Research Article Design and Performance …downloads.hindawi.com/journals/ijrm/2013/293486.pdfResearch Article Design and Performance Evaluation of a Very Low Flow Coefficient Centrifugal

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2013 Article ID 293486 12 pageshttpdxdoiorg1011552013293486

Research ArticleDesign and Performance Evaluation of a Very Low FlowCoefficient Centrifugal Compressor

Yongsheng Wang12 Feng Lin1 Chaoqun Nie1 and Abraham Engeda3

1 Key Laboratory of Advanced Energy and Power Chinese Academy of Sciences Institute of Engineering ThermophysicsChinese Academy of Sciences Beijing 100190 China

2Graduate University of Chinese Academy of Sciences Beijing 100190 China3Department of Mechanical Engineering Michigan State University East Lansing MA 48823 USA

Correspondence should be addressed to Yongsheng Wang wangyongshengietcn

Received 1 August 2013 Accepted 16 September 2013

Academic Editor Masaru Ishizuka

Copyright copy 2013 Yongsheng Wang et al This 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

Very low flow coefficient centrifugal compressors are often applied as the last stages of multistage compressors Due to the lowervolume flow rate the flow channels in the impeller and diffuser are so narrow that friction loss becomes the main factor whichleads to lower efficiency than that of other stages in the same compressors In addition most of design methods are generally basedon medium flow coefficient centrifugal compressors Taking on researches on the low flow coefficient centrifugal compressors issignificant and necessary One-dimensional (1D) code consisting of design and analysis parts is developed in this study to providebasic geometric data and predict the entire performance of centrifugal compressor Three-dimensional geometry of the impelleris built CFD simulation is carried out as well to be compared with 1D prediction With the continuous geometry adjustment thefinal performance of the centrifugal compressor will be fixed once the performance discrepancy betweenCFD and one-dimensionalcode is acceptable The details on the flow field within impeller will be presented through CFD

1 Introduction

Centrifugal compressors are usually designed in the flowcoefficient range 120593 = 001 to 016 Often the need arises todesign very low or high flow coefficient 120593 radial stagesbut the aerodynamic challenges are not easy to satisfytherefore a designer usually has to make certain criticaldecisions Figure 1 shows the typical application area ofvarious compressor impellers and the typical areas wheremost industrial radial impellers are designed Narrow radialimpellers (smaller 120593 and smaller eye diameter) have non-twisted blades Wider impellers (larger 120593 and larger eyediameter) by contrast have twisted blades for better adaptionto local flow conditions

In very low flow coefficients stage design the majorchallenges aremechanical configuration anddiffuser stabilityand most significantly the aerodynamic and parasitic lossesare substantial as seen in Figure 1

Lowflow coefficient compressors are used formany appli-cations in industry and generally found as the last stages ofmultistage compressors where the volume rate is lower Two-dimensional radial impeller is usually applied Compared tothe centrifugal compressors with considerably higher flowcoefficient the range of very low flow coefficient centrifugalcompressors is between 0007 and 002 It means that alow volume flow rate passes through the impeller relativeto the outlet diameter and peripheral speed This causesspecial problems in obtaining an acceptable efficiency whichis mainly affected by three sources of efficiency losses skinfriction disc friction and leakage These compressors havegenerally a very narrow impeller channel that leads to highfriction losses The power required to overcome disc frictionand leakage is in the same order as a conventional higherflow stage Therefore efficient design of low flow coefficientcentrifugal compressors still needs more great effort from thedesignersMoreover in themost recent years the demand for

2 International Journal of Rotating Machinery

Axial flow compressor

Common applicationarea of the

radial impeller

Mixed flow impelleror

radial impellerhigh flow coefficient

2015

100806

04

02

01

001

002

004

006

008 0

1

02

04

06

08

10

120593

120595

(a)

120593

0 005 01 015

Aerodynamic losses

Leakage lossesDisc friction losses

Design point efficiency

0

02

04

06

08

120578

(b)

Figure 1 Application areas of high flow coefficient radial impellers and typical industrial compressors [1]

such high pressure ratio compressor operating at lowflow ratehas risen Nevertheless little information is available con-cerning low flow coefficient centrifugal compressor designin the open literature The following descriptions reviewsome research work regarding design of low flow coefficientcentrifugal compressors

Rusak [3] briefly presented three types of impeller designfor low flow application low flow conventional stages with2D impellers low flow high pressure coefficient stages withradial impellers and very low flow stages with wedge-typeimpellers Casey et al [4] designed and tested several low flowcoefficient centrifugal compressors with high head and lowoutlet meridional velocity component Koizumi [5] designedthree different compressors with straight blades S-shapedblades and almost radial blades In the mean time theinfluence of the viscous effect on the slip factor was studiedand the stall-onset condition for rotating stall in the vanelessdiffuser following the impeller was investigated as wellParoubek et al [6] tested six conventional low flow coefficientimpellers The effects of inducer hub to exit shroud diameterratio and channel width parameter on the stage perfor-manceswere investigatedDi Liberti [7] designed several low-flow-coefficient impellers and analyzed them numerically toinvestigate the effect of the design flow coefficient of theoutlet width-outlet blade angle combination and of the bladeleading edge position on the stage performance Biba et al [8]used commercial CFD codes together with in-house programdesign and analyzed two different impellers with low volumeflows and high pressure applications

Based on the above research results this study aims todesign an efficient lowflow coefficient centrifugal compressorwith wide operating range MATLAB is used as a tool todevelop 1D design and analysis codesThe three-dimensionalgeometry model of impeller is built by ldquoBLADEGENrdquo OnceCFD indicates that the performance of centrifugal compres-sor is satisfactory and has an acceptable discrepancy with 1Dprediction the final geometry of impeller will be fixed anddesign process is terminated

2 1D Design and Analysis Methods

1D code developed in this paper is divided into two partsdesign and analysis parts Design part provides basic geomet-ric data of impeller at design point Based on the geometryanalysis part can do the performance analysis using empiricalloss models at design and off-design points [9] and export thewhole performance curve of the centrifugal compressor It isnecessary to keep adjusting geometry until the comparisonof CFD and 1D codes shows that discrepancy is acceptableand design goals have been reached The 1D design andanalysis scheme is illustrated in Figure 2 Clearance loss andrecirculation loss are excluded in this shrouded impeller Infact hub-to-shroud loss and choke loss are not activated aswell which will be discussed in the following section of thispaper

In the case of this paper mass flow rate (119898) inlettotal pressure of impeller (119875

01) inlet total temperature of

impeller (11987901) rotational speed of impeller (119873) impeller

outlet diameter (1198632) and hub diameter (119863

ℎ) are given The

working medium is natural gas and typical compositionis used while setting the fluid properties in CFD In thedesign part the values of 119881

