Scaling Friction and Inertia Losses for the Performance Prediction of Turbomachines

Transcription

1 Scag Frco a Iera Losses or he Perormace Preco o Turbomaches Peer F Pez 1 a chae Heß 1 1 Char o Fu Sysems Techoogy Techsche Uersä Darmsa Darmsa, Germay Phoe: , FAX: , E-ma: peerpez@su-armsae NOENCLATURE a C a P p Re Y=gH spee o sou power coece ameer roughess hegh, reae roughess hegh scae acor ach umber roaoa sha spee power sac pressure Reyos umber p cearace hegh, reae p cearace hegh oume ow oss srbuo acor specc wor Gree symbos α expoe hcess η ececy emac scosy κ geomery scag acor λ rco acor ψ pressure coece ρ esy φ ow umber Subscrps u rco oss era oss ea oss m moe mache mech mechaca rs rea sze mache p scous w reae eocy INTRODUCTION I pracce measuremes urg he esg process or or a specos oe cao be carre ou o u scae maches osy s oo cosy o bu a ow es rg or each mache, so es rgs wh saarze ameers a measurg equpme are use Somemes he ameers o he rea scae maches are so arge ha a measureme s uery mpossbe Exampes are waer urbes or as use or coog power pas hese maches may hae ameers o seera meers a requre a brae respecey re power he W rage O he oppose maches wh ameers o oy a ew mmeers o o aow he operao o esabshe ase measurg echques such as Po-ubes, e hoe probes or ho wre aemomery because o he sgca uece o he probes o he ow e Hece measuremes o eerme he perormace are carre ou o moes whch are ow or up scae As was show by Spur (199) a u smary bewee moe a prooype mache s he coex o urbomaches possbe as og as he ach umber s sma, e he compressby ca be egece SIILARITY To proe raseraby o measuremes o scae moes o he rea sze maches he smary bewee moe a prooype has o be cosere Scag he geomerc meas scag o a mesos such as ameer, bae egh, chor egh o roor a gue aes, surace roughess hegh, p cearace hegh wh he same scae acor '/, where he moe aa here a he oowg are cae by a ash Ee so u geomerc smary s possbe, he smary reae roughess / a reae p cearace / are somemes sacrce Foowg he Brgma posuae, o he absoue sgcace o reae magues (190) u smary s reache whe he reae magues o a sysem, whch are cae mesoess proucs oay, are eca or he moe a prooype sysem Oe exampe or a geomerc reae mague s he reae p cearace / Oe exampe or a emac reae mague or mesoess prouc s he ow coece 4 / I he rs exampe he p cearace s o measure mmeer or ch, bu mupes o he mpeer ameer I he seco exampe he oume ow rae s measure mupes o he mache ypca ow rae ~ O he oher ha he roaoa spee s measure o Herz or reouos per mue bu mupes o he erse o a scose uso me /, whch eas o he Reyos umber Re / Foowg he Bucgham -heorem (1914) he oa specc wor Y gh, a,,,,, s equae o, a,, where here a he oowg he roughess a p cearace are reae quaes: /, / For he ow ach umber case, e eary compressbe ow, a / a pays o roe a he pressure coece -1-

2 : gh / s ge by,, (1) The sspae eergy per u me P s a uco o P P P,, a,,,,, whch s equae o P / P P, a,, / P For a pump, a or compressor he ececy s ee as 1 P / P a he power coece s C /, P s mechaca power raserre by he sha o 1 he mpeer For a urbe he ececy s (1 / P), a he power coece C P eoes he mechaca power raserre rom he mpeer o he sha Aga or sma ach umber he ececy boh cases ca be wre as,, () Equaos (1) a () uy escrbe he perormace o a urbomache Hece, as og as he mesoess argumes,, are he same, moe a prooype show he same pressure coece a ececy Aga he moe s eoe by a ash he raos, or scag acors '/ /( ), Re' / Re hae o be oe Ths eas o he sysem 4 o equaos P Hece, prcpe compee smary s accessbe I pracce seera probems arse Assumg a geomerc scae acor o κ = 1/10, he roaoa spee a he power ceupe, whereas he oume ow wou be a eh Esmag a usua urbomache wh a ouer ameer o 5 m 1 a a roaoa spee o 500 rpm (crcumerea spee a p 10 m/s), prog geomerc smary wou requre scag he reae roughess a reae gap wh o a eh o he rea sze mache I s possbe o achee such sma mesos, bu s a eas ery cosy Usg Eq (7) o (11) by assumg compee smary wou ea o a roaoa spee o rpm (p spee 110 m/s) respecey a oume specc power he area o seera GW/m or he moe mache whch s mpossbe o achee The meoe probem o ureachabe compee smary ca oy be aoe by gg up smary o a mesoess proucs I hs case aoher scae acor ca be chose ree Usuay he smary Reyos umber a reae roughess s sacrce a he pressure rse s ep cosa As a resu he pressure coece a he ececy s ere moe a prooype mache For exampe ', Re', Re ', Re', Re or Re' Re a LOSSES 1, 1, () (4) 1, or see scae acors, where hree ca be chose arbrary For he same u moe a prooype = 1 hos Wh = / = κ as he geomerca scae acor hs resus : 1, (5) (6), (7), (8), (9) s (10) Wh hose scae acors, he scae acor or he hea chage becomes (11) gh The scae acor or he mechaca power resus 1 Hece he oume specc power o he P gh mache wou scaes as 4 (1) P/ Fg 1: Loss coece or ppes a ere Reyos umbers a reae wa roughess Sacrcg he Reyos umber smary meas, ha he smary he reae bouary ayer hcesses s sacrce Hece or Re' Re he reae bouary ayer o he prooype s her comparso o he moe Hece scag he moe o he prooype he reae rco osses are ecrease whch resus a hgher ececy Ths s a as og as he scous sub ayer / w / (Spur, 008) s hcer ha he surace roughess The wa shear sress scaes as w ~ (1 ) wh he we ow rco acor ( Re w, / ) show Fg 1 / w ~ (1 ) Re 5 (1) he rco osses w be epee o he Reyos umber For he hyrauc smooh case he emprca Basus aw (Spur, 008) 1/ 4 ~ Re or / 5 (14) s a approprae approach For hgher Reyos umber ow, whe he scous sub ayer becomes her ha he surace roughess, he rco osses w become epee o he 1 whereas ormer meoe waer urbes or power pa as cou reach ameers o 10 m a more --

