axial turbines

7/27/2019 Axial Turbines

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5S i m p l i f i e d m e r i d i o n a l f l o wa n a l y s i s fo r a x i a l t u r b o m a c h i n e s

I n t r o d u c t i o n

T h e l a st tw o c h a p t e r s h a v e b e e n c o n c e r n e d w i th t h e c e n t r e - li n e o r m e a n r a d iu s d e sig nof tu rb ines , compresso r s and f ans , enab l ing us to def ine a s ing le des ign du ty ( th, if)r ep resen ta t ive o f the per fo rmance o f a g iven s tage . We were ab le to exp ress thed ime ns ion less ve loc i ty t r iang les , n o rma l i sed by the b lade speed U = r I~ , in te rm s o fthe f low coeff ic ien t ~b , the work coeff ic ien t ~ and the s tage react ion R, leading usd i r ec t ly on to a r a t iona l ana ly t ic app roac h to b lade p ro f i le des ign l inked to s tage m eandu ty (~b, ~p).

In r ea l i ty o f cou r se the s tage per fo rmance wi l l be de te rmined no t ju s t by i t scen t r e - l ine sec t ion bu t w i l l be the average o f the who le f low f rom hub to cas ing . Ther e a l f l o w in a t u r b o m a c h in e i s t h r e e - d im e n s io n a l a n d i n d e e d e x t r e m e ly c o m p le x .Cond i t ions may vary cons iderab ly f rom hub to cas ing and the b lades themse lves wi l l

u sua l ly be bo th tapered and twis ted as was i l lu s t r a ted in the in t roduc t ion to Chap te r2 and Fig . 2 .3 .

W e m o v e o n n o w to c o n s id e r th i s p r o b l e m o f h o w to a n a ly s e t h e t h r e e - d im e n s io n a lf low in tu rbomach ines . The s ta r t ing po in t fo r th i s was a l so g iven in Chap te r 2 wherethe ' cascade ' and 'mer id iona l ' f low s t ruc tu res were in t roduced main ly to p rov ide amanageab le des ign f r amework . As i l lu s t r a ted by F ig . 2 .1 , the fu l ly th ree-d imens iona lf low can be t r ea ted fo r p rac t ica l pu rposes as an ax isymmetr ic o r c i r cumferen t ia l lyaveraged 'mer id iona l f low ' , and a se r ies o f super imposed ' cascade ' f lows to def ineb lade p ro f i les a t se lec ted sec t ions f rom hub to cas ing .

Now so f a r fo r ax ia l mac h ines we have as sum ed tha t the m er id iona l ve loc i ty Cs i scons tan t fo r a l l mer id iona l s t r eaml ines as i l lu s t r a ted fo r an ax ia l f an in F ig . 5 .1 (a ) .

Downs t r eam o f the ro to r , however , the f low may be swir l ing qu i te cons iderab ly ,r esu l t ing in an inward r ad ia l p ressu re g rad ien t . Th is may wel l r esu l t in some rad ia lsh i f t s in the mer id iona l s t r eaml ines as i l lu s t r a ted in F ig . 5 .1 (b ) . As a r esu l t o f th i sthe m er id iona l ve loc ity Cs wi l l va ry f rom hub to t ip wi th co nseq uen t mo d i f ica t ion o fthe ve loc i ty t r iang les wh ich mus t be taken in to accoun t befo re the b lade p ro f i les a redes igned .

Mer id iona l f low ana lyses to hand le th i s des ign p rob lem are ex t r emely complex .In th i s chap te r we wi l l con ten t ou r se lves wi th on ly the s imp les t t echn iques knownas rad ia l equ i l ib r ium ana lys i s, Sec t ions 5 .2 and 5 .3 , and ac tua to r d i sc theo ry , Sec t ions5 .4 and 5 .5 . F ina l ly in Sec t ion 5 .6 these ana lyses wi l l be ex tended to dea l w i th thedes ign o f a comple te ax ia l f an o r compresso r compr is ing severa l b lade rows and

in c lu d in g t h e m e r id io n a l i n t e r f e r e n c e b e tw e e n t h e m . Be f o r e t u r n in g t o t h e s e ,however , we wi l l beg in in Sec t ion 5 .1 by cons ider ing the spec ia l case o f f lee -vo r texd e s ig n , t h e s im p le s t a n d m o s t p o p u l a r m e th o d f o r t h r e e - d im e n s io n a l d e s ig n o f a x i a lt u r b o m a c h in e s .