11989821198802 1205732119887

[10] and 1198811199062

1198811199062119887

are assumed within empirical ranges With these parametersimpeller outlet calculation can be completed Then in orderto calculate impeller inlet aerodynamic parameters the ratiosof 11986311198632and119882

21198821must be chosen to relate outlet to inlet

Since the inlet flow coefficient is 0022 impeller inlet shouldbe located in the radial direction which means that inlet hubdiameter is equal to inlet shroud diameter [2] It is because thereturn channel upstream of low coefficient compressor in thereal application guides the flow to be more uniform Inducercan be neglected which reduces flow path in the impellerand produces less skin friction lossMeanwhile impeller axiallength (119871

119911) can be determined as well Number of the blades

(119885) comes from original slip factor formula of Wiesner [11]Analysis part only includes one input variable which is

mass flow rate In the following analysis procedure fluiddensity at impeller inlet is initially approximated and requires

International Journal of Rotating Machinery 3

Given paremeters

Outlet analysis

Inlet analysisOutlet design

Inlet design

Des

ign

part

Export impellerGeometry data

Mass flow rate(variable)

Ana

lysis

par

t

Impeller performance

(design and off-design points)

Internal lossesIncidence lossSkin friction lossWake mixing loss

Clearance lossEntrance diffusion lossHub-to-shroud loading

Choking loss

Blade loading loss

loss

Parasitic lossesDisc friction lossLeakage lossRecirculalion loss

Figure 2 Flow chart of 1D code

an iterative solution procedurewith the constraint of iterationerror being less than 01 The same iterative method isapplied until convergence is reached on the fluid densitycalculation at impeller outlet in which internal losses areinvolved and used for computing impeller efficiency totalpressure static pressure and density The equations used inthe analysis part will be introduced and the related statefunctions and loss models are included

Specific heat at constant pressure is given as a constantprovided by industry in this design case No preswirl exists atthe impeller inlet which makes 119881

1199061= 0 and 119881

1= 1198811198981 Inlet

gas density is initialized by the following equation

1205881asymp 12058801

=11987501

(11987711987901) (1)

According to the fixed geometry data and given designparameters the values of 119881

1198981 1198801 1198821 and 119882th can be

easily calculated Equation (2) below provide the rest ofthermodynamic parameters

1198791= 11987901

minus1198812

1

(2119862119901)

1198751= 11987501(

1198791

11987901

)

119896(119896minus1)

12058811015840 =

1198751

(1198771198791)

(2)

The inlet calculationwill be done once the iteration showsthat the difference between two density values is within theerror range shown in Figure 3 Before moving on to theimpeller outlet calculation fraction area blockage at impeller

exit (1198612) has to be initially approximated first and updated

later Then 1198811199062

can be given after slip factor calculation iscompleted

1205882= 1205881(

1198802

2

(11986211990111987901)+ 1)

1(119896minus1)

1198811198982

=119898

(120587120588211986321198872(1 minus 119861

2))

120590 = 1 minus

radicsin (1205732119887) sin (120572

2119888)

11988507

119890lim =(120590 minus sin (19 + 2120573

2119887))

(1 minus sin (19 + 21205732119887))

119890 =1198631

1198632

If 119890 gt 119890lim 120590 = 120590(1 minus ((119890 minus 119890lim)

(1 minus 119890lim))

radic120573211988710

)

1198811199062

= 120590(1198802minus

1198811198982

tan (1205732119887))

119879and

01= 1198791+

1198821

2

(2119862119901)

119879and

02= 119879and

01+

(1198802

2minus 1198802

1)

(2119862119901)

(3)


Start

Choose 1205881

Choose 1205882

Input variables m[mmin mmax]

No

No

No

No

Inlet calculationVm1 V1 W1 P1 T1

Yes

Yes

Yes

Yes

1205881998400

Internal losses

Calculate 1205882998400

Stop

|1205881 minus 1205881998400|1205881 lt 01

|1205882 minus 1205882998400|1205882 lt 01

No parasitic losses120578totimp

Calculate P02 P2

Outlet calculationVm2 Vu2Wu2W2 T2

V2 T02 1205732 1205722

Check if Mth = 1

Calculate Mth

Parasitic losses

Include parasitic losses

No parasitic losses

Calculate 120578totstg


Calculate Δhvld

Calculate P4 1205884 Vm4998400

Choose Vm4

Choose n

Calculate Vu4 V4 T4

|Vm4 minus Vm4998400|Vm4 lt 01

17ndash20

Figure 3 Detailed flow chart of 1D analysis part

So 1198821199062 1198822 1198812 and flow angles can be easily obtained

based on basic impeller outlet velocity triangleThen (4) willprovide 119879

2and 119879

02

1198792= 119879and

02minus

1198822

2

(2119862119901)

11987902

= 11987901

+11988021198811199062

119862119901

(4)

Then all the internal losses and parasitic losses calcula-tion can be finishedDetailed equations regarding lossmodelscan be found in [2] 119861

2can be updated by (5)

11986121015840 =

(ℎsf + ℎhs)

11988222

+ (03 +1198872

2

1198712119861

)119860119877211988721205882

(1205881119871119861) (5)

120578I =119862119901(11987902

minus 11987901) minus (Δℎinc + Δℎbl + Δℎsf + Δℎmix + Δℎdif)

119862119901(11987902

minus 11987901)

(6)


If parasitic losses are included a revised formula (7)should be applied

120578II=119862119901(11987902


119862119901(11987902

minus 11987901) + Δℎdf + Δℎ

119897

(7)

Finally density at impeller outlet is updated and com-pared with the initial one

11987902is = 119879

01+ 120578I (11987902 minus 119879

01)

11987502

= 11987501(11987902is11987901

)

119896(119896minus1)

1198752= 11987502(11987902

1198792

)

119896(119896minus1)

12058821015840 =

1198752

(1198771198792)

(8)

The code will continue running until density at impelleroutlet converges If vaneless diffuser is involved in the analysispart the corresponding loss model shown in Figure 3 has tobe added

Internal losses mainly include incidence loss blade load-ing loss skin friction loss wake mixing loss hub-to-shroudloading loss entrance diffusion loss and clearance loss [2 12]Generally parasitic losses are not involved in CFD In orderto make an efficiency comparison with or without parasiticlosses disc friction recirculation and leakage are also takeninto consideration In this case clearance loss and leakage lossare not considered due to shrouded impeller

Besides the above descriptions how to predict stall andchoke point in the analysis part must be mentioned herebecause these two points determine the operating range ofcentrifugal compressor directly In Figure 3 mass flow rateas the only input variable has a range coming from designpart directly The initial value of mass flow rate at chokepoint is predicted in design part by (9) developed by Dixon[13] Actually (9) always overpredicts mass flow rate whichis because aerodynamic blockage at impeller throat is notconsidered Therefore it is appropriate to apply (9) to obtainlarger mass flow rate for choke point first The initializationof mass flow rate for stall point calculated in design part isto use critical absolute flow angle at diffuser inlet which iscomputed by (10) Pinch between impeller outlet and diffuserinlet is also considered if applied [14]