3 Reyos umber Hece or he hyrauc rough surace he rco aw (Spur, 008) h ˆ c, (0) g 174 or / 70 (1) s approprae Sce or he purpose o up scag oy he reae chage rco has o be cosere, s easbe o wor wh he aboe meoe rco aws Aer hs reca o he gs o Pra, o Kármá a Basus oe has o rase he queso wha o o wh hose cassca resus he coex o scag Sce pressure coece a ececy boh epe o he Reyos umber, reae roughess a reae p cearace so has he power coece C (, ) / (here or he exampe o a compressor, a or pump) Neerheess or axa maches he power coece s epee o Reyos umber, reae roughess a p cearace whch was corme by expermes oer a we parameer rage (Hess, Pez, 009) Hece '/ '/ hos or he same ow umber or a a, compressor or pump a ' ' hos or a urbe Thus here s oy oe scag meho or pressure coece a ececy Oe approach o ga a scag meho s a physca aayss o he possbe osses They are reae as ae o Reyos umber or scosy epee, e rco osses a Reyos umber epee or scosy epee osses whch are cae era osses:, ψ, ψ ψ, (16) Acere (ühema, 1948) cae he rco osses scaabe osses a rouce he oss srbuo acor (,, ) : 1 (17) whch was se by Acere o he xe aue o 05 Bu rom (16) becomes cear, ha s a raher crue assumpo o x he aue o 05 I s he ma as o hs corbuo, o ga a physca moae oss srbuo uco, whch w epe o he Reyos a ow umber Beore og so s worhwhe o scuss he era osses ee urher They ca be sp up a oss par ue o he ow bewee casg a p a a par ue o cece osses: ψ ψ, 0 (18) The rs par was rs scusse by Aber Bez (196) or axa maches Foowg hs houghs he ressace s ue o mxg a a uce ressace cause by he p orex Ths uce ressace has o be a ee uco o he ow umber or gog urher has o be a ee uco o he ea mache perormace a has o ash or zero p cearace Hece he asaz s, b ψ (19) s moae by hs assumpo or sma reae p cearace Ee hough we o o use (19) he oowg mgh be a eresg hough o scae he reae p cearace The heoreca mache characerscs s obae by he measure pressure coece a ececy, h where b,ˆ, c are mesoess mache cosas Fgure shows he ay o equao (0) or a axa a From he measureme aa he cosas ˆ, c are easy eerme Foowg hs scusso he p osses are maxmum or sma ow umber a he sope o he osses approach zero ψ / 0 or 0 The seco par o he era oss 0 shows a mmum a he esg po 0 / 0 a 0 o he mache For par a oeroa he cece osses crease a rs assumpo proporoa o ( ) 0 FIRST ODEL FOR THE LOSS DISTRIBUTION FACTOR To ga a oss srbuo acor, whch aes a eas he epeece o he oss srbuo wh ow umber o accou, s, Re promsg o suy he shape o he ececy cure whch ca be wre as: 1 1 ˆ c For sma ow umber 1 he rco moe h (1) (, ) ( ) ~ ( ) or 1 () s a reasoabe approxmao Hece (1) s or hs approxmao equae o ( Re) ( ) 1 ˆ c Para ereag wh respec o φ eas o Re h / c 1 h h c 1 () (4) Rearragg a egrag rom φ op where he ececy s maxma o φ yes o ( ) ( op) h c1 (5) op Re Iegrag by pars usg he reao ψ = ψ h η resus ( ) ( ) ( ) ( op) op c op (6) Hece by oog oy a oe characersc cure (ececy a pressure rse) a oe roaoa spee, e Reyos umber s possbe o eerme he oss acor he orm : ( ) ( ) ( ) c op op op (7) Fgure show a sgca mproeme he scag o he ececy by usg he oss srbuo acor (7) comparso o a xe aue o 05 as Acere Oe major aaage o he resu (7) s ha here s o g parameer use The oe rawbac shou be meoe cocerg he aboe erao --

