High Reaction Compressors

8/2/2019 High Reaction Compressors

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The s ~ % h @ r;vi@h@s@@wr@@ad@ S ~ ~ C B ~ P @p p ~ @ ~ f ~ t i ~ ~ %o .Dr.

&esHa. 23s heare of awm~agkimeat,N b c e . m d helpliulcuittcism were invalublo during the course of this work,

rnmagcdpL.


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Az ady818 af a mdtrl- stage a x i d cox231arecieoz *~fPeEz3igl3 Peat-rim blsdlng i s c a~ r i e dut. The uslaraja 2~3 : tb.aiis at' l iaea~9zfng134 sgw-tione, for QOW@%roughxe cobmpnreseos fa2 G%ia6me of Ma*%$. Aw~rmerical sdu t i a q of the noa-linesr eq~l;tkioaa9 won"2sed through fopdf-design ope~lteim. Flow velocities a r m g h Lbe erix stages arecaleuliated for the off-darsiga laow rate8 in orr3e;r Do inmsur~hat the Plowangle@ ar bcth rotas sld stator are ca lcda t~ t l ar ihia repeatiag Bowcondition. 'r31e oB-deefm iactdennce aardae $Em @&tops at Pow flicwrate@adicaw a poersgble deteriorafioa of ePf8ciency.


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SUM P FOR PARR-Tm e ressure rise per stage In an a x f d now cosnpresgor is pzo-

porttonal to the square oP ~e incidenrce velocity relative to %a blades.a m c a t dl axfa c6mpreseors for gae m r b i n ~ ~ere deeigned ivfth thewhirl iis tha direction as &he rstos blade speed. This choics of&sign f low gave high effgciencies when the blades w er e desigaed prop-erly. Ua t i l Ule introduction of t~ansaolicblading, a high afiiciency waeobtained only when tPle Ma c h number relative to the blade waa lescl thmunity -- usually in the nsfghbo~hood f 0.75. R e l a ~ v e elocitiss sf thismapltuda did not ~equirewhirl opposite to the di rectfon of gotor bladeopeed &ad. in fact, ehe matching with a turbine was easier with whirl i athe same direction as Phe rator speed.

Sut axid connpresaor for a gae cf low mclecular weighte POP in-, caa ba designed with blading sirnllrr~~ t h ~id compres-

aor fcr air. However, Ule pressure r ise per stage i s prolpcrtioad tosquare of Che Mach number af &e relative vellociw. The ro tor blade

rotationaal apead c at be increased beeauae of strength linaitalio~s,andhence ehe aquare of the relative k4ach number and $ 2 2 ~etlore, the prasslure

w ratio are decrairieed by a factor approximately prop~o~tionblo the ratioof molecular welghfs. Thus, the number of stages required fog ia corn-pzeesor u e b g helira~nwould be approxin3ately seven ti s &hat sz air

e averall pressure. Clearly, higher relaavive velocities ararequired tc decrease the lnumbar of s tages far a gao af lawar molecularweight. Arm obvious method of doing this i a introduce the whirl in theopposite direc tim to rotor blade speed.

A vortex flow pattern was chosen ao the d e t l i p condition: thenthe flew i e t r d y two-dimensional; tha& s , t h @ ~ es no radid shift of %be

*a %8SBO%etrearolfnes as the nPlid proceeds dolw~lstresm h r w g h the co, p


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This considertkbly simplffiee the a~lalysie f Uler flaw at the design ncpwrate. Ofl-design cdculationcs r u s t still be carried out in ordlsr tc in-veetigate e$e compressor performance and tc indicate how rapidly l ~ a s e amight increasrt as the compressor flow rate increases or deckeases fromthe deaign value. For conventional m d t f - stage ccmpreesore d t h rela-tively l a v i d e a pressure coefficients, Chfs analyeis is asuallly carriedout by linearizing the governing equations f0%khret d i m e n a i a d fio$vthrcugh the campressor. and is qutte batisfactoryo However, dt31whirlsgabst the rotor blades and i t s correspoadingly higher preasure coef-ficient, this me!f linaarizatton PaLls compketely. It is necessary to dl-low fcr the streamline shift to avoid singularitlee fn the ~oluticn. l%us,the full set sf na-linear etquatiana, t e n s nto a c c e u t the sadial shiftof the etreadiines, muae be used. in order to calculate the zepeatiagnow pattern characbristlc of multi-atage csmpreasora, it was necessargrto numericdXy dkctermins the flow pattern through the first four or sixatage@.

Although deailerd calculations of efficiency are scarcely psssible.the reeults of ihs araalysie can be uaed to give indications of probablelosses. The degree af turning required by the blades appears to be with-in perdssfb1,le limits. The off-desig3 incidence mgl:lea f o ~%e~ o t o s sdevfate very i fas from ths design valuss. The s a e o ~ncidence anglesare more areware, and might prcduce a ose in efficiency over part of thestator blad~.