119898choke = 119860 th1205880111988601[2 + (119896 minus 1)119880

2

11198862

01

119896 + 1]

(k+1)2(119896minus1)

(9)

120572lowast

3119888= 1205723119888

+ (1702 minus1484119887

2

1198892

)(1 minus1198873

1198872

) (10)

In the analysis part for stall point prediction two meth-ods are adopted The main one used by majority designersis to check if 119882

1119904119882th ge 175 which seemed as an inducer

stall criterion The second option is to estimate equivalent

10

12

14

16

18

20

22

02 03 04 05 06 07 08 09 10 11mmchoke

Deq

Figure 4 Equivalent diffusion factor

diffusion factor (119863eq) Lieblein [15] found that blade stalloccurs in the axial compressor when 119863eq is bigger than 2Later on 119863eq is generalized to radial and mixed flow bladesto evaluate the blade loading This criterion is also foundto be an appropriate blade stall limit for impellers [2] Therelated calculation equations are shown below119863eq is checkedat every iteration step in analysis part to see if it reachesthe criteria or not If neither 119882

1119904119882th nor 119863eq indicates the

compressor stalls the initial mass flow rate value from (10)will be considered to be the minimum value

119863eq =119882max1198822

119882max =(1198821+ 1198822+ Δ119882)

2

Δ119882 =212058711986321198802119868119861

(119885119871119861)

(11)

Finally in this case 1D code points out to 119863eq which isdetected beyond the critical value first The distribution of119863eq over mass flow rate is shown in Figure 4 Based on thecase in this paper it is still hard to determine which methodis suitable to stall point prediction so far According to CFDsimulation and test data further evaluation of the 1D codeneeds to be done through huge number of impellers whichwill be focused on in the future research work

As for choke point calculation iterativemethod is applieduntil relative Mach number at the impeller throat reaches 1Generally choke mass flow rate from analysis part is smallerthan initialization based on (9) In this case 1D code predictschokemass flow rate at 80 kgs andCFD shows it at 8155 kgsThe difference is less than 2 However the initial value is8954 kgs larger than analysis code prediction as expected

Since mass flow rate in analysis part is increased byspecific growth increment the real operating range of thecentrifugal compressor will be fixed after choke point cal-culation is done The code will plot the entire performance


3

4

5

6

7

8

05 06 07 08 09 1 11

Test

120587to

t

mmchoke

1D code

(a) Adiabatic efficiency

076

078

08

082

084

086

088

09

05 06 07 08 09 1 11

Test

mmchoke

1D code

120578

(b) Total pressure ratio

Figure 5 Performance curves between 1-D and test

of the centrifugal compressor automatically including designand off-design points

1D analysis code was calibrated by the test data of one ofthree Krainrsquos impellers [16 17] In Figure 5 it is obvious thattotal pressure ratio and adiabatic efficiency show similar dis-tribution trend and the discrepancies between performancecurves are acceptable Compared to the test wider surgemargin is obtained by 1D code Overall based on this case1D analysis code is functional and satisfying Also a detailedflow parameter comparison of 1D code and test data of a highflow coefficient centrifugal compressor have been carried outwhich also indicates good predictive effect of 1D analysiscode The related results are not shown here

3 1D Design and Analysis Results

Main impeller geometric parameters from 1D code are listedin Table 1 Design parameters like mass flow rate rotationalspeed impeller outlet diameter and shaft diameter areprovided by industry according to practical demand

Since leading edge and trailing edge of the impeller havethe same streamline slope angle in this case hub-to-shroudloss model is not activated The criterion in the chokingloss model is not satisfied and the choking loss is also notconsidered as well Recirculation loss is neglected due to theshrouded impeller designed in this study

Therefore internal losses actually used include wakemixing loss skin friction loss incidence loss blade load-ing loss and inducer diffusion loss Figure 6 shows losscoefficient distribution of impeller It is observed that skin

Table 1 Impeller parameters

119898 48 kgs119873 12800 rpm120593 0022119873119904

0071198631

02172m1198632

0381m1198871

00123m1198872

00082m1205731119887

289∘

1205732119887

40∘

119871119911

00193m11988721198632

00216119885 13

friction is the main loss source in the impeller as expecteddue to narrow flow path of low flow coefficient centrifugalcompressor and increases gradually from surge side to chokeside Wake loss appears near surge point and presents thesmallest loss coefficient Minimum incidence loss occurs atdesign point The distribution of incidence loss is prettytypical Either increasing mass flow rate from design pointto choke side or reducing it from design point to surge sidewill causemore incidence loss Higher pressure ratio is alwaysaccompanied by accordingly higher blade loading loss whichcan explain why blade loading loss distributes like the trendshown in Figure 6 An inducer diffusion loss appears onlyfor applications of less than 63 mass flow rate which is


0

01

02

03

04

05

06

036 040 044 051 057 063 073 078 086 093 097 100

Loss

coeffi

cien

t

Inducer diffusionBlade loadingIncidence

Skin frictionWake mixing

mmchoke

Figure 6 Loss coefficient distribution of impeller

0

002

004

006

008

01

012

014

036 040 044 051 057 063 073 078 086 093 097 100

Wor

k co

effici

ent

LeakageDisc friction

mmchoke

Figure 7 Distribution of work coefficient due to parasitic losses

the design point For parasitic losses or external losses onlydisc friction loss and leakage loss are involved They presentsimilar distribution trend seen in Figure 7 Overall 1D codeprovides asmuch information as it canMore importantly the1D analysis results look reasonable and acceptable

Note Please refer to reference [2] to look up the equationsused for calculating loss coefficient and work coefficient inFigures 6 and 7 which are based on the results that theanalysis code provides

05

06

07

08

09

10

11

12

13

14

000

015

030

045

060

075

090

105

120

135

004 005 006 007 008 009

Test CFDTest CFD

120578120595

120593

Figure 8 Performance comparison between test and CFD

4 3D CFD Validation

Many impellers coming from either open literature or indus-try have been used to evaluate the accuracy of the CFDanalysis before It is confirmed that our CFD simulation isreasonable and reliable An impeller similar to the case inthis paper is chosen to present our CFD analysis accuracycompared to test data from a cooperative company Thisimpeller also applies two-dimensional blades and is usedin the last stage centrifugal compressor of the multistagecompressors It can be observed in Figure 8 that CFD has agood agreement with test data

