overall stages119867-119878 diagramThe super-script lowast represents the stagnant state The second numbersof subscript are as follows 1 represents the stator and 2represents rotor Taking the first stage as an example theintroduction of design process is as follows

(1) In the stator the steam expands from state 0 tostate 1 Lines 0-1 represent the actual expansion and

lines 0-1119904 is the ideal expansion In this process thepressure energies are transferred to kinetic energiesThen the thermal parameters velocity and geometryparameters can be calculated by using

isentropic expansion 1199040 = 11990411119904 = 119891 (119875lowast0 119879lowast0 )ℎ11119904 = 119891 (11990411119904 11987511)

energy equation ℎlowast0 = ℎ11119904 +1198882111199042 = ℎ11 +

1198882112

11988811 = 1205931198881111990411990411 12058811 = 119891 (ℎ11 11987511)

continuity equation 119866 = 120588011988801198600 = 12058811119888111198601 sin1205721

1205941 =1199061111988811119904

= 12058711989911986311out6011988811119904

(1)

(2) Then in the rotor the steam continues to expandfrom state 1 to state 2 Lines 1-2 represent the actualexpansion and lines 1-2119904 is the ideal expansion Thefluid kinetic energies are transferred to mechanicalenergies whichmake the turbine outputs shaft powerThe thermal parameters and the velocity triangles canbe calculated by using

11990811 = radic119888211 + 119906211 minus 21198881111990611 cos1205721

sin1205731 =11988811 sin120572111990811

continuity equation 12058811119888111198601 sin1205721

= 120588121198602 sin1205722radic1205952 (11990821 + 2Δℎ2119904 + 119906212 minus 119906211) minus 119906212

energy equation ℎ11 +1199082112 minus 119906

211

2

= ℎ12119904 +1199082121199042 minus 119906

212

211990812 = 12059511990812119904isentropic expansion 11990412119904 = 11990411 = 119891 (119875lowast11 119879lowast11)

11987512 = 119891 (11990412119904 ℎ12119904)

11990412 120588101584012 = 119891 (11987512 ℎ12)

1205722 = 90∘

11988812 = radic119908212 minus 119906212

1199051198921205732 =1198881211990612

(2)

(3) The rotor actual outlet density 120588101584012 is compared withestimated density 12058812 If |120588101584012 minus 12058812| lt 120576 (120576 is


No

Yes

No

Yes







lowast0 PN n G





12 minus 12

lt 1 times 10minus6

12 = 12







120578 = ℎ0 minus ℎ12ℎ0 minus ℎ101584012119904

(3)






h

s

1

2

0

N

0lowast

ℎ0 minus ℎ12

1s

2s2s

Δℎ2s

ℎ0 minus ℎ12s






1st

2nd

3rd

c1 c21 2

u1 u2

1

2

w1

w2

c1 c2

1 2

u1 u2

12

w1w2

c1c2

1 2

u1 u2

12

w1 w2










Mean camber line

Suction side

Pressure side


Leading edge

Trailing edge


Figure 4





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (4)





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (5)





M-prime (LE to TE)0000 0025 0050 0075 0113

(X1 Y1)

(X2 Y2)

minus55

125

375

625

835

Ang

les (

∘ )


01

25

50

75

100

125

154

Nom

al la

yer t

hick

ness

(mm

)150 30020010000 35825050M (LE to TE)

(Xt2 Yt2)

(Xt1 Yt1)


Figure 5









minus848

minus500

minus250

00

250

610A

ngle

s (∘ )


00750000 0025

(X1 Y1) (X2 Y2)


200 40430010000M (LE to TE)

03

50

100

176

Nom

al la

yer t

hick

ness

(mm

)

(Xt2 Yt2)(Xt1 Yt1)


Figure 6














1st

2nd

3rd
















5 Conclusions




Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










No

Yes

No

Yes







lowast0 PN n G





12 minus 12

lt 1 times 10minus6

12 = 12







120578 = ℎ0 minus ℎ12ℎ0 minus ℎ101584012119904

(3)






h

s

1

2

0

N

0lowast

ℎ0 minus ℎ12

1s

2s2s

Δℎ2s

ℎ0 minus ℎ12s






1st

2nd

3rd

c1 c21 2

u1 u2

1

2

w1

w2

c1 c2

1 2

u1 u2

12

w1w2

c1c2

1 2

u1 u2

12

w1 w2










Mean camber line

Suction side

Pressure side


Leading edge

Trailing edge


Figure 4





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (4)





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (5)





M-prime (LE to TE)0000 0025 0050 0075 0113

(X1 Y1)

(X2 Y2)

minus55

125

375

625

835

Ang

les (

∘ )


01

25

50

75

100

125

154

Nom

al la

yer t

hick

ness

(mm

)150 30020010000 35825050M (LE to TE)

(Xt2 Yt2)

(Xt1 Yt1)