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Cmventima subsonic axial-Baw cormpressors designed useair ss a.wo~Mnglrgd are ncl: wee ssarayadtable fcr gase a af lowermdecular wefgkst. For a particdm clor~ed ycle system, wbese aworMng; BuPd c&er than air might be desirable, the raumi4ar af sUrljsa~srqui~edg i vary p@&y \Kith ethB C ~ @ ~ C Bf gas. The number @&atagesi e dfrecely p r o p a r a d to the specific heat at cmertanf pressaxe and PBinvereely p ~ o p o r e i do tl%eMade srpaed and f i e c h a g e in twgenti&valccity ob gas throagh a sage. Tlse blade speed i~lxerl!3y t h ~strength of tlt;e blade rnat~rialmd i s gsnerdxy heid &@ high a e posaibie.WItPl present-day air comprassors, t t ~ r es a mivdmula amouat of

geaUal velociw change that can be accolnpliehed sfficleaay. Thue,for a W i c a L compressor designed to use air, the numbor of @@age@e -quOred w d d be? lacreased approximately five times if a gas evch ashelium wew ueed and the same empsra tu~e atlo acroes &e compres-sor were ta be rndnMaecl. This rapart t E e ~ c ~ i l wsb l ab design t2at

h ~ e f d l q r e h a eais high umber of otages required far heliunl eoagprcutima%lyone-hdf the number presently used,


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stator blade: i s such ss ta give aa accele~ating low Gxroingl~PLo satoxPOW@. m e rat02 Mad@@hen %eraEia flow back t a the aIi2e~tiaa*This me of design has basn used in same sin@@-stageans. T2a aota-tlon to be used in this repart i s i l lusrseted %a Figure 1.

The fdEotlowtng l i e t of symbaler wtll be used t3ilsoughowt t3d.s rrjlaart:


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aa@e beheen axial direction slnrl absolute veBocStiebapl@ile betv~:vveen xial direction asld relative veiscftiestip radiusBp &padan@ar v e l e ~ c i ~lf rotopradius measurad from compreesor cen@plk~s a t % cgrsesuracbensi6y (aasurned conetmt th~f0~1gholPt)

in Figure 1.uLwo-dim@ntoiaadHow thswgh an axial campreeoor, the

*@ugh it is s la&r t b t at d s s i p for vortex blardfng, tkc Bow i sWo-dirnenarfonal a d he pressure gradfent~ re properly accounted for.

t h i s dasrim &rises, we vfU d i scuse i t brielly here.The blades are a& the free vortex deelgn. TRw

(z ~ I l r .A dlnncnsfaarless r&*aus s defined by

T11e pressure drop ac.cross a etator rwfI i s given by


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The p~asouwriare czoa& a @%age ie then

sure r f r sc s acrosa the stags,

A work cocffieioat 6s u e d l g defiaad as

Xq@wbv@r,ri is not unrsatricted. It has been f o u d experriolen%&yr


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where i s bemean 0.5 and 0.6 . Defining a aoa-dimannoima sow

ntal set-up prescribed the hub ratio g I = 0.6 . T b r s i amothor rclistPict9m o~ he velociMee *aaL i s aot eaeilLy describedintical form. T&is is tb amowt cl W m b g that can be ~acccp~2plLial1@dl-lficieaay Pn the bl&e r %Eue to lowep t~l&espeed ize %ha habm.d tieadveree preeoure gradient in the PO~QT, thi~ ondftfon d ~ oo mastsevare at tke rota^ harb. f3 was chosen aa 0.6, and v dw e of the furnfng

and @ r 0.58 vJarc subsequenay chosen. The value QE $ c mk a s e deubstmtiaPly fcrr a compressor with hub ratio in the r a g e 3.7 ta go75.

Sa fa2, no dSfie1PIe wWl3 t h ~q7o-dhraeasloaal l x ~ o ~ yas arisen.TWlo i s sat u a q e c + a dsince, as morxtjossecl before, vortex bl,la&n; i eexacelly ksro dimelasioaal at the design E b v rate. klwsever, we have yetto coagfdsr the oa -des ig~ onditiow v&ere a radid compone?n%f ye-


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531g at the and ad ePle compreeeoa, acluatly l o neccssa~gr n 2LU stape)thc f i r a t &'ew becausa a piling lup oP prossure at eome ragikua i~

fmpoesiMe far a very lalqgeaa~"~$be rf atagego TEBUS,@%isrestrkct$anwill b ?apticslt 02S -dee i ~ ialk~ a k s also. "?73ena

Ranne3nberitsg&at c $ O for aff-deeign canditioa~, than

A first apprdrnarion can be de that t;sny and ta 3 , earrespond-h g o tile leaving aazgles of f iov~, aFe iadependePlt: of the inlet m@c ie aadhence ape indapeni?e& af now rate. If &%is i s dona, the L o c a l vdue +