Note Vaneless diffuser and return channel are also involvedin the test and CFD

For the case in this paper before running CFD impellermodeling is built in the commercial software ldquoBLADEGENrdquoaccording to the fixed geometry from 1D design codeFigure 9 shows the geometry and meridional view of theimpeller It can be observed that leading edge of impeller is inthe radial direction and flow channel is so narrow which arethe typical characteristics of low flow coefficient centrifugalcompressors Inlet and outlet computational domains haveto be added to avoid reserved flow and improve accuracy ofcomputation in CFD Extended outlet computational domainacts like practical vaneless diffuser and makes impeller runclose to the real environment

Three-dimensional viscous RANS solver and Spalart-Allamaras turbulent model are applied in the steady simu-lation with inlet boundary conditions of 119875

01= 13602 kPa

11987901

= 424K and axial flow Static pressure is set on the outletboundary condition and increased gradually Near lower flowregion using mass flow rate condition captures stall pointinstead of static pressure condition The working medium is


(a) Impeller geometry

80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view

Figure 9 Impeller design

100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke

(a) Total pressure ratio

010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578

(b) Adiabatic efficiency

Figure 10 Simulation results with three grid plans

defined as typical natural gas All solidwalls are assumed to besmoothwith roughness of 10minus6m In order to reduce the effectof skin friction coefficient affected by surface roughness samesurface property has to be employed in 1D code whenmakinga comparison between 1D prediction and CFD simulation

In order to understand the analytical accuracy of theCFD on the impeller of this case three grid plans shown inTable 2 were tried to check grid sensitivity Figure 10(a) showsthat total pressure ratio distribution curves from three gridplans match very well Adiabatic efficiency curves seen in


Table 2 Three grid plans

Circumferential Streamwise Spanwise Grid numberGrid 1 27 87 39 185367Grid 2 35 107 49 295421Grid 3 43 127 57 435309

Figure 10(b) also present a good match However it seemsthat larger stall margin is obtained by applying less gridnumber In consideration of running safety in the test thegrid plan 3 with smaller stall margin will be selected as areference for design because designers can not just pick theoption with the best evaluation

For adiabatic efficiency of impeller 1D code can presentthe results with and without parasitic losses However CFDdoes not usually include parasitic losses unless the exact prac-tical structure of centrifugal compressor is built That is thereason why Figure 11(a) shows three distribution curves forefficiency In addition to evaluate 1D code through comparingwith CFD without parasitic losses the efficiency includingparasitic losses close to real environment is needed to bepresented as well 1D code shows wider operating rangeand further stall point which means that comparing withCFD 1D code predicts that the centrifugal compressor canrun at lower mass flow rate Itrsquos shown that steady CFDsimulation cannot get an accurate stall point Transientsimulation should be done after steady calculation which isnot completed in this paper According to the comparisonof efficiency CFD has a little bit discrepancy in adiabaticefficiency on the right side of blue line However differencein efficiency becomes bigger and distribution trends showdifferent as mass flow rate continues to be reduced

In Figure 11(b) total pressure ratio from CFD is overpre-dicted but has same distribution trend with 1D predictionSince empirical loss models are used in the code and CFDcannot show the exact results with real values we think1D code works well and has the function to finish the per-formance evaluation within acceptable discrepancy Furthervalidation with more impellers needs to be carried out in thefuture to see if a constant coefficient for total pressure ratioexists between 1D code and CFD And if this is the case theprediction accuracy of 1D code can be further improved

Note Equation (6) presents the efficiency without parasiticlosses However parasitic losses are taken into account in(7) As mentioned before these two equations are used to becompared with CFD and test data respectively

Relative Mach number distributions at 10 50 and90 spans are shown in Figure 12 It is very clear that alow velocity region exists near pressure surface and becomessmaller gradually as blade span increases Because high staticpressure near pressure surface accompanies low velocity thelow magnitude area can be expected near pressure surfaceMoreover narrow passage introduces large viscous effectsand the typical velocity distribution of the bend comes upjust before the leading edge of the impeller which makes itobvious why Figure 12 presents relative Mach number like

that Fortunately no flow separation is found in the flow fieldof the impeller

Figure 13 shows 119881119898distribution on the meridional view

No low flow region and separation bubble exist whichindicates that this design is rational It is seen in Figure 13that velocity is gradually increasing from hub to shroud inthe extended inlet region which is a typical flow distributionin the bend High velocity means low static pressure andvice versa High pressure static pressure near hub regioncreates centripetal force to balance centrifugal force In orderto reduce the effect of this kind of flow distribution onthe incidence angles along the span of impeller inlet lowercurvature of shroud in the extended inlet region which cancause higher flow area and reduce hub-to-shroud velocitygradient should be applied for performance optimization inthe future Figure 14 presents the idea that static pressuredistribution is smooth through the impeller as expected

Due to the even absolute flow angle distribution atimpeller exit shown in Figure 15 vaneless or vaned diffuserhas perfect inlet condition and can be easily designed ifneeded Blade loading at 10 50 and 90 spans distributesreasonably which can be checked out in detail in Figure 16The blank area between pressure and suction surface staticcurves represents the load that blade withstands which doesnot vary too much through the blade at different spans Thatis what designers expect

The main performance parameters comparison at designpoint is listed in Table 3 The power calculated by CFD isapproximately in the middle of these two values computed by1D code Compared to CFD the efficiency is overpredictedby 152 in the 1D code which is really satisfying The totalpressure ratio in CFD is 305 higher than 1D code Inaddition parasitic losses drop efficiency a lot based on 1Dprediction

5 Conclusions

1D code is developed in this paper and divided into two partswhich are used to design and analyze Based on the code alowflow coefficient centrifugal compressor is designed and itsperformance is predicted In comparisonwith 1D code steadyCFD simulation has been carried out and provides flow fielddetails in the impeller

Comparing to 1D code CFD overpredicts total pressureratio of impeller but shows the same distribution trend whichis encouraging Further validations through more impellershave to be done in the future to confirm if a constantcoefficient for total pressure ratio exists between CFD and1D code For adiabatic efficiency 1D code without includingparasitic losses matches very well CFD in the operating rangeof near design point and choke point However 1D code isoverpredicted between near design point and stall point Thedifference of mass flow rate at choke point between CFDand 1D code is less than 2 Compared to steady CFD 1Dcode presents stall point at lower mass flow rate No flowseparation is found in the flow field of impeller and even theabsolute flow angle distribution at impeller exit is shownThe


02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10

CFD1D (with parasitic losses)1D (without parasitic losses)

120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke


Figure 11 Impeller performance curves

0 01 02 03 04 05 06

(a) 10 span (b) 50 span (c) 90 span

Figure 12 Relative Mach number map

120

100

80

60

40

20

0

Vm

Figure 13 Meridional velocity and streamline distribution

15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e

Figure 14 Static pressure distribution

0

5

10

15

20

25

30

0Axial length (mm)

minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1

1205722

(∘)

