Cavitation and Separation in Pumps and Turbines Citation Wislicenus, George Friedrich (1934) Cavitation and Separation in Pumps and Turbines. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/28qb-kc88. https://resolver.caltech.edu/CaltechTHESIS:11112024-181310694 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mechanical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Mechanical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1934 Record Number: CaltechTHESIS:11112024-181310694 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11112024-181310694 DOI: 10.7907/28qb-kc88 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16844 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 13 Nov 2024 00:33 Last Modified: 13 Nov 2024 00:34 Thesis Files PDF - Final Version See Usage Policy. 35MB Repository Staff Only: item control page

CAVITATION AND IN PilllPS . George AND Sfil?ARATION TURBil'IBS Wislieenus. F. In Partial Fulfillment Of' the Requirements for the Degree o.f Doctor of Phil.osopby~ Calii"orni::a. I:nsti tute of Techri~logy • l?~,3adena 1934 Cali forni :9, 13 2 T.A:BLE OF CONTENTS• Outlin& Chapter I. On. the. Application o:r Aero:foil. Theory · a»d Experimen.ts to Hyd1-aul~c Runners. _ • • • "1 The Ch~.n.g~ o:f' the Basie. Pre.ssure in Hydraulic· Runners.. • • • • •· • • • .. ~ • • •., • •· Chap•er :tII. Cavitation in Hydr~.ulic Runners. •~•••• Chapter IV. Sepa,ration i .n Hydraulic Runners. • • ••• • 15· • 31 S~paration ia RulU.ters of the J?:t"o:peller Type. •·• • ••,,. •• • • •• • 35 Separ~.:t.io:t1 illl Radi;;.J; Flow Ru:"lne1·s. • • • • .... • • •• • • .. • • • • •• 42 The Influene,e of' Separ~,tion o• the sc~ie E:f:t'eet:. • • .. • •., • • • • • • • 50 List of' Re.f 'erenees. •••,._ 0 ,, 0 • 0 0 • •· • • · • • . . . . . . .. . . . . . . . . . . . 56 Appendix., Se~tion l. Some thaoretical facts a.bout axial flow :runners. • • • .. • • • • • • • • • ·• • •·•· • • ,. • • • • •· • • Seetion 2. Comparison of' the giTen pressure formulae with experimental results.••••• Sections 3 to 5. Derivations of aome formulae used in the text. • • • •,,.. •,. • .• • • • ... • • I VI VII 3 v~.nes of hydraulic runners ~.ct on the surrounding fluid in a . . ·.:. •. , . . . ., ~ .. ·· . manner eimil~r _ to- the sffeet or the wing e:f' mt. a~ropl2.ne., . Theore· ,1e·a 1. . . ·:: : -.• · . ,. . - ·· .. . -. . . the:r~d·ors may be applied to hydraulic 1-u1u1ers, p1~~titing the ~~onelu... sions are :modi:f'i~d ase:ord.illg to the new .eondi tion~ in sueh _runners.,_ ·which eerta.i:nl.y di!fer from th.ose around an aeropl:l),ne wiiag; The :pre<';! ee:at pa.yer deal.s with these modi:f~eatio:ms eon_e arning the pressure . . . behilld ~- si.ngl@ aerofoil a :r~ equtl,, while t.he . .·eorrez_po:adi::ag pressu:r~s . . -. . . . - . . . . . · ·. . . . ·-- . . , a.t the intake and the diaeharge se_e tions o:f hyd.r:ia:ulic nuu1ers . of' the reaction type are di:ffereirt~ This d.i:r:t:~;~enr1e :mu:at be taken into -_ account for al.1 phenom~l?l~,, . which depend. ?Xi _the ~;Coaolute pressure or . . its dir3tribution ;along the Y.B..11es"' 'f.b.~ tw«l moI~t important phe:nome:n a of this kind are·: eavi tP.ti,')n and :se.11a-1:~.t!on" \vhi~h., therefore,. form the m9,in objee.t of' our _,:Pr(!-:sent i:nveiistig~;tio:ns. First . we have derived ei:mpltr.., appro:dm:ate expressions for the "basic :p:resSl:lre~' { defined. on :9agea 8 a.ml 9) ~r ~Yerage statieJ pressure- for runners of the propeller type. The latter ex_preBsion3 are then used for deriving ·theoretieal eri teria for the- eavi tation limits Qf runners oi' th-' propeller type., taking into ac~ount the distribution of the b~,sic pressure., Due to the fa~t tha:~ for pumpa the- point of lowest pr~ssure lies near the ~.mction edges _of th" blades., and · therefore in a region of oomparatively low basic pre$sure a.nd for turbines near the other end of the vanes in a region of high b.,1,sie pressure,- there exists an important. dif:ferenc~ in the cavitation features between pump and ,turbine runners (which, for i,nstance, has not been considered in corres:pon-. ding recent investigations by S.:pannhake). J?urthe:nnore we have computed the relations between the cavitation li:mi ts and the specifit1 speed for . a definitely . • defined series- of pump runners, and these relations finally have been brought into the fam.bli~.:r form of the chart, • a.dopted by most ;pump manuf·~.cturers for the "Upper limits of speclfie speeds for dm1bla suction,. ai,ngle , stage centrifugal pumps: I11 , order to realize the effe,::t o:f the change o:f the basic pressure on "·sepa:rtrJ,tion° or aat::,,llinguc of air:foils. or vanes .• it ia impc,rt.a.nt to :note that this form of separa--· tion is ca.used by a. pressure ine::t•1~a ae ~long the low pressure aide of the vane or airfoil-, and we may consider the value • of this pressure r ise a s a cri te1'ium of the commencement of separation. In order· to derive from the se:p.J,rB,tion or stalling· limits o:r a aingle airfoil in the wind tunnel. the same 11.: mits for a corresponding vane in e. hydraulic runner, we add (algebraically) the change in the basic pressure to the pressure riae on the equivalent airfoil in a straight and parallel flow. If the resulting pressure risereaches a value which, if occurring on a aingle airfoil in the windtunnel,would lead to separation, we may assume that the same 5 phenomenon will -take place on the vanes in the runner. Ba.aed on this principle we have dari ved simple expressions for · the separation limits of hydraulic runners. For axial. flow runners the resul.t3 may be expected to be fairly accur.;;,te.,: while for other types of runne1·s considerably more experi- mental and theoretical data will. have to be obtained before the given r.esulta me,y be used for ar1ything beyond a :preli- While cavitation-sets an u.:pper limit for the specific speed, separation - vd th respect to pump runners. de- termines a lower limit primarily :for the :gni t .. speed ( ~D. )_~ which al.eo lea.ds to a _practical lower limit f'or the speed- fie speed. For turbine runners euch a lowe)r lind t doea not exist according to ou:r Rpp:roxima.tion,. ~L'his limit of the spe- . cific speed is lower- for radi&.1 flow than for axial flow pump runners,. due to an iruportant d:tf:fer:n ce between .thane two t yr,es o:f r unne r s: wlth re s p e ct to s epe.ration ( see page 39 ) "' ,. ' Since m&,y standard p ump runners must be expected to show separation trl th corras.ponding energy losses the question arises whether these losses ft.Xe large enough to warrant the considerable departures froL'l sta..nda:r.>d desigl'.ls which would be necessary to avoi'd sepai~ation under all ci:reumstances., Recent invr~Htigation:s he,ve . shown that. the separation or stfl,lling limits of airfoils are apprecia'bly influenced by Reynold~' number or the scale. Hence we nw,y expect a eimilar 0 scale influence 0 :for hydraulic runner3 whenever 6 the runner vtmes are working close to the separation limit;. vs '. "rhe given fonmlae., therefore.,. allowvto estimate whether or not :for a given ease of model testing the- aea,l.e. is likely to play an :i.mporunt pg,rt. CHAFrER I • ON TR,1:r! APPLICA'I'ION OF · AERO]'OIL TID~ORY AND EX?E.RILIBNTS TO . HYDRAULIC . RUITN.ERS~ Aero:foil.s,. a.s used in aeronautics, · and thevanes · of · hydraulic :runners,, . both serve the purpose to deflect : the-i _/: surrounding fluid~ Theoretical :and , experimental . informa.tion. obtained aboiit. the former therefore may .be . applied to the . vanes of hydraulie machines. Itis not _surprising that.,,mor-e exact and detailed infonnation . about .. .the behaviour of such . defleeting sur-fac_e.a has be,~n obtai:r1ed. i .n aeronautics than .· ii-1 hydraulie.s, :firstly,. benause the aeronautical e:nginee:i is mostly dealing with but: one vane instead of a. whole system of vanes (like a runner)., ~~o that hi~ p1·oblem is inherently simpler, . and, sec~ndly, because __ of system.:s or v~nes. we oan us.~ a very simple apprrndro.at.ion - the Itularian theory,: oon-,-: si d.ering the vanes as infin:i tely close together · - · which yields fairly good ove:tall r es~lts, while 51.lch: e..n . approxf...; mate theory . is not a:vailab:te fo:r •an inq.,i.vidual ae1·o!oilf from the v ery ·beginning the aeronautical ,e nginee1·, therefore., was :i,n need of a more exact .t}1e-~:ry and more detailed experimental results. In recent . years, however, _·i;he develoy- ment of high speed . runners h~s forced the hydraulic engineer to consider the vanes o:f hydraulic machines rather like individual aerofoils th~n as memberB of an infinitely elose syate:m of va.~es, and it is quite natural th~t the large 8 amount of theoretic~-1 and ex:perL~ental inform~ti~n:,. accumu..; lated in aeronautica. a.bout the behaviour of' ind.ivid~c>,l aero~foils,. is used for the va.."les e-:f _hydraulic runners•:-·, Thereexists, ho·w eve-r, an important _dif:fe:rence-,. .which must be considered for· this application~ The aero-nautical- ertgineer- ·< -. ·__ - is :primaril.y interested.· in the . beha,1tiour . .. of'-: a.It aerofoil: if" ·i·,:"., ,. ,, '. ' placed into a. i'low which, origin~.lly,., i~ ::ttFaigh~t-·: pai·tl1al:,; and infi.nitely _extended.:, _and _h4-a reanTts are. either: obtained undar thene co~ditions- or ~mmed.l~.tely ~onverted : t? it (••Wind- tunnel correctlo:nstt) ,., For the vane of' a hydraulici nu:mer;,· howeve:r'.,. the ''basic flow eondi tions" ,- i,.e. those flow_ con- di tio:ns which would e:r.:ist without the i1'l:i 'luence of the parti- · cular ·vane or a.ero:foil., _whieh we consider., are greatly in- flueneed by· the other- guiding surf:.i.ces f vanes, et c.)., and -the ~b~sic flown certainly is not ~,ny longer-- straight. or parallel ~s ~ssumed before., - The ehe..nge of the gemac trieal · shape of the basic flow has been considered in a ·very ·- excellent paper by Betz (reference/ j and, a,ppendix., section l. }., This-paper gives.- a :practical method tc· design a vane of a. _para.Jlel syatem so th~t it hl!{.5 the same effect as a certain given single aerofoil in a parallel., straight flew. The method fI.ppliea directly to the vanes of runners of the propeller type, and - to stationary vanes. In the present p~per we shall consider the ~hR,nge • of 9 ' :. the wba,sic };)ressure 0 defined 9.S that pressure which would exist without. the · influence of the pe.rticular vane or a.erof'oil which we ccmsider. The ba.5io _P:ressure of'_ t~t! :flow. conditiens to whi,e~ the rssult3- of Jh,e_ usual aero1oil t~_e ory · or windtunnel test.s ,ge:nera-1:ly _apply_Js,. ?f course;.. _e on:3tan-t . (s tra;ightt- :parallel.; : ~;nd -_infinitely: , ~~ten de_~ .b~si e _flow J• while i t el.e~r th~t the. . ~,s:lc pressure · in- a pump irie-rease 1 - ,.; . will. . -· . -: -- :·:· : . - .• . . , ·.. .. ' . • . . . -. • --. . .. --:- . ~ •- .. a.nd in a >turb-i:ne it wil~ _deere~se (in -~~e. di:rect1011 :<?:f th~ / -. main fl.ow _thr()ugh ~:ne machi!l~J d!:.t~: ~o..~:r-!1~ ,:vel!'y _nat.u~e o:f the~e _two typetl of: hydraul:\e n.1:,2-eh~nea. !~~e1: i~yeiit~~a.:ting these ch3;nge$ _o:t the_ba~ie- .J?ressµr~: we sh:all: t.ry _te: express vanes~ i.e. we shall :modify the tll~ore-tical ~,nd. ex:f)e-rimental. information regarding these jhe:nomen&- on ~,n indivi.,iu::,J. va..vie PRESSU.B3 IN HYDR~ULIC HU:Nl~RS- -• 'l"l}ie fttllowing results about the cha:nge of the basic. pressure in a runner ;,;re ba~ed on the Eulerian· ~-sav.mp-tionthat the van~ circ,...1lation or? in oiher word~~ the. change 01' the angular mo'.!lli:mtu:m, is uniformly distri buteii along 10 concentrie circles t),rou:nd the a--tl:a of the runner,, while·' it actually is concentr.;i.ted at the vanes_ This :means.,_ in the · - being -equ~,l to .the .s tatic, 1>ressu1·e .. ..... in.. the runner;. Our. . assum-o. . . . .. . • •. , ·- . : . . ·. . ~ -· tion will lead to quite a.ccui·at~ results in a :.region -: a13o~~ hal.:f way be.tween the su~tj,():tl am.l t:b.e J;?:tel3-inire- side of'·_t.lle: runner~ It _wi~l _p-e le~s, ae.cui·ate with re~pe~t,: ~o: !,he - change of the basic- J,)ressu_ :n.l ,.;i.long t.he ·v~nes in ae; _f2.r_ ~s it ag.sumes that the total: _pressure: change takes place in _the region-- be- twean the suction and prt}ssure enda of' the vanes.,. while,,,_ actually~ th:is change _starts in :lA certain dista.>1ee (tJ:~eoretict;,lly in an infi~i te dista:n~e} in f":ront-.,. and is e,:>ml)leted o:rdy in a corrt:::-sp-onding distance behind t.h~ ruri.nc:c• In .s ection 2 oi: by a_ppl:,dng t'h.e above ~ssUD1_pt.ion ~-:re em:u_pa.red _with -the ex- -dis t:ri- agreement i $ better than one may be •incl ini~-d to expect_ from .. the com_parati v~ e.1..11deness of our apprnx.i.m.~.tion.,, Md our theol'etical results art,- cert~_,inly aocurate enough for tbe practic,J.1 requirements- of the hyd.r3,ul.ic engineer,. l> (x) ::; 2 2 V-- V _ - _,,. -1.),._ -·. + ll where the aubBcript o ma.y :a_pply to any convenient . point • ••. - · - ,>.-· . .... cf reference. z,dong the mai:n flow_ ~hr:eugh the sy_ t3~~in .. For a rotating sy:.item. we find that : p(x) where a,,nd V is thl! relzi.ti ...,e v eloei ty of . th_e fluid .. ip-. .th1-~,-·-~--ru11nCi ..,. : :. _· :.:. ,. ' : •. • .• . . . . u . the peripher;,1 ve16elty of the runner· at · the~,,;poh1-t . . indioR1·~d JY . th~ 3.11bacri:pt ~ _TJ.le der-J va~iort _o f , eqna:ti 01! :: (2L·<· is gi~en in __Daugh~rty: . ''}f,ydraul~c Turb~netJ_" ,P8,ge_:·. 