Figure 5









minus848

minus500

minus250

00

250

610A

ngle

s (∘ )


00750000 0025

(X1 Y1) (X2 Y2)


200 40430010000M (LE to TE)

03

50

100

176

Nom

al la

yer t

hick

ness

(mm

)

(Xt2 Yt2)(Xt1 Yt1)


Figure 6














1st

2nd

3rd
















5 Conclusions




Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










h

s

1

2

0

N

0lowast

ℎ0 minus ℎ12

1s

2s2s

Δℎ2s

ℎ0 minus ℎ12s






1st

2nd

3rd

c1 c21 2

u1 u2

1

2

w1

w2

c1 c2

1 2

u1 u2

12

w1w2

c1c2

1 2

u1 u2

12

w1 w2










Mean camber line

Suction side

Pressure side


Leading edge

Trailing edge


Figure 4





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (4)





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (5)





M-prime (LE to TE)0000 0025 0050 0075 0113

(X1 Y1)

(X2 Y2)

minus55

125

375

625

835

Ang

les (

∘ )


01

25

50

75

100

125

154

Nom

al la

yer t

hick

ness

(mm

)150 30020010000 35825050M (LE to TE)

(Xt2 Yt2)

(Xt1 Yt1)


Figure 5









minus848

minus500

minus250

00

250

610A

ngle

s (∘ )


00750000 0025

(X1 Y1) (X2 Y2)


200 40430010000M (LE to TE)

03

50

100

176

Nom

al la

yer t

hick

ness

(mm

)

(Xt2 Yt2)(Xt1 Yt1)


Figure 6














1st

2nd

3rd
















5 Conclusions




Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Mean camber line

Suction side

Pressure side


Leading edge

Trailing edge


Figure 4





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (4)





= 119891 (119883120579[12] 119884120579[12] 119883119905[12] 119884119905[12]) (5)





M-prime (LE to TE)0000 0025 0050 0075 0113

(X1 Y1)

(X2 Y2)

minus55

125

375

625

835

Ang

les (

∘ )


01

25

50

75

100

125

154

Nom

al la

yer t

hick

ness

(mm

)150 30020010000 35825050M (LE to TE)

(Xt2 Yt2)

(Xt1 Yt1)


Figure 5









minus848

minus500

minus250

00

250

610A

ngle

s (∘ )


00750000 0025

(X1 Y1) (X2 Y2)


200 40430010000M (LE to TE)

03

50

100

176

Nom

al la

yer t

hick

ness

(mm

)

(Xt2 Yt2)(Xt1 Yt1)


Figure 6














1st

2nd

3rd
















5 Conclusions




Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












M-prime (LE to TE)0000 0025 0050 0075 0113

(X1 Y1)

(X2 Y2)

minus55

125

375

625

835

Ang

les (

∘ )


01

25

50

75

100

125

154

Nom

al la

yer t

hick

ness

(mm

)150 30020010000 35825050M (LE to TE)

(Xt2 Yt2)

(Xt1 Yt1)


Figure 5









minus848

minus500

minus250

00

250

610A

ngle

s (∘ )


00750000 0025

(X1 Y1) (X2 Y2)


200 40430010000M (LE to TE)

03

50

100

176

Nom

al la

yer t

hick

ness

(mm

)

(Xt2 Yt2)(Xt1 Yt1)


Figure 6














1st

2nd

3rd
















5 Conclusions




Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus848

minus500

minus250

00

250

610A

ngle

s (∘ )


00750000 0025

(X1 Y1) (X2 Y2)


200 40430010000M (LE to TE)

03

50

100

176

Nom

al la

yer t

hick

ness

(mm

)

(Xt2 Yt2)(Xt1 Yt1)


Figure 6














1st

2nd

3rd
















5 Conclusions




Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














1st

2nd

3rd
















5 Conclusions




Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Periodic boundary

Inlet



Figure 9

Stator Rotor

1st2nd

3rd


0minus002

06

05

04

03

H(m

)

R (m)


Figure 10

12750368

11592074

10433779

9275484

8117191

6958896

5800602

4642308

3484014

2325720

(Pa)




11

08

09

07

06

05

04

02

01

00










5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














5435

4529

3623

2718

1812

906

01

(ＧＭminus1)


0

2

4

6

8

10

12

14

16

18

G (k

gs)

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

01 02 03 04 05 060 = PNPlowast

0

Design point




0

10

20

30

40

50

60

70

80

90

100

(

)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point



Nomenclature



0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1000

2000

3000

4000

5000

6000

7000

8000

W (k

W)

01 02 03 04 05 060 = PNPlowast

0

n = 5500 ＬＪＧ

n = 6500 ＬＪＧ

n = 7500 ＬＪＧ

Design point


Greek Letters


Subscripts


Superscripts




Acknowledgments



References














RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









aerodynamic optimization design of a multistage...

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