108 S imp l i f ie d m e r id iona l f l ow ana ly si s f o r ax ia l t u rbom ac h ine s

y _ l l -( a ) ( b )

F ig. 5.1 M eridional f low through an axial fan consisting of a single rotor only: (a) free-vortex axial

fan; (b) non-free-vortex axial fan

5 . 1 T h e f r e e . v o r t e x a x i a l f a n

I n S e c t i o n 4 . 7 w e c o n s i d e r e d t h e m e a n r a d i u s d e s i g n o f a n a x i a l f a n c o m p r i s i n g a

ro to r fo l lowed by a s t a to r , F ig . 4 . 9 . The ra t iona l des ign p rocedu re adop ted to a r r ive

a t a e r o d y n a m i c a l l y s u i t a b l e b l a d e p r o f i l e s c a n b e s u m m a r i s e d a s f o l l o w s :

Selection of o ptimum duty points (~b, ~,) from mod el test data

Construction of dimensionless velocity triangles

Use of computational fluid dynamics to select suitable blade profiles which

will deliver the velocity triangles with stable and low loss flow

Obv ious ly the f low cond i t ions a t t he mean rad iu s a re l i ke ly to t yp i fy the genera lp e r f o r m a n c e o f t h e f a n b u t w e a r e s t i l l l e f t w i t h t h e m a j o r t a s k o f d e s i g n i n g t h e r e s t

o f t he ro to r b l ade wh ich , as we wi l l see , wi l l va ry cons iderab ly in bo th du ty ( t h, ~,)a n d c o n s e q u e n t b l a d e g e o m e t r y f r o m h u b t o t ip . W e n e e d t h e r e f o r e t o c o n s i d e r

c a r e f u l l y w h a t a e r o d y n a m i c l o a d i n g c a n b e c a r r i e d b y e a c h s e c t i o n o f t h e b l a d e a n d

w h e t h e r r a d i a l v a r i a t i o n s a r e l i k e l y t o i m p o s e e x t r e m e a e r o d y n a m i c d i f f i c u l t i e s i np ro f i l e se l ec t ion o r o f f -des ign per fo rmance .

A s a fi rs t st e p t o w a r d s t h e t h r e e - d i m e n s i o n a l d e s i g n o f o u r f a n r o t o r l e t u s i m p o s e

t h e r e a s o n a b l e c o n s t r a i n t t h a t t h e f l u i d s h o u l d b e g i v e n t h e s a m e s t a g n a t i o n p r e s s u r e

r is e f or a ll m e r i d i o n a l s t r e a m l i n e s . F o r i n c o m p r e s s i b l e f l ow a t r a d i u s r t h e E u l e r p u m p

e q u a t i o n ( 1 . 9 b ) , w i t h z e r o p r e - w h i r l u p s t r e a m o f t h e r o t o r (Co l = 0 ) , m a y b e

wr i t t en

l~ = _1 ApoE = Uca 2 = rl'lca2 (5 .1)P



5 .1 T h e f r e e - v o r t e x a x i a l f a n 10 9

m

w h e r e W i s t h e s p e c i f i c w o r k i n p u t a n d ADo E r e p r e s e n t s t h e s t a g n a t i o n p r e s s u r e r i s eo f a per fec t (Eu le r ) f an wi th f r i c t i on less l o ss - f ree f low. Thus we a re ac tua l ly adop t inghere a cons t an t spec i f i c work inpu t fo r a l l r ad i i . Equa t ing cond i t i ons a t t he mean

rad ius rm to any o ther r ad iu s r we thus ob ta in

r m C o 2 m = r c o 2 = K (5 .2)

w h e r e K i s a c o n s t a n t . T h e r a d i a l v a r i a t i o n o f co m a y t h u s b e e x p r e s s e d