9._I . ·" -'.· . . The difference in ele,r~,tion jz0 -z{x) J . -' ;:?ih~ch_~?-'·is :=· th.e o?d.'!-? ef ma.gpi tude of' the -rertie:tl dimensions of the runner need.a - to ¢1)~$idered. only f 01" very L;;.rge. 1i.ir...r1er3,. 'J,'o .. . ., ..· . • ,•. . . •, . ,• .- '• -- . . . _. . press.ure wt th respect;. to the sus.ction pressure, _-~O: _that With this eonventi011 and disregP.iding differences in dr:mlic runner of &.rbitrax-y eh:?,p.e m~,y b~ ~::x:;p_::-e:,;H,,.~d as :follows: + ) p(x) 2 2 UA ~ - Uo (2a.} -::.., in :pressure due tt) the change in cr!')~s~ction of the p~ss;Rge:s . .: ., 7i-:----iR-,.---__:__.'L• ptx) - - H ~ + - ~··.~: X"· ..svclion · • .sid~ Fig. la B C a C Fig. I .b • ,q, =/?:s 12· 'between the . vanes, and the. second te:rl!I the ch.:,;,i d{½.f \ !'.Jies_Su;~ d,ue to the . c.~ri_trifugal forces. .~:.·::;:;:)\:~/' In .the ,Sl)ecia~ ~~;Se of :purely- ~ia,l ::flow rtlnners, ...to; which mo:,t o:f our f'ol1owing conaide~tions - will-:_a pply" the ·.c/·: • • • • • • • • • • • • • • • • • •• • • • • • ~ - • • • r , • • ch~.nge o:f the basic :pressure_ ean ,_~~ ~xp:rease~ _by~ileaI.J.~' ·o:r. -· '. : t!1e. lift. ;o~;flcte:n.~:_JCJ,-l and_ ~he :· sp~_c 1~g ~o f;: bal~,des,- .whi~h ·is .. . . ' . . • .. "· .. ' . - _th~(~4~'~rf:::\>_ ~f. .~~e~.t &~;~nta~:.__t~~ .i;he -• •;~-J..lo~j.ng·. :a~pli~•, . . ,. . . .. , · . .' . ' ·;· --: . . - . -- . . •· - : - .... . . ~ •. - . .. : . ·' ' . ='"· ' . cations. Th1s:.- rel:a.tion f'ol1ow1> .:from -the •;simpl~ f;1,ct - that fur. ::- . ,. • .-, ~ -- -~ - ~ "' . • ,, . . : , · ·· - · •. . • ,. - • . - · ·. -· - : · • · · -· . .· _: •. . . · · .. ,·. - - · ;. , . •, - ::··· - - ·- · -·· . ·. , ,-· • • ;. ,. ,... . ._ .. : - r., - .- --; ·-" ·_ ; . : : . -.·, • the ~,xi~J. . momentum., so-. . t)1at. the . -ch~nge in-· ..statle,:Qres.rnure -•. .... • . -· -·: . -· . . . . .. . . .. ' •· . . . . . '. . . . .- . -· ~.· ' .......· -· . : .. _ must be eouir:l .t.o the a.x.ia.1 component of. , the pressure d;tf:fe. · .. ,. . . •-- · .. · .: - .....- . ·· .· . :· .. ··· .-: -- - ·- ... • .. . ··•· _ .. ·.- -- - ·. . ... re.nee between the . two- sides _o:t the V_9..neS'.:,.. or, which is the · . . : • . .'· .. _.· same, e-qu;;,(L_ to . the _pre-ssurec _diff:erence _ae_ting on th~ axial projeetion :~f the va.ne9 as shC)wn on ±~ig. l:l.> • VT~, therefore can write :for any eoncentri. e r:bJg ~-J~ea : : . ·; . ,·· · . , -:-._'" ' \_-'. : · -:·;-,-_c .:• _._,·· ·._. , •·• : · :. ~ ~ .. ~- ~ ., . . -:•: . or 'Where la.v is the average pressure d.iffa:renae between the two sides o-f the vanes., and e = 2 _1(-r Irr , 1T being the number of v~.nes o:r the runner. Aeeording to the usual aerofoil theory the average presaure difference on the vg,nes may be expressed as follow:s. : 13 . ._ .,. _- wh~re eL is ·tl1e -1.ift coeft~icient a,nd . v the • "mean effective relative veloeitytt as defined in seo~ion 1 of th~~ appe~dix ..__ (v- , ia ~.p:pr~~ately equ9.l · t.o ..- Vp ; .v~ -f~_: . ·..... · Introducing fu!,tr:tecrmor.a -~}le ._.·-'~ r _a:i;:L~ . ~; .O'!e~lapp'i ng 1• •. ; ~. ~ · a -·· · :N ·t co~ 0• · j ;,.,; -- - ·. -~ -. .', ------=--,;;;.- -:.}t • • .;.__ 2 x . • ,. rr we f'ind . f'o1t - the rise af . . . . · ... . -~-.. :., ·:: -·--· . , .·; - the static _pressure in -an a.--d:aL ; ~ :..·-~ ::..~· :· · --- :; In o:rder to find the. rise o.:r the static _pressure (basic pressure) up to a-11 inter:medi9.t.e- point or plane gi"Ven by x • in f'ig. 1. , . we. must . consider tfl...e distribution of i:h~ pre~sure di:fference .dp. -a.long the vanes - (in of . the .· . . . the direction . . - . . -... . . .. .. . ..- . - . · .. ·-· __ X {~(x_}. • _p 8 J c _~ - j AP 8 t bei ng . the chord of the ~ero:foil and dt an infinitesima l _part of it. Inst.. ead o-f preseP-ting the presRure dtstri bution o:e a cert;dn aerofoil at ~. given velocity v ~irect.l y.- it ia custo:ma.riJ to plot the pressure on. the V?.J"le or .?..erof:oil divided by the at;.,gn~tion pre3sure .. 14 of unit length. Putting: we oan replace the -above integral ov~r the vane preesu:re by an integral.: of the _custom~ry pressure diagr~m ~,a follows: 9onsequently ~e eon write :--;'}-: X •~ .- ! Ll).. r dj _ or (6} The influence · 0f tha finite t.hiek:nes:'5 o:f the inmes on the ba3iO- pressure will be neg1-e~ted,- bee.~,use rel~,tivaly - very thin v~nes are in ttw~-~ vase~ to· bi;, pre:ferred f'or hydr:t~ulic runners. "W11iie we have eho~$en the pressure on the·• su ction edge . \ of the v:mes for a. basis of reference,- it is ele~.r that the. p r essure a.t ;my oth~r con,reni. ent crossection ~hrough the- - runner may be used, if x is defined correspondingly. On :fig. 2 we have shown t..'1e cnr-..:re for the integrr.tl in equs.tion ( 6) for an lL\CA profile 6406~ if placed in a tur·bine runner, and on figts. 3 .ri,nd 4 the s;;;,,me curve :for. a. l:e tz-.Tou.kowsky por:fi le ( refar-c nce 6 ) c?.nd for HACA profile 6406 if plri<.ced. in a :pump ru~"'ler. ----- ------- .... . ~ ~tl, I ~ '1 • Il . corrticfton , Fig~2 NA CA profil~ turbin~ ru~r-Ht:r. 6406 pl~~~d in a e:L ,9 -- .1 <>~, ~-. • "-'-,.:..,;..--=- . .,..,. . .. --~df• •• . B~tz- Jouko-w~ky profil~ 4506 0 8 pl:1.aed in & pum:p ru1tner,, { 4% ~amber, 50% bid.ck from le:iiding edge; 6 0 8% thi ~kn~HJ,.) a~gl~ of ~tt~~k; O; @L ~ ~· "' "'('. . . ~ . :' • • \1 : C corr4!ction N A C A profile 6406 pl~ced in a pump :runner., 15 CHA.."PTER III. CAVITATION IN lfYDRAULIC RUNNERS. -·: ;. f:: · . ,.. . -~ :"'·- ... , The :phenomenon of cavitation ha.a been discussed. e.xten--· sivel:, _with i-espee.t . to hydraulic_ me~chi_ne~,.,_ ao . ~ha~ we, shs.iJ.; ~-, _-· • restrict our inve~tigat1_oris to the problem ~f :tindi~g :the·o..., :: : retieal c riteria. ror_ .the commenct')ll1ent of' ca.vi tation in ' -• :\ ~ -'. · hydraulic runners.., -;t:t is exceedingly dif.ficult .to sol.ve;:': .: : this problem_ theoretically with respect __to r.afi:iaI how and . mixed now •runners, of' a gene:r.ai form, which are common . in • -· ·_ • ·. .... ·.. ~ ' :: ,·. •, • a region of medium and. low g~_peel fi o s peeds• · Fortunately there existi, a. suf':ficient supply of · te_s t data on _th~ ca.vi-· tation limits of auch runna1·s, so that., to $Ome<extent., the- ·.. problem can be _so-1.ved em:piricall.y. An outst:i;.nru.ng e:xmnple :f'o.J:> such a solution is offe1~ed by: the , weJ;1 known - chart i0r a ngle s t age. cemtrifugvJ~ _pump stt; adopted by most · pump m~,nu ... f'acturera.; On the : other hand we find that in the :field of exceedingly high speed :run:nel~s,. namely :(or those- of · the · propeller type"' where test da;ta a re n ot so plentyful as yet., the theoretical. solution of' the problem does not seem to offer appreciable difficulties-. The latest. att~ek on this problem has been made in a re~ent paper by Spannhake ( :reference- 4 ) . The present investigations., al though carried out independentlyi naturally hatl to follow aimila,r lines;a as those by Spa.nnhakei they differ., however~ f'rom the latter in the following points: 16 ·.:.-."'.: i . . ,• 1) Spannh3.ke considers •but- one very specialva.ne-'- ·.' ·, . <· :i:\:/ ~:r~.-/:~_-:·: profile. for which _the- minimwn presfJUre alwayi{j,J;ttrs in- the center_ of' the vane~ and• consequently,- the ~alue of the ba.sie _preesuri, had to be determined f'or- thi~ ~poi_~t <?n;i/_ Due te this symmetry o:f the pressure distributio-fr ,the ,r~sul.ts> 'by Spannhake - 5hew .bu~:l.i_ttle: ·cH.ff'ere:~e, bet~e~::- i~1~: : J : _ -- • •- - • • "' ·· · :- .._, _-~. ,_. _, ::" · · · ~-. . . . .~.~ C•· - ~ ~ --: , ~• :•·: . ·:- ;.· ~: .. a'. : .--. , ': - , -., , ) . .·; ;_ • tuz·binea:_ rega.rding -_the eavi:t.3,tion , limit. In accordance with., • • - •• • • • •• .• • ~ .•• -, ~ •• • • ~., . . • -: r • the influence of, the ba.sic pressure il'l .- •• · · ··• ~•·. ·•· ..- _,.. • ~-·- . · ,, -.• ,· '•. • . . •-.· • • ,- • •- · -· • •• ,'.;. ~ - ·•: ·./ . . - - • .a more ~general niannel", - .. , •• · -··•.,. _... ·_,_,.:.,. __. ;_ .. - -·.·: :..-: \.> -~., · ·· ·:· •- .. .:···. ~- ·. ••• : •.:·.- _.: ..._._.., - -. -.~.· . - ·- . so th~t the reeul t!k D'la.y be- applied to vane :profiiesc of ar~ -... - "'• · · · - -- : . . •· ··•····- . •:- . . - .. · ·- ~- - •• _ ,,._· . . .... . · -. ·- . ·-- -- , . ._': · ·· .. ·••'.-- "' ) : - ·-•, ..... .. ._..... -.. ._·- .--' ;(:·;·' ·"•,:··; :,··: : - ·'. =·-- ;_,~-~- - .. . bi trary flha,,:pes., -Thereby also ~- marked ~lfferen~e .betweeni • • • .• • . - . . . . , - .. . , · • •• •· -- •• . • ., ··. --.- .. -• < · . _._ .. .... .- . : •. , . . .. , ._ 2} We have earried our computa.tions -so- fei.r that for a. particular eJro.JTI1)1a -the ea.vi tation-11:m::tta et,uld , be -shown in- . . .. . . • . . . ,. . •- - ea.refully a:vo.ids 3Uch a SJHN~ial :representa,tion in order not to erea,te the imp::r,es.s ion that there might.... be bU:t one definite . -- · - · - .. ' - . . rel~,tion ~be-tween .the e_peeifie speed an_d the, cayi,tation 11mi t; whi.le., 2-c.tue.llyJ'thi.i;-1. :relation must. depend o:n spe_e ial cholcr~e f'or the design of the runner. Although the sc.ientifie, justification of OIIJ'ttting therefore _such a final re:presenbition cannot be doubted., we believe that the viUue of' these theo;.. l't)tical inVe$tigations will hardly be e.p:preciatecl., unless the re::ml ta ... thoug}1 not quite general .,.,.,. are repre- sented in a practical, useful fo:rm. Such a definite evaluation 17 seems the !]lore: permissible, as eh0,nges o-f th,~ _:<legigni:ng , :fea.:.. • ··1·~- • • • - . - . • tu res' within rt39,:lonnab1e limi t3 will not chang~Yth~ res~lts-_ ·. • . •• . -.:,_,.- .~_•· ·-···· ·.• . -- .~-~~ . ·. :';· very. radiacally • -- 3) _S pa~ake ex:pre~aea the specific speed_ pr~a.rily . by flov, -characteristies,- while we have used form. charac.- -_.:,, ••_- _- ~ . • • -; , : ·•- - '.7 · · · •,• ✓ , -- • . ~ - -- · ··:·- - teristics in.i seie:ntd:fica11y mora correct., -- the _:representa. _-·::f-~-- -~ •. : ·· • '. . .• • ' ·:::··' ··· : ' . • • __.._ .. :· ,:. :: • • • .. ·: __ • • • : • • _, ' ' " .. , .. ..:. i;_.,:1·:·:• J>: :--·- • . - -·· • • • • ' ' • •-. • • • • , -· -- .- • •• _ . , ., •.- •· · ·· . • __tion by :fonn cha:ra,eteristi'cs~ on the eorres-, " . . ., ... . ··~. . - ·•.:-. other -hand:.~. " : ,,_•.•. ·.-. "'. •, . . -- -· .. -.-. ·. ·;- , "\ ... .. .. :·· -- ·. , ·•.;. -· .. . -· •. --. :, •,.' , . . .; --·--• , • _.,.__ ·- ·,_ .. - ~ • . ponds to the :pra.c·l;ieal _procedure of aesaigning _to one pe.r- ·- •• .- --:, ,- . . .. .. . . . - •• _, _. ·- -·. - • '.-• - - . - -- .- -· :· -- _, , _- :- · · ' ; -· , - - .-, -~- ~ ' ! · • • , • • •• • ·· - .;:--· -- -- - - • _specific speed.,. _while- actu~.lly . the -~:pec:htte _speed :Will,. - of _ eourseJ! yary s~ the oi,erattng poin~ :_} .s _shi1:t e_4 along _the , ehara.cteristie:_ curve,, Our spe-ei:tic speed i'orm.ula"' therefore# muat "bf:t understood__as applying to the operating point. :for which t;tie rtume:r has been_ ~i~aig~ed,. • _ 4) _;rn orde! . t~ avoid t:ompl_ication~ w-e have diBregar. de d f~loti on _l ossesa Th i ~ Jeem~ juet~fi ed i n so~far a s for wind tunnel will hardly apply·_to th~ : van~ of _th~ :_runner; where the end eff'ects- (wall friction) will play a deciding part. In this connection it is important t.o keep in minde .rfv that the p:rofi le drag of the V."8.nes in may cases tends rather_ to increase _than to decrease the desired effect of' the vanes.,. while the :friction on all stationary wa.l.ls will. alw8,ys de-· · c:reaae this effect. The influence of friction,. therefore., 9artly cancels out., which tends to make- all calculations regarding friction in high speed runne:us the more uncertailn. . ,- ·- 5} _Sparmha..~e has earried out _very valuable . investiga- ._· tions on the : influence of certain changes . in. ..the .:flow .. ·· ... .· - -:· . . •• -, .. conditions,, considering especially a rotation of _the ._:flu! d:·. • to . on t.."ie suction side of the. runner. •.. In order not distr~ot• ._• .. .. . -·· . - :·.• ·. ----~: • • -~ • ·.•. ~ -. ·: • ··. ·- . _.,. . . -. :have e ousi der ed_ t h e _sdmpl ~st :fl ow ~onditions only, _ ~~pee i a lly ··.>,> • • •. :., .' . .. • . ·, ;. :·. •.. ·- · · .: • . • . ---~"<·· . : · . - -::>- ·· - - . purely a.x.ia.l and constant :fluid v:el_o ci ty_at __tlle sue t i on side . o-r the runner ~. Be:forE?! p r e sent i ng the main part of ou r i nvest i gations -·· 1 t will be necessary to define. wba. t we. mean by· the fJcavi ta-·., . . .- - . •.. . - • :- • ' . .• - . ·· ·-,·· - . - · tion limi ttt. Theor.e tical: :tnvest.igation~,. _naturallY1- can detec1only the very . beginning o:r _ cavitation.,._ when _a t the point of absolutely lowest _pressure _the v :a_por -or gas pressure. ot: the- liqµid has Jus t been ·rea.,;l1e de. _Considering .ci. typi~al_: ca.v i t ~,tion~te st dtagram of' a s c rew punip., _we mtiil.y say that we consider- as i~ca:vi t at i on limit '~ always the point where the charaeteristic curve starts to devi a ta from its ·normal cou rse, not the poi n t of complete e a.vi tation, where the characteristic curve suddenly drops 'to zero. It 1s qu.ite possible that the ea.vi ta.tion limit~ which 1:ve c ompute, will lie still in frent o:r the point. where the eharac:teristic curve shows first indications of a drop below its normal courae., ·which would mean that . locally re~t;icted cavitation phenomenaa do not a:ppreeia,bly influence the beha viour . of'. the maehine. 