A COMPREHENSIVE ONE-DIMENSIONAL STREAM LINE THEORY BASED ON EULER, WITH PFLEIDERER'S MOOIFICAT10N FOR SLIP AND HYDRAULIC EFFICIENCY,

SUITABLE FOR CENTRIFUGAL PUMPS WiTH PREROfATlON

H. Sprlng, Suparviror of Hydraulic Derlgn fran%america Oelavaf Ino.

Trenton, New Jersey

ABSTRACT

This paper is w r i t t e n t o f a m i l i a r i z e the American pump designer wt th what was probably Pf le idere ra a g r e a t e s t con t r ibu t ion t o the a r t and f ledging sc ience of c e n t r i f u g a l pump design, namely h i s modification of t h e Euler equation for s l i p and hydraulic e f f i c i e n c y and the P f l e i d e r e r s l i p equation.

P f l e i d e r e r s work and t h e work of Stepanoff f2) a r e brought on a common denominator s o t h a t they com- plement each o ther r a t h e r than t o appear a s d i f f e r e n f methods. It is hoped t h a t t h i s paper w i l l explain t h e Pf le idere r stream l i n e theory s u f f i c i e n t l y t o make i t a t t r a c t i v e co t h e American user , and t h a t its use w i l l become wide spread a s i t is i n Europe. His stream l ine theory w i l l remain t h e corner s tone of p w p design f o r many years before science w i l l surpass i t.

Also discussed is Wiesnerts s l i p equation and i t s appl ica t ion i n p u m p s . ~ s e E u l impeller f l u i d v e l o c i t i e a a t e defined i n the frsme work of t h e stream l i n e theory.

A - S l i p coeff i c i e n t (Pf le idere r ) B 2 - Impeller discharge width BWP - Break horsepower BSHP - Back s e a l l o s s e s on mul t i s tage impellers C - Absolute v e l o c i t y CU - Tangential component of absolute v e l o c i t y OI - Meridional component of absolute v e l o c i t y - Impeller diameter DFI( - Disc f r i c t i o n , HP F - ( l+P)/?h 8 - Acceieration of grav i ty CPM - Pump flow, gallons/min H - Head, FT N 1 - Prerocat ion head Hcorr - Corrected he& Hth corr - Theare t ica l cor r head - Theore t ica l inf i n i f e head (Euler) Hthw H i - Inpuf head '

MPL - Impeller hydraulic l a s s e e L - Length LK - Pump leakage f r o n t wear r i n g and balance f l o w ~ H P - Pump mechanical l o s s e s , HP

MST - S t a t i c momenl: of center atream l i n e NS - Specif ic speed NSC - Corrected s p e c i f i c speed P - Pf le idere r s l i p f a c t o r R - Radius SPE - Specif ic g r a v i t y T2 - Vane thickness U1,UZ - Par iphera l v s l o c i t y W - Relat ive v e l o c i t y W H P - Water HP, o u t of pump discharge Z - Number of impeller vanes - Absolute flow angle - Relat ive Flow angla, o r vane angle - Head soef f i c i e n t ,

t'h - Hydraulic e f f ic iency & - Wiesner a l i p f a c t o r .b, - Capacity c o e f f i c i e n t - Vane cont rac t ion - Condition on impeller i n l e t 2 - Condition on impeller o u t l e t I t - A PPleFderer quant i ty

INTRODUCT EON

The following descr ip t ion shows the c l a s s i c a l developsent of t h e Euler equation, named a f t e r the Lath century mathematician from Basle, The Euler equation zepresen ts a foundation f o r a l l turbomacfiinerp However the equation, as important aa St is, i s only a mathematical. concept, cornparcible t o a machine with no losses and no s l i p . ' The l a t e Prafeasor Pfeiderer & of Braunschweig Universi ty , Gemany, modified the Euler equation t o make i t a u i t a b l e for a c t u a l machinery, i , e . far the inclusion of s l i p and hydraul ic e f f ic iency , Ha did t h i s i n a p r a c t i c a l way t h a t is usefu l fo r the pump designer. This add i t ion i s universal ly used i n Europe and is, unfortunately, very l i t t l e known i n the U.S.A. Pf le idere r ' 8 modification i a discuased i n d e t a i l . The development of the maln dimensioning equation i a a l s o shwn,

The w r i t e r introduces t h e concept of "Head car- rect ion" for plmrps where pre rc ta t ion is allowed a r the BEP point. Also shown i s the corrected head coeffident,

This add i t ion is necessary i n order to dea l with BEP prerotat ion and make the d e f i n i t i o n f o r ca lcu la t ions uniform.