Sineat under ouar apg~rodrnatlan twBZ - ksyl) e coa@ka3g m d IB great-as&= *ga, we @@@ that $02 ? 3 O 0 ?2 the cw f f i c$e~a tt the reat

gci! with now &peedand this e ge calculated by caacado theory,


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a%*

for iaatancc, &e Wo-c$na~sfonalanalysie af tBlB Plow l e unsatisfactory.d that i f B cadd ba puahed as high a@ 0.72 ina a

s t ~ d a r damprssgor, there w d d ba &@ satno difficufw. Hoemdlgr ,

t%ii;raan,d a d e 5 P t M u e?g8h %,@@-d$m@~@f& l 8 ~m d p a l s cxf the compressor dwra lnot contab thie rrlngularity in t b .

phy@fcde app4)aPB $0 e,e ~ u t i o a s

af motion There seterne tobet nowayof cdeulat lng the ids& now pat-terns~ xcept Zly a r i c a l pgocedure w i a a racaial cf the stream-

t a s n b t aThe aesumptfcuzs made in the &re@-dimensgod low

&P@ :(1) incomprassible perfect nuid;(21( 3 ) eqweime are written far coaat ion~ar upstreasll and fag

psfesxcep% tv@am f @tae@~rows and ehe cevena subscripts refez it@egatfaaa farrowca (sae Pbm~ure2).

t h ~ o a g h s t a t o ~ow is

The B ~ e m d 1 iquation for Row arough a


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The equatioae fez the pretsswre gradis& the radid direction aroJ

By u~lOng 15 ) to elinnhate the pressure! t e ~m e , quations (13)and (14)


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-9 ,Coslefdwriag a oyl%~driczrf,t ~ e amube, the emtfmity c?qua%ieaan bewfi%&a$a d iae~az%t iaa ~ m

t l 3 ~ectpa of the assamptiansil paedously mentioned, aeee sgm-t i a s are qui%~ emral. ft i s now agsamed that the labath g t a t ~ ~d O~OP cone d t h rerapect go sow rate and givpl

kte the dagfgn flow rate. Thus, evau t i ng tha LeavLng =&en

Sinnfluly, w~ obWn for %he rotor leavhg w@a!

sqma~~tFbP the

The eqltlakioa for nov ~hrough a rotor rovv becomes


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0 (22)

The cattaptie relatiom&are+bme ~ n -~82%- P 2ad&tlr @an+ ~g2n+ eg~ a + (231

Tke wc pd r s e f rmn-liineas eqwElons (21) and ((22). anrl ( 2 3 ) casl besnlved a p p~ admae l yor +m a f~slcliond Qm for m = 0.1,2,. .d rn , et f 4 l m a @sag@&@$as a@ ~trea~4faeW"k no r;plillctEvelocity

p l l f i c ~ ~ ~ sa t the free vbrWx design has boea used so freqwnlly. S X m -ra$es, no such simple solution exfstn, evert

fern:~e free vertex c l ara fp

cats ibr ern off-da~ignconslmtaxla iim (ire approaching ehct firset ota-tor. ]In hfe ~~bse *age must ba $&en & applylag sgwaomr (21), sincethe a~sumptioasf & whirl velocity i s iacorporatcd in the rigl%t-hand tde


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abstihufag from equation (1 1, the v o f ~ c fy dfagr mzatfma, a d ie

+I +a eEqua@fns (21) (22) are appXica+bleto all af the oaer blade

TOW@

e,4 i f g 2+2

@tm*

evBr. the htegratian for these must be dona nu-msricaity. Also, mether mathematical diftfculty arises h Ule saluaon

ing stage@. T b ked~tence f both Pnrdepeadaat


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4 2 -vazlablee on the right-hand side af both (21) anrt (22) re@re anft+ratSonprocess. The pprocedure is to first let gZn+&= 6 2n a d olvafor +2a+l. corat indv rslaeioa can novJ be irrtegrated fsonn G F to6 a reiation beeween Zn+Pt u ~@laHonh%oke difZes%atidsquaeon for +2n+l yteldo a re?ftwdvslue for 4;?111+1 Fortunarely, the correcl;fanto the Bow ewSficiea i r snrot lazge, md ths fteranon process i s genardly sot wceerpary.


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pscreeor w a ~ar r i ad out for Wo aff-deolfpa cmaaons. The blad~s e r ~chomn such thaE % 0.57791 and $$ 0.70003 . The amrage nowrate for &e dg-desiga ~eaClitio~~6&B t&4n as + = . 4 0 0 0 and ( ;:.49000. E'iprss 3 %braugh @haw the variaaon the now ceeSfificfentThe sow field acr f9 \ 4 * e ~ h m d e b l i ~ ao grate 1s eonverglng fairly rapbpidlyta a steady-st* cowtlon, whilLe at atthe higlrrjr Bow rate the convergencadoes mt appear to bs as good.