KC02 -- __

r(5 .3)

and w e see tha t t he swi r l ve loc i ty c0 2 dow ns t re am o f t he ro to r i s i nver se ly p rop o r t io na lto r ad iu s . Th i s co r responds to t he wel l -known c l ass i ca l f l ow genera t ed by a l i ne vo r t ex

as i ll u s t r a t ed by F ig . 5 .2 . In t h i s case we have show n the f low f i eld in t he (x , y ) p l a ne

i s t ha t i nduced by a l i ne vo r t ex o f s t r eng th F ly ing a long the z -ax i s be tween z = + ~ ,fo r wh ich the induced ve loc i ty i s

FCo = 27rr (5.4)

In consequence o f t h i s t he type o f f an wh ich we have se l ec t ed i s ca l l ed a f l ee -vo r t exd e s i g n _ I f w e w e r e t o d e p a r t f r o m t h e i n i t i a l d e s i g n c o n s t r a i n t o f e q u a l s p e c i f i c w o r k

inpu t W a t a l l r ad i i we cou ld in f ac t se l ec t a very wide range o f vo r t ex des igns andindeed we wi l l r e tu rn to t h i s i n Sec t ions 5 .4 to 5 .6 . Fo r t he p resen t l e t u s s t ay wi th

' f r ee -vo r t ex ' , ' cons t an t spec i f i c work ' des ign and see where th i s l eads .F o r i n c o m p r e s s i b l e f l ow t h e w o r k i n p u t c o e f fi c ie n t ~ , E q n ( 4 . 4 ), m a y b e r e d e f i n e d

as fo l lows for rad ius r :

APoE/P ApoE

~, = U2 - pr2f~2 (5 .5)

L e t u s a s s u m e a l s o t h a t t h e m e r i d i o n a l s t r e a m l i n e s a r e c y l i n d r i c a l a n d t h e m e r i d i o n a l

ve loc i ty i s t hus cons t an t and equa l t o C x . The f low coef f i c i en t a t r ad iu s r t husb e c o m e s

Cx Cx~b - U - r f~ (5 .6)

Th e d u ty coef f i c i en t s a t rad iu s r may now be ex p resse d in t e rms o f t he va lues ( thm, ~m)

s e l e c te d a t t h e o u t s e t f o r th e m e a n r a d i u s rm . M a k i n g u s e o f t h e f r e e - v o rt e x E q n ( 5 .2 )

w e h a v e

6 = 6m , ~t = ~'m (5 .7 )

a n d t h e r a d i a l v a r i a t io n o f s ta g e r e a c t i o n R , E q n ( 4 . 41 ) , b e c o m e s

R = 1 - ( 5 .8 )



110 S i m p l i f i e d m e r i d i o n a l f l o w a n a ly s is f o r a x ia l t u r b o m a c h i n e s

\

/

I '

C o = 2 r r

x

F ig. 5. 2 Flow f ie ld induced in the ( x , y ) plane by vortex of s trength F ly ing along the z-axis

At th i s po in t i t wi l l be he lp fu l t o i n t e rp re t t he consequences o f a l l t h i s by cons ider inga n u m e r i c a l e x a m p l e .

E x a m p l e 5 . 1

P r o b l e m

A fan i s t o be des ign ed wi th a m ea n rad iu s du ty o f ~b = 0 .5 , ~ = 0 .3 . I f t he hu b / t i prat io h = rh/r i s t o be 0 .3 ca l cu la t e t he du ty coef f i c i en t s and reac t ion a t e igh t r ad i a ls t a t i o n s e q u a l l y s p a c e d b e t w e e n h u b a n d c a s i n g . C a l c u l a t e t h e v e l o c i t y t r i a n g l e d a t aand p i t ch /cho rd ra t io o f bo th ro to r a nd s t a to r assu ming a d i f fu s ion fac to r o f 0 .5 .