The nondimensional discharge t r i a n g l e shows an organized way t o c a l c u l a t e many q u a n t i t i e s f o r vo lu te and d i f f u s e r design. The nondimensional t r i a n g l e was f i r s t : introduced by Stepanoff, f o r s p e c i f i c condit ions 3 n l y m . Our text w i l l show how t h e generalized aon- dimensional. discharge t r i a n g l e can b e drawn or concep- t iona l ized f o r c e n t r i f u g a l pumps with d i f f e r e n t s l i p and hydraulic e f f i c i e n c i e s . We w i l l a l s o show how t h i s t r i a n g l e and t h e main dimensioning equation a r e re la ted mathematically.

The Pf le idere r s l i p equation is discussed and c w - pared t o Wiesner 's s l i p fac tor u). The w r i t e r exphins why he thinks t h a t ehe Pfleiderer s l i p equation is t h e bes t too l f o r pump designers . Also shown is how the a c t u a l s t a t i c head f o r impellers is calculated, a s wel l a s the der iva t ion of the pump input head. It w i l l be shown how the input head is r e l a t e d t o the t h e o r e t i c a l head, but the two heads a r e not e n t i r e l y equal. The paper ends with an abbreviated procedure f o r pump analysis .

Equation 3 +s Euler ' s equation.

DEFINITION OF VARIOUS HEAD TERMS - MODIFICATIONS BY PFLE IDERER

Euler Head This is the t h e o r e t i c a l i n f i n i t e head produced

along t h e center stream l i n e

Theoret ical Head The theore t ica l head is produced by a pump t h a t i s

subjected t o normal s1Lp, but has no hydraulLc losses (100% hydraulic e f f ic iency) . P i s the Pf le idere r s l i p f a c t o r .

DEVELOPMENT OF THE EULER EQUATION Head - The a c t u a l pump head is produced by a pump that

has normal s l i p and hydraulic losses . @is the hydraul ic ef ff ciency

Hthoclis the Theoret ical I n f i n i t e Head produced by a pump with 100% hydraulic e f f ic iency and i n f i n i t e number of vanes (no s l i p ) .

Hstar- is the S t a t i c Head produced ( i n f i n i t e ) by same pump a s above.

The ~ a i g Dimensioninv ~ q u a < i & n From equation 6 , the h p e l l e r diameter can be

calculated:

N s t a t w c o n s i s t s of a pressure r i s e caused by c e n t r i f u g a l forces: ~ 2 2 - ~ 1 2 and the pressure change

2g caused by t h e change of 31 to W2: w12-6122

2g .

Rearranged according t o the following geometry:

' F I G . # I

Solving f o r U2 y ie lds t h e impeller diameter 1

02 = U2.229.18 RPM

The Concept of Head Correction f o r Prerotat ion

Head H can be corrected f o r prerotat ion (~ ,or r ) i f prerotat ion is allowed a t BEP. From the square root of equation 7:

The corrected Head Coeff icient The corrected head coef f ic ien t is c

use when working wi th t h e nondimensional

V c or r % ~ y J g uz

p l o t it on t h i s v e r t i c a l l i n e . Now connect a l i n e from t g p 2 on the abszissa t o '+torr on l i n e and extend l i n e to i n t e r s e c t with o rd ina te . This y ie lds 1/F on the ordinate . This procedure is mathematically r idg id and i t can be traced back to the main dimensioning equation, provided the pr inc ip le of the corrected head i s invoked. (See the proof on the end of t h i s sec t ion)

( 9 ) I t can be seen t h a t f o r a given impeller discharge angle p2, a flow c o e f f i c i e n t and a head c a e f f i c i e n t Ycorr, a s p e c i f i c value of P is required or v i c e versa.

onvenient to The '#cart. t h point is obtained by dividingYcort. exi t t r i a n g l e . with t h e hydraulic e f f ic iency and p lo t t ing i t on t h e