19 In aecordance with the ususal proeed~r~ :or:·:P }t:th~?tY'. for runners- o:f the· pro:peller type ... we.-~--ru;srune that(-·-.•:,_:1;he:: fiow. . , . . . . . - • ·; . -. . • ., . ; .- . . -·- -- • through the . runner .:proceeds a.long eyl.indriea.1,,,. eoaxi~.l stir~ . ' • · • • • • • • '\ •• • .• • • ., ·• • • • • • , • • •. " • - ... •. • ' ... ( • :· ~ ' • ·: · I . C- ··• • • < one of ·the_s e .~tream sur:fa.ces ..". Furthermore. we tuisume -~that.·<the-·.-, . .: . • • • -·· ·· f • •• • • ·- --'• • ' · ~- : - · ' • • --·. .- - ·• -·.-· :- ,- ·•. : :··~ ··, -:- -:;·.;.!:· ·' _ ·r.. •. . :-- ~: .__:·. •• _ .· ___ . •. change -· in _~ngular: momentum ts.. •~o:n13:tari~ -~o_r _differ-en·t ~1stEl,n°cee _. -· ··· . . - - -- . . .. . -·. .. . . .. . . .. . ' -· . ' • • ' from the axis _~f th~· _,runner ti --~.-' .. . , . ·•·.. . ' . . ~ As mentioned before we }}'lust investigate eondi tiona :·'~-r the poin!, of·. abm,1-ut~l;f l,<,wes:t : :pr~ssur~ ,,in ,\h~ } ~~llbti . ~,nd/ ~~ no.t : a?Ca~~l:t- '~<i:~n~i~:al. Witli this poin~ will be. near~Jo# • •• • •• • ·- · • -~-- · .·:·· · •• . ,. ., • •, • ~- :• ..: • . ,• •• ···: • ' .• ••. , . •- • .• • • • . •: .. • .' . . ·- · : =_ , · ' ,. · ·: · ; .. ' f" • the point of lowest pressu~e: on. thri va,ne or · a,e:tofoii i f ' ' :-, ,· • • •• • • ·•• , , . ... •• •••• •,;•: • •• '"\ • •,:.~ . .-• , •• \· -•:, ' • , ,, .. ,_.!, \,,., '' •• •• •• -~ •., ,, •; , , '.•\ ._ ,., , ~••; •• • -• ,•• •, •F•• • .~. ,•• •• investigated in a. : straight, paral.lel" ar1d :infiniteflow;: ,.. •• . . ·-· - • ·. .• .• • ,,. . . . • ·-·. • .• .•. .• .··. ·:' · '··· · . .. -\ the exact lo.:~tion t,f the point _o f l~west pressure_ in the " runner will be ·eunged by the- _fievi~,-tio~ of the: - b~;r~: -:p;~;~iure> ·:·. . .. : -:. ... • - - : .. ... -. ·- ·---· -· · ··- ·· ··_: •.. •. • · . . .. · • •· ·:. · .. determine this ,change . in ,_ the · tocatio:tJ: ~f -'the- point _-_of lowest • pressure;,: aasu:rnfngs ' however;.·· that. at. •the new· 1ow i:r es.sure" • . . . ~ , . ... • . . .' point the pressure drop below· the basic pressure. , is the • same as the maxix.rnun )?ress:ure drop below the eonstB.nt bazie ·• pressure f'o1.· the van_e in the atre.ight,- parallel flow., Design9,ting this extrerae pressure drop as " P-extr .tt, .so that :. (?) we find with reference- to the usual :pressure distribution diagram ( see fig's. 2 to 4 ) : 20 P-extr, V v~. 2 Vie are J.)a!tieuhirl.y interested in the ratio betvreei1 this __ ex~rem& pre~sure ,_drop and _t:O.e avl?rage pressure 4if;ference on the ••va:ne~< :f'or- which we 1.1.ay ,·write_ ~, ·· ,.., :· ·-·:: ·>~~:"/:.,.'" •,:, -- -.; · ··· -. ' . ..· . , -.. _·-~·-. .( •· ·:· · /.~\:__ ;· . . . . ., _,·.'-::·. , · . /-'·-_ . • .· . .P..;ex tr.t.. :·:.. - ·- ... extr _ p _ ivt ·tr' respect: - ~<>.i:' tJ-1~ ,:il;ariger of ~aV'ftati.91:r:-.: ;::On:j11:e _d~agf.a:-7rf ,on _ ' :fig • . .s . we :;resent. . -~·-·:-:-· k values· for:- a ·., number of. ir1~A<~ero.;.. ··· ,-·-, --. . -, · , . · ,- • ·:~ - · ; _·_· : ,,,· ~ ---·-· '. . · ,. :fc,ils { _refe:i-~noe f} taken _J r_om_. theore_tical pres{v,r~ :~ · _: · . ·· . . , .-~ .. . . . -. .. • .•· .• -. . . . ....... ... · - . ·.-- . -~-- - ':· · ·- ·, - :-.•·.: • • ··· -':· · . distri.but1.:on _curves. The dot.tad extenalons: indicate k values whifjh m:;.y actually be _ expeuted in a, _ , region v-rhere :B J 4;'i06.,8 g:t,res the ·- _Je • value for the )3e~z.;..,-Jcmko;Nt3ky pro.file for which the ;primeil :fo:rm cha,ra.cteristics .and the_ theore0 tical pressure dist:ribu-tion are given on fig. 5 ~- the angleof at tack being equal to zero O _ This. :i_1osi tio:n approximately yields wh~.t - in the lru'lgu~1ge of the hydraulie e_ngineer , ia calle<l the condition of 0 shockless approachn, and, there:fo-re is :favorable for obtaining low values foi.~ k • It is ea,sy to antiei_pate th-3,t this eondi tion- is particularly important when- ever the danger of cav:ltatio:n becomes the deci ding factor. ~ ~ l-0 • . ~1 l( u~ i-----t-·-1----+---t---+---+-.;...,.~-__j!-\~-\L---l li),e - 21 We have . still to determine the position of' the -poil1t·'·. ·. -- •. of the absolute pressure minimum. Thia _means that !le _have • to find a p~int.,. where _the- curve . of" the sum of __:the_diff"eren- .· · tial aerofoil pressure ( curve a on fig':,.? to 4}- ~;d the> , baaie pressure (curve b} _, i.e. the curve· c _ on :f1g's. 2 to . 4 ~ . - .· .. has a horizontal. _:tangent., This: point., however~! hcts of course · •• _.,. the s~une posi t lon ( al ol'lg tlfe __aer<:>foil.) . a~ -- a ;poi?rtl': : ~1her'3·. . the • . ;· . . ~he nega~i Te yai~~e <>:f . the slope . f.l:f: th~ } ~n~e fi:>l _C!:f.. the. ~asic . . . pres~ur~. ~ht:) __1a:tter,.: :however_, _}s, _a c~oi~ng _to:, ~q11a.t.ie>,!! '.'o_.:· •• i :! written· i n the :form i equal to the total dif'ferenee- in prt:HJJsure on the aerofoil times the ratio o:f overlapping · (j) i) if the eherd of the aerofoil ·or vane (t) is pu~ equal. to unity .. The poiD:t .satisfying by a :pr oces s o:f iteration» startingf:rom _the' point where the normal A ... eurvs has_ i ta :minimum;· generally t~e _:f~~st ~tep of this process-,. _wh:i.ch has been indicated in fig's. 2 to 4, is sufficient. We are now ready to formulate the condition for the commencement of cavitation as i'ollows: Denoting by p0 .. that underpressure (pressure below atmos- pheric) at v1hieh cavitation \Vi.11 just. begin,. we ea.n zay that the relation : 22 gives ua the limit regarding the :p:re:ilsure conditions in~ the runner,- ·. at which eavi tation will start~ ·· Aeeording to _equation (6)., _. the .. basic pressure iS:given : . 0 . .. -• .. ........ >· .t._; .·-·';_ ._ ,..,,... .,; _ . : ~.. ·:~,r ~~;: :, -', -" .'·-~'. f where the . poi_~ t, _. x should be ~ha't, e>:f' the abso~~.t e pressttre : ,.· . - . ·, . - , .. ·- : .-:·:· ' : .- ;. '. minimum as deter.mined be.fore., according to fig's .. 2- to 4 ·, ·• ·,~- ' : ' . .· . . . . .: .l. .: ~:~ ,'. ;: .. -~-: : • • .·I . . · , : •~ ·: · .....• ..· ,. ;•. .. • • •• • , -, . . -. · •• • . • . • •· • -· •• ''.·; .. • • • -. _.: • - • • • •• - : • • .. - •• : • . .. . •. On the suction side of' the ruruier the· pressure aboveatmos"'I : •• pheri c. pressu l'.'e i s i h s being the ~ta.tic head on the f.mrd;;ioR side of' the· runner · (measured in :f~et)i ·and. · Va, th~ absolute velocity of the . fluid: a.t •t he sa.me p laee ,. . A::3 :mentioned be:fore: we shall . a,ssu:me t h~,t,. f or our- :furthe r e~.::tcul~tions., •. V ·· · • _. • ;-· i.e j that • axJ.al · · ·. · · ;;;. V · ·s • the- fluid has . no rotai;tion orr the suctions:ide of. the runner• . . . ·. ... . . •• . . .. . •. So :rar \'le have negle c ted. tbe. diff.~rence· in elevati.o n • from the . Buetion side of the ru:rmer to the point o·f • lowest pressure (see- chapter II). Whenever the dimensions o:r the runner wa~rant it , tliis difference in eleva1~ion ean eas ily be taken . into account by mea.suring h not to the suction . . 3 . . . side o:f the runner but to the point, where the a.bsolute presiaure minimum .is expected.,. i.e. vie put :: 23 h8 =: d~fference in elevation petween the ::free. water:: -~'.'<//i"•. J.owest pressu;-e _~n the runner,. ._ _h 8 ~a posit~!e_. is ;positiver ,ii: the- .the free eu1. tion, level . . 1 · .. :&af::::1te;::::::~·:::e::~::t:::: ::··j[!~l~J~"t~=/. of 1owestpreseu:re wil1sl'ightlydlffer . .. ---from that .. . .. . . . determined ' . . :. · · : -.. . .. . _, ·- ' .. . .. ..· - .. - ., - - , .· - · ·,· · --. - ·- . abo-ve_. duec-_:./·.:_:>/:·· t ; \ the addi·tional> presJ~'; e gradient. from the .,_ ,;-.. ·_,_; . -·• , . .. . -: . . ' ,, ,. .. .. . -- . ·-, ._ / ·.. . .· . ---•,. dif:fer~nc~~\ i:n:. elevation.- This . i~;tluen~e;._ howev;r/ eari. be <' •~'--·~_ -: · ··. -. :../{,>(),~:-.( .<: :- "·_:·_." • • • ?ieglected a:Sc long • : . ... . , . · .. •- ' . ' . • .••- __. . _. - - ·\ :~;~~?;.:--: - ·_.. -_ ~ -~---~(:~'--~~:;:_:_·,.::·~:·- __ . . ~ _··_·_ . - as-· the- ~xial. !d:i,mensions. . e:f' . -· •the runner . .. ,- . • . . • . . .. _., ·: -. . .. It one ·wishes tg, consider -friction lossesin the . .. .. . : . sucti,.on liner .J:her_e ?value {in .;fee~) should be .:~,ddeci :to h$ ·, i-n. the case; o:f . turbines, ~,nd subt:r-19,cted front hs-. ;· Jn the ·'· ,' l?utt ing Pc- ,.,., w h e , w& obt-ab1 f r om. e qu8,ti6n (9 J : ;:;: the value gtven by tha and after substituting for equations (7a.) and (8), we --find ~"t,2 · "axial + 2 g 2 v . 2g 2g ... X j • !=. J A;\ El J} 8 (11) .. . ·., . .. . ·- ·-· Since all.. pressures- in the runner. ~re.,. :, for otherwise similar > conditions., proport.ion:.tl to the total head ._ff•< ,corisumed \or . • • • • ~· • •• ••• • •• : "• • • • •• •• •• • • ' • '• · • _. .. : :•_;;·:,.·.·f.,--.:-_ .-. • ~, .· ·.:. ·. ·.·:; ·• ":" . • I • values o~.. Ps + h 0 _ _in a.___ ~ime:flsi_onle~~-- form,. 1?:r.J~~d~~~~ tr R~ In se~tion 1 of' the appendix we have deriv_ed for, the- ..:total?> . • • • • ..-,: · ·:."- . • . _- · · _. •. •.•• ' ,_' · ····c - ·.•~ • _ .... :· ·:· · · · -.-. -·- · ········· · '· :• · ····· .· ···',· _ _ :-:_. ••··; · · : ·····"'-:-;•.:~ .·-..< -_ ••_ ·,:i . '·· : : _::.;: ···· • he-ad . o:r a.n;r cylindriea.1 Se(ttion. thi·o,ugh Rn axial ' :flow runner··.. the expression i • ·: - •:··· · •. : •• •· :"-:' X _,. r _ .. ·. /i!;\dJ. _· s _~ _} '· • This eguatiol'!· is fund~ntal· f'or .,:mr method of' computing ~he ~!~!--1 tai:l.~!:!___1!!!?.-!~~:- It l!!\ :lm_p orta nt _to note thv.t :the _r.ight side of · this ~quation i~ entirely tletennined ?Y tbe form of tht, runner and by the direction ·. .. · · o-J: - t.he · relative fluid velo- _ .. - ..• · . ' . . . ... , . • .' . . , . city on· the auction may . be consi. . side of· the runnel"$--.. which ·- . . ; . . ·•· · . .. .. ·· ·· ·· •· . .· . . . Analysing equ~,tion (13) we :f'in<l that the ' first factor comes from m~ing on the left side. the tot9,l head of the eha.nge in static pressu_re~ o~ly:, replace H H instead l.e. if we would by the oha,nge in static p1•e-s sure, this fi:z-st • factor would become equa,l to one. In the second factor the first term comes :fro1n. tha pressure drop due :to the g,xi;..-.1. velocity {which is the larger the steeper the vanes ~~ i.e. the .larger;1...}, ·._ •. ·.:·, :':~\:.·-,;.: ·::·:_/)? the second term :from . the pressure difference ofr: the ••• vanes> and the thi 1·d term from the ; presa1.1re --ir1cr·ea:se between, the suet ion side of the runner and\ the.. ;point : o:t- loweat::: •• : .; . ~-:..·: :·. ": ~ :· . . •'.. :· .. .: .; ., .·; • since thei, tt~ioua terms l>ar.. beef diVi<led 1If r .' • - • • • • • • ~~, ;~f u •-• - o~;>i . : -- •- • .ej; ~•' • The J.ast te~: __;fl-:.isl~:p~,:. tliet, ,:t'f1>t~!? ..:t?.e-t~e~~:_.JJ?-:e:\sflad,ea::: ~t~~~/Jfi:: :-'} <:·•: '· .- , . • in fig;·s ., 2 · to ·4 : _and the .total. area of the ,Pt' es~ure: :'i ia.gram., •,• , . • • ,• • • ,· -· • ·- '-4 · -~ - ·· :~. ··.1 ... / ,· · -· . ·- , ·-·· ··::/~ , · -- ·: _~ - •• -: ,,, • • • • .. · -~ • . • _.. . .. . •• : - • ' - · . . . ~. •, . _.. -- -,- · ·-··: • , •• , .. -.• ~ ' ' . : • ~- ~· .. ·, -.... :, :~ -- · -' · .. -: ., ~ ••:' ·, .._:: ~ . :-. - .~- • • ••• ·: ,'· . : .. .-··. ' --~'. . It ·:·is . ,clear. from equation {13) · that one will try to' make }tliis ....• · ,.' · - ::. . ·.~'. ~-:. ., :. • .. .•.·.·:,. • • .. , ·--··· --- ~- - . ·.··.' . . .... : ~ • ·:--· · ~ · ,· - · .• • •.• .. . • ; ~ .. . ~ •,: ,~ •· ,ratio aa larg~: ~s p~ss:tble,.. _:w,hich means physl<,ally_ that t}ll& : ·; .-_:. . . . . . .• . . ' . . ,• possible into .~ . _:region of hig}1. "baaic. 1>ressure ; ::since ~.o°! __ standard aero:foil.$ the pressure minimwn. &1J.way13 :;tie~ nea.rer\ . to . the leadtng edge. ·- th~n to: •·the •trai1ing edge, the _~. f. o:rnie:r· . • •., . . ·. .-·· . ... . . . .. . .... .·-- •. • : .. . _. -· . .•. :: . - .. - . . . • ·, - -· --~ -- - • ~ . . . bein.g in a turbine . the the . . high pressure. . end#: .. . __.. ~.nd in a pump . . . .. ·- '. . .. ~ . . . low :Pressur,~ en~ of the va11es1- . w_e flnd that it .i~ easy '. to. ; _• . . . a turbine ·1 _a:l?out '' .9) 11ihiJ.. e :fo:r a ,tm1u_p it w111.: hardly -be • . . possible to i11erease 'it~ v~ue be;londt · about • 1/~ ·.,- · and ·.· generally we- •iy havei to · :figure· with . v1.du:es ar~:p_;md · 1/~. (com.pare fig .. 2 with fig'5~ 3 ~.nd 4 ). This f~~t s;ppears to be the main di:frerence be-tween uunrns and turblnes re~ar------~------·---------"1'"-----..----------...:.B....__ .Qing e:a,vi tr,,tion... It is pa._rtly eounterbaJ.2..};teed by the 0ther fact that for. vertical ahe,ft runners the l~ad.ing edges f'or turbine vanes lie at a higher elevation tha..ri • for- pumps . (:for the same position of the runner}~ but the :former . e.ffect will in most ce.ses be -the domina:ting one. ( On the other ha.nd 26 . ~- - - turbines usµ~lly need longer dra:rt·\ubes to ·reoov~t--. the; . 0 v _e loci ~y _he~i( l~._ ~he 1-atter;. ~is -~;-;qui.r es _;~ilt['~'.;l:i:;r;~~~ •, , > " f _o r :the run~er,: a.bive. _ .• ·· . ha•- there:f'ore~ will ' ·.-,;. -- ·- " • I : •, · . · · ·(' " " - .. ·- • ~•; -;_· • • • ' :~:-_; _·:: ~ -:;/.:_.:._ .. ' :wh_i ch eoneer~ _the ,l'.~-nner _o~ly, _,th_i~-- ~ii":fe~e:~H;~: ~!i'_- n'!_t-· a.p:psar: • -:-,· .-·. \ , • 1 • .- ~ >: :,., .• • ~ ., • ~-- <;.~:<- . • ......•. . -~- ~ :· .. ~ ~ _·· _- :- .'° :'· .,: . ., .,, , __ ,· _:., ,.::.-:·'i\:/;:\: \-:/ :_i..'