Figure 15 Absolute flow angle distribution at impeller exit


0Axial length (mm)

10 span50 span90 span

10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)

minus20 minus18 minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2

Figure 16 Blade loading distribution at different spans

Table 3 Performance comparison between CFD and 1D code atdesign point

CFD 1D(no parasitic losses)

1D(parasitic losses)

Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257

blade loading at three different spans distributes smoothly asexpected

Since the low flow coefficient centrifugal compressordesigned in this paper is not the usual type and empiricalequations used are based on medium flow coefficient com-pressors further research on the improvement and validationof 1D code needs to be done by experiment and CFD toincrease prediction accuracy

Nomenclature

119860 Area (m2)AR Area ratio (119860

2sin(1205732119887)1198601sin(120573th))

119886 Sonic velocity (ms)119861 Fraction area blockage119887 Blade height (m)119862119901 Specific heat at constant pressure(Jkg K)

119890 Impeller mean line radius ratio119863 Diameter (m)119863eq Equivalent diffusion factor119896 Ratio of specific heatsΔℎ Enthalpy increase (Jkg)119868119861 Blade work input coefficient

120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2

119871119885 Impeller axial length (m)

119871119861 Length of blade mean camberline (m)

119872 Relative Mach number119898 Mass flow rate (kgs)119873 Rotation speed (rpm)119873119904 Specific speed

119875 Pressure (pa)119877 Gas constant (J(kgsdotK))119879 Temperature (K)119880 Blade speed (ms)119881 Absolute velocity (ms)119882 Relative velocity (ms)Δ119882 Average blade velocity difference (ms)119885 Number of the blades

Greeks

120572 Absolute flow angle from tangential (∘)120573 Relative flow angle from tangential (∘)120578 Adiabatic efficiency120582 Workenthalpy coefficient Δℎ119880

2

2

120587 Pressure ratio120588 Gas density (kgm

3)

120590 Slip factor120593 Inlet flow coefficient119898(120588

111988021205871198632

24)

Ψ Inlet head coefficient(8544119863

2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)

Δ120596 Total pressure loss coefficientΔ1199010(1199010minus 119901)

Subscripts

0 Total condition1 Impeller inlet2 Impeller outlet3 Diffuser inlet4 Diffuser outlet

119887 Bladebl Blade loading119888 Criterionchoke Choke pointdf Disc frictiondif Inducer diffusionℎ Impeller inlet hubhs Hub-to-shroudinc Incidenceis Isentropic119897 Leakagelim Limit119898 Meridional componentmax Maximummin Minimummix Wake mixing119904 Shroudst Staticth Impeller throat


sf Skin frictiontot Total-to-total119906 Tangential componentvld Vaneless diffuser

Superscripts

lowast Correct value1015840 New valueand Value relative to rotating frame of reference

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work there is no professionalor other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thepaper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China under Grant no 51010007 and theChina Scholarship Council (CSC) The authors would like tothank Dr Marco Vagani and Christopher D Bolin for theirdiscussion and advice

References

[1] T Sato J M Oh and A Engeda ldquoExperimental and numericalinvestigation of the flow in a vaneless diffuser of a centrifugalcompressor stage Part 1 experimental investigationrdquo Proceed-ings of the Institution of Mechanical Engineers C vol 219 no 10pp 1053ndash1059 2005

[2] R H Aungier Centrifugal Compressor A Strategy for Aerody-namic Design and Analysis ASME Press New York NY USA2000

[3] V Rusak ldquoDevelopment and performance of the wedge-typelow specific speed compressor wheelrdquo Tech Rep 82-GT-214ASME 2009

[4] M V Casey P Dalbert and E Schurter ldquoRadial compressorstages for low flow coefficientsrdquo in Proceedings of the IMechEInternational Conference on Machinery for the Oil and GasIndustries 1990

[5] T Koizumi ldquoExperimental studies on performance of centrifu-gal compressors with very small flow coefficientrdquoMitsui ZosenTechnical Review vol 117 pp 28ndash39 1983

[6] J Paroubek J Kyncl and V Cyrus ldquoExperimental investigationand performance analysis of six low flow coefficient centrifugalcompressor stagesrdquo Transactions of the ASME Journal of Turbo-machinery vol 117 no 4 pp 585ndash592 1995

[7] J-L Di Liberti Design and development of low-flow coefficientcentrifugal compressors for industrial application [PhD thesis]Michigan State University 1998

[8] Y I Biba D A Nye and L I U Zheji ldquoPerformance evaluationand fluid flow analysis in low flow stages of industrial centrifu-gal compressorrdquo International Journal of Rotating Machineryvol 8 no 5 pp 309ndash317 2002

[9] M R Galvas ldquoFortran program for predicting off-designperformance of centrifugal com-pressorsrdquo Tech Rep TN D-7487 NASA 1973

[10] K H Ludtke Process Centrifugal Compressors Basic FunctionOperation Design Application Springer Berlin Germany 2010

[11] F J Wiesner ldquoA review of slip factors for centrifugal compres-sorsrdquo Journal of Engineering for Power Transactions of theASMEvol 89 pp 558ndash572 1967

[12] R H Aungier ldquoMean streamline aerodynamic performanceanalysis of centrifugal compressorsrdquo Transactions of the ASMEJournal of Turbomachinery vol 117 no 3 pp 360ndash366 1995

[13] S L Dixon Fluid Mechanics and Thermodnamics of Turboma-chinery Butterworth-Heinemann 4th edition 1998

[14] Y Senoo and Y Kinoshita ldquoLimits of rotating stall and stall invaneless diffusers of centrifugal compressorsrdquo Tech Rep 78-GT-19 ASME 1978

[15] S Lieblein ldquoLoss and stall analysis of compressor cascadesrdquoASME Transactions Journal of Basic Engineering vol 81 pp387ndash400 1959

[16] H Krain and B Hoffmann ldquoFlow physics in high pressure ratiocentrifugal compressorsrdquo in Proceedings of the ASME FluidsEngineering Division Summer Meeting June 1998

[17] G Eisenlohr and H Krain ldquoInvestigation of the flow througha high pressure ratio centrifugal impellerrdquo in Proceedings of theASME Conference Proceedings 2003

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



Axial flow compressor

Common applicationarea of the

radial impeller

Mixed flow impelleror

radial impellerhigh flow coefficient

2015

100806

04

02

01

001

002

004

006

008 0

1

02

04

06

08

10

120593

120595

(a)

120593

0 005 01 015

Aerodynamic losses

Leakage lossesDisc friction losses

Design point efficiency

0

02

04

06

08

120578

(b)