$ l ine . Draw a line from tg P2 thru Ycor r rh and extend i t t o the ord ina te t o obcain 1/(1+P). Now we h o w chat f o r our example., with a given hydraulic e f -

and f i n a l l y the capaci ty c o e f f i c i e n t is

THE: NON-DIMENSIONAL DISCKARGE TRUXGLE FOR TKE GENERAL PUMP

4

FIG. #3

The non-dimensional (o r un i t i zed) discharge t r laagle is an impeller discharge t r i a n g l e where a1.3 v e l o c i t i e s have been divided by 02. In t h i s way a l l quane i t i es a r e l e s s than 1. The discharge t r i ang le presented here d i f f e r s From Stepanoffs triangLe inasmuch a s is adopt- able f o r var ious s l i p and hydraulic e f f ic iency

f ic iency , a fixed value f o r t h e slip must7ex is t , othec- wise our pump condition would not be s a t i s f i e d .

If we mult iply '-Pcorr t h with ( l+P), we obtain V c o r r t h @ . This point l a y s exact ly on the l i n e

from t g p2 t o 1 on the ord ina te , i .e . y c o r r t h c~ is an Euler quant i ty . W cor r t h belongs t o a pump with 100% hydraulic e f f ic iency but has s l i p . Final ly,

Ycorr represents our r e a l pump. A s seen, there is a f ixed xe la t ion between these head c o e f f i c i e n t s and the Euler quant i ty . If the pump has no pre ro ta t ion a t the BEP, then ~ c o r x equals \Y , and H cor r equals H.

It would be wrong t o use the un i t i zed discharge t r i a n g l e t o obtain hydraulic e f f ic iency and s l i p based on some assumptions of P2, and (Vcor,. For new designs s l i p and hydraulic eff ic iency a r e obtained and v e d i e d by other means. But a l l f i n a l assumpeions must f i t i n t o the cons t ra in t s of the uni t ized discharge t r iangle, otherwise the desired pumping condit ions w i l l not be s a t i s f i e d .

The uni t ized discharge t r i a n g l e can be used f o r ca lcu la t ion of many important pump q u a n t i t i e s i n add i t ion t o the ones already shown. Quantities with a quotat ion mark t' a r e Pf le idare r v e l o c i t i e s or angles and a r e associated with the Y/corr t h posir ion. See Figure 3.

This is the tangent ia l component o f the e f f e c t i v e absolute discharge ve loc i ty C2". CU2" is t h e bases of a l l volute and d i f f u s e r ca lcu la t ions . If t h i s va lue i s noc properly ca lcu la ted , the best ef- f i c i e n c y w i l l not be a t the desired capacity.

This is the Euler tangencia1 componenc of C2 Eu Xer .

C2" is the abso lu te average discharge ve loc l ty a t a point c lose t o the impeller e x i t . CZ" is Large enough t o take i n account impeller and volute losses , i .e . f o r a c e r t a i n hydraulic e f f ic iency , the lmpeller discharge ve loc i ty must be CZ".

W2" i s the average r e l a t i v e discharge veloci ty occurring under the inf luence of s l i p and hydraulic e f f ic iency .

conditions. The construct ion of the cr iangle i s as follows: 42" 1 absolute flow angle a r discharge

Connect an o r d i n a t e point of u n i t 1 t o t g p 2 on the p 2 " = r e l a t i v e flow angle a t discharge abszissa. .At6 e r e c t a s t r a i g h t l i n e p a r a l l e l t o the ordinace. Ca lcu la te the correctedhead coef f ic ien t and 2 ;. vane discharge or Euler angle

P is the Pf le idere r s l i p . The def in i t ion of the Pf le idere r s l i p s impl i f i es t h e ca lcu la t ion procedure.

The foXXawing der iva t ion i n d i c a t e s the connection of t h e 1/F point on the t r i a n g l e with the main dimen- sioning equation:

d iv ide both s i d e s bv U2

s ince ?Icorr = korr ' u22

R

Inspection of the non-dimensional t r i ang le shows t h i s t o be true.