-,\··,···· _ 7 1 ~Eqt1;at1o'n C_(l~ s .o far ~,pp1ies -t o\ a11 aoa:id;1j<"e yilnd1d.c;i :- _,.: exp1ici -~~Y"c};_J(t( _. --;fi. , _, ';. <(j/(/"',_ -· . ... . ' ... -~- -·;-. . .•. • .• :: _. .-\.· ·-:; ·_: .. eitream surfaces- through the rur,n~rt ~nd,._.thereiore.,'._i shculd:··:·• ' ::&:::::t:~lf=~:e:~:•:a::t:::n:u::::e:~:&~*!! !f~t;h~f;~~-• x--eachas a · nfai.:idmu.m: > lt is possible;;. however.,. to _estimate, ,i:n· • a~~ance wher~;ihi: m~.nmm w1~r ., - .- ,: . . ... . .. :- • •~ • ~~~ur by _uatng e~1;p_fio'.~ (;:}: ,:·--. .- • ,. •.• • .. • ·. ·,.· ? ."}.:~··- of , chapter lf_:Jn the ;fo_llcntir1g 101:m ·: t For axi aJ. flow J:'t1m1era of high· apeci~ie speeds ( if the vel?~ _ _ city _on the s1!c}Jon -_s:lde: i~ purely axi:al _} _the · pha.?ge in _~tatic- pressure ( Py·_:~ Ps l mostly .e:o:n.&ti tut es __ t~.e greater part ·of'-'.. the total hea,d • 11 ;: andr there:rore,- _is _ne~.riy oonztant o.crcss the whole runner-. He~ee t APav: • ~ constant constant or The region where j {14-) j J is a minimum,- t}iere:t'o:r-a.,. has in :first ~p_p:ro.xi.matio1L the-' greatest- p:reesure> difference . on the irane s j and :1nder ~th~ri::ise constant conditions thereby the greate~t_ danger of' eavi t:13.ti o~ .. 27 Further eonsiderations reveal that ·the mo:;,t imper~ant , t:levia~ion~ -fro~ ~ur. _first a:pp!"o:d~~tion_- - ~~ ._-~-~ ~ly< the· : eha.nge in (:P:9, - :Ps) JI_ _ which becomes emm.ller _to~~rd~ : . . • . 0 • th~,. .. ., . oentef •of the runn~r~ ~.ml the d1 a.nge in ·the distri:bution• • -. - - -· .. : • •• - .. . . : . ' •. • : ·•· .. _ ,. , . of the vane ni-essuro, 'which :.. iowards<:··:::o ..· .. . . .,,,. . . - -· utsually . . is-_less: unif'orm. . 0 •, -· · ··- ., . -. .. -. . .. ,. . ._J>;~:~'~ii~;t~_~fi~~;~,/;t_: :_,_·; .-•.~e _. :<:_en~_e_r .__s9 .?h.~t... ::_..lt __, •·-~n~rl'a,~~s :,·:"~'.}_hay_e_ and~ the~e?:.O~t~; f?('),rlCel_ eat;h t,ther~ 't;~. so~~- _e:x:te:ri~._T.p.~ first _.·.". app1.·c:z:i:m.p~tion~ -therefore/ can be con-aidered · 8-5 su:ffie-i~nt·;,, . --. . . . .. . .. - ·. :-_.,. ·-··. _ ·• . ,_. ·, '. , ;- . . -- -· - .• ·' :- -··. · - . _, :· ., , . _, - ; -~ - -- .. : ,: . . • .,• .· . -· ··,.·· ••• . .. ·- -.· . __. .:>.... ... ·:.:- ]'rom . th~ fol.lo.\ving :t'or::m.2lc1, -:tor th~ - s_p_ec:t':tia ."$J)ee4.:we .·;., · ;,~~ ~~e t~-titii; ;~.;1;~ ; ,tt..il0 ; :_ ~~ ;c\;ec:i~ duc; ,?;, j ;"' .. ..•.• 0 must ':tnc:r-ease ·thehu~ o:f -the". runner.,: We;-- alwa.ys ' can :~,: . . ·. ' . .. -.. .. · ... :·to·warda -. - · . -· · - . . . . . . . . .· .. .. .-. .. . . . : . ' ·'· · -· : . -. . .. . _,. ,. ., ; .• .. sa~~i"y thi~, condition for a. given·- speci:fie inerea• . .. ~. ,·. . . . , ~peed :by : .• • . ·' - ··: · ·-, -. .. •. . ., ··~ .' . •. . ., . ... ·: •. .. . _. · ~ ~i:ng· . .J ~ tovra:rds: the -~ enter _of tne; _: rur.mer.;. -so.:'. tiiat;: tne: a.v~rage vane pressure .difference . a.11d the-l:eby the danger. of . . .. . '. .. .. .. · . . . -- . .. .- . eavitation· becomes: · greatest a.t. the. .outer periphery o:f ·.the . ._. •. - · .. • , . ' .. . . • .. . . •. . . ·· -· As our next anti last problem in the chapter on ea.vi- · ta.tio:e 'W8' shall. cor.ilpute the ca:rltatio:n limits for a cert~.in se:r ies o:t" axia,l fl{~W PlllilP runners lltS a :funetion of their For the outer periphery of the impeller the specific speed of axial flow pumps can _be expressed by the equation: 28 where . C = 8160 if Equation (1-f? l-. j s deri v ed ·-iri sec-tio:n 4- o.f t he ~.Pl).e ndix '; .It .• .. , .. .• · ·· =-. ·---- .: ... -. expresses the: epeioi£i.e $_peed aom_plately· by-·-·· form: chara,~:teris~: . . -· -- . ...... -- . . .. . ··.···: .. · .--. . ., . ... -. ------ .·,. . . .. • · . -. : ,_ . '.• • • ·•-··-- · . .: • . . . .. : -: • tie$ __of." ~~he:_impel!er~ _: i~ _ ·(3 5 : . ~ . :( .~ ~'-:tli~ ~~~:r~.~t.,i?n.- ~{ ;_th.i / .~_-•. initial--: now. ·-----in front. af the- runner; :·1 s given~ ~-> On t}~a ,.~ther-.- ·- - ·. . -. · ·.-- - . ... .. . - ,:---- -_:-·-- ·, . -··· ----·:· :. ::::,· .~·::~:_ ~~~: ~·-::: .· ... .. . . ' ." •' ·•-. " ' -,: ••_-- .-. · :, ·- ·-· ·· --; _ ··" ·::-~_-· :-·· •• · .• - • .:: • • .· ·- -~ - - h and:, wa lfa:m.e:rrib fir 't h :3;t ' -~ he_ uh a,~ac-t~J:' isti~ :f i ga:r.· f:;:" ~<;>l~_cayitation; . . . . . . •· . . . ,; ? · · . ·:. , : .:·. _ ;: /s ;.-·.' . ~ ,. .- :,. • ..-.- -: ' . . . (see equa~ion :l~} al.so was ~omplete1y deti,rmtned;_ by_.the .~orm · of __the 1:U:~_e:;-: ~Hi l>;y •ea , ., .W~ ·therefor~ ~-an- re·~ ;~~~nt~<-- Ji~ _; ~~ as a function of ·the il:p.eci_ :ft,~.._-,. .~,speed.,. if: ' . .we-· - -r:.'1&.ke . the :'·..:form o:f" . .•. . _.· ,· . . . . . ·. . - . . . . ... . ··. . . . -~ -- . ·-. . . . -. . -· . --. . . . •,• •::, -.. : .. . ,• . . • ·. . . • •' .• .. the wa.y_in': _which : th~ :form eha:ra~teri~tics 9:f::t}:11:J _ru:nner shall cha~g~ , w~th : tJ:i~-. s:peei~io spee~l, because d:tfferen.t- ty-Jea- of runner~ may have the aame specific speed.,.- or.,. :in_othei. . woi·ds.,. different vaJ.uea for or,7 .oi j, /:l , 3,nd q/D0 .,. if subeti~utad into ~qu8,tion {15) m.ay le-Rd -to the same value for . n$p• The most influential facto:t• is the :produ.et "L• j ~ a.nd therefor" we have inti·oducgd at fir3t definite as.sWll:ptio:ns about j, er,, a,nd D1/D0 as :functions o:f OL.J • . Theee assUID.ed relations are given by the diagram on :fig. G 1.0 .8 r-- . . I Fi,g. 6 29 '. t . For our. compu ta ti ons we have ].JU t . • '... -.. . .. , f3s ~ t> · _which: . 6.. •,. • • - ; . ' fH?e~s . - ... ~: permissi bla,. since the difference between ~h8:-. Jw._o.. _n ,nglesjt. which is- equal to half t-he deflection of the flui~ plus the . angle of: attack, 1.s always very small. at th& out•" ; .1>e~~~ery 01 ~ore~ pump impe11ers. With these a..asv..mp:tiona we- can now no.mputa: nsp :function- of. (cL • jJ :for- different.,, eonstB:nt ya:t.ues, • :_ and on the diagram: on :fig.,, 7 · w.e have plotte<:l the•_inve1·se functions,:: n&."'lely values o:f the 1f'B,n.e ~ ~ . j af3 a :function o:f l Hngle , <:) ._,, .. , . r These last curves give us _fo_r _every ~ho~c~}or- £1: ~e.fi~ ni te f'or.m characteristics as a funct,ion or _ ns;': W?ic1~_,c_a:1_ _. be substituted into the right side of equa,tlan_-(13.}~ leading finally f'oi; _~v~:;-y ye.:t~e of' ~: to d.efinite ..1alues of as a. f'unction . of the s._peoific _speed. Slnee h3 + -1¾1- R -er; does . not rea,ch very high. v&.lue s_» we may a f;s u..m:Ef that the ?....ngle of atta,ek • will .be :zero; which leads to pressure distributions along the vanes similar to that !Bhow:::1 on -f'ig·. 3 ·; we,- therefore, . have chosen for. the evalua.,tion of equation (1.5)' k ;;: 1 and X f.t1A dJ/cL .. .,2 & The corresponding results :for also have been plotted on the dia.g:ram on fig. 7 hs + he H . This dif1.grara :cepresents the cavitation limi ·ts :for different types 1 of propeller pum_ps as far as. _the stee:pnes~ of the vanes ((3 ) is concerned; it is., however, not by a....~y means- generally valid, since other form characteristios remain restricted by the choice represented on fig. 6 • :&'or other assumptions ',:?( fl:$ rfic H . . ..... . ~I -t . •CH CL·J •4 l\) (:) <:) C) ~ 'l'\ ..... lt) ~ • -t '1 t <:.) ~ ~; ' \. ,,~... '\o \ \ ~- - \cS> \ 't. \ u. -1\ ~ ~ (,) (,) U\C, \\ ':> \ U\ \ ' :;~,., I ,·:~ I Cl} \\ \ \ \ \ \ ~ ~ ~ ~ \ \ \ \ \ \ . ' :50 about these f'orm characteristics the curves .?.~ •fi~.-7 , would ~~· a;t.tered, these .· alterations,, .Ji.o~e"!~:.r:~;::;:~ ~\f )~at be·c:ome•.:·:\: -~ '·:' · _·. ··:..·-:_ :~·;~;,; : .···.:. _. • -._: -,: _ •• • - - ... ver-J r,a.dieDJ. , a~ long _as . tllEl f~nn_, _ c ~.a:t:_8,e.~ eri:st.: tcs.::t.l.:r.~ ehanged ;-' • •:f ::•• :'-: •,• • : • • • • • •• • ._ • •:•= •, •, •• • .• ~·:'. • • •k• • , : ._•: , : . •• within _~~~so.nna1>1e, pra.etical _1 -~m!ts ~n,l~~~J:h.~;._d i_a~?-M. ~·n. .••fig~ •.7 .•. . ...~h<11'e.f o l'"..has . some: ge11e ~al :dsn.1nc.1cci ?i ,?o~ , 1o obtain finally the greatestpermisaible:: hea d: H . as .-,:.: ~->·. :>/·: ~:,''.-i(: ;- --.: _.,,.:::-~ _:'•···: · -· -,. - ·-. •• , . _.. , ... _. _• • __ ...... . _, :' • •• , . ,. _.•-.- :: • ·•!._ . . :_ , {. . . .~ - - • --. "_;~ . • ·_: ·· · · ·. : :: : · :- · ·-· _ . a tunction o·-r t.he s :pacifi.c speed and .t he suctf.on-i head, ,'. ~h i ch·. . • • ·• . -~ , : •• - , . • .. · :;_.._. :· · - ·- - . _ . . --:·:·.· . · - . - : ~ · ,_ -· ~ - ~ - • - - - -_ - . - - - - .. .- • . • • • - - . • - • • . · , - ~ \ : / . . : .. -_ .- : . -, .. : : .. : : · · · _ · _: ·! : > ,-·- . ' . _> :- : : , : · .. ·<-?r:-~\:_.::_;.::< .. ·_:_-.- ·:·j.:.: --. ·-·< ~orr esponcls •· to .t h e we1 1 k nown cavitatlo.n cha.:i-·t a.d.optetl'. f or· . • - • • . • • . '. • • • :: ·: ._ , ·, t ' .... "· •. • • • • .., • ,:· · . -- ··: _.-. : · ··_:- .• :•,:.- • • •. • : •. : , • • double . SU!ltion pumps., . it. i s necess ~:ry :to make'· a . final, cho:tee .• F..gar~;;~w<>.:-:ane angle ¢; ~cl• ·.u, ~:~:nittiE>'.J {~. i al :e ·• o r .. . ho ~ tri~.- .1.?'t~e..r . S}~S,lll~ be done . eX;pe.r~~~ntally~- :~ in~ec.c~;}~~~ -e,:;, exists a multitude of influe~cea ·-:fo~ th~. ...: .. . ~-~t~~tc~mmen~;~ent . -~- , . .. ' ~ . . ... . . '· ;t • . .• in-vestigatior1s.,. The :figure cho5en for . h ... . .· . . .. , · . . .. . . . ··. . .. .. - . ·- . . .. ·· ·. . o ' . • : also should in- - . ·:··· . ·::. ·=· ·- •... , , . _._,- .. ·. . . : •. elude _a _ certain mrir.gin of s~f'~ty t(? aeeou n~ - for,: deviations . · ··-. . ·: . van e s~ rhe curve a _~ n ,· . : . . h B,;11'~ ·1foen d ra wn f'n :r '• ~· ==· 12.0° _and·- :ho-. ~ -~6 - f~et of water. For specific sp,feds not' exceeding about · 16000 -- · · fig. 10 G 7. • a nd sho'7S that the inmenoe. of - ~hallgi;:~ l 0 t etwee~ • •··•· •. ••• 20 . is sma.11,. so tha t . the, ~iagram Oi'l f'ig·. B . ·~ be valid for all vane amgles. of' this rage., · if should. n 5 p < . 16 ooo. The isolated point ma,rks the condition o:f' the- _p ropeller pump developed by ~fleiderer { re:ferance II Page 319: }~ which agrees wel1 with the statem.ent:1 of his text. The re&,son why our diagra..."lls have not been e:xtented: to low~r v~.lues of nn:p will be g iven in the following cha.pter. 10000 2,,0000 ..specific Fig. 8 speed 30000 40 .000 3l. CHAPTER IV. SEPARATION IN HYDRAULIC RUNNERS~ By _the -te:rm 0 Separationit we :mean the phenomenon tha,t .. - . the ·stream lines of a fl_ow along a rigid body ,cease to follow . the surf'e,ee o:f th_i s body., forming 1:>etween·:,,them -and:._ -. . . . the body : a region ~:r_ highly disturbed., y;hirling fluid 'motiona 8 _ . . which do not show any preferred direction ( ttturbulent . • . . J.·~ •Wajret4 - ' .• ' Separation . i~ observed chiefly und~r the :foll.owing .three conditions- : • • · : • - .... --~--- ·· l) In e:, diffusor,, i:f: t he angle between the sides exceeds . certain limits (. see :fig's.9a and9b) •. In. thi_s ; f'o:rm the ':_ . .. . . . . . ~ .. , phenomenon ha.s been e-oiwidered extensively by hydraulic- en- ginee:rs •. 2) ln the ease of tbe :flow ar,n1nd a sharp corner or with the size of the erossection o:f the flow (like in a, · short-radius ·elbow);; - e-e,:pecially if the :flow is deflected 3-) On the suetion a.ide of an aerofoil or any vaneJ, i:f the angl~ of attack is ine:reased _byyond certa.in limits. (3ee fig's. tOa and IO~ ) • Thi3 form of the phenomenon., in which we are chiefly interested here., has been the object of'm.a.ny investigations in ci.ero:nauties., where it is referred to as the 11 ata.lling" of aeroplane wir1gs 0 With respeet to hydraulic .. -- ..... 7✓7/~ • . . • . -· . .· . -------~ -~ -. •·. • •• • ._.·. ~ ;-· , •· •• ) ' : ..• . b ·-.·•·· Fig~ 9 '. a b : .- . machines the f'ormation of a turbulent wake on the suction aide of the vanes has been observed in the hy<i!a~:l ic labor~ ~-. . to:ry o:f Thoma in :Munich - ( reference 3 _ ) ~--and_,Jt s eff_e c:t on •• -· the behaviour of' the runner has been considered_in _a _chiefly, _ quali ta.tiv~:~anner by Pfleiderer (re~erence U _>: page -106) . ... , Numeriaal criteria :for the eo:mmenc~ment o:f separation in hydr~.ulic. runnerss however, _have not yet been derived .. To • _. . establish _such eri teria,,, therefore., will be the main ol>ject -_ o-:r this chapter. • No ~ttempt- will be ma,de to reach this goal by an exact., ma them.atica.l: _procedure, __al nee not even for _a _single -•aerofoil this phenomen~n ~. can be des-cribe(l coropletely _1:>y theoretical means. Ins_tead of entering into_ rigf,:rous- theoretical _investigations,- _!le __shal,l try to eov~r _only _the most f'undamental : facts by means of sim_pl~ and plaua~ble a,p _p:rcxim~,tions. The foll owir1g con s ide ration o:ffers 1. tself a.s first step: For an individual aerofoil •in _a. straight ·a nd parallel flow the limit towhich this aerofoil can be _u.sed without 5howing separation ,1>henomena ( "sta,lling") may be expressed by the greatest lift coefficient (cL)., which ean be reached before separation occurs. ( The phenomenon of separation or stalling is experimentally indieated by the fact that from a certain value of the angle of attack s,nd the corresponding lift coef- ficient on the latter suddenly does not increase any more in many ~ases it even decre3,ses ... as the angle of' attack ("" ~ •. · de:fine for the vanes o:f a hyd:rfJ.ul~e -:rna~hin~ ::a\ ~fiit _co~f1fcie:atn'.; ,, •• :··, .:.; ,;_,.. ' • , . . .• . . •· -. .• -~·-:.".-. - in the srune -- ml!J.nner ai!,S for a single aerofoil. (the.i'only : dif':fil(_ ':. . . ' • ' ·- ' ' , • •• :..