S o l u t i o n

Fi r s t l e t u s r e l a t e t he me an rad iu s r m to the hub / t i p r a t io h = rh[r . By def in i t i on

r m = 89 h + r t ) so that the rat io r / r m n e e d e d t o e v a l u a t e E q n s ( 5 . 7 ) a n d ( 5 . 8 ) m a y b ee x p r e s s e d a s

r _ r ( 2 ) ( ~ t ) 1 . - ~ (r ~ )rm l( r h -I- rt) 1 + h

I t is u sua l ly mo re conv en ien t nu me r i ca l ly t o u se the t i p r ad iu s r t t o non -d im ens io na l i se

l o c a l r a d i u s r . N o w w e m a y c o m p l e t e T a b l e 5 . 1 .F r o m t h i s e x a m p l e w e o b s e r v e t h a t t h e b l a d e s e c t i o n s b e t w e e n t h e m e a n r a d i u s

a n d b l a d e t i p , 0 . 6 5 < r / r t < l . O , a r e l ig ht ly l o a d ed ( 0 . 1 2 7 < ~ < 0 . 3 ) a n d als o h a v el o w f l o w c o e f f i c i e n t s ( 0 . 3 2 5 < t h < 0 . 5 ) , t h e s t a g e r e a c t i o n l y i n g i n t h e r a n g e



5 .1 T h e f r e e - v o r t e x a x i a l f a n 1 11

Table 5.1 Duty coefficients and stage reaction for a free-vortex fan

r/r dp ~ R

0.3 hu b 1.083 33 1.408 32 0.295 84

0.4 0.812 51 0.792 20 0.603 90

0.5 0.650 00 0.507 00 0.746 50

0.6 0.541 66 0.352 08 0.823 96

0.65 me an 0.500 00 0.300 00 0.850 00

0.7 0.464 29 0.258 67 0.870 67

0.8 0.406 25 0.198 05 0.900 98

0.9 0.361 11 0.156 48 0.921 76

1.0 t ip 0.325 00 0.1267 5 0.93 663

Table 5.2 Velocity tr iangle data and pitch/chord rat ios for free-vortex fan of Example 5.1

Roto r S t a to r

r / r t f l l ~ f 1 2 ~ ( t / l ) R ~ 2 ~ ( t / / ) S

0.300 42.7094 -20 .65 26 0.597 27 52.4314 0.276 83

0.400 50.9061 14.3469 0.490 84 44.2748 0.618 82

0.500 56.9761 37.1789 0.865 75 37.9542 0.938 17

0.600 61.5571 50.1039 1.567 08 33.0239 1.242 020.650 63.4350 54.4623 2.008 10 30.9638 1.389 68

0.7 00 65.0952 57.9415 2.500 86 29.1241 1.535 10

0.800 67.8906 63.1343 3.628 40 25.9892 1.820 51

0.900 70.1448 66.8242 4.933 02 23.4287 2 .1 0 (} 3 4

1.000 71.9958 69.5861 6.406 69 21.3058 2.376 00

0 . 8 5 < R < 0 . 9 3 7 . A c c o r d i n g t o th e ' S m i t h ' c h a r t fo r 9 0 % r e a c t i o n c o m p r e s s o r s t a g e s ,

F i g . 4 . 4 ( c ) , t h e t i p r e g i o n o f t h e b l a d e s w i l l t e n d t o h a v e a l o w e ff i c ie n c y . T h e r e i s

n o d i f f i c u l t y w h a t s o e v e r i n d e s i g n i n g s u i t a b l e b l a d i n g f o r t h e t i p r e g i o n b u t l o s s e s

w i l l n e c e s s a r i l y b e h i g h t h e r e b e c a u s e o f t h e h i g h r e l a t i v e v e l o c i t i e s .

I f w e c o n s i d e r n e x t t h e i n n e r r e g i o n , 0 . 3 < r / r < 0 . 6 5 , p r o b l e m s o f a d i f f e r e n t k i n d

a r i s e . W e o b s e r v e t h a t ~ r i s e s r a p i d l y t o t h e h u b s e c t i o n v a l u e o f 1 . 4 w h i c h i s f a r

i n e x c e s s o f w h a t w e w o u l d e x p e c t t o a c h i e v e w i t h a s i n g l e s t a g e f a n , t h e f l o w

coef f i c i en t ~b = 1 .083 33 a l so be in g qu i t e h igh .

L e t u s n o w m a k e u s e o f E q n s ( 4 . 4 3 ) t o c a l c u l a t e t h e v a r i o u s f l o w a n g l e s ( s e e F i g .