row. h rroz of cnly approxims~tefy parcerat of Q8 at th@ mid-heighlatof the blade i s iatroctuced by the ffrst wprodmatian. ft is fe l t realnuid ef lects will be greater than this

fn FLlyre 8. Hoaca f iat f i s very nsarly W@penda~%of Uzo radiu~,

tha ghree-d.lnr?cnsf& eguationef. Theye i t s assumed & k t h ~ +1from a @ t a w ~pproxlmatrsly e q d ta .uGxeE ~ P G Fu2 from a rotor.the procedwre is f a find a sdutlonfop.$vh i~h + Z - + l I (+2+91) TQthe zeroth ode^, tlds rceulfs in a Itnear compxesscr grrforcurve fa, &a $arm


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The two-dimensional theory d e o produced an eva t ion of thfe tyge,since h hat case rg t was taken identically equal to (P.Z a

helium compreesor, consider the ~pproximste erfo me@equation

For free vortex bladhg, @ ~rpati@ n25) @c@m@@

Ulen e q u a o n ( 2 6 ) can be writ ten

Typfeal vdues of Qd and A+ for a conventfonal compressor allow theterm containing A$ to ber ignored. However, for the compressor pro-

2posed in t h l ~ eport, $d is approaching 25 at the hub, a d he termcontaining AQ c m no long@r be neglected. Figure 9 erbaws the per-farmanee curve@ f bo* types of compsessors wieh and withcplt ehe in-ciueion of the A+ temm. The, lrmadequacy of the two-dimensional

of the graph* t , ~ o dliference can ba observed for the canvelatioad eom-pP@B 020

Figure L O shows the t o t 4 pressure varia t i c~ oag tkm bladesafter tke now has esoentidly sett led inlo a ateads/ pattern. For lower


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-15 -desim rates, mose work i s baing done by the tips, whtle at

higher sow rages, the rocta give the graatei~ iee. The cumula%$vei f -fer@%~@fter nlx stages i p l s t fa leas ~iaanhalfthe esignpresouge riaefor a @in@@ @tag@.

It was mentioned previously that it i s desirable to have

W1This ity is plotted in Fia;ure 11. Bo@sibledifficdties age indicatedat lower &&- & a i p f low gates. A higlxalr. l2u.b ratio w o a d elimfaibbe $Idadmiculw. The corresponding ratio for the stator is plotted in Plgure12. Siace $he etators are tn a ield ci decreasing pressure, no dfffi-~Ul t i l idsare expacted hareo

The oft-deeign eateriag an&@@ fog Che stator and rotor are plot-tad h Figures i 3 and 1% zsspeetively. The staters axe very criticcaland wW requi~e are in their design. Sharp leadiarg edges af ehe sta-to%@w a d probably caraae high losses. Tim off-desi&nZlcw incidencem%gladeviatians for the rotcrs appear saesfactosy.


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1%-as been demoastrated that high rcactim3 blading cara be usedin mati-stage azl t l compressors t o produce a sipiflcant incrsase %aproosure rise pei. ~tzr.geover conventioxd blading. tme af blad9agcovld be u ~ e dn $ closed-cycle hofium cos3pressor a ~ dreatly reducetfie total number of stages reqdred. The analysis also imdicatee %I&a hub ratPo greater than 0.6 wod8 poaaibly be a more eff ic ient design.since statcr incidence flow mg l e devfations become quite large near the

It waa d s o aha~l*mat the u s a Pbaaqbri~atfa ef $;ha ~ I F G B -dime?n~ionalqutians cf'aoxv i e not val id f o x this ma of blading. How-aver, a aumericd sawdon allawing for the ~ l d g tf $Em atreamdims doesconverge to a st;ea3y, repeating Bow psttern dter the first five ar sixsctagas fog f law@as much as hvenq per teat ufi' d e s i p .


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mFEPaNG@3 FOR FART fA. and Bulllck, R. U., Eatoss, g% a r o d ~-maby Camgreseors, P4Ab;A RM E S b ~ 0 3 1956).


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Figure 3 . Sta tor Axial irelocity Coefficients, = . 46000 .


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-Figure 4. Rotor Axial Velocity Coefficients, C : 46000 .


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Figure 5. Stator Axia l Velocity Coefficients. 5 . 69000 .


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Figure 6 . Rotor Axial Velocity Coefficients, 7 = . 9 0 0 0 .


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Figtlre 7. Streamline Radii Upstream m d Downstream of a Rotor.


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Figure 8. Local V d u e ol $ Versus Non-Dimensional Radius.


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Figure 10. Total Pressure Variation with Radius.


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Figure 11. Kinetic Energy Conversion Through the Rotor VersusRadius,


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Figure 12 . Kinetic Energy Conversion Through the Stator 'JersusRadius.


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Figure 13. Stator Flow Entering Angles.

20 I I 1ENTERING 7th STATOR

lo0 -

0 . d , = + d

-lo0 -

-20 - -ENTERING 5th STATOR

- 400 ---------" -----" I0.6 0.7 0.8 09 1.0


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EN T E R I NG 1st ROTOR--\

E N T E R I NG 6 t h ROTOR6 . 6 9 0 0 0 -1 I I 1

&Figure 14. Rotor Flow Engering Angles.