-- ::::-. • • • ~- •. . : - ~: . • . • • • -"r culty peing : the ~mi.table de:f!niti .o- n for.. an aver;;;_g~( ~elative :: ·;',. . .. ... •.. . .·. ·_ ... __ .. ..... . . " • •. -·· • .. -. . • . • ••• .· ,.. . _. · . •.:._ •• •• ·, velocity·· o:t< the; flu-i d . )' . • : ._We~ ~oll~d ~hen• _s ay ~:~.l ;~~/ se;ea~a_t ;_~~- ::,;:.·~;:~:; will occur, ··if the ·lift· coefficient 6.f the ·vane : is inc1·-~as~d to a val~e., :for whieh Se;Earation would oceur on a; single aero_.... <-:, .·. _(_-·.-•• foil o:f similar. h;y~rg__!ly?'lamic · features) _In ' ~hi.s.: iµanner' '·th<'3 '·.;· . . i . ' ' . .. ... f.!_.epa_r~i.i_on .._;timi::t, llas - •~-~ • f~r, J>een : consi_dered __ res_peot .-· . . . . . with . . . .. . -.to •• : ~ ~. :·. . · .. ' . . . " .. ' . ' . " ... :.. _ ' .. . .. •.• •_runner.~ : o_:r, _t_~ _~\ propeller • typ~ as i)ir ·_as )hey 'have ~·• bee~ ¢om~. . • •. •. . - • . . ·. • ' ' • '' • .' • . .• , • ' ," . - ' • . ·- ' ,· t·' • : ,• puted by ,_means: _ . of _ the ._ a.e-rofoil; ._theo!Y.: _For _rup.n~rs ·-· of -: a · more . .. . . . . ' .. ' ' . ' . . . ,. . .. .·- . . general shape , _ t he _co:m;J;)utati<;>'ll (?f a ' "l~ft _eoef:fi~ie~t 1~ • .f or.: __ the. vanes-.. ,, and .··. the deterrilill<"ttion of: a -eorrespond.ing __. se:pa~ · ··: . .. ,-_. ·-.- . . . .. .• . .· :. : .· . ' The :forgoing a:ppro.ximation ie incomplete for the :following . • ·· · r •• ., We have no right to assume that· a va,ne · as ,Part of s, whole • system of vanes {J,;tke . . . . : .. . . . . • separation} at the same 11:f-t coefficient~ _f'or : whi~h ·a ttcorre~-'."' > ponding" &ingle aero:foil (~ee appendix., aec,Jion 1)- in a straight_. pa.ra,llel flow will stall• This possible influence of the ar:rangemei1t of aerofoils as vanes in a system., which _w~.s the main object of our p:reYious investigations about eavitation.,will again be in the forground of our present investigations o:n separation. In order to attack this problem it will . be . necessary to consider the cause of separation, because on1:, then we wili b; in the pord tion'' to' decide~ . whici-t;~ ri~~:ngea:tn/::· the ge·ner~l / fl.ow__conditions may ,h J;e an_ ~ffec1?··· t~:~\te ~he~~ (/ men on-· of' 5 epaxati on• • .. . .. . -- . . _- .- .. . . - ~ ••·_ :;)le ~l'!dam~ntai idea of the present . expi'a~a:tlon'.: i'oi on . on:. ;i,a:ro.foiis '\::::sepa:rati ~---{.'" • -· • . ·.•-· . •. ' ·. • ·. : :fclllows .': >> •.. .. . -·,.: ,:. : ..._.: .. .. . -/:":'~~ ,: •• ·xhe_f lui~_- p art i~l.es nea~ to . the -. sur:face will. ' b~ :_·x et~r,de.d ·bY.· the ·t~ietion -~i :);he · 5U~:fao~'\::·.:c} '.t¼esi:, __ ___ _ •·: pa.J!_ti~;te.~__:~o~e-/ ~11f?/ a_': .region:wheJ:'i ::~~e·._:pre~~U;; ._~ o;ef•ithe---\;~1 '. . •· • · inc~~a.;~·s':·_:i n , th~ -_ ,d ire~tion: o:C ihri :· 1'10~~-- ~ey .YIJi~'. f ,/Jue \0:;~}~~1r:.:, ini.n..ii{itJi~ ::J.~ci.tY' \~ ;;~~JJ[d ~Y ·th.e •· pressUre .. sr;;u .;,;1; ·• . ·_ much· quicker . than:: the, fast~r ' moving partieles! 'i n . . . ••• •• -· • • .• •... ··•·'·:_ ·,; ·· · ~- • . -· - -- · ·•' .•, . .. - . .-· a greater • ·.•, , -~·:· ·- .... distan:~e f:rorn- \the wall/ The layer .of 'the· n~id n'eareat' to .:t he • • • • • •• • • •• . • • • _, • v , • ,• , • •. . •• • .• .. • •' • • , • • • • • •• .•,• • • • •' • < • •• •, :• • - • • • • .. wall/ ·. there:fore., will' eventually by brought to• rest. a..'1d even . :· - . . . . - . • • • . .. ,· • .- - . ••-- ' . .. . . ie sri:f:ficientl.y long and steep., while the fluid ·in a greater. ..• • • '• •• -, •• , • •::- •- ~ • . ••• • • •- , •• , , ..... • • •• • • •' • • . - • • • ,S - • • I ciistm1oe f r om the wall w-i l l stil l r e t ;i.h1 i t s orgina l d irect ion ~f mo t ion. :__Separatio n fl-ta.i·ts .at-- a :Eiolrat;· where the li;,yers,: neai _to the -wall: ''have< eom:~:\ to-''a;'_'_; ato:p1,' dr' _ha(ta:::'. f t ve.r,sea,}·h ~Ir", lilo~l°.n., ... The ea.use of' this-- 'form o:f seP-a.ra.tion., there:fore~- is a P-i-essure • • • ( ·1· increase- along a · f'ixe!l_boundarl (li~a. on the suction side of an aerofoil after the _point .of. lowest pressure}. -, • It is h1ta1·esting to observe:_ that according to this theory the reason :for saparat.ion _i:n a dif'fusor is exactly the :same as for separation ()n a.n aerofoil,. On the other hand it Oft!l. probably be _said thB,t the separation ,phnomena in the ease o:f abrupt cha.nges in the d reetion o:f' the flow cannot be explaine d 35 in this llle..nner..,, .~d. :are very likely due to a ..certain _instabi- . • . .. .. ·, · . '· i~)l_~u~-~-t:i_ff~~; ;: iiis _f~:r~~o:r 11~:y__o_f.. :t~~\: :force_ distx:ibution. •. · ➔ • separaticni i~ _~ot. .very eompl:et~l.y ~xp~ained. ~s-_y~t,•_)so. -th~t - . .-,,-_".:' it dot!s , :n.ot. :seem po,ssible __to __eonside-r it. ·.with .. respeet . to< . : ... . · :·.·_.~ .. ·.:·.. . : _ .;~ . • .. • ' ' ••· ' , -~ :--,- •• •• • . . · · :··-.. . - .· .. _ , ~ - ., hydrauli~ ,- ~nne.r s : . .~)1-eoreti:cally.,- -~xcept_:1n ~"i>~~~:±; :~~E1-li ...·:: . . -~-~- - . · : .. , -. - • • · . - •• • '" . _.. . • ... . - ...., -~ '. : ' : , . ' · · · · · • ·· ·· · -~·-- : · ~· . ... ... - ••,-_,:~ ·: ,_. · ' , •, .·•-~-- -::·:. - . : .- .. -, ' =,~' . ' _ . ,· . . - . • _. tative ma.nne.r in so fa;r~ as the tendency for se;;,r~,tion pro.- - -. -- - · · -- ,. , ••• • • •• ·· _ - ·_ , --- - ➔ -. ', · - •· ··· - • • • : • ·. ,-! , •• ., , . . • . . •' · • . .• • ' ... - . • • - -~ .~ ~ - - bab.1:r_J ;1!C.!.:e ase~ -~ .. t~e cu~~~1u:~e _o_i' t.ll~: ~bsolute : :h,:~vi: _th!O~~ '. ; ,. .- ;:::;:~:t:~:::r::s~;;:::::~!::t:::~e;~;;~fd:;: ~s: ed .; . usual the devele:pmen:f; _of -the cylindrical s. _tream surfaceas . ' .... . '. ... . .. ... . . . . . . . . . .. . _.~ . ._- . ,,. . ·\ .· , - ~ ·- . ' . .,- ·,~, . - . . , , ' ,, .. . . . . .. . • ' . . '' . . :· -- ... . .. .. -- . . - for: a·:· single aerofoil now.,. and : we ar·e the.r efore . . . . . . . - in .a- - straight . . . : -:-;· . -, . " ' ,• '• , .. -. .· ·. ·. .. • , • .. .. :- . ~ _- , ... . . . · , .-·. .• •. -~ ·. ' .. . ·.. ·•· . . ·,. ··--~: • ... • • • :' , ·.-. • · ' : · . ·:·.: ; • t· . ; '·: • • : ~· . . ju13tifi~d_to , a.s_sume _in th_is case .·that the _phenomenon of sepa- ration i1:S:_ °-'o_ntroll~d by the prea.sn.1-r~ incre,t:tse along the vanes, or!ly. We ·shalJ.· damo~st:r:a.te' our_ eo:m:p11tations about- St~I)aration · at first" on· ru:n,ne.rs of the 1Jropelle1•' •typ~i where ' these i:nves ..,: tigations cotn be e_arried through in a straight..::rorwa,rd and, si:ruple ma.nner. tie have seen be:fo:r.e that the p1·essure :tis-€f on the suction side of :J.ll a.er of oil i.3 re-sponsi ble f o:r the n s tallingn • narati on •. This -pressure :rise is e01_rnidered as equal - to .r:- aa tlefi:ned by equation (7). i.e. sen _Jl'-extr (Theo:eetieally one would h~,ve ·to add to thia ~.>wunt the stagnation pressure., but the :presau:re 36 .</(i ·. / . ~.- -· -- ria e: :from the "basic :preasure", ' i.e. the pressure at infin:tti· -ror a single aerofoil in a straight_ ~.nd parallel· flow, t~ : the ·.·. stagnation pressure i $ hardly ever: realized· at the, trailing _edge o:f• the ae.,;o:foil..} Aeeordi:ng to the equatio~ - (7~f and (8) we have: .. For the :fu.rther computations wa lntrod.1 ee: the f'o;tl.owing - - ,. -··- .. --_; .. : ·- . , _. --~. -- . •, • •. .. : ·• • :· •. ··- . t ·- ·... . -~ . . . . ··- fundamenta{ assumnti on -. : ,: .-,.-- •~--- ' -. . The pressure · :rise., v1hieh determines the da~er of~~~~... ._.. tion on -ava.nein_a. hydr~ulie runner is equal to the pressure rise on the ..Ueorrespo~ding ae_r~foil a( see ~!::HH'!ix, section · l.J in a. straight, 12arallel, and.!n:tinJte flow PLUS the tota1 . change o:f the basic pressur.e in · the l.~mner~ the latter being · _;eosi tive in the case of a pump,.and neg_a.ti:;te in the ease of' a · turbine. Dart ef By thi1o assu..mptio:n we, t here:fo:re., · oonsiderthat - - -· • •. the shange in· })Elsie _pressure., which oe:eurs before . the point of P-extr$ on the vane is reached ( i.e.,, between A and Con :fig's. 2 to 4 ) as also contributing towards or against separatiin in the fill awing region . of increasing vane :pressure·., 0 This seems justified, since we are considering all-the tim~ the same 1ibounda.:ry layer 14 ( aee :references 7 and 8 )., whose !'ut.ure behaviour may well be expected to be influenced by- the :pressure changes in its very first part. 37 ~hio}l ¢'on-troll~ _se:para_tion~ __by~; LlP5e .~-- ~ .•.: :, : .. . .· .. , · .···. . '· .. . · . .. . , . .. ; · . to 0-Ul?' a'bcrva a.$~"Uruption : ,. '; - .P . . •• • . .;.;. -· , :-,;• --· . ' •. -·,:· .,· ; ·.~.-: ;:~;-:,:/~ {ff~j.fr>:;_:~-r.:~').~·-:=.'.:- ::i ,: ~negatf":~ > ; _·_. , ·.'.. . . . . $ign .to ·~t:.,r'oine1~~. :: _given ~;: e{lu~ti on . ,:3t10~it.~~u tl.ui !o~,:._·p; ~f.ro~'. #J;~"tt)hnt tr'"" the .~~uatf" t'i~1 ·obtair1 ; .· · - • ...... ·- duce a~eoe-ffi~ic~mt or· ae:parstio:u" &SJ, wh'.tt)h has t~e , e~:me ?'.e·lrz., tinn -_ t_.n ittp;;,~; • tn ~';t, sy$t.-em o:r. _ '\'tU'le~ :as. ~ - ~- • h.aa to . -'O·..-.eAt:::·.,·· • •. • ~~,1 · ~ ... J.J £or ·a n aaroi'oiJ. in a straigl1t~ and :p!i.nllel flow., _:By a..""lal_o gy with equ.ati.)n (l.5} 'V!e t.ne:refnt e de'.fina the coe:f:fiei~nt o:f aeparaticm . oy· th.~ rialat.ion ~ LlL\,ep &l!" Ca ~ £"'2. ., k' • O' • a 8 = ~:t2 2 ilPse12 k• 38 where k' is a new k value taken from the di a.gr ~ . on· -fi~-;.- S oo?Tesponding to the v a lue of c 3 in the_ ~l~e : ;:r:'..~L ·_• • • . Sl.tbstituting into f:!quati;o n (19a,} _for ~Ps~;:;\h" •;.; ~~:e :·· • . . . ' given by equation__{18}~ we obtain : ' ; . .. • ·. ' , . ..: Since the-:_~o_unda~J layer theory as yet does · not. give uz: - sufficient·."" ~-.information about the. influence sha-oe 0! -: . · -of:the · -:-/~--.. 0 • .., · .. .- · •.• . • .. ~ ... -· ~ - ! -·~. ,. . the- pressur~ ' cu~e on sepa.r~-t i;n to decide which .value we, · .. shoui d , cho ~se fo r k. " B'. we ·-- s,ha.,ll _us~_ :tor our :further eonsi. ~:-:--.. ... deration the simplifying _a.ss_1 lllll}tion •. . , • • • • • • • V • • •• • •· • •• : , -•• • • , • • , k' • • ;. = k which 1eads. to the ~quation_; (21) vrhere we s houl d kee1.1 in mi n dc the.t we h ave to :pick our · for The · upp er limit of the eoeffici ent of sep2,-ra.tion c 5 ls in first ~.: pproximation the highest lift coefficient, which can be r eaehed in the windtunnel :forsimi.la.r a erofoils ( a nd for the sa.:m.e Reyn?lds number}. To obtain a..·•1 e::cact cri terium for the commencement of s~paration it would be necessary to deter.mine the values for n3 and for k experimentally. 39 '· •,' , t"' •• .- We: have . seen ~e:fore _that for -the same ·_veloclt~,:co11- ;;· . •.·. di tions· _the he-?'.d eonverted, in the _::runner ia eh~efly;'de~ermined , by .t h~ produ,~t . ...oL ...... · • .,·j . .(eee . .. ,equation . . . .... . .... (12). ...._in . .·,.~hapter ' . :.-. : ·.• . _...:IIIi/. .. .. .:·.·:_• . and section 1 ., of the appendix) .; In : order to judge _the limits . . . ..... . , o-r this value-' with : respact to separ.ition we re~it~ -equB,tion ,, ·:· • • • ' - · · • • • ~ • • : • : , . ·-· ' · •• , • .,, ·, • • • • • • • • -~ • • • ,.- . • • ..... •• • p - • , ·, in the, f'orm t ! (22) .+ : ! "'! k j • i ~i~.:-~6.Sil?.~(: ~P .::tur~~-n~-~--(n~_g ative:_ s~gn} .. ~~l:a:-~.qua,tior1 ex~ p.re~ses : tl1e ::fac.t.:, t11.nt. ~or . .t ~ -k , the _va.lue , CL·'.":J .!' ,and ,, . there_b Y: ~~~ :.l?-~-~~da whiC~_.. ee,n be.· conve!.ted .ir~_~he ,_n:1nne:r without . se_paratf.cm,, 1.~ .u :alimitet.¼~ J?hysieally this mean:;s _that _u~der .. " . these eonditi.o n~ <the _drop of tha ha,sic preiasure alw:;i,ys ,ca,n- cels the us1.u:1J. pressure: ri :e e on the suction side qf the vanes to such . ~,n •ext_e nt/tha:t :no .~epai~tion __ ean _take : _plaee,regardleaa how far :.eL may - JJ~ :b-1cre.rn,t1~d { see :ff.g. 2> )-. • (' ~In the e.;.;se j > k . equat:lC1n _( 22) T o?.ses}: t e . physic:.:.q : mecj.ning ..'} .• • With respect to :-grJp 1:unner3 (positive sign) we :find 4 that - e-r •j remains definitely bounded if we increase •• ..l..:-4 . ' ' •" ? ' j • arbi- (23) In the di :;,,gram: on fig. // we have plotted the v~,lues -o f' 1 !. + j (22a) l k •a,a a function of the ratio of overlapping j ~ showing that ... ... ,, ------,.-----.-----..------,-------,-------r-------,------,------. " ~ - - - + - -- - + -- - -- - + -- - - + - - - - I ~ "" ~ • <lO • • ~ • \0 • 1() • >::, 1~ ',::, ....V • t<) • · . '<i • •. • Q 40 an in:crFJ~:tst; in i .,e. &n increa,se of the 1~ngt11. of th~_:_':: . . j , vanes _eo.?!'-:Ps.red with th~ir di.stan~th eontrioutes ler-a3 •~.nd 1.e;:is; • •• • . . '•" :--···· •• ,••••••• -- • ' -~_.: ···.-~-:.~-.->.. ' -• -• , _:,_ . . -towards an in<n·e~.$,i, of. th~ highest posai ble. • er,"..J :-;i~-~ •j ~_·: ~a i _ncreHi.130~~ ~-. '.Wllicp -~i~ ylausible.,. __ 1>~eitUS_"'t ,.a.s. :w~ fmn:e~se ·: j,wec - ·, , · ~ : - ; • : ' ' · • ' , · • • · _. ,. , • .• -~ •• • .' , . - - • · ,- , ,. -r . . - : t -· ,.., 1 .- · , • •• , . :- ,i. . 1 inerease the .ris6" in bR,si~ pressure~ .al'ld therefore must .de- • • '.•'" •~ • • • • • • • ,• --- •~•: '• • •• • • • • M • •- • • • , , • • • :• • • -··_:, ·· : crease er.h in _orde r to _k eep . the, total pre~sure _ r ise __Oil the .: 7· . - , suction 8ide of th~ 'va..vics ·vvithin -pexrnissi'ble. . l _imit~{; ..... . .. . ·•;.. - ~- . ' . . .. . ' . These- considerations -also yield lower , limit. s . ,.~.. - . definite -- .. ·. • .... · ~ ' .. ·- . .. : .. · , ' . ,.- . '. for the spe~~:fic speed - of pumpS >O:t..' the propeller:· ~yl)e. ; -- ----, - · · --.