4 . 9 ) . A s s u m i n g d i f f u si o n f a c to r s D R = D s = 0 . 5 w e m a y a l s o c a l cu l a te a p p r o p r i a t e

p i t c h / c h o r d r a t i o s . T h e o u t c o m e i s g i v e n i n T a b l e 5 . 2 .

F r o m t h is w e o b s e r v e t h e f o l l o w i n g a b o u t c o n d i t i o n s a t th e b l a d e r o o t s e c ti o n"

( 1) E x c e s s i v e l y l a r g e fl u id d e f l e ct i o n s a r e d e m a n d e d o f b o t h r o t o r a n d s t a t o r ,n a m e l y e R = ~ 1 - f 1 2 = 63 .362~ es = a2 = 52 .431 ~

( 2 ) E x t r e m e l y s m a l l p i t c h / c h o r d r a t i o s a r e r e q u i r e d t o a c h i e v e t h e s e d e f l e c t i o n s ,

n a m e l y ( t / 0 R = 0 . 5 9 7 2 7 , ( t/ /) S = 0 . 2 7 6 83 .



112 S i m p l i fi e d m e r i d i o n a l f l o w a n a l y s is f o r a x i a l t u r b o m a c h i n e s

A l t h o u g h i n t h e o r y w e c o u l d d e s i g n c a s c a d e s t o a c h i e v e t h e s e d e f l e c t i o n s , a l b e i tw i t h p o o r a e r o d y n a m i c p e r f o r m a n c e , t h e b l a d e s w o u l d e x h i b i t s u c h e n o r m o u s t a p e r

t h a t w e w o u l d b e h a r d p l a c e d t o a c c o m m o d a t e t h e r o o t s e c t i o n o f t h e b l a d e , t h e

c h o r d o f w h i c h w o u l d b e o v e r s e v e n ti m e s g r e a t e r t h a n t h e c h o r d a t th e m e a nrad ius r m.

O n e p o s s i b l e s o l u t i o n t o t h e d i l e m m a p o s e d i n E x a m p l e 5 . 1 w o u l d b e t o s e l e c ta h i g h e r h u b / t i p r a t i o . T h u s f r o m T a b l e s 5 . 1 a n d 5 . 2 w e s e e t h a t q u i t e a c c e p t a b l e

(~b, ~,) du t ie s and veloci ty t r iangle s are o b ta ine d for r / r t > 0 .5 wh ich ind ica t es a su i t ab l e

va lue fo r h . A secon d so lu t ion m igh t be to ado p t a l ess dem an d in g (thm, ~ 'm) du tya t th e m e a n r a di u s. T h e r e a d e r c a n m a k e u s e o f t h e s im p l e P as c a l p r o g r a m F V F A N ,

t h e s o u r c e c o d e o f w h i c h is p r o v i d e d o n t h e a c c o m p a n y i n g P C d i sc , to e x p e r i m e n t

wi th th i s . The mos t app rop r i a t e so lu t ion in p rac t i ce , i n o rder t o main ta in low va lues

o f h u b / t i p r a t i o a n d t h u s t h e m a x i m u m a v a i l a b l e a n n u l u s f l o w a r e a , i s t o a b a n d o n

our in i t i a l a im o f des ign ing fo r cons t an t spec i f i c work W inpu t a t a l l r ad i i . In s t ead

w e m a y p r o g r e s s i v e ly r e d u c e W a s w e m o v e r a d ia l l y i n w a r d f r o m r m t o r h a n d t h e r e b yredu ce the p ressu re r i se coef f i c ien t ~ , t o acce p tab le va lues , l e ss t han say 0 .4 . Th i sw i l l h e l p t o m a i n t a i n m u c h l i g h t e r a e r o d y n a m i c l o a d i n g i n t h e b l a d e r o o t r e g i o n b u t

a l so p i t ch /cho rd ra t io s i n excess o f 1 . 0 and hence a wider s t a l l - f r ee r ange ( see Sec t ion

4 . 8 ) . U n f o r t u n a t e l y t h i s a p p r o a c h w o u l d e n t a i l a d e p a r t u r e f r o m f r e e - v o r t e x f l o w ,

resu l t i ng in a non -un i fo rm mer id iona l ve loc i ty p ro f i l e , F ig . 5 .1 (b ) . Complex

ca lcu la t ions a re r e qu i re d to eva lu a t e t he va r i a t ion o f Cs wi th r ad iu s a nd two o f t hemos t s imp le ana ly ses wi l l now be p resen ted in Sec t ions 5 .2 to 5 .5 .