STATIC HEAD AT MPELLER DISCHARGE

The s t a t i c head is used f o r t h r u s t ca lcu la t ions and f o r wear r ing leakage ca lcu la t ions . The computa- t ion a p p l i e s t o the cen te r streamline. To ca lcu la te t h e s t a t i c head a t the impeller discharge [ a t BEf) the following equations can b e used

(H-W1.P) HST ----- - c2"2 - c1c2 - lMPL '7h 2 g

(18)

Equations 17 and 18 a r e numerically i d e n t i c a l provided tha t the previously given d e f i n i t i o n s a r e appl ied. Lf rhe h p e l l e r suc t ion is from a stagnant body of water, ~ 1 ~ ~ / 2 ~ ) must b e subtracted from NST.

Equation 1 9 is est imate only when impeller l a s s e s a r e not known. It is assumed t h a t t h e hydraulic losaes a r e equally divided between s t a t o r and impellet..

THE INPUT ilEAn

By d e f i n i t i o n from Fig. 3 we have:

The hydraulic e f f i c i ency is a l so defined a s

W f l l - L K ) (21) ?h" BHP-DFR-MLliP-BSWP

Set equation 20 and 21 equal to each other and so lve f o r Bth corr . _ Hcorr (BHP-DFR-MLHP-BSHP) 3961 Hth 'Orr H * GPM-SPG . ( 1 + LK) a b p u t Qz)

H

B E G I N n

-,

100% CAP - 0

F I G . #4

Since t h i s equation is derived frcm t e s t r e s u l t s only, i t is re fe r red to a s input h&d. The input: head can be calculated over a l a r g e flow range f r w t e s t da ta . It is i d e n t i c a l t o the t b e o r e t i a l head (Hth corl

down t o t h e flow where the ro ta t ing s t a l l begins. At t h a t point , Hinput and Hth carp depar t from each o ther . The input head must not be confused wFth the Euler heac The input head r e f l e c t s hydraulic e f f ic iency and a t lower flows a l s o the s t a l l losses , but not d i s k f r i c - t i o n , nor mechanical losses . The Eular head, by con- t r a s t , represents an idea l pump with no s l i p and 100% hydraulic e f f ic iency . See Fig. 4 ,

SLIP EN IMPELLERS

The previous sec t ions d e a l t with the r e l a t i o n s h i p of s l i p and hydraulic e f f ic iency with t h e discharge v e l o c i t i e s , From t e s t s such s l i p f a c t o r s can be ob- ta ined. This can be done very read i ly provided t h e BEP pre ro ta t ion and the hydraulic e f f ic iency a r e known ( see l a s t sect ion) . We want, however, t o b e ab le to p r e d i c t the s l i p of a new impeller in order t o f u l f i l l a design requirement. This requires some addi t iona l too ls .

These tools a r e the many s l i p equations t ha t have been developed over the years. I t is not: t h e purpose of t h i s paper t o list and compare them a l l . This has been done elsewhere.

For the pump designer tuo equations a r e most help- f u l . F i r s t , P f l e i d e r e r ' s equation and, second, Wfesner's equation. Before discussing any of these t o o l s , some general remarks. Designers know that the s l i p increases with preeottation allowed a t BEP. If counter r o t a t i o n i s allowed, the s l i p decreases.

Also the s l i p can d i f f e r between vo lu te and d i f f u s e r pumps. The s l i p even changes when the d i f fuser / impelkr gap i s changed. Everybody knows t h a t the s l i p changes with discharge angle and number of vanes. The s l i p presents a t r u l y complex problem. The s l i p does not changemuch i n t h e s t a l l f r e e regions along t h e c a p a i t y a x i s . Inves t iga tors have shown t h a t t h e s l i p v a r i e s around the Srmpaller circumference, even a c r o s s the impeller width changes i n s l i p a r e observed. S t r i c k l y speaking, l o c a l s l i p condit ions e x i s t around t h e i m - p e l l e r . Our t ask however i s t o f i n d the average or g loba l s l i p , a s i n g l e number t h a t descr ibes t h e o v e r a l l ef Eective impeller s l i p ,

Nobody has been a b l e t o c a l c u l a t e s l i p d i r e c t l y f r m fundamental physical condit ions. The ex is t ing ca lcu la t ions a r e only simulafions, and we don ' t expect: any break-thru In t h e near fu ture . However f o r every pump chat has been tes ted , a s l i p f a c t o r can be ex-

,

t racred, t h a t is v a l i d f o r the prevai l ing design. Having s u f f i c i e n t t e s t s a v a i l a b l e permits a c l a s s i f i - ca t ion of s l i p condition. In t h i s regard P f l e i d e r e r s equation exce l s , a s we s h a l l see .