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I 1 1

7 -

EN TER I NG 4 t h ROTOR

EN T E R I N G 1st ROTOR

1

Figure 14 (continued). Rotor Flow Entering Angles.


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FART I . A DESIGN STUDY 2F A BQUZdTI-STAGEAmALCO&fpRE%$QRWITH BLADING OF EBGFI ASPECT RATXG


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ABSTRACT

A me t h d fop das8 ng blades of high aspect ratio in a. mdt f -compressor i s investigated. Th i s m e ~ c dequires a cal-

culaam of %haBow tadaced by all blade raws. The meaod i o lllustratadfag the limitfag eoaditton of hub ratio equ4 Laout for &a maze realistic case d inite hub raao.

pla of B blade design for a pzra~ticdar law is carriedoat. 'Sha results are compared wi th those oP fn. prevfous design for thesame sow condiLTa~,ug based on a theory applicable far $Paang cf lowaspect ratio.


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The fiow @%rough ost pretsent-dayyead by applying a three-dtmensioaal Chea~y h a e based

on &e sessumpWon of bladtag w i a low aepact ratio, Even LElwgh existingcompreeclcrs iClo act have what could ba called low aspect ralio blades.Ihe theorp pzectictiane and expedmentd reoulte show rearrsaea agree-ment,

Wksrlre weight considerationca are aS primary concern% tha possi-bfll(y of rerluclng %he e i ~ ed a compressor by shortsning i t s lengUl 1~

reaeon why b l a h rows ctt, their sarmd height but shortened in cha.ald. Them i s . !nowever, a dif-ficulty in %pip% gj the aforementioned ana lys ie . T h e ~ e , t i s assumed*at equilibrium now paettarntil ex i ee at the leadiag and t ra i lhg edges dthe blades. TUB i s idantical to assuming low aspect ratio tonr the '[email protected] they m o ~ te wide enough to aPlow & chmgee in tha! now to cccurentirely withtn the blade row itself, Several extenstons of the t h ~ o r yhave been develaped to accowt of the norm-equilibriwn Bow Bear theblade rows. ari3e oT these a~ia lyses s sme d smdll vorticlly ehroughoutthe l i l av field m d alee a emall change invortfcfty thrcer& '& a Made rows.

now field.The tf~eorgrpme~iawdn t h i s theale is aa extea~lonnf thae latter

app~oklch,a d s based osl the assuanptim?~ f closely spaced blades ofhigh aepect ratio. The equatiorms of motion are linearized by a p e~ t u r b a -

&e blade rows. To flluetrate the m e a d , the calculaDim i s giret


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-34-restrlcterd to tke limiting: case cf hub ratio close tomore involved problteln far any hub ratio.%@nvestigated. An examplei s worked.fcs lades d aspect raao about three. Here, there appearedlitatle duereace bw e e n the &easy applicable to blades oP low aepeetratio a d W theotly oE b l a d ~ s f low aapeet ragio.


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The theory af three-dtmensloasl aav in the adal compreseordescribed in Part I of this Chesis i a cane~rned nly with naw conationsfew upstream and fag dappkable to ixladea cb varv low aspect ratio. Tkm dfaerenee b tweenthese v d w I 9 prof'j188 and the profaee, near tke ieaang and trail-Sngi edges of $&a, blades was assuaned to be negligible. It was on *hiebasicr L%eeneerbg fino wa&es far tlxe blades were determiaed. If"the aspect ratso is not ve r y lovv, it ie possible to obtafn more exact in-f~rmation bout tl%e now pattern Paear the blade row@by making use ofeome of MarBla9a raaults (Ref. 1). This has beau dam by BcweaeS&~r~3lcy, dverstigatore farrA%ethe basie of t h ~lmory go bet discuesud in aknls report.The ~apgroach~ to c ~ n e i d e ~he distu~b'bancesue to: rows of bls*rs c&high aspect ratio placed v e ~ y loee tc~gethez. Rere, the Row in an)p par-tlcular blade ow i s idusaced by neighboring blade rows. Yn Ule three-dimensional a%eoryf Part f, the change@in the flow pa tem were as-sumed to =cur ontffelgr within the blade row itraelf and one bl&e row didnot &feet i t s neighbor.

The analyeits bagins with the calcdaticn oL the disturbmces &44nitt?stmally thin disk. These dia~%rbances an Glen be super-

imposed to farm a blade witk finite a d d exteat. TI@ ffect cf matryb1ade rawre e w be date ined by the suparpositicn of the d i ~ t u ~ b a n c e sdue to the f in i te row. In. this way, the Oow pattern for a multi-stageaxid turbomachine can be constructed. For a repeaang now paeerw,ths aummatlon ob the ekfeeta of dal blade rows i s posefblc.