5 . 2 R a d i a l e q u i li b r i u m a n a l y s i s fo r a x i a l t u r b o m a c h i n e sF i g u r e 5 . 3 ( a ) i l l u s tr a t e s t h e m a n n e r i n w h i c h t h e m e r i d i o n a l s t r e a m l i n e s s h i ft r a d i a ll y

i n w a r d p r o g r e s s i v e l y u n d e r t h e i n f l u e n c e o f t h e r a d i a l p r e s s u r e g r a d i e n t d p / d rg e n e r a t e d b y t h e s w i r l i n g f l o w d o w n s t r e a m o f a b l a d e r o w ( a s t a t o r i s s h o w n h e r e ) .In consequence o f t h i s t he re wi l l be a s t eady g rowth in t he s lope o f t he ax ia l ve loc i ty

p ro f i l e Cx. S o m e d i s t a n c e d o w n s t r e a m o f t h e b l a d e r o w a t s t a ti o n 3 t h e r a d i a l v e l o c it yc o m p o n e n t Cr wi l l app roach ze ro , r esu l t i ng in t he so -ca l l ed rad ia l equ i l i br ium f low.A r a d i a l m o m e n t u m b a l a n c e i s t h e n a c h i e v e d b e t w e e n t h e r a d i a l p r e s s u r e g r a d i e n td p / d r a n d t h e a n g u l a r m o m e n t u m o f t h e f l u i d r c o . Our f i r s t t a sk wi l l be to der iveth i s r e l a t ionsh ip by re fe rence to t he equ i l i b r ium o f a smal l f l u id e l emen t a t r ad iu sr as i l l u s t r a t ed in F ig . 5 .3 (b ) . Here we a re adop t ing cy l ind r i ca l po la r coo rd ina t es(x , r , 0 ) wh ere the x -ax is i s co inc iden t wi th the ax is o f ro t a t io n o f the tu rb om ach ine .

The mass o f the e l em en t , wh ose s ides a re o f l eng th dx , d r a nd r dO, i s g iven by

d m = p d x . d r - r dO ( 5 .9 )

I f w e n o w e q u a t e t h e r a d i a ll y i n w a r d p r e s s u r e f o r c e o n t h e f a c e s o f th e e l e m e n t t o

i t s cen t r i fuga l acce l e ra t ion , we ob ta in

(p + dp ) ( r + d r ) d O d x - p r d O d x

( 1- 2 p+gdp2

d r d x s in ~ = dm c'b (5 .10)r

= p d x d r . r d O c2~r



5.2 Rad ia l e qu i l ib r ium ana ly s i s f o r ax ia l t u rbomac h ine s 11 3

p+dp

;I. y

p-~-ap/z..~4~ ~a r ~ + d p / 2

r~ IP

- /

/

/

/f

~ d x

J

(a) (b)

F ig . 5 .3 Radial equi librium of a smal l f lu id e lement dow nstream of a turbomachine blade row : (a )

meridional streamline shift and axial velocity profile development; (b) pressure forces on a small fluid

element at station 3

w h i c h r e d u c e s t oldp cEo

= - - ( 5 .1 1 )p dr r

In rad ia l equ i l ib r ium f low the rad ia l p re s sure g rad ien t i s thus un ique ly re la ted to theswir l ve loci ty co i r re spec t ive o f the type o f vor tex f low. For the spec ia l ca se o ff ree -vor tex f low, in t roduc ing Eqn (5 .4 ) , we no te tha t Eqn (5 .11) reduces to