Figure 1 Application areas of high flow coefficient radial impellers and typical industrial compressors [1]

such high pressure ratio compressor operating at lowflow ratehas risen Nevertheless little information is available con-cerning low flow coefficient centrifugal compressor designin the open literature The following descriptions reviewsome research work regarding design of low flow coefficientcentrifugal compressors

Rusak [3] briefly presented three types of impeller designfor low flow application low flow conventional stages with2D impellers low flow high pressure coefficient stages withradial impellers and very low flow stages with wedge-typeimpellers Casey et al [4] designed and tested several low flowcoefficient centrifugal compressors with high head and lowoutlet meridional velocity component Koizumi [5] designedthree different compressors with straight blades S-shapedblades and almost radial blades In the mean time theinfluence of the viscous effect on the slip factor was studiedand the stall-onset condition for rotating stall in the vanelessdiffuser following the impeller was investigated as wellParoubek et al [6] tested six conventional low flow coefficientimpellers The effects of inducer hub to exit shroud diameterratio and channel width parameter on the stage perfor-manceswere investigatedDi Liberti [7] designed several low-flow-coefficient impellers and analyzed them numerically toinvestigate the effect of the design flow coefficient of theoutlet width-outlet blade angle combination and of the bladeleading edge position on the stage performance Biba et al [8]used commercial CFD codes together with in-house programdesign and analyzed two different impellers with low volumeflows and high pressure applications

Based on the above research results this study aims todesign an efficient lowflow coefficient centrifugal compressorwith wide operating range MATLAB is used as a tool todevelop 1D design and analysis codesThe three-dimensionalgeometry model of impeller is built by ldquoBLADEGENrdquo OnceCFD indicates that the performance of centrifugal compres-sor is satisfactory and has an acceptable discrepancy with 1Dprediction the final geometry of impeller will be fixed anddesign process is terminated

2 1D Design and Analysis Methods

1D code developed in this paper is divided into two partsdesign and analysis parts Design part provides basic geomet-ric data of impeller at design point Based on the geometryanalysis part can do the performance analysis using empiricalloss models at design and off-design points [9] and export thewhole performance curve of the centrifugal compressor It isnecessary to keep adjusting geometry until the comparisonof CFD and 1D codes shows that discrepancy is acceptableand design goals have been reached The 1D design andanalysis scheme is illustrated in Figure 2 Clearance loss andrecirculation loss are excluded in this shrouded impeller Infact hub-to-shroud loss and choke loss are not activated aswell which will be discussed in the following section of thispaper

In the case of this paper mass flow rate (119898) inlettotal pressure of impeller (119875

01) inlet total temperature of

impeller (11987901) rotational speed of impeller (119873) impeller

outlet diameter (1198632) and hub diameter (119863

ℎ) are given The

working medium is natural gas and typical compositionis used while setting the fluid properties in CFD In thedesign part the values of 119881

11989821198802 1205732119887

[10] and 1198811199062

1198811199062119887

are assumed within empirical ranges With these parametersimpeller outlet calculation can be completed Then in orderto calculate impeller inlet aerodynamic parameters the ratiosof 11986311198632and119882

21198821must be chosen to relate outlet to inlet

Since the inlet flow coefficient is 0022 impeller inlet shouldbe located in the radial direction which means that inlet hubdiameter is equal to inlet shroud diameter [2] It is because thereturn channel upstream of low coefficient compressor in thereal application guides the flow to be more uniform Inducercan be neglected which reduces flow path in the impellerand produces less skin friction lossMeanwhile impeller axiallength (119871

119911) can be determined as well Number of the blades

(119885) comes from original slip factor formula of Wiesner [11]Analysis part only includes one input variable which is

mass flow rate In the following analysis procedure fluiddensity at impeller inlet is initially approximated and requires


Given paremeters

Outlet analysis


Inlet design

Des

ign

part



Ana

lysis

par

t





Choking loss

Blade loading loss

loss





1199061= 0 and 119881

1= 1198811198981 Inlet


1205881asymp 12058801

=11987501

(11987711987901) (1)


1198981 1198801 1198821 and 119882th can be


1198791= 11987901

minus1198812

1

(2119862119901)

1198751= 11987501(

1198791

11987901

)

119896(119896minus1)

12058811015840 =

1198751

(1198771198791)

(2)



later Then 1198811199062


1205882= 1205881(

1198802

2

(11986211990111987901)+ 1)

1(119896minus1)

1198811198982

=119898

(120587120588211986321198872(1 minus 119861

2))

120590 = 1 minus

radicsin (1205732119887) sin (120572

2119888)

11988507

119890lim =(120590 minus sin (19 + 2120573

2119887))

(1 minus sin (19 + 21205732119887))

119890 =1198631

1198632



radic120573211988710

)

1198811199062

= 120590(1198802minus

1198811198982

tan (1205732119887))

119879and

01= 1198791+

1198821

2

(2119862119901)

119879and

02= 119879and

01+

(1198802

2minus 1198802

1)

(2119862119901)

(3)


Start

Choose 1205881

Choose 1205882


No

No

No

No


Yes

Yes

Yes

Yes

1205881998400

Internal losses

Calculate 1205882998400

Stop

|1205881 minus 1205881998400|1205881 lt 01

|1205882 minus 1205882998400|1205882 lt 01


Calculate P02 P2


V2 T02 1205732 1205722

Check if Mth = 1

Calculate Mth

Parasitic losses


No parasitic losses



Calculate Δhvld


Choose Vm4

Choose n

Calculate Vu4 V4 T4


17ndash20




2and 119879

02

1198792= 119879and

02minus

1198822

2

(2119862119901)

11987902

= 11987901

+11988021198811199062

119862119901

(4)



11986121015840 =

(ℎsf + ℎhs)

11988222

+ (03 +1198872

2

1198712119861

)119860119877211988721205882

(1205881119871119861) (5)

120578I =119862119901(11987902


119862119901(11987902

minus 11987901)

(6)



120578II=119862119901(11987902


119862119901(11987902


119897

(7)


11987902is = 119879

01+ 120578I (11987902 minus 119879

01)

11987502

= 11987501(11987902is11987901

)

119896(119896minus1)

1198752= 11987502(11987902

1198792

)

119896(119896minus1)

12058821015840 =

1198752

(1198771198792)

(8)




119898choke = 119860 th1205880111988601[2 + (119896 minus 1)119880

2

11198862

01

119896 + 1]

(k+1)2(119896minus1)

(9)

120572lowast

3119888= 1205723119888

+ (1702 minus1484119887

2

1198892

)(1 minus1198873

1198872

) (10)




10

12

14

16

18

20

22

02 03 04 05 06 07 08 09 10 11mmchoke

Deq





119863eq =119882max1198822

119882max =(1198821+ 1198822+ Δ119882)