····· · -- . .,. -- :' .· . . .,. ,... , . . ..., • . ·, . .. . .-_ -· . . .- _.., . , . ·-· . .. . . . . .. ,Correspcmding _to equation (15} _preeente¢i in<phapter III, . . - -·· we can _a.,"'{press _tht-) specific speed als°: as, a :fUi:l<!_tion ,?( the ::, form charaeteristics at the inne-rm~ cylind:ric:;i;r streru.,';.· • :' surfaee of· the runne:::. 7his ex.p:resJ;3ion also has~been .derivsd in section 4. of the appendix: .-.... ~ . .,_· .. where w~ have to t.-Jce the allglea and other form characteristi c s: :from a cylind:rie~,1 section with the in.nei~ di a.meter n1 •. . C ,has the s;:me value as before<? The only dif:ference between equation {24) $)',nd the corresponding equation {15} for tna outer surf.;i.ce is the :fact, _that 2 the value D0 /D?l. - 1 has bae:n embsti tuted in place of' l . ... DI/~. Sinee the new value is always largt:r than the old one, we must,. in order to obte"t,in the ea:me value for the speeifio sp e ed, :mak~ 41 ~•j con:'$itl.erably la.rger ,at. the :Lnside -.than the .value: o~ at __the puter .:r;,eriphery.. This is ~11ysJ1Ja.l;ly_v~~.,/ ~~~i: to-,> comprehend~ . considering . th~;t. near the -hub ..re ·fu;?.ve to. --.- · ~-• :>' . c-on,;/ . ~.~ .. ·--·~ .· eity or ' rel.;ti11e _veloelty (se~ _3,lse1 ~quatior1_~(l?l)~<H~ice · . . . -· • .• ._ .. .. ':' _. . we :find· tha,t ,~epa.ration O!ti the v~~~:s: (3.ta1i1n~).J~_111ore :, • difficult . to ~V<>id at: _the ine~de •than. _a ~ .the ~ou.t~1'.' periphery : . . ' ·: .- :, . . • • ' • .• ·, • . - •• ,·· . ·_ . ~ '. o:t P.n .. axial ; fl.ow runner.and that: . th·:::.. lower· li:n:i.t-~ for "th~ -.. . ,. . -:; ~-::. . . . . . . . ' . ~ . . .: .. , . .. -· . ,.... ... . ., , .. • --·· . .-- -· -· .,_-,, ~ --. . ·. •• -.- apeci:f~e_. ~peed _haa to be· d~te:rmin~d from the. -,~onditi~;~r>- · '·, . .: _.: . .. ·.·:.•, -• ,•. the previous cha.:pter :for the. series o:f r:popell_er pumps,.. which we eonsidtJred., -we ehO{H3~ fol' the form ehax-a.eteristiee. -. -_ , of the inne-r- :st:ream· su:r-fa.c e the follm·dng valu~s : Di/Do = .5 r-i•-~~ ,40~ ·• • Wi.th cL•j ;;;. I. we -obt a.in •.;i.~co:tding to equation -(24) for the s.9~cifie ~psed. , -·..:: -aooo . ' ... •since . . ,· :n . t,]! . ~ecox•di:ng to the diagr~.m on fig~ // cavi ta.ti on eharts., eon~idtn~ed cL · •j _;:;_,i: r~q_iii _r_. it._i;_S_; _· alre~tdy _a :ratio of over- <-· nap ;;;: l:O 000 as praetieal lower li:mi t fo~ the speeifie. :3:peed of :such: runners,.,• For -turl:,ines '3ttc-h a. lower limit for- the s:peci:f'ie- speed' . . . .. - ' ~ does nor seem to exist ( from the point of vie•t1 of' separation _. . and wi thi!'l the lirui t:, of our prt~sent approximations), sine~. we have set?n that for such runners GL • j c:a,n be- increased indefinitely without danger of separation., Sepnr~,.tiori in r~~~ow it.mners. Wl th ~espect to rf.l.di.;t,l. _:flow. ( ;:md mi)(ed fl.~vz) runners • we can only hope tct inke r;;i.ther c:i.-udc es-timates: aa to ·whether . wall k.now:n that the rel~,tivc motion of' the' W:J,ter ia -alw~,ys •: ~ • • • • • ' ~ .- • • ••• • • ~. '• • • •• • •: : - • ••• . .,.;~,- . • ., :•. • ~I . . . drop,·. that sep~,:r-3.tiO!.l_ -~:,1n al:"."_lay3- _·:'0~ .~:"Oi_d~:d~ :8,!1:~, ,;{e.:: sh.,,}11., . . . . . . t1rnr.efore~ •restrict .our :following_ in:veatiga;t;ions to pump :. -· - runners ·• only~ . Alt.h ough :for r:'l'.di.,d flo>:i runners we do not have su.~h ...· . -.-. .. • . , a sir11_ple :relation between runner v~..nes .• . .aml . . a. . ."eorrez:po:nding" ·. . ' . - ::. . aero:f'oil ~-s in the ea.se of ~.xi~J. flow .r uruiera~ . we; a~·! ye·t · ·· i. n the J?.o:-:li t1 on to . eom_pute : for the v~tnes of' such runners . • a 0 lif't coe:f1~ie i entu defined in t..11.e, sane :.n.;a,:nner 2.a · for · a single aerofoil" Thereby we eot_a i.n a _p:.rimit.i Ye ide~ :-i?egH.r(Ung the danger of separat5.on. ~for- · the ,P&,rti cu1a.1'.' :Y&.~lt; ~--. :following f'o:r:mul#, :for . th~ tfli:ft eo&ffi eient n of· the V9nes, of ~, r~.<li~_l :flow runner 'tUlder the ;,,ssumption that. the vanes are eurved as log~.ri thmic s_pirs.ls ( always assuming no rotation of the fluid at the suotion side of the rumier). 2 j v: ~ V p At cos i.., . ... C .A . Fig. 12. .1: where _j _;;:. as de:f'ined by :fig. /2. , AB and .AC being. meaaure along _the outer periphery o:r the impell.~r~ , .. . : (It is intertts_ting to note _tha_~ equation__(25} is _identical with• the corresponding equation :f'or axial now fUnners., . . • ·:- /:.\/.; /-: ' • ••• ••. • except that -j 1 •• • ·-· · " and the velocities hav-e to be de:fined in ···\:.> ·.n : :. a di f':f ~rent _manner .,J __ _ For the investigation, about the influenee of the basic pressure· on the danger of separ~tio,n it is _of greatest ,impor..... -, ._ . • • • -.- - "·#, ' - . ' • •• • .- • in the baste - ta.nee tG note", that that part of. the in.crease • • - • • :· # •• · • . -: • ••• • ' • • , ' '." · •• ·- - - • • pressure, .. :W:hieh is due to the ~entrifuga.1 forces,,. _doe~ not _ contri b~te :t~wards _s~pa:ration_. _This :fact was first po~nted -~on _K"arman _ on ground _of' the petr:feet equiyalanee out by . between a field o:t- cen·trifugal :foreea . a..nd a gravitational . .. . ,. ·;• . . -~ . . . ·. .. ., .. . . . . .. . . field, since the pressure :i.:ncre@A!h, due: tu grav~ tation ( for • • • • • I instance in a. yertieal pipe~) _ doers 1wt le~vl to sepa,ra,tion; • A fu rther ::tnv es ti.gat:l on imme dl nt e ly xcv ea.1B- the f ;:;i, ct. t h at~ the pressure increase du e to centrifug::;.l. forces cannot contribute towards the retardation of'' a certain particle, of' fluid# whieh we consider# since it :ts balanced by ... - and numerically .identical. te . .., the centrifugal force acting on the same partieJ.e. Having seen already that the :pressure changes due to gravitation (differences in elevation) do not have- any effect on the possibility of separat-ion _-ei~er., we:find that accordiniJ to equation ( 2a) ( chapter II} only the pressure. changes iue to the changes in relative vleocity 0 -;t the fluid., i.a. the q.iffusor e:ffect of the channel between the vanes has to be taken into account. ; . ·,: :: .. ·~ .,. •. '. . For theJ)ressure increase a.long the vanes which has · · an infiuence on . the commencement of · separat.io_~ ~- we,-: _th_ere:fore, :. •• may write : · :: .·. __ . .- . ,. . ~ where ··•• y- ·. ,is: ~ i n the '°mean eftectiye vel~ci_:ty_1: __ a.5_ defined..:.-.·:- • in .. section ,1: o! .the a.ppe~di;; · thi~: defin~~~~~.. .. . . .... •if' we extend . . .. .- , , ' · •:' • · ·· · - - ~ •' • .· , • · -: . . . -~ , -~ • •' • ~-·~.,- -··to .the · cend:i.tio~s ·of raditl~_ :f.low_~n~rs. . " . ,: • .. •• •-. . Bf. ' · - ' - ·•· • .. .. . . - :· . • . . .. . -~·, :; · • :.._;J . • : . . ,. . ... , . • ·· ~ : -:._";.."'~ .. analogy:,g1: tn our pr~viou~: procedur~ _·we compute now (27) or i'inaily i -•• 2 (27a) + · k Substituting for _eL the value gi-ven b;y: equation {25) we obta.i~: 1 + k (28) : ... ·. .• . ~ . ' '. ~ ,: . : _only to runner_s _with ve.nes curved as. l,e>g;ri th."rllll s:piraia. We _. shall, ho~ev_~r, use equation {28) in _C_oimection-_ ,;;i t~ :i~~i~l .• 0 ·:r1ow and mi~e3d flo,w _im;P~ller.tJ__p:f __ q~it,_~ _gene37a,_~,_f .o-rr!i, characte... : . :ristics.,,-~ eon~ide~ing expression. (25}: simply ~s .deiiniti~n >..,: , ; ,. · ,-, . , ·-. ··-· . . .. _···. :·.:. ~-- --·. . .. .-· ,.--··.: ·- . ·.- • ,• . , .. . . ., , f'or ths l~f~ co~f!,Jcie:nt s~apes~·>.__ F~r - :-.~- . .. ., . t}.L ·._ ··.,. •'• •. · . -· - . . • ·:_ . . ' 8 of _rannar vanes of arbitrary .f _w~ ?aY.: choos~ _il'l . this. f~.se ___-t~~-•- - ~-ye,r~ge . ~n.cli- -~ .::; . , . _:; . , ,- . .~ ' ' · :-.. - _ . ,-;._ :·_,': . : . , . ·· ; . . . · , . . · , · ·nation__of : _:tll~ -_van~,.._J .e. the inclJ:r1~,t_io_n of i ~ l9gari thm..i.~ : - spiral. drawn through the two. ends of the_ vane. ·. • •, '.•"·• • _ . • •.- , , • •••• •• •• • • • •.- • ; . ••• •· • •• -•- • I• > ._ ., • • ·, •, •• • '- •• • •• "-,' •--_(Note that :<equation ( 28) .is equally· ~alid :for· a.xi~ flow .. ~nners i:(_·: j , i0 i11terpreted__eorres_pondingly.;, and Vp i:n • the first tenn is. replaced _by v .. ) - Since in _the n:w,jority ot prg,ctic:al cs,ses we do not know the pressure- distribution along the va,nee., . which a.etermines of e 8 $1.l'!. .:P9,r_tieula.r, ·would rf'Jquire methods :fof t~~ o'bser--- • vation of' separatlon on the ve,nes for a given_ma.chine · t>f • ge ... - neral. design. Such -method$ are not available as yet, although promi5ing attempts have 1)eP-n mada in. thi3 dire-ction. 1Totwithstand.ing this present restriction., equation (28) per.mi ta us to estimate for a given cv,se., whether· we can reaaormable expect aepf?;X-8,tion, be~~.use we know that the lift. · coefficient of a single aerofoil can hardly be ex:pected to exceed 1.5 very much, while 1 and 2 • k will probably lie between Due to the eo:mpe,ratively long pa~sages enclosed 4o --. between tha-J va.nes and the sharp curvature of' ti{e fJow . in· <> <t.he ease _or/ iowe-r specifie __ sp~edSr / e3max·:-_ f'('.)~( i n1P ~_} ;lers i ·~ ' ':must be exp~~ted to lis somewha:t· b;low the eorr~s~ondin.c-r · . ,, ·_ ·. -· : . ; · . . .. ,. . . . ·. . . • _. . . . -· · :'."' · : ,. : - . .. • · · ... :, , - . ·- . ~-. . .. . _ . . .-, · ·· · · · . _. __. , . ._ . _,_._ . o __ , . . · . .~.,i kiues • .clnt~'- f'or .Jndi vidt1a1' ~a.~r~f.'gil:-s. ; .. _.,:'..'. --:.(/:~;.';,;/, _ , •::: ·:. '. ._;'.:::;t ·"·' T~~re.:_,J ~~ : a.-.:. :f~! . ge~er~~ :..~.e.~?.'.ri ~ t~ .be . ma_dfl~:';. ith: respect.~,.: .I o equatj_o_ll}:'(j7a)_ J1nd (?8) : • ' •• }:':~~;: • •-~. , ·.-; . r~; ··•/~d: •:;, ,·,:,,_.,· •• ' ~ one • must· e"o nclude 'th~i eepa1.. atio n does actu.ali;i' take place •§, •~~e ,,;.,,e,~ Co( e:~11.: ~ne~;, (Thi~ a~reeft .wiih· th/ ob~er" · ·-· -.; ations , ma.d~{-by Thoma j,... see refc,r~nci.~ 3 _) ... SepaicJ.tion on _ = • •• .,,. •: · • ., '••· .· •., : •• ·, _·• .· · • • C• . • • •, . . ••· • '• , :d;• ' . •, , • . ·,l, •,, • • :, • ,- • • .. • • •. - _, . • , • • . , • , •. • . pump runners; "thare:fqre.,, does }lot seem _to ~.Ve such a . , • ·: ' • • • • -: ~ • ' • • • •• -- • .. • - · • • • • ,.. _ . . " •• - ~ • • . .. ·:·- · .. ' • ~ - - .":, - ~ - . < . . ·, - ·. •. . • -- rad!acai :for h~dividu.?,l .aerofoils: (see, norma,l li:ft . - . ... e:ffect .. as . ·--:·-. . ::·. . ... . . . .. • .. . :. -~ ; . .:, . . ._ - - •; • . .. • . :., . . .. .. ·; .' .. . ' . ·,.,,. --. ' ., . ' -.- . ··. . . :- . .,. ,; . . ..· . .. .. . no t surprisi.ng, si nc~- in- the c a~,e: of sep a. ra.tfon fro m the >: - • . . . ;: •. ,;_ . ·- . -· .. ·. . •, . . . . -··- . . .. ·'" '. . •.... '. "·· -. ·: . • ,• . •, ~_ ; . ·, . . -- . .. . .- '; .. ' . :. .. -.suction side v ane . ·t·:..;·;· -. of' ' .·· ...a.- ,.:pa.rtieilQ.l" -· . : ·- vn.ne.,/ . ,... ,' -. _. . ' .the followi~g - ...' will . prevent thei VI~l<e :from becoming very wide,-_ and the flow will -. , ... . .· ' •' · -· . .. . . - .. . •. . . . ··· - . . , ,'. :not be able to devia1,te very :ra.diaecl.ly from it~ .ideal · diree..;, • • • - • • • .. • •, • ., -- • ,. • • •• • • ·: • . • .i:- tion., However.,. one would ex;peet that the formation o:t any . · -· : ..•- , . -· . •. . ,: w~,ke on. the.. Yanes. must cause eome energy loas~s, .· so th~t . . . . • .· --·· .· : . . -· . . sepa:r9,tio1? should. ba a-voided. where very •high ?fficienoies .. ' The most. in1~luential. :ra,ctor in equation {28) is the .ra tio Vup/,,p- _especi~,lly since an increase in j ( length o:f the va.nes) 47 c~-n har4ly _._ 1?,e __e.xpe9ted ,t o ef:fect.ively preyent _separation due to the, cor:responding. incre:;uH:!__.) n _waJ.l _f?7.~~:t,ion_~_ Hence 6 -. .·· .. • . .· v?PIYp .1'~1? cor~~$l}O_n ~i,ng _! :+l!li ts ~·{ · .w~_~ h__:r:e:f.e::~en~~ __:to.< ~lle _ "!l()e~ty_: d~air~ ,,on .fig~ .- ~~ - ~-t, -~~ -P:llt. __ :,~!p,. .aPP,fO~~~~te~y <,:. . equ8.l to •· u ~ > . . . Vup- _:thereby obtainl,ng f ?r . _the r~'.~~~$ •· · VuP/.vp . . _____ ; ,:. .~ ~ -,: _. - (29} TJ:1e rat.io: · 1:-/Vup. -- ~ hower~er,: is pJ:oportional to-- the sq_u&.re of' t.h~ ..uni:t: latter. b~ing . defined as -• . .. -~·speed; .. the -· . .. . . . . ... . ... . . . -- . . n . = I . •, •.. . · -· . n ll.' · - .. 1 ' ~ ---· - - e· · below certain limits . -~-----~-----_~bove ~- corres:p~~!)cli...£&..l.1?..:,;e :r l inti.. t,. . _Xh~ uni,_ t _s~~~<:l,.._ ho:•H!V1:~r; eanno~ _b_e incre_ased very f;ir without inere;;i,~in_g ~~~c _s?eei~ . fie apBed,: Ginoe othe_rwts_(:t the- impeller wt>uld assume· a hyd:ra:nlie~lly u:nf;tvor.~."hle shap~~ Ona, the1·~:f"ore., obtains the result that., as a gane-re.l rule., ~-lso- the epeei:fie speed muet be kept abov~ ~ .eertc1,in lower limit in order to _prevent. . - . si,pa.r::1.ticn. This eonclusion ia -the :more prtib;;1;t ~Tn;e ii • 1ncrea8-e in speaifle speed usually ia acc:>:mp~nied- -~J- a dser3 ase in friet1on surfaee {wider c-rossections), which :must be beneficial for preventing se_parntion., si11ee we ha,re seen that t.his phenomenon is caused primarily by friction on a rigid 48 prezcnt impossible :t;_O' give definite _figures fp~c<th~ lpwest a;peei:t'ie sp~ed ~t_whieh _ee,p~n,•a:tion__ ea.n_1:>_e ~v0:!de4,r -.w :~ch -• · - •.. :. - -- ' . ' wou.:td .be : t~¢~_ m()r._e -~if°~ieul t.: r5_ in<.:_e, .fr~~.. . -equat~pr \ J2?} -w~ o~ly _-.·• .. . . • • . . . . -. . .. . . . • ... . . . . ' . . - . ' ' ·• · · . .