1 dp F 2= (free-v ortex f low) (5 .11a)

p dr 47r 2r 3

and we wi l l l a t e r show tha t the ax ia l ve loc i ty Cx i s cons tant for th is specia l case . Fora ll o the r vor tex f lows, fo r which rc o i s no t c ons tan t , w e nee d to re la te the ax ia l ve loc i tyCx to Co and p . For s impl ic i ty l e t us f i r s t cons ide r i nc ompre ss ib l e f low fo r which we

may de ' f ine the s t agna t ion p res sure Po th rough

P- 2~ P c2 P -- +c 2 c2 c2~ (5 12)

o - 7 + T = o T + T + T

Diffe ren t i a t ing th i s equa t ion wi th re spec t to r and pu t t ing Cr = 0 , we have

_ dcx dco1 dpo l dp + c x + C op d r - p d r - ~ r d r

I n t r o d u c i n g d p / d r f rom Eqn (5 .11) , we have f ina l ly

(5 .13)

w h i c h i s k n o w n a s t h e r a d ia l e q u i li b r iu m e q u a t io n f o r i n c o m p r e s s ib l e f lo w .



11 4 S i m p l i f i e d m e r i d io n a l f l o w a n a ly si s f o r a x ia l t u r b o m a c h i n e s

A n e q u iv a l e n t e q u a t i o n f o r c o m p r e s s ib l e f l o w c a n b e d e v e lo p e d b y m a k in g u s e o fthe fo l lowing therm odyn am ic r e la t ionsh ip wh ich l inks tem pera tu re T , spec if ic en t ro pys and specific en thal py h to p and p :

T d s = d h - 1 d pP

= d h o - l d p oP

(5.15)

wh ere the s tagna t ion en tha lpy i s def ined as ho = h + c2 /2 . D iv id ing t h ro ugh ou t byd r and subs t i tu t ing fo r (1 /p) (dpo/dr ) in Eqn (5 .14 ) , we have f ina l ly the rad ia le q u i l i b r i u m e q u a t io n f o r c o m p r e s s ib l e f l o w :

The axial veloci ty Cx i s thus a func t ion o f the r ad ia l d i s t r ibu t ion no t on ly o f rc o b u talso of ho and s .

5 ,3 S o l u t i o n o f t h e r a d i a l e q u i l ib r i u m e q u a t i o n f o r t h ei n v e r s e a n d d i r e c t p r o b l e m s

Two types o f p rob lem may be iden t i f ied as fo l lows :

(1 )

(2 )

The ' Inver se ' or 'Des ign ' prob lem. In the des ign sequence ou t l ined in Sec t ion5 .1 , once the ve loc i ty t r iang les have been se lec ted ho and co a r e k n o w n a t a l lr ad i i as par t o f the des ign spec i f ica t ion . The r ad ia l d i s t r ibu t ion o f spec i f icen t ropy s may a l so be ob ta ined f rom a f i r s t e s t imate o f the lo sses o r r / x - rf rom model t es t da ta . So lu t ion o f the r ad ia l equ i l ib r ium Eqn (5 .16 ) theny ie ld s a new es t imate o f the ax ia l ve loc i ty d i s t r ibu t ion Cx a n d h e n c e a nupda t ing o f the ve loc i ty t r iang les and thus f low ang les p r io r to b lade p ro f i leselect ion . This is the des ign prob lem.The 'Direc t ' or 'Analys i s ' prob lem. W e m a y p o s tu l a t e t h e o p p o s i t e p r o b l e min w h ic h w e a r e p r e s e n t e d w i th a n e x i s t i n g t u r b o m a c h in e o f k n o w n b l a d e

geomet ry and asked to p red ic t i t s f lu id dynamic per fo rmance . Th is i s theanalys i s prob lem.

Theore t ica l ana lys i s to dea l w i th these two r a ther d i f f e ren t p rob lems wi l l now bep resen ted wi th the he lp o f numer ica l examples in Sec t ions 5 .3 .1 and 5 .3 .2 .

5.3.1 Solution of the inverse radial equilibrium problem

This i s bes t i l lu s t r a ted by cons ider ing the case o f a se t o f in le t gu ide vanes wh ichare to be des igned to genera te a so l id body swir l ing f low.