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B s f o ~e lroceading directly into the Zhree-dimensional a e o r y 0%high aspect patio Mading# it i s inlornnative to invcstigata first the limit-ing caee of Mgh hub ratio. TMs problem i s simpler because the radlzlpressuw gr&&entsvanish far from the b%ede. The analysis i s baeedupoa tl%acliekurbancrte due to a disk at x 3 0 (gee f%gum1, and the

These assumpdong are: ( 1 ) there are no circudase;;llial vaieia$.tfons;BOhat isr ( ) 2 O . Thie i s squivdant g a disk v J i a 31% i d n i t s

number ~ f b l ad ao . ( 2 ) The now i o inco;.npressible, the density p being


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-37 -held ceastant. Tha c6n&uftY and rnoirecsba~ c,uatians then become

where u . v , and w arrt the garJ ve3lacitJr car~%poneoatonthe x , , andz directions, ancl p le the ~tSLtiepressure.

The boimdary cmdf t ims to be imposed on kb4s s ya t e~3ape asfdX@ws

( I ) v r O at y = O and y = b far all x. since the W ~ Bf f i e

( 2 ) u i s caa%inucplrst x = O from cwaalsuity.%hatl.oughlac acW-aging disk.

(3) v Ps cmtinraous at x 0 due ta $he con2ponant of g r d b ~ ~ ~beiag zero fn the y dfrec~on.

4.8) w i e &owed to chang~ locomblauously across x - Q .( 5 ) A.0 n - a

u = q y ) euo(yf.v = O $

ID.*.w 3 wfy) - rw,(y) ,oilp = p - EpP , = cmslaat.

( 6 ) At x = + EO ,U G{y) + iluo(y) @v 0 ,


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Thertle Isst twa boundary candieaas are writtan in ia t o ~ mo aa to con-form wiLh &e fallmirag perturbatPcn esherne. Far upstream ;and fa^

d he disk, a@ icw is hdapendont of x ; nry dism~bancedue to the a s k h~viw een ernoath~d u6 GI;-loseto &e dig16 e l e f i av~srdisw~bady a sn& amomt, che fatlowing pe ee~rbatimci r ~s-

d, where tb @ ol E I s carried ody ts denoe the assumed

Notice that 5 m d pa are aeeumed kn&pe&en%of either x o~ y a&thaf the lneanvdua sf v 98 sera. The tog e i s n are far - o~< x c Osnd the bottom sigw are for O < r c 4 or,.

Intrcdacing equatian (5) into the eqaaoa of motion flalde f i e


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B.The ffirst three oP the preceding equations (6. 7. a d ) carm be

&v O n d i t i ~ a ~hat v = O at y = O and y = b . A solu-tion to eqwHon (PO) that sat i s f i es these b a y caditions i s

with boundary condl.tiaaer

wb r e the pdma denotes dffbe-ransationwith retspect to y . Notics thatthe bowdary condiLians aa. u are satisfied at x .r - oo a d x r + a, .


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Since ti& ua are to be chosen and the shape ol t h blades Ps Lo bedeterdnfned, $$ 5s possible at thfe p o b t to choose An = O Lol: n 3 1 .

X I e % .Nots &a% lit le not neClsosaFy that f i ~ s e : articula+ choice&be mada. Pt.i s ddns here wJy to educe ehe amcmg of dgebra in the nest cd the Y-a @ t b ain@&our laks~ asse choice@ g p l&deed ~ ~ p ~ e t e e n tthe now th~ough cogvention& eompraesos quite well. Therefore, %eperturba~on eioeiseirils become

and with the use of e g ~ ~ a71,

The~esr!are sovera rogui~em@a$@at we can now fmpoee. FIzshwe c m require: the t e a ppesgure rise across the &ek to be a. emstLnf.TMs wiU Snsuro bh& tlow pattern will repeat itsel& This is a similarcmatian to glwt set up- the we* eoofftficioat .tin Part f of this paper. To


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cacda te *e c h a g c h ot& preirsrtre, m c seeds chmge wk.li%land hher chaage in stiteic pressure. The c h a g e in wh f ~ l8

and uslng ti?;qwaonar 5 )m d (15). become

a 5.4 P througZrt &e ask* the change 1x3 tow

- -ceibns crP y : a, uo, w , auzd w , d WO esmeanks and pd Qa q m ~ c n i l6) ralatLng then%. APsh ginca we ara la teres t~d nly

in (he pregeruge rise h a compreaeor, v ~ e re not immensely coslceraedwith ehe meirn preersure , . The pressure changa :oacPaee the diek i s2p0 , m3dthfs is the more fmpor iky*far upstream ELnd far dmmstreitm be cmsWd f ~ o ma,l: to tip* No&


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pressor d e s i s m d @vea two more relaaons cannecthg our Pour un-

The?@@can$maklon Qf equaeians (18) s n m chally the saa e a q W m 16). rt;Che a s k wars act a@ a stator, &en (16)w d d