P = MST *: Z Pf l e i d e r e r s s l i p equation (23)

An examination of P f l e i d e r e r s equation shows t h a t a l l q u a n r i t i e s have a physical meaning In respec t t o the impeller, except t h e s l i p c o d f i c i e n t A. I f an ana lys i s of a pump eesf yielded tha s l i p f a c t o r P , then the equation 23 determines the value A. The co- e f f i c i e n t A served a s proport tonal ly f a c t o r i n P f l e i d e r e r s d i s c r i p t i o n of h i s equation. He gave a value f o r A fo r ' d i f fuser and v o l u t e pumps. The w r i t e r has seen v a r i a t i o n s i n A fram .5 t o 1.4. Bowever long term obsetrvations permitted a reasonable c l a s s i f i c a t i o n of A a s a parameter af sane impeller geometrical pro- p e r t i e s , r a t h e r than j u s t a p ropor t iona l i ty fac tor . We found t h a t A diffexs between d i f f u s e r and volu'te pumps. The w r i t e r s company designs d i f f u s e r pumps with various amounts of p re ro ta t lon , from'small amounts of counter ro rac ion t o perhaps 450 pre ro ta t ion . Most of our vo lu te pumps have a small amount of p re ro ta t ion . We separated s l i p c o e f f i c i e n t s i n t o groups f o r 0°, JS0, 450 f o r d i f f u s e r pumps and a f o u r t h group f o r vo lu te pumps. Zn a coordinate Eiald of corrected s p e c i f i c speed versus s o l i d i t y , A was p lo t ted as a parameter.

NSC RPM -w " C d 2

CHORD S o l i d i t y = PITCH

P i t c h * Davg -T Z

Chord is defined a s t h e a c t u a l d i s t a n c e ( th ru space) .from vane i n l e t t o vane e x i t on t h e cen te r l i n e . See Fig. 5. Davg occurs where wane is a t half length.

CHORD = (~12 + ~ 2 2 1 4 C

(27)

For the a t a t i c moment see Fig. 5.

CENTER

S'REANLIN7'

It w a s found t h a t f o r a low NSC and low s o l i d i t y the c o e f f i c i e n t A i s small. For a low NSC and a high s o l i d i t y , A was la rger . For a high NSC and a low s o l i d i t y A was large. For a high NSC and a high s o l i d i t y , A was la rge . A var ied between .5 and 1.4. For an i l l u s t r a t i o n a sample of v o l u t e c o e f f i c i e n t s a re added I n t h e appendix. Our pumps a r e of highly de- veloped hydraulic design. A graph f o r simpler hydrauxc designs might look d i f f e r e n t than our example, but not too much. We were a b l e t o prepare for each of t h e above groups of pumps meaningful graphs. Needless to say t h a t t h e ana lys i s work i s tedious and requi res pro- cessor computer programs. The processor program were not ava i lab le for t h e s o l i d i t y ca lcu la t ions i n i t i a l l y which accounts fo r some incons i s tenc ies . To some de- gree incons i s renr fes a r e probably a l s o p a r t of t h e described system, bu t o v e r a l l t h e system permics to i n t e r p o l a t e the s l i p c o e f f i c i e n t b'etween known impeller designs. It should bbe mentioned t h a t i n t e s t ana lys i s t h e amount of pr.erotatian must be known, otherwise che s l i p can not be obtained cor rec t ly . The prerotat ion (flow f i e l d measurements) probably is bes t measured th ru a i r model tests. b

Also with t h e Pf le idere r s l i p equation and the parameter pos i t ion of t h e d i p c o e f f i c i e n t A, t h e ~ c is no requirement t h a t the angular distribution along the main stream l i n e be Logarithmical. We modified P f l e i d e r e r sltp equation is the b e s t of the s l i p equatf on a v a i l a b l e today f o r t h e pumps designer , be- cause the whole equation can be r e l a t e d t o physical ' impeller dimensiors, including t h e s l i p coef f ic ien t . .