· .. . ·- . . . ~ eould de.ri-ve- ~n rapproximata lovrer. - limit for the - ..,_- unit s:pe"ds . . . • • -·- - ' • . . -· .. ·--· ', ..:. , ._. • ·. ·- , . ' .. ' ·- .· .. --, ' -- . So e;n~would: r:1ave to vuu:e-~ ad.di- tiona l as sumptions a,bou.t. ,· th:;..t . -··. --- . :. ·--:. '. . -:·:,:,· . .•. •-- . . . . .• : .. • -, . _, .. . , .. ·• ·- · .• , --- . , .. -..,.. .. . . . .. '• .. . -.. . .. . ··: -. · -_ . __ "'· ··· :· ' the renw.ining eharaeteristies _(unit eaps.ei i;y) .- · From -th~ :-: , · ·'.· • ••• , • .• • ;., . - .• • • . . •.-· . • ,_. • __ , • - ~~-•_-. -·· • •. ·,.. r· - •• . . . :. ·.:·:'··. • ', • .:.- .· .- ,.- • - ~- .·" • ·>:- .- · • •, •--·: . ." · · - ·, ~,---;;· _. p;rcsent state or. infor.-aation one ·!n:~rely e~,n e-fitimat~ that: -. · ~ • ·, • • _ ,:, . · , .,:. • • • • • , • · . . -, • " • : •"" • ' •"'• • . • , • . - • ••• . ~•• • • •• 0 _.:. • . . ·~ • --, , • • •. • : • , • • : . -• • ' · ; _p••. the lower lirAit for the zpeeific sp~ed; • • • •• ., • •• • • • • , , . ' •• • • ' · .• : ,• • ~ •· • • •· A • • •, > ' • · • ,, • • •f • •• • • • • •• • • •. - .?.,t • •. ••' • •• • • : • • • ;-••• • • •• •• ••• A• • w11ich sepa:r~,tion ' • _. • • : • • :• '.' • . • • • ~ • • • eaA _ b ~ airoide.di _lll~ _J.~e f11 :the yic;i nity _of, _4509. , 11nl _e ss c · · o:a-e uses _sp~e_ia~ Ille;;i.ns _fl;;)r _preventhig se:parat~_o n, : corresponding to tf1ig1'1 1:lf't_ d"'vieeatt .. on a.::lro:pla,n ~ wings, _O;F )>e!-: ha:ps 'by $u.bdivi-ding the i?npe ll ~:r-: into two o:r ruore _;,uec.essive . . ~ their a pr,l ioa.biiit:1 f£.hould ·b~ :se~r.i cu oly •studi e d, J?rovided . - . • . . . ' it. ca.?1 be -.shown en . -thtit.t · the . energy .losses d.rte .·-.-to~ aep.!:rratiorr . • ' :. ·• -. -· . ·· . . . _ -· ·- . . . .. ' . a:re ·1al'ge >- V:;lI3e3 o:r ~tanda:rd 1-v.m1ers _of _lew . spa_cit.l e- speeds . . en.ough to wra.:r~,m.t sut,h depar tures :fror11 the con"'rent:tonal- dezigna. . : . ·, . .• •' . _' . . .. . - . .. . . ,' . ' ~--·. . ·· - .. .. , . . ' . By this rt}ma:d~ t.h0 _writer :wl:nhe,; .· to imgg~s.t that. ~he energy . , comli tions in high head pmn1J rrmners rn~.y hav1~ to bo considered in fl, ve~y ~if':r.,r-ent manner f'rom thosGt of the :flo'l'T. ttbout ..an- a1"e ~.ppliet:tble to hydraulie runners. 49 . ~ ,;:; .: 3}_.: ,_'.rhe ;:f~-~:t that" tha presfflre:- :incre&.ae, ,.vihi,en, i_:is-.-Aue . . • ·- - ~:· : - : • ,• ·,- !' -~ --. :..,· __ .-. ~·. - .: ' ta the .eentri-fug~,l foree, ,: does not . eontribute..:-;,towll,rds :-seu~--~~:-: •>·:">.:_::/•.,: ·: : . ". . · • •• , . ... " • • · . • · •: • "' ' .; • ;" "• a. • ··- • • .... , • ' '•••., • ~ ·• , :.~:/ :~ · · • • , _. . . • .- ":,~-'•'>··. • '",•·•·•:•: :W • • ,• . .•· ~ r~.tio~ ;Jsi·---~~!.t.~~-~ d -- ~~e~ _ln91.:~·J 1Y-:.~;qu~~it>.~·- ·_( 27 ~:)\ in\ st>-_: ~:;pt ;:·.. :. ·,-· , • .•,• ~reates. stntic.·,}1ead , du~ to the, eilntrifugr),1 •:fo:ree,s.• Su~h __::a :··.:, . .• , . .... ·~··· ···•·· ·· ·••··· .,... '··· ··· -'·' ' •• ·--~-·- ,... , , .. .. , _.,., _~ . ...·..._.::···- ···"•":•····., .... _.· -. •· . ·-:_-~ :-:.-- · .~ :.'. :·, :.· :__., . ; i. eonditiori,..-,0:f course, , ts:_impossible for :purely arltt-1 ~:€'1ow._,.,.:~.: : ,_ ···._. -• •• ··:· ... . . .. . .. ·· ··•;4 ': :_..:·· ·:·: :-~,-.. • • ••• ·.· -~·-·. ·· · ~· · •' . --· : _,. __. • •• , ~ - - _ • .,_ , ••• ••_ ••• •• • • -: ·· / - -•..- .• · •. ·- · ·-·:·:.: . ·._:--• ._ -_\·. ~-:-::· ! - ·-· ··· . -·_· ··. ' . ·. . runn~ra,.·: For -.the eom_puta.tion -0:f the runner thiZJ -means-.: tlw.t . ' .. ·: \'~·:,: ~~~:1~:·:;: ~~;~~ ~o~·~~-:~ -t~; •. :_ - · -· ·: • • • . , ~ - ,. • • · -· ··: · . •••• • , o\_ • •• . ; ----~ • • •• • • > .- - •••• • • : - : •-" • -· - • •• ,. • •• • • - ;e·i :1:::~J~ii~:~- • --~~ - ·,,.--u~~;; • ' I:"·· ·.- . : •• • • :· •• • •.•-· ••' \. • • ·,:·.~- • . ~---=~- ; ,:~ .,.:· - -_:' ':; :~_~---~- . ·. ·:·•.;- . •· :<'.--. ·- . ·, \t _iri1- _-~ e~.n. P~::Jc?~~f~:: ~s: ,~q1:1a;t , :<>1~ -~Y~1;\ ;t;~~ge!,i __~h~_ n ,_ :es~,. ~~ie!1_ ••· l,)r.>~sibl:e· tor :,Purelv axizl flow pu..11!,P. runners, be~a.use- here,,. the• •, •• , • - -- ~ . . • . _ ., ... •· . . ·-'"''·· ·• · ..--:. .. ·- · - ··- -· · : · · ·· ·····-- •••• . -..... . .-.- ; ... ·- · . -. : •-.· · :•• t ·J·>··.-: ·. :····· . - ~· -:·; .- ;. __- decre.~e-~< in- :i-~ la.:t~~e. yel:oe~_t.Y:;.;~-- ~ ~ a r : l .-:f'o:l:' _ ; lm;(JlllCre~se . .' . . . ". _. . - .. _. : . - - '.'" .. . ... . . . . . . ·) . . . ~low ·pump: ru.nni:?l'1J &lwa~ work. under. :&"ela:tively m.~re . untavoc» : . • - · - • • ,• • • • . .•. ·· : : ·, -,,. · ·· •> •• - . .. ,. • ••.,.. • •• • • . .• . • . •• - · • · · · · · · · · • -- •':• , · ·."" , • . . .•. . --·:. ·· •• . :·: . -:: ~ ~.:.: . ·- . _ • 7."a.b le , p r e sst·c~•e ._~HH'J ~l:i tJ.:011~ ~an ~~- ~ingli~_:_ :ae J:"oi"o~I>- in a 8trai.gb.t ·· -. . . ' • . • . .~ and para llel iL0'11~ v1hinh is :not 1'1.ee,es::sm.ri.ly . the cam~ : for::_}. • • • • • I • ' ·._, : ' ' : • ,: ••·· • • •• • • •· ' • •• •, ·, • • •· • •·• • ' ' ' •" • • , . · ., . . • ... · •./'"'O,',, •. . .."'-. ., ,...... --Thie di:ff'~r.en~o betwe~n:-radbd emd. axi~'.l· .f.lov, pum.p . n.:mners · is one the--most raasons hmv impel ... -. - of .. · . .. .. . . . . .. fund3.ro.ental . . . . ··- . .. . .. ..why . . .. . radial:. .. .... , • .· _ .': •·· .. . . -. •·. le:rs are- tr>_ 'be preferred __for _;pum_p s of low speeifi~ speeds~ 50 . - . -: e;,_ .n . b _· e aesu."!le.d to,· 1:-~,. _' •.very high~ .: a ::>;:--· flha.'rlg~ : in:. th~ ; s-eal,f 'of". • ·,.. - ·- -- • , . ,,... ___ . -..... . , --. ~-••• · : •• - ·· · .- ... - •- •. ··.- : •• ·-': : ·"·~-~.:- •:.- ,. . -.- . . ...... . -· ·-- -- : ·- . _, _, ;~ ;. . ', - ,~~~ -~OW':, or,- more. _e xaatly.,.. e. ~ha.nge _ :ill Reynoldsr ' muube:r;,ti\>• \h:Tb~h;;~:~i~):- .. . ea:nnot bP- (eX!)e-; t~~ _to· ha,ye_' a __:ire~t -•e·r ;~,~t- o~ . of' .:the l~l? : o;r .turb_i,n~ &''. This _Jao t •h~~:;: ~o f o¾.r:: be_~ n, f~:\rf;~·_e.a; , 0 in most , cas~~ where the. ---so ale: e:ff'e.c:t eoulcl. be' obse,rvea:':·c:z:.,. : ··--··· -·· .. .. . ..... . ~: •· . •. - ·• . -. . --· . .• ... .; . ' . ~ -· • - .• .gener~.lly . based on.. _tl1:bi .· / •• • . . . . . . -· .a,$stim1-,.tlon·~ ···-· . .. . - ----. . .. . ····• - ,· ··: •. ,• -· - ' . .• •·.•·· · ._ ._!-{e0.en~ :~~;P_~r1::m~_nt ~:,: ho~~:e y~r~ _e~:ried outc_.eit •the CiTi:for ... .. . . , nit?,:. _,InatJ1-:u~et- _9:r_·t~:Ch~lll~g;r (s~!'l_re:f~r~!l_e~~:· 9 :, ~ - .__ )._)a~ve'· . . . '. ,. . . . ' ' . ~ . . ~- . - shown'- that. the liaits :fo_r se1)_3'1,r~t.ion· . <?~;-~~1:ofoil~ :;r~~;· a.ppre~ c-i:a.bl_g . infiu~neeq ?Y:_th~_ 'lfv11t<~ !):f .Reynole.5.' 1IttmJ?_~r {~tl b;/ •• t1le __{legre_a ·_ Q_f': ~ur1Jul..en-,e) .. Tl'.ti~ :me;,,~a ., thfi.t Ol~~:. '!:fl(~,t e:q;>ef}~G- ·e. • - 51 CHAJ?TER V. $UMM.A,.-qy AND RESULTS. ·~:::- .. ;.. ,-~ • : ._.. . _._ or aerof'oil. ,?,3 that pressure., whiei"l w:ould exiat ,,1:icthou~ _th~. . •,",';> ~~.J{~~ii/J;;i~?/1~ ;'ei'feet of . the particular vane . or: aer.ofoU w~,i,,;,., 2)_MO$~ ·_information about aerof'oila pert;;,ins to conditions ..· . • with eo~atant b~sie pressure •. The_main objee~ , ~i'-Jhis ,_paper -i the modif'ication o:t such result3 regarding cavi,tation and · : _; _: -~e:pare.tir:m,·~ ;~ / aerc,foila,, ·._. so .that •~.pplfed.: , hy: . . they may. be :-·.· -. .. to . ·:- . .. ,. • ·. -. •• • • .. . . .. , - .. . . '. . • . ·~ . . ·' ~ ~ ;. d.rau1ie •:runttiu~'s~ wher<:, the. basi.~ pressure , is. not·:.~o~s_t_an_t; , : -~ 3) In chapter II _we derive simple-: :formulae ::for the basie. . pres$1.tre :Jn -a runner, considering the basic. ;pressure iil,S equal . to the s~~ti11t pressure in ·t he .ru:ru1er as eomputed actilording t9' .' . .. • ' the Eulerian _theory :for hyd.raulie. :runners (infinite number of ._ :vanes). p:ressure ~-s found on a single. aerofoil regarding its v~,lue · as -well a,s - it.s loeationalong the ~vane •. J3oth eha.ng.e s are- deter-. , .mined iit . ~:bapter -III with reapeet . to . rod al. n O'W! runners, •an~ ths results are used for computi~g ca:rlta.tion charts (:fig'~~ 7 and 8 ) :for propeller pumpsGI Sinee in pu:rop runners the point oi" lowest :pressure lies in a regien o:f' l.ow basie pressure, . a..nd. in turbine runners in a region o:f' high basic pressure, there exists an import9,nt difference between the cavitation limit• of these two types o:f runners. 52 ! 5} Sinee -aeparat.ion an aerof'oiJ..s . begins i:f . th~/ pre•s sure· t - l ,· ••.- : . - - ~- :: --· •-:.·:- rise along ~he suction side O·f . the: '.;?,ero:foil exeeeds.· :certain •. . if :· ths l)ressure xise# ·.--whleh - a.. ·vane .with. .• ..the 69.Dle•lift.::eoeff'.i-: •.•• . . . . . ····. .. : ... ,._._-..::·.. ~_ :_.. . • ·· -- :\ ::' "' . - ~;;· ;. · .. . . .. • · , . ·.;. -~'.~~ient · aS;·:th~~: ::~tmn~r :-!ane .would _~~~;:: i!l:::a :·5.tr~fk11_~~~~~f1~:- ~~f ~~~1<:_:: . nows .PLUS th~: 'r ise · :in :l>asie p;e·:; ·s~r& -exce~d-'. t~;fl.;J;:eJ_. itr~lt_~:{;.:.,: : --· . . . . ( The •1i:ti;,:c,oe:ffioierrt ,:for Tun:!'let.': y~nes;p. ~here:fot'ef:; li.ad Jo .. be, . .- . . de:i·ived :~ci£ :rtmners.: of · 9,rbi trnry sh,ape, : a1s~~l: I!t:.~lJ-_a:Pt_e ~_-: ·;y--: . . ·we :present: :£o;mulae< :for:: the separation limits ~:for°_:· pum_p,: ruiu1era ... -.~hape.·::·~~~,,;~;_;p~~,t:~o~'.: ,..'. .··.· · oi>:: th~-.:. p;;~;1~r:-_ ty-pe _:.and _- o;::: ~en~r~•. . ,.:·: 0 limits praet. ica.11y .:leierm.ine lower limits for · the ;unit · sp eed;. -,; = ' .. • . • •. .. •• ··-·. ~--~. . . • . . . .• • . • • • •• • • ~ ::.,OQ • of.·pump ·runners.-, turbin~ runne:rs_ sueh .....se:paration.-li.:mi te:L .i ... . . . .. . . . . .. For . . . . .. . . ,. , ,• . · .- ,. . . ··. mostly . ..dq· .not exist. ·0 .. . . ·.• " · 5} , Sin~·e _the . beginning o:f' sepnl.rn;tion . is,. co:n_s ;d.~rably: -·~ . _ influenced by Reynolds' . munl,e.r ,- cur formulae allow . to esti:n.,ate., . - . i .a whieh cases- such an . i.nfluenee; must be: · expected for' hydrau• • •• • • - -· • • • • • • • : lie runners {~eal~ _ef':fect)c ---··-~~---. . . . • •• • •• • - -. • •• ~ " . # • • • • • • • • • • ; 53 List of' _1fotations. W'.aile '. the 8 Amerioan Stand~rd . Symbo1-s !'or ,Hyd;~~licSJ'~ - were llSeHi _wheree:ver possible., . the subjee.t: requil:'ed the . .-lntroduetion · of . a .•number of_ ~pecial. . notations·;• D!~ensions ~ ... .. . • - . .. ; ~ • • . .... . · , . ' . -. - - ••• • • •• • : • • , .-- --,~ _-. ,• :~ .- •• ·--:"-. .. ::.~-~_:-;: _ · ~:.: ,; · ~· -=-.- ... - -- !·< ·· . ; . •. a:re gi,,r.ea · oru:y .if' tH35ential , in _the _place where the p~~ti~u... .. ·- ·_ ~ • I" • • • • • ' • • .... ·- • ••l.a.r .nota.tiona are u s ed,. ...-~L .:.'D -l.i:t'~ eoef':fiefent -t il-l)penai.x~ i p~ge tiamater . •. • .- _D._ outer diameter of an- axial. now :ninner •D inner diameter or· ~ - axial flow :runne·r {di:un. c:f hub) _ F f'orf?~ _ftot:al pressnra): .: . g gra.viatatio·1uil. a,{jeeleratioJt lI total, _head of a n..in:ru,l." : ( !'~et) h.3 _ "auc_tion 1 he~,q•_• as _d,,fined: on pg,ge 2:.5 1ke;a·d ·- ctr·ol) l; o),, cri1r· ;74~trtr(JS,p } :u~ r f .~ 11':r e s tJ1J.. :C -!? a. (:feet) g.,,t vrl1l~J1 . ea~_i t ~tio~ wJ.11 jiist 'beg~n _(~~et} ·-· j -· . -. ;eres~~re drop on 1:1..n aerof'oil below,_Pr~ssure ~.t infin_!.~- averagepressure _t.iiiierenee _hetweenthe two aides of the vane -- N n number of R -.P M 13peeifie speed unit speed .e!\p n = pressure !lif:t"eren.tH: on the vanes , , •. ·- - ., 54 teet3/seo- ) 9. rate e:f now r radius t ehorfl or length e:f an• aerofoil. or vane • u peripheral. velocity of the r,unnar {:feet/sec}, V -a.bsolu.t-e :fluid: velocit3" . (:feet/see) ( :- .-· T relg,ti ve fluid velotti ty (feet/ae~} (without suba~ript: «mean e:ffet:.tti ve relQti·ve veloci ty st - -~Q$ . da:t:~ned _J:a . th0.. ap;pend_ix page. II . :fig •.~.) w density (w1aigl1t pi,r_ unit of' volume) •:x •oordimatc _in the axial direction o'I .a,n axial .floi.v 1mm1eJf. z elevation {:feet) angl~ between: the l'te:l5l.tl'Ytl ·velaei.ty and the :peripher3.J. direction {fig~ A l o:f.' the ,~:l)pe:mUx) . , ,. (without subscript it refeJr$ to the i'mean effective -. ,_ relative veloc:}.ty 0 as d~fint:d i:n the appendix, p.II) (3" angle between th~ var11:, a110. the pei--:f.pheral •direction ..... • r . , ..-.. ..... • • .. . ·-•· .. .. ,( !~ge, 1 1: ) eir cu l a t i. 0)1,p iif1Jfi xH%l t.i,s t h -t l brn :t1nt.~gr.;;i. l .of the ~vt)1Q=, eity eompone:nt p~lnt.ing in the diri:H:-tion ef the close cl path of' irstte gration (l)ia.ge I of' the a,ppe:zH.U:it) • _. Ta.lie pressure divide by the stagl-!l:ation J;)resstire of' tha ":maa.11. effective rel.;,tive veloeity 0 • • dif:f'arenee between the pressures on the two sides of· the Ta.ne divided by the same st~,g:nation :pressure as bef'o:re~ angle ©f' attaek oeordina.te al.ong the vane or a.ero:foi1 divided. 'by VI/ 'g W ( speeifie mass ) a.ngul~.r veloei ty t 55 . . . Subscripts:- ., _: a refers to the axial di:recti~n of' a» a..""titl flow runner { except suction \TJ: of' thit:' t1.ppen~:\.x) ·-. • :, _•'> ..>.> re:re:rs to the high. pressure : side, . o:t. .. the/ ~ ~ r: .. . . .-_.. - . • . ~-. . . . "' ~--;-.. :-... • • . :· ·, . -,"· . ;" • . ;_ _ :refers .te the low. pressu:r:e ,-side of the rti.nner • _(except _in the ·e onneetien· ·es y;hich: denotes the· aoef'fi.elent o:f e... ~_p arati.on) •· ...-... •-.· ' .'.••:-... • .. . - ... ' . :• • • . . ~ ,. : :•- . : - - • --~ ; - / A :few · other notations a,:re explained in the text~ ' .... ~: •• _, -·-···-- :. ~- - ~· -· ·,. ·. - - ~----- .