Since i s a cone and from (18)%uo + wOG = Eh2 , the last

&careat y = 0 , h@n i s constag must: be aara. Thus, with asstotal pressuge, wa have four o ~ ~ mu2lcaons at y

and two cmd.itionsupon them. Hence, for example, we c ~ d dick u


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has been doae. t3e g f a 2 e t5e 13135?=,0 L?~ bc f ~ t t ? .To der%vaihe @Beet0s the f l a ~ ~u3 to a blade oi ia9te ~ d d a

h~tght b * we can @urnrom r = 0 to x 3 a U%eastuLioa8lce dars toa d%sic.Zocat.edt. ~rI, zaa"3,ges~ P O E ? ~ $0 a . PJaace, hoiv-ever, Ivoger must m&a a choice of $ha s h ~ ~JF the laUle bSiMe. That is , we ma~ f squfraW each disk corpt~bato omafractfon f ( g ) q af 06e d l ~ t a b a c ( h ~f tbe blade. h G%epleoMsm p ~ o -pasled repa~h g@~ r 9t IPbe~ws,pllck the fwctPoa f(5) . WB

ca t . kn case, %he ading PigactUm i s in geslezd a fugtction of bat11

aven %e fmetQoa f(&). we can proeeed tsl dete @";ke a-%octtt.ttesu md w $hraughoa%% compressor. a 7 R ~mi$elm E(6) OsFor a Tmire stater af MsI@x a , height b , nd leaaa%g dge at s = 6 ,

a3 velaciw far x O becomes


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Th@ qwaam~ior a ~Dtor re ~ lmi lazath anly the slgegeb~aic ib= cf

For a rewatlng Bow paB&rmq &e vaEacSC&losare same 1/J %:haewe$@m%ern,we mwO m%&mpoee the nowe EZue ta mmy seatora a d marmy ro-

d e a d%a ad l 1 "w aged:ag 0 viffd& Clf a ~ t&%arO%Y

Md& Cd a s a c g ir,wc, = gap balwean, 8tlES0~ d @tag

f@(&)dg fr;$pw$:$oaf dfefE(&)elracllon dLs

% =t mk =he%@ m 0 0, + 1, 4- 2, a * .- M-4 jtoaang edges& Iha rotors aro at


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Tbe WMF?rbB1m3Cle h a a rotor i s d e g ~ r d by chmghg &e sig3 d xia

c lb , s xh, ,w tb case may be, aaG%EeLtme &rmgE, s s c ~ d F ~ Ere


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retained, t h s vs%oclltHesssducs to

f s t i;i gatoro Xn b s ~aGsa

T b Pltaring and leaving Bow an@&@ for bath @statorm d aolon.blades can new be calcdated. The defining erlurjcns for &ese angles

tan y 1 = uQx 0 )

where y l i s the stator entering angles and y Z is the statop leadng-glee For the rote%,

tan =ugw = 0 )

In summary, glvcsa the blade loading, blade aapeet ratio, &aaxial valoeity through the compressor, the whirl valoepeity amone ~ a d i u sof a blade ro~v, he desired pressutc rise acress a stage, and the de-sirold degree af reaction, one can ccmpate ehe shape of tb blaaies suetjest


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*.48-to the rsstrictfan Lhat they be of high aspect ratio and very clcaselgrspaced.


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The a n a l y s i s for the liwriang case: of hub ratio equal to cmne isextended to the axisymmetrlc case of finite hub radius with aady slighlmodUieatfons. The y eocrdinate goee over into the radius r . Thevelocity in t-he F ddirecti~il emains v . The mao r change here is thatthe r a a d preasure gradients m u ~ ee accomted for properly.

The agweitons of motim in the adsymmetric csss

As before, we assume a perturbation approach of the farm6m.au = a(r)T euO(r) cul(x, r)

Notice t h t and po are now functioas of elme radid variable, P .The resulang linear equatiane are


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and also the r a d fd pressure equilibrium equations- 2w

P

Eliminatfng t32e athe dependent variables. the equation for vl

An aplprqrfab soluti~iae of the form

Substfwting into equatfon (35)yields,

w b r e &e prlme indicates differentiation with reepect t o r . Ib Far) ischosen dentically equd to zero, tbxn (36) educes to Besse18a equaiptLrion.If we fien take only the n ~ s term af the sdrutioa for v l , the bouadary

Y = O at the hub a d he tkp gives, fox tlae f i r a t aigen-

the first eigenvdue does not chmge subs tiaUg* V Ia w l l aow c~,o@@@P(r ) identically s q d to zero and later conaputc its ac tad vatu@to


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-51-2compare wieh (X f b )

Proceedfag as in the case wSa infinite hub radius, the peseurba-

Tha conation that the toeal pressure rise across a disk be a

Agda chocsfng the specid condition 0.f canstant t o t d pzessugo in th eradial directfosl implies $0 girst order in F: -that

et=lp + p (w'w + u u o ) = K Z e8 0Then. as in tile simpler case, squation ( 38 ) applied aczoss z se tor re-quires K2 Q . Gombfcwad%t%sna