Of i n t e r e s t is a l s o the s l ip equation published by Wf esner a. H i s equation is a convenient form af the d a t a given by Busemann 0, which was obtained mathe- mat ica l ly f o r impeller vanes of logari thmic s p i r a l f w . The f u l l Wiesner s l i p f a c t o r g i v e s good r e s u l t s Par v o l u t e pumps with no pre ro ta t ion , On d i fkuser pumps with p re ro ta t ion the Wiesner s l i p f a i l s . There i s also another disadvantage, If a pump has been analyzed f o r s l i p , the r e s u l t a n t P can not be reproduced by the Wiesner s l i p equation. The d i f f e r e n c e may be small i n case of a volufe pump -with no pra ro ta t ion . On pumps with p re ro ta t ion , there is no way t o ob ta in a corre- l a t i o n . And i f the angular d i s t r i b u t i o n is noe a logarithmic s p i r a l , no good r , esu l t s can be expect& because of an over-emphasis of t h e exit angle,

WmSNERS SLIP FACTOR where

Where 6 a (- 8.16 ;sinf!2) (30)

RJ. i f - < 5 , oplit the t e rn i n R2 I I

If d i s t o be used i n the main dimensioning equation, it has f i r s t t o be converted t a P. Wiesner defines d as:

I c e -. C5-I CU2 a i 2 and CU2 - UZ - - e g g 2

CM2 U2 - - 4. l + P = tg%2 and f i n a l l y P - 13) - .]

dl u2 - - tg P 2 d- -

(32) Now d can be converted i n t o P and used i n the

previously discussed manner.

PUMP TEST ANALYSIS

The systematic approach Eor t e s t data reduction requires a l o t of thought and i s beyond t h i s paper. However we assembled here the basic procedure t o ob- t a i n ' t h e s l i p P f rom a pump t e s t .

WIIP CPMBEP H - SPG 3961 (33)

WHP BKP = - E r n

For the ca lcu la t ion of d i s c f r i c t i o n , see Ref. (1) or (2) . Do not f o r g e t t o include h p e l l e r s ide - p l a t e thickness Ref. (1).

The s l i p can be obtained from the main dimension- ing equation.

y = Twist of vane, see Ref. (1)

or simpler

and f i n a l l y from P follows A:

As previously discussed, the s l i p c o e f f i c i e n t A can be grouped for var ious ranges of p r e r o t s t i o n on d i f f u s e r and vo lu tes pumps f o r NSC versus s o l i d i t y . With s u f f i c i e n t da ta on hand, a secure means of e s t i - mating new impeller designs can be thus devised.

CONCLUSIONS

The scream l i n e theory presented here i s f o r the mosc p a r t s not new, but t h i s paper o f f e r s a "one source" d i c t i o n a r y f o r many def in i t ions . The equat ions given here pravide a r i g i d background f o r one aspect of himpeller design. The m i t e r f e l t the re was a need t o present a comprehensive one-dimensional stream l i n e c a l c u l a t i o n theory, tha t can handle p re ro ta t ion flow f i e l d s a t the impeller i n l e t and a t the same time explain PEleiderers s l i p equation i n t h e Englf s h language region. Thestream l i n e theory i s a l s o a must f o r people t h a t use i n t e r n a l impeller flow programs such a s Katsanis 's MERIDL and TSONZC codes. These codes can not be successufl ly appl ied, un less ' inpur i s pro- vided t h a t is based r i g i d l y on the. described stream l i n e theory i n a three-dimensional way.

REFERENCES

1. Pf l e i d e r e r , Carl , "Die Kreisel-Pumpen fGr Fl i iss igkei ten und Gase," Springer Verlag 1961

2. Stepanoff, Alexander, "Centrif ugal and Axial Flow Pumps," John Wiley & Sons Inc, 1957

3. Wiesner, F, J., "A Review of S l i p Factors for Cent r i fuga l Impellers," Transaction of the ASME, Paper No. 66-WAlFE-18

4 . Busemann, A . , "In hgewand t e Mathematik und Mechanik," Volume 8 , October 1928, Pages 372-384.

For appendix see next page.

APPENDIX

N S C X 100 CORRECTED SPECIFIC SPEED