4 easuremes show, ha he bes operao po o a urbomache shows a sh o hgher ow rae coeces wh creasg roaoa spee, e Reyos umber (see Fg ) Ths scag eec wou o be coere usg Eq (7) Hece s worhwhe o scuss a aerae way o ga he rao := ψ / ψ reae p cearace / o 01% a a reae roughess / o 6E-06, Re = 06E06 respecey 1E-06, Re = 65E06 The expoe α s se o 0 SECOND ODEL FOR THE LOSS DISTRIBUTION FAC- TOR The crca assumpo oe so ar was ha he rco osses are epee o he ow rae coece Essea or he osses he bae passage s he reae eocy, e he erece bewee absoue a crcumerea eocy: w c u Hece he square o he eocy mague s ge by w² = u² (φ² + 1) Sce he rco osses are ue o bouary ayer rco, hey are o he orm 1 ( w /, / ) b 1 ( / ), b (8) wh he ow mg behaour ψ ~ (φ² + 1) Re -α or 5δ ν a ψ ~ (φ² + 1) or >> 5δ ν The mesoess cosa b s aga a mesoess geomery quay escrbg he mache Hece he rao := ψ / ψ s ow ge by 1 b ( / ) (9) 1 Usg he meoe measuremes by Hess a Pez o eerme he cosa b eas o a aue o approxmaey 11 Fg 4 o Fg 6 show scae-up or ececy respecey pressure coece usg Eq (9) Fg Comparso o ececy scae-up usg Eq () a (7) wh measuremes a he scae-up meho by Acere Ahough he cacuae aues are s smaer ha he measure oes a crease o accuracy compare o Acere s obae Fg shows scae-up o he pressure coece where he same eecy s sbe I s assume, ha he power coece eeps cosa wh creasg Reyos umber (Hess a Pez, 009) a he pressure coece behaes he orm / ' / ' GENERAL SCALE UP ETHOD We ow compare moe, eoe by a prme ( ) a prooype mache From (16) a (1) oows 1 1' ' ' ' ' ' (0) Wh he acor = ψ / ψ we e up wh he scag ormua 1 ( / ) ' 1 ' ' ( ',( / )') ' Re (1) For he speca case where he reae p cearace or boh maches are he same ψ / ψ 1 he resu Fg Comparso o pressure coece scae-up wh Eq () a (7), measuremes a he scae-up meho by Acere 1 ( / ) 1 ' 1 1' ( Re',( / )') () s obae where s ge by he xe aue o 05 (Acere) or ow by eher Eq (7) or Eq (9) The absoue aue o he rco acor λ Eq () s o requre, sce oy he quoe o he rco acors s eee Assumg smary he reae roughess Eq () smpes or he hyrauc smooh case o: 1 Re 1 ' 1 () 1' Re' Fg compares scae-up wh Eq () usg Eq (7) o eerme he acor wh measuremes o a axa urbomache by Hess a Pez (009) a a commo scae-up meho by Acere (ühema, 1948) The measuremes were carre ou wh a Fg 4 Comparso o ececy scae-up usg Eq () a (9) -4-

5 wh measuremes a he scae-up meho by Acere Sprger, Ber Spur, J H, Ase, N, 008, Fu echacs, Sprger, Ber Fg 5 Comparso o pressure coece scae-up wh Eq () a (9), measuremes a he scae-up meho by Acere Ee goo resus are reache case o ower ereces Reyos umber show Fg 6 Fg 6 Comparso o ececy scae-up usg Eq () a (9) wh measuremes a he scae-up meho by Acere O he oe ha a urher crease o accuracy compare o meho oe or eermg he acor s sbe O he oher ha measuremes a ere Reyos umbers are eee or eermg he mesoess cosa b Reereces Bez, A, 196, Über e orgäge a e Schaueee o Kapa-Turbe, Hyrausche Probeme, S 161, DI-erag, Ber Brgma, P W, 196, Dmesoa Aayss New Hae: Yae Uersy Press Bucgham, E: O physcay smar sysems; Iusrao o he use o mesoa equaos Phys Re, 4 (1914) JD Deo, 199, Loss mechasms Turbomaches, ASE Joura o Turbomachery, o 115, pp Hess,, Pez, P F, 009, Comparso o easuremes o Axa Fas a Perormace Preco usg Commo Scae-Up ehos a Par- a Oeroa, ASE Fus Egeerg Dso Summer eeg, Paper FEDS ühema, E, 1948 Zur Auwerug es Wrugsgraes o Überruc-Wasserurbe Schwezersche Bauzeug 66 Jahrg, 4 Sorer, JA, Cumpsy NA, 1994, A Approxmae Aayss a Preco eho or Tp Cearace Loss Axa Compressors, ASE Joura o Turbomachery, o 116, pp Spur, J H, 199, Dmesosaayse er Srömugsehre, -5-