2

Δ119882 =212058711986321198802119868119861

(119885119871119861)

(11)





3

4

5

6

7

8

05 06 07 08 09 1 11

Test

120587to

t

mmchoke

1D code


076

078

08

082

084

086

088

09

05 06 07 08 09 1 11

Test

mmchoke

1D code

120578










119898 48 kgs119873 12800 rpm120593 0022119873119904

0071198631

02172m1198632

0381m1198871

00123m1198872

00082m1205731119887

289∘

1205732119887

40∘

119871119911

00193m11988721198632

00216119885 13



0

01

02

03

04

05

06

036 040 044 051 057 063 073 078 086 093 097 100

Loss

coeffi

cien

t



mmchoke


0

002

004

006

008

01

012

014

036 040 044 051 057 063 073 078 086 093 097 100

Wor

k co

effici

ent


mmchoke




05

06

07

08

09

10

11

12

13

14

000

015

030

045

060

075

090

105

120

135

004 005 006 007 008 009

Test CFDTest CFD

120578120595

120593


4 3D CFD Validation





01= 13602 kPa

11987901




80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke


010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578


















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Given paremeters

Outlet analysis


Inlet design

Des

ign

part



Ana

lysis

par

t





Choking loss

Blade loading loss

loss





1199061= 0 and 119881

1= 1198811198981 Inlet


1205881asymp 12058801

=11987501

(11987711987901) (1)


1198981 1198801 1198821 and 119882th can be


1198791= 11987901

minus1198812

1

(2119862119901)

1198751= 11987501(

1198791

11987901

)

119896(119896minus1)

12058811015840 =

1198751

(1198771198791)

(2)



later Then 1198811199062


1205882= 1205881(

1198802

2

(11986211990111987901)+ 1)

1(119896minus1)

1198811198982

=119898

(120587120588211986321198872(1 minus 119861

2))

120590 = 1 minus

radicsin (1205732119887) sin (120572

2119888)

11988507

119890lim =(120590 minus sin (19 + 2120573

2119887))

(1 minus sin (19 + 21205732119887))

119890 =1198631

1198632



radic120573211988710

)

1198811199062

= 120590(1198802minus

1198811198982

tan (1205732119887))

119879and

01= 1198791+

1198821

2

(2119862119901)

119879and

02= 119879and

01+

(1198802

2minus 1198802

1)

(2119862119901)

(3)


Start

Choose 1205881

Choose 1205882


No

No

No

No


Yes

Yes

Yes

Yes

1205881998400

Internal losses

Calculate 1205882998400

Stop

|1205881 minus 1205881998400|1205881 lt 01

|1205882 minus 1205882998400|1205882 lt 01


Calculate P02 P2


V2 T02 1205732 1205722

Check if Mth = 1

Calculate Mth

Parasitic losses


No parasitic losses



Calculate Δhvld


Choose Vm4

Choose n

Calculate Vu4 V4 T4


17ndash20




2and 119879

02

1198792= 119879and

02minus

1198822

2

(2119862119901)

11987902

= 11987901

+11988021198811199062

119862119901

(4)



11986121015840 =

(ℎsf + ℎhs)

11988222

+ (03 +1198872

2

1198712119861

)119860119877211988721205882

(1205881119871119861) (5)

120578I =119862119901(11987902


119862119901(11987902

minus 11987901)

(6)



120578II=119862119901(11987902


119862119901(11987902


119897

(7)


11987902is = 119879

01+ 120578I (11987902 minus 119879

01)

11987502

= 11987501(11987902is11987901

)

119896(119896minus1)

1198752= 11987502(11987902

1198792

)

119896(119896minus1)

12058821015840 =

1198752

(1198771198792)

(8)




119898choke = 119860 th1205880111988601[2 + (119896 minus 1)119880

2

11198862

01

119896 + 1]

(k+1)2(119896minus1)

(9)

120572lowast

3119888= 1205723119888

+ (1702 minus1484119887

2

1198892

)(1 minus1198873

1198872

) (10)




10

12

14

16

18

20

22

02 03 04 05 06 07 08 09 10 11mmchoke

Deq





119863eq =119882max1198822

119882max =(1198821+ 1198822+ Δ119882)

2

Δ119882 =212058711986321198802119868119861

(119885119871119861)

(11)





3

4

5

6

7

8

05 06 07 08 09 1 11

Test

120587to

t

mmchoke

1D code


076

078

08

082

084

086

088

09

05 06 07 08 09 1 11

Test

mmchoke

1D code

120578










119898 48 kgs119873 12800 rpm120593 0022119873119904

0071198631

02172m1198632

0381m1198871

00123m1198872

00082m1205731119887

289∘

1205732119887

40∘

119871119911

00193m11988721198632

00216119885 13



0

01

02

03

04

05

06

036 040 044 051 057 063 073 078 086 093 097 100

Loss

coeffi

cien

t



mmchoke


0

002

004

006

008

01

012

014

036 040 044 051 057 063 073 078 086 093 097 100

Wor

k co

effici

ent


mmchoke




05

06

07

08

09

10

11

12

13

14

000

015

030

045

060

075

090

105

120

135

004 005 006 007 008 009

Test CFDTest CFD

120578120595

120593


4 3D CFD Validation





01= 13602 kPa

11987901




80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke


010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578


















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start

Choose 1205881

Choose 1205882


No

No

No

No


Yes

Yes

Yes

Yes

1205881998400

Internal losses

Calculate 1205882998400

Stop

|1205881 minus 1205881998400|1205881 lt 01

|1205882 minus 1205882998400|1205882 lt 01


Calculate P02 P2


V2 T02 1205732 1205722

Check if Mth = 1

Calculate Mth

Parasitic losses


No parasitic losses



Calculate Δhvld


Choose Vm4

Choose n

Calculate Vu4 V4 T4


17ndash20




2and 119879

02

1198792= 119879and

02minus

1198822

2

(2119862119901)

11987902

= 11987901

+11988021198811199062

119862119901

(4)



11986121015840 =

(ℎsf + ℎhs)

11988222

+ (03 +1198872

2

1198712119861

)119860119877211988721205882

(1205881119871119861) (5)

120578I =119862119901(11987902


119862119901(11987902

minus 11987901)

(6)



120578II=119862119901(11987902


119862119901(11987902


119897

(7)


11987902is = 119879

01+ 120578I (11987902 minus 119879

01)

11987502

= 11987501(11987902is11987901

)

119896(119896minus1)

1198752= 11987502(11987902

1198792

)

119896(119896minus1)

12058821015840 =

1198752

(1198771198792)

(8)




119898choke = 119860 th1205880111988601[2 + (119896 minus 1)119880

2

11198862

01

119896 + 1]

(k+1)2(119896minus1)

(9)