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the dtsturbarices due to the disks in o ~ d e ro produce a finite blsde aqdatlmr over all b i d ow@to derive the complette Row pattorn

for the compressor, proceeda exactly as fn t!ro limiting case, d d yr a e d t a be #given 3ere. T>x'ibh:the rlcffnitio~

r

z

the velocity earnpaneats for &@ ee~atorsare

The blade agierei are determined exacay as befosa with the uoe of aqua-


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Before proceeding with an example that will all- us to vecuy&is theow a s t an eldsting compressor. we d I l indicate the eteps

1. Choaae ehe b l a b loading function f ( 5 ) for both the rotcr and

3. Sdoe equation ( 3 6 )for t h ~igedm~ctiolae nd determine tlzeeigenvalues by application d he boundary coaditiovls or%the ~ a a a l e-locity v . For mmy s~pplicatfons, 21x9 can be t & @ ~q u d to eero mdall consta~ts n the eerie@sdugion for v taExenz e q d to zero except thabf+st cnres. Fcr such ap#icationo, the magnitude of h wm be approxi-mately n .

4. Gornf;h.aaethe summation grocecluraa that reeult in relalicna

5. Pick 3 ii& uo . Notice uo must isatisfy ths conelitio92 that*of=I r u0 dr = Oa,

where ri ancl re are tho innerr anti eater bowdaries of tl* cyliwclricd

e from @%etrse of ecguaaoas (40), There will

7. Choose the preasluge rise acroso a stage m d the degree af re-action &sired. The ppreesure rise across a stator 2p0 thas be de-


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-54-8. DelermPne wo from the secead of cquarioas (40). Tho m e -

i e thaa svaluawd by use al the sec~ad f eqaa#ms (39) .9. Evalwte f i e 1Eunctiaa t# .PO. G d c d ~ t eW a d d and wMrB vdociass for bo& the atawr

aad rotor by wing f i e equations derived &a s b p 4.11. Compute the blade an@@@ by ucse of eqaatgons (28) Etad (29).

1%was decfded tb use this theo~yo compare with the blade, declsigpgiven h Re% 1 fog glelgd body blading. The ~ltelpewbU be taken as LfrrWdabove and thir zeedlc~ reeented hare and in Table 1 an page 58.

g funerion was t&aa tQ be COPIB fog eaaacompuaeion. A trianwNardistdbution m d d be more;.nea~Iy arrece.

23 TPz raao s /b wa@t&en %ab 4b / 3 .3. F(r) was t&ea aqusl to zero md x fouad ta be agproldmately

3,18, The Lim F(r) was subsaquenay checked m d f awd negll-

4. EquatS~nsl41) and (42)wcsPe &en correct as etated fin the

5. z u were then cl%aseno approxfmate as closely a@pas-Qsfble ths a&al flow uoed fn the design af tlue, sol id bocly bgadPng in Refe 1.

where 6 ie, a non-dinaneiond radiue baaed the tip radius ~uadu i eTtka rator tip @peed.6. Eqwtion (46) was used ta deMrmtne in the t c ~ m


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& 2) .342e-.1578 - 7 .gcl want determined by ehcosiry., = .385 at g ::0.8 .7. The i dea presraupe cwfficlent for t138 C O ~ ~ ~ B ~ S B F~ Q.40.

Thela by deffniaon

where nT is Ehe blade Elp spead t h a E heea been used tkrcughout thfs

par cent reac~cln a t s compressor oeeursed at g 3 0 . 8 . Siacs

8. The aqueSion Ecr wo sea, izrecaxlc!

9. 'The uPldioni + was ev&amd raurnerically aad s ifs$ed i oT&P@ 1.

HO, m d I f * The v d u e ~ u m d vg age lhabd 532 TaMa 1 fwdiffegang value@ . Stmilarly, the entering an$ ea.ring angle8 y .yZ f3I and P2 are t&aa&d. The vduas of thsea an@ea used in tlmo r f a ha design z t ~ c sale@ tabdated.


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s t coazpseasor deeigns oceurpiag in practice, ehe aspactrario af the Made@Oe neithez very lcrw nor very high. It f aUe BOW-where in betwee& The aeory & Piast X 02 %$@ &@st@ s applicable forbladfng & low iesp~cf aeio, where i t i s ass

exlst a% leadfag a d ratng edges of blade srotvs. ABlade r o w e not afPected by neighhorfae,blade rows exc ep t Wlz~aghhetnddent velociw sllstrlbutions. The t b a r y of P u t IU bop oh highaspect ralio cwtains $ha uence d isturbances ob potential $me tslectcan ba felt upstream a@ weU ass d

The close agreement between *he result^ 8f the two G%eol-.iee araft aspect ratio of k m e indtcaten &%at there now babveena Wades f svary near t a equilfbriml etate at $he eading anrl trafling edges,


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2,

Report on Research Gonciucted under Coatract 4 t h 5b.e Cace ofMavd Research, C d i fo d a asti&% a& Teehndo$y, Pasadena(J~WP~P.Y,949).


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High Reaction Compressors