• ,, • I • •' ,.- ~·- . ...• - .. ~ ."., .. ·-- ,, ;• •:-- __ .,,,,.-- 56 ·. .1) :Beta ., Di.1.g::re.mme zur Eereohnul!:g von Fluegelreihen ., "° - Inge:aieurarehiT, , vo1 9 II; 1931,. pa.ge . 3_5 9_: - • .. . - ... . _,:; . . __ _. 2) Prm.ndtl - .Betz .,. . E:rgebidsse der Ae:rodyn:;,,.;ruischen VersuohrJ~ -- <·· >, 9.J}stalt zu. Gcetti:ngen .... IIIQ Lie:ferung • .. 16 Untexsu~hu.ngeA'l· • • ueber . Dru@kve:rteilu~g Fluegelgi ttera. pag~ 132~ : , . .an g~st$1:f:fel te~_ . . . - ' • ·.-- --,•--· - ~ . . - ~ . .· . ' .. ·• - " 3) Fischer :;·::..·,:, ,Untersuehu:mg ·_ der St:t·@e:znl.Ulg in eine:r , Zantrii'ugal~:. •· pumpe. _~ _in .· De _Thoma .-;,-_ lritteilungen des hydr~,ulis~hen · · ,?·: ;, ·1nsti tuts der teeh:aisehen Hoohst,1,iu.le ~e~ehe1t, Heft 4., _page '1 ~< 4) Span.Make Problems . of' Tv!oder,a Pu.mp ·and Tu:rbi¥Ie Desig1t~· ~· • A.s.xia:.~,. A"9ril }9341 ve~ 9 56., NG.4_ - (HYD-56-1) Tr~.naaetio:m~__ oi the 5) J"aaobs - _Ward - l?iu.erton ... Report· 460 • of _the. National - • '.,' . Advisory Committee :for Aero}l~,u-· tias~(N,tAeC,,A.) . • •• • , ._ •• _ • .. • •• • . • -, •· • - -. . .. . . · · · · · ·· :' , •• - ' , _ ... . ·· · ·1 -·· :··· <·.--··· • .. 6) Betz Verallgen::u~ineru:\'¾g der J'81.tkowsky Prt,flle· •~ Zei tachrif'_t _fuer Motorluftsehi:ffahrt ( z.F.,M~) 1924 • . 7) Ton K)u-m'an - Turbulexu:e :a.nd SldM :l!'"'ri~tion ... Jou:t1ml o! Aeronautical Sc~_ence~ __ .,. v0l.l; N{).,1 1 J~..nuary 1934• . YOl\. Karui),~ .., . Th~ori~ . 6.0a Rei bu1:1gfft;i-ldt~XB'i:.~l.'.td.eB i:n . t Hy~r.-.,ulis~he l):1.~obleme d!'!s _S chiffsa:a:trie'bs - - H-,..mburg 1932 e ·. : ·-··,,_ . _. . c.B.,Millik.a,n and A11;L.Klei& "" - .· The ]!ffe@t of Turbulence ... Airaraf't Enginesri:ng . .. " _..,. August 1933.., . 8} . •• • •• '· , .. . ' 9) . - . . ·"' .. . ··:: •,,, • •. ·- , . . . .• . .. . '. . ·- Text Books a?.td Hand Book!_: 10) Daugherty ... Hydna.uli~ Turbines - 11) Ffl~iderer ~ Die Kr~iselpumpen - 2•d edition 3a edition The:r.a exists, of @ourse, za. large number of other valuro le referenee books; we have quoted.; howeTer.,· anly thosa whieh w~re US$d directly i:ra the present paper 0 - - - --- I APPENDIX Section 1. TO TP:ESIS Some theoreti~al facts about axial flow run~er~Q . The fl.ow i9 assumed to :proseed in the runner cylindric'3.l., coaxial str~am surfa~ef! 0 The ehaltge .in ~ngular : momentum, or the "vRne cireulation" la .:i.ssumed . •to ht,,ve the . ' . ' . • .• .. same value for every one ,of · theee atre.:1->n surfaces / The flo1r 1~ investigated in the develo_pmente of the latter, . fl,nd the ·· res::tl ting flow pictu.reB are tre;;tted Ra two-dimensional, so that the hydnwlic problem is reduced to that of 2.n~· i .n i ti~.lly p:irallel flow through a stra.ight, p"1,rc1.llel, and infini t~ly • long l .B -ttica. The rel~,tive motion is irrotational, i~e .. the 11 :Bernoulli constant" or total energy of th~ flow hRs the same v.,;lue for evr-ry · point in the field {irn:t merely :along · the same atream line). w~ are therefore considering primari~ ly . th ~ r~lative motjo ns ~f tha fluid. The· 1)12.des of the lg t tlce ;:,,re computed like aerofoil~ expos~d to the "mean effective velocity" T _., defined by the velocity diagrams on fig's. Al. and A2. These di~gr~.nrn ahow relative velocities only. If the fluid h,s no ;i.bsolute rotation on the auction aide of the r~Rner: Tu,I (for pumps)and turbinas) (Al) With r~ference tq the sontour ABC D • fig. Al on we find for the circulation 9rournl one vane th~ expreasio:n: r 1 - ds= y,..( v. ASC!) ::::z :vr r .,r 7T Vup (A2) II Fig. Al Yup flt,.s Fig. A 2. Fig's A 1 s~d A~ show conditions p ~rtPining to pump runn<'"rs. Th~ cor:rt°sponding pictures for turbine nrnn~rs c"J,re obtrd_ned by rev~rsing u ~,~ cUrections of all velocities, and by intcrch~nGi ng l e ading Rnd trailing on the v:;i,nes. III Th~ circulation about an aerofoil, in gener~l, has th~ following relation to th~ "lift coeffici~nt" cL ; • r . ~ . ,,,. ; ~ .. z r .,.L _ __. or t t,I' (tho lift coefficient is defined by the •lL rel~.tion : Lift f!. •• .z.. .:, Ci., o 2, t• wid,1/J the "lift" being the forc:e ~-ction between the vane perpen~icil1r to the (relatiTe) velocity From equati0ns i .) ( A2) ~nd (A3) we obtain: • Yvp = z ( ;;i,bout the definition of Ep V T V j e".s 13 1 (A4) ~ee chapter II.) The head converted in the runner (:neglecting losees ~~nd ~-sauming again thz-.t the fluid hSts no abaolut>~ rot;;.tion on the ; f.Picti on rddF) i s exp r~ s:aed by thl~ folJo wt:ng rnomt~:ntu:m equ.;;::tJon: u. 7/vp ~ .-.-- H:1 Substituting for Vu p ( A5) .5 the v~.lue expr~-ssed impl loi tly by equation (A4) l~ade to th~ following relRtion: -2!]V ti ( A6) In order to express th~ "me~n effective velocity" (T) and i ta direction /3 gi v~n by and v8 as. a function of th~ ini ti;:i,l fl oN con di tiona /3 9 we deri v~ from fig (Al) and (A4) the relation: = O A~ ,9nd equ~.ti o::is ., . ·, ..,. ., ·, _~: . ·--.. ' = cos ;S', we . obtr.i.in: U3ing the approxim~tion: V.s V • For turbines one may prefer to expreea T ' vp• By the same approximation as before we Vp - · For the . direction of the -'' mean effective in t h~ same ma nn~r: . /3s · (11- Ej,1-). .si11 /1_ . ~ . .sin : .sin/3,p • ( /- 5:j,1-) So far we h~ve considered the whol~ v~ne in the ayatem as being exposed to~ constant and straight flow with the velocity T e,nd the di reotion • ,/J Actually the r~lative 0 v el oci t y ch~ ng es steadi l y ~l ong the T Rn a. We ~hall Rssume th;a t for ::i,11 our compu t Rt :l. ons this ch;.,nge is taken into ao .... co:rnt, by the method de-.;elop~d by Betz ( reference / • ) , whioh ie baaed on the following idea: We start fr011 in aerofoil which in i:. . stra,ight ~,:nd :par~-llel l:flow would have the"" deaired lift C<?~fficient, pre3eure dietri b'..ltion, et 0 0 This ?erofoil ie then defor:ned in l.'5U~h :.. m~nner th~t under the changing rel~tive flow conjitions i2 the system it will have the s~e di3tribu.tion of the tot~l forcee (not of the specific pressures) aa the original aerofoil in a straight and parall~l flow 0 In our text we refer to the l~tter as the ttcorresponding ~~rofoil" & V All our data about the vanes of axial flow runne:ra, lik~ th~ chord (t), the direction of tho chord {t3'), the ratio of overli;r.J2Eing • ( j), The Valne • preseure ( AP, AA, A), and the • · lift coefficient (eL) pertai~ to the "oorr~~io~ding aerofoil" ·which would h;::ive the deaired hydraulic properties. if e:cpo~e~ to a flow with constant velocity T and conetant direotionftt - they do HOT pertain to the aotual vane ehs.~e ·in ·the2_tye~ ·-·· after the J3~t• deformatio~- ha;s be~2!_ applied. It -ia the . "eor1·e-s~ . ponding Rerofoil", not the flnal vane shape, for which we eelaeot standard profiles ~c~ording to their known properties 0 .Just this possibility, namely to o».rry out :;.11 oonei deratio:ns ae if the "mean effective relative Telooi ty 11 wer~ con5ta.n t over the whole length of the v~nea, 9,nd to .?.pply the neceei5.ary • • corrections afte,rwa1·de when dt11~ignlng the n.ctui:a,l T~iie, thie. appears to be the gr~ateat practi~al arid theoretical advant;a.ge of the Betz eorrection me·thod., . It m;1.y be mentio::ted thAt th~ Bet z c orrecti on e b e 0ome n~ gl i gibl e for am~ll valu a B of ( about ½ ) 1 while: they !lre quite im:portant for 1~.rgt! j ( 1 or more) eepeci:a.lly if f.,' j ,r~lU'l!l!-s is t:small,(> The contr~diction, that we con~ider the ehan~~ of the "b~.5ic pre~sure 11 a.long the vanee, wh:lle ~oneidering th~ relative velocity (basic v~locity) ~s eo~3tant 1n the a~~ r;tnge, val'li~hes, if the Betz · oorrection i2' applied afterwart~ 0 · Both, the velocity and the pree9ure eorrectione, :ipp~d\;l" t~ ha1'~ Zt,pproximately the aa..me order of importia.n~e .. VI ~~~erime~t~l r~8ult~. The result!I of equ1tio:=i (4) i11 clllll.pter II haYe beel! gom:par~d with the ex_perimemti:tl reaul ts gi Tel'! ill reference 2, pages 132 to 138. The inTestigated ayatem correaponde to~ turbine runner (aaoeler1ted ralia,tive motion). The given pressure diagrams are drawn with the 3ta,gnation preseure · of ·- · the initial flow (Tp) not- th~ 10 m:,an effeet1Te flow 11 (v) ·aa unit., whieh St,mpli ca t~e th~ @o.mp~ri:so~., ·Another di ffi@ul t:; liee in the fact that the e:xperime:at had, of aouree, oarried out with ~ eh:-.pe for the profile t,r,i be placed into dif- .• .. fere!\t positio~es 1 while our f'ormul~~ t15sume that th~ Tane.s , are aorrected by the B3t~ method ( seati Oli\ 1), whigh f~r a· fixed "corresponding aer&foil" Blmpe would ,::~11 for di ffere:rtt actual vane 5hapee in different poaition~ in the ey8tem 0 Imo der to minimize thi~ effect we hRTe conaidered only thos~ following result~, Teat aeries ehange in b~eic pres~ure ( "Ver~uoh.ar~ihe") theoretioal OL • j I II III IV V VI .24 .34 .44 056 071 • 76 from the e::;q;ierime .19 • ?.6 030 037 060 • 76 C ona idering the uncertaintj' of th a 0om:pr;1,ri :eon i:r. itself (the diagram~ are Bo ~mall th~t values taken from them ea~ ~ 5% to 10% accurate, · ne h~rdly be better than .. ding~ for the leading or trailing edgee being give~) -&h~ . , . reaul te are auffici e:titly conahstej,l'lt to juetify the ua e of the given formulae for practic11.l deeigning work _and zimii1.1.'r .· •· computati one. If the influence of the neighbouring V#.nee .. . . would be taken into account more accurately, the iesuite . j well be ex~eeted to approa~h, as incre~aee, racy in a more coneiBtent manner. Some tr.-..n~form;ll.tion:3 for the eavi t::t.tion formula Seeti on 3, From equation8 (11) and (13) in chapter III we obtain at fir:,t : h.s r b& ::: H V 71 = -1!u (Alo) Aecording to fig. Al in eEotion 1 we derive the relJt..tlo:a. i :: From the 3ame diagr~m we find that: C0.5 A F" u ::: u v+ ¼e 2. CASt1 if the fluid hr-rn no absolute roti,tion on the auution ~ide of the runner. Eence: C(lS /3 'DI - I + mI l,J V I/ _L 2 CU3J.3 sr co..s;3 mJ. .!/ue p - VJL VIII 2y equation (A4) in section 1: wh-.ence : or putting (Note that "L .,j alway2 has a low value at the outer p-~riJ)hery of the runner, to which the follow'ing equ~tion g~nerally applli!s.,.) . ' .·, ~ ' ... . Substituting the aboye values into equation {Al.0) ' w-e - obt~j-~·: Jt . ?f 4J - .1,.:/dj cm)" = (!- 4eu.S ..9:.:4/J,}( .fiC111• 4 Defin1t101t . f- • {Al4) i WPP[ f1m a Pl Yax 2.1.2 H;;;+ - where n (RP1!) .-: Sub~tituting theae expr~;:,s3iona aa well a0 ex_pr~s::sion (A6) fQr into the ape«::ific apeed formula, we ~btain (u11d r.g th~ fir~t R -l% ·L! (W•~)=U , we hia:Ye to {v; ui:,. y1,tg. IJ).1!- = YV. v~..,. From ~quation (All) BSection 3 J1 ¥ - :-; '. , we put .· i . cc~A' + ~:L r 4 Ct>S/J' obtain ~tt expression aontaining form oh~racteriatic5 Aft~r the::s -~ transform;;a.tions w~ fin~-lly obtaiJ.t : . . .3. ,/.ifiiii; • CO$ ~ , n.rp= C . ·c~1-·j)w;-. : . , I • •.•.: .· : ·:t/4 ' • vl-{f;) ,'tafJ+~d;/3· C, • -) ~ . . ., . . (Al:} , . 3 ~ c= H (29J ~ ~ 2.1.2. = s160 for Q, we find: _ n.sp - C ~ .• t:'!.~~, ,J;~,~T'(cosA'+ .foL, )~ tc ii¾ y1,.r;/ ~ r' 4@.~/J'/ <~w - - Equation (A15) ap:,plieB to the · outar, and equation (AlG) t ~ epirala._ w~ ithall m,t .:i.tte:m:pt to give a). rigorous deriv:-.tion. X 0f a formula, which J?rim~.rily iu.s derived for ~- stradght, . .,-·_:_ par~llel system ( equ~tion (A4},section l ),to a rRdi~l fio; s:,rstem by means of the following simpl~, phyaic~l conside- ro1.tions : If we eompare the radial flow syst~m, with tte '¥.-.nes v1ell as the rel-. ti ve flow lines consi d~red .as log~,ri thmi e : -• ._ I ' - spirals, with a eyatem of straight ~-nd _pnr.;;iJ.l~l van~s ( eorr-es-:--:;. · ponding to fieeA 1 ), we find: l) Th~t equQl di visions on the par;;,,11~1 V8.r.es corr~sp~rid. to divisions along the spirals which ar~ proportionAl (iri ~lengi to their distanc~ from the center of ~he syata~, providin~ radi~l li~es in the radial flow syatem sorreepond to parRllei· • lines normal to the general direction of the straight system. 2) _The relative (basi@) -.,-~loci ty., which is eonetRnt in the dintanc~ from the center of the radial flow system. ~~nee: The products of the lengths o~ the divisions, whi c h we camp~re, with their resp~ct1ve relative velocities are equal in th~ two syst~m~ 0 As the diTi3ions become sm~ller tothe e~nter of the re,di~l flow syet"'m the bale rel~tiv-e 'HD.rd:3 vel~city inorease3 @orroapondingly~ Aa~uming that the differe~ce in the actual relRtive velocity on the two sides of the v~nea { vo rti ei ty or. t;-;e v~.nes) is the B~Y'le at corr~r1ponding po:L!1ts of th~ two syt»:me, ..,~ find th ,.. t. n,e force distril>Lttion is f:llso the erune iN the two systems b e eause it can be proven thAt the en~rgy equ~tion: ( va. and T the vane::,., 0 b~ing the rel~tive v~locities ~-t th~ two .;i,r.d Ll t . one of the divisions of the v~n~) ' h~i~~- for th~ rel3.ti ve motion in the with ~ + J1b 2 - V = basic rel~tive points <tf the two systems., tr..e al,ove st,:1,tement f-.bout distrio,ution followa 0 With the force distribution {and the vorticity)2long the v:;ines being th:'! san1e in the two eyst~rna., -r:re ~::>.lao may g;ill the ''lift Co~:::~fieient 11 tl1e ::Hur.e for the two syet~l:!~ 9 whioh flow syste~s, ... q. e. d 0 The new definition for j follows from the above e3ndition r~g&rding the relatiofl betwea~ ~oarre•pending~ divi~ions ~long t h e vanes in the tw~ syRtems, if etsirting the oonsi der;,;i.tion from the out'!ide i?f th~ r9.di~.1 flow runner; this i!J ~.lso the rea.san why v in equ~tion (A4) h~s