120572lowast

3119888= 1205723119888

+ (1702 minus1484119887

2

1198892

)(1 minus1198873

1198872

) (10)




10

12

14

16

18

20

22

02 03 04 05 06 07 08 09 10 11mmchoke

Deq





119863eq =119882max1198822

119882max =(1198821+ 1198822+ Δ119882)

2

Δ119882 =212058711986321198802119868119861

(119885119871119861)

(11)





3

4

5

6

7

8

05 06 07 08 09 1 11

Test

120587to

t

mmchoke

1D code


076

078

08

082

084

086

088

09

05 06 07 08 09 1 11

Test

mmchoke

1D code

120578










119898 48 kgs119873 12800 rpm120593 0022119873119904

0071198631

02172m1198632

0381m1198871

00123m1198872

00082m1205731119887

289∘

1205732119887

40∘

119871119911

00193m11988721198632

00216119885 13



0

01

02

03

04

05

06

036 040 044 051 057 063 073 078 086 093 097 100

Loss

coeffi

cien

t



mmchoke


0

002

004

006

008

01

012

014

036 040 044 051 057 063 073 078 086 093 097 100

Wor

k co

effici

ent


mmchoke




05

06

07

08

09

10

11

12

13

14

000

015

030

045

060

075

090

105

120

135

004 005 006 007 008 009

Test CFDTest CFD

120578120595

120593


4 3D CFD Validation





01= 13602 kPa

11987901




80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke


010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578


















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120578II=119862119901(11987902


119862119901(11987902


119897

(7)


11987902is = 119879

01+ 120578I (11987902 minus 119879

01)

11987502

= 11987501(11987902is11987901

)

119896(119896minus1)

1198752= 11987502(11987902

1198792

)

119896(119896minus1)

12058821015840 =

1198752

(1198771198792)

(8)




119898choke = 119860 th1205880111988601[2 + (119896 minus 1)119880

2

11198862

01

119896 + 1]

(k+1)2(119896minus1)

(9)

120572lowast

3119888= 1205723119888

+ (1702 minus1484119887

2

1198892

)(1 minus1198873

1198872

) (10)




10

12

14

16

18

20

22

02 03 04 05 06 07 08 09 10 11mmchoke

Deq





119863eq =119882max1198822

119882max =(1198821+ 1198822+ Δ119882)

2

Δ119882 =212058711986321198802119868119861

(119885119871119861)

(11)





3

4

5

6

7

8

05 06 07 08 09 1 11

Test

120587to

t

mmchoke

1D code


076

078

08

082

084

086

088

09

05 06 07 08 09 1 11

Test

mmchoke

1D code

120578










119898 48 kgs119873 12800 rpm120593 0022119873119904

0071198631

02172m1198632

0381m1198871

00123m1198872

00082m1205731119887

289∘

1205732119887

40∘

119871119911

00193m11988721198632

00216119885 13



0

01

02

03

04

05

06

036 040 044 051 057 063 073 078 086 093 097 100

Loss

coeffi

cien

t



mmchoke


0

002

004

006

008

01

012

014

036 040 044 051 057 063 073 078 086 093 097 100

Wor

k co

effici

ent


mmchoke




05

06

07

08

09

10

11

12

13

14

000

015

030

045

060

075

090

105

120

135

004 005 006 007 008 009

Test CFDTest CFD

120578120595

120593


4 3D CFD Validation





01= 13602 kPa

11987901




80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke


010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578


















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3

4

5

6

7

8

05 06 07 08 09 1 11

Test

120587to

t

mmchoke

1D code


076

078

08

082

084

086

088

09

05 06 07 08 09 1 11

Test

mmchoke

1D code

120578










119898 48 kgs119873 12800 rpm120593 0022119873119904

0071198631

02172m1198632

0381m1198871

00123m1198872

00082m1205731119887

289∘

1205732119887

40∘

119871119911

00193m11988721198632

00216119885 13



0

01

02

03

04

05

06

036 040 044 051 057 063 073 078 086 093 097 100

Loss

coeffi

cien

t



mmchoke


0

002

004

006

008

01

012

014

036 040 044 051 057 063 073 078 086 093 097 100

Wor

k co

effici

ent


mmchoke




05

06

07

08

09

10

11

12

13

14

000

015

030

045

060

075

090

105

120

135

004 005 006 007 008 009

Test CFDTest CFD

120578120595

120593


4 3D CFD Validation





01= 13602 kPa

11987901




80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke


010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578


















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

01

02

03

04

05

06

036 040 044 051 057 063 073 078 086 093 097 100

Loss

coeffi

cien

t



mmchoke


0

002

004

006

008

01

012

014

036 040 044 051 057 063 073 078 086 093 097 100

Wor

k co

effici

ent


mmchoke




05

06

07

08

09

10

11

12

13

14

000

015

030

045

060

075

090

105

120

135

004 005 006 007 008 009

Test CFDTest CFD

120578120595

120593


4 3D CFD Validation





01= 13602 kPa

11987901




80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke


010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578


















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











80

100

120

140

160

180

200

220

240

260

280

300

0 20minus40 minus20

Z (mm)

R(m

m)

(b) Meridional view


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

120587to

t

mmchoke


010

020

030

040

050

060

070

080

090

100

02 03 04 05 06 07 08 09 1 11

Grid 1Grid 2Grid 3

mmchoke

120578


















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















5 Conclusions




02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02 03 04 05 06 07 08 09 10 1100

02

04

06

08

10


120578

mmchoke


100

105

110

115

120

125

130

02 03 04 05 06 07 08 09 10 11

CFD

120587to

t

1D code

mmchoke



0 01 02 03 04 05 06



120

100

80

60

40

20

0

Vm


15E + 07

145E + 07

14E + 07

135E + 07

155E + 07

13E + 07St

atic

pre

ssur

e


0

5

10

15

20

25

30

0Axial length (mm)


1205722

(∘)



0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0Axial length (mm)


10E + 07

11E + 07

12E + 07

13E + 07

14E + 07

15E + 07

16E + 07

Pst

(Pa)






Power 1778400W 1656740W 1857339W120587tot 121 11731 11731120578 902 9172 8257



Nomenclature


2sin(1205732119887)1198601sin(120573th))



120590(1 minus 119898cot(1205732119887)((120588211986021198802)(1 minus 119861

2))) minus

119880111986211990611198802

2





Greeks


2

2


3)


111988021205871198632

24)


2)21198621199011198791((11987521198751)((119896minus1)119896)

minus 1)


Subscripts





Superscripts




Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Superscripts




Acknowledgment


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









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Research Article Design and Performance …downloads.hindawi.com/journals/ijrm/2013/293486.pdfResearch Article Design and Performance Evaluation of a Very Low Flow Coefficient Centrifugal