Example 5 .2 'So l id body ' swi r l in le t gu ide vanes

Cons ider the case o f f low th rough an in le t gu ide vane b lade row, F ig . 5 .3 (a ) . In th i scase the swir l veloci ty co i s to be p ropor t iona l to r ad ius a t s ta t ion 3 a long wayd o w n s t r e a m o f t h e b l a d e r o w . W e s h a l l a l s o a s s u m e th a t h o a n d s a r e b o th c o n s t a n t



5 .3 S o l u t i o n o f t h e r a d ia l e q u i l i b r i u m e q u a t i o n 115

t h r o u g h o u t t h e f l o w r e g i m e , n a m e l y

c o = k r (where k i s a cons t an t )

ho = cons t an t

s = cons t an t

P r o b l e m

D e r i v e a n a n a l y t i c a l s o l u t i o n f o r Cx as a func t ion o f r .

(5 .17)

S o l u t i o n

I n v ie w o f E q n s ( 5 . 17 b ) a n d ( 5 .1 7 c ) , th e r a d i a l e q u i l i b r iu m E q n ( 5 .1 6 ) r e d u c e s t o

d c x c o d ( r c o )0 = Cx ~ ~ (5 .16a)

r d r

w h i c h m a y b e r e w r i t t e n

d c 2 2 c o d ( r c o )

d r r d r

a n d h e n c e , a f t e r i n t e g r a t i o n , a t r a d i u s r w e h a v e

C x (r ) = K 1 - 2 - - d ( r c o )r

(5 .18)

I n t r o d u c t i o n o f E q n ( 5 . 1 7 a ) f o r t h e s o l i d b o d y r o t a t i o n c a s e t h e n r e s u l t s i n

Cx = N/K1 - 2k 2 r 2 = V ' K 1 - 2 r 2 (5 .19)

T h e c o n s t a n t o f i n t e g r a t i o n K 1 c a n b e e v a l u a t e d b y a p p l i c a t i o n o f t h e m a s s

f lo w c o n t i n u it y e q u a t i o n . T h u s t h e m a s s f lo w rh t h r o u g h t h e a n n u l u s m a y b ee x p r e s s e d a s

i n = p C x 2 7rr d r = p Cx 2 z rr d r

s t a t i o n 1 - ( e n t r y ) s t a t i o n 3 - ( e x i t )

(5 .20)

w h e r e C x i s t he m ea n ax ia l ve loc ity and thus Cx = Cx a t e n t r y to t h e a n n u l u s . A s s u m i n gi n c o m p r e s s i b l e f l o w a n d i n t r o d u c i n g E q n ( 5 . 1 9 ) , E q n ( 5 . 2 0 ) b e c o m e s

C x ( r 2 - r2h) = 2 r V ' K 1 - 2 k 2 r 2 d r

1= 3 --~ [ (K 1 - 2k2r2 )3 /2 _ ( g 1 - 2 ki r2 ) 312]

(5 .21)

B e c a u s e o f t h e c o m p l e x i t y o f E q n ( 5 . 2 1 ) , K 1 c a n n o t b e e v a l u a t e d e x p l i c i t l y a n d c a no n l y b e d e r i v e d b y su c c e ss i ve a p p r o x i m a t i o n s . N e v e r t h e l e s s a r e a s o n a b l e a p p r o x i m a t eana ly t i ca l so lu t ion may be der ived as fo l lows .



1 16 S i m p l i f i e d m e r i d i o n a l f l o w a n a l y si s f o r a x i a l t u r b o m a c h i n e s

A p p r o x i m a t e s o l u t i o n m a t c h i n g C x a t t h e r o o t m e a n s q u a r e r a d i u s rm s

= 1 2 r 2) T h u s E q ne t u s a s s u m e t h a t Cx = Cx a t the r .m. s , rad ius , namely r m s V ' ~ ( r h + .

(5 .19 ) y ie lds d i rec t ly an e s t ima te fo r K1 , namely

K 1 = C 2 + 2(Corms)2

= C 2 + 2c2~(rms/rt)2

= C 2 + c2~(1 + h 2) (5 .22)

Thus f ina l ly , a t o the r rad i i r , f rom Eqn (5 .18 ) we have

c x ]C x ~ (5 .23)

N u m e r i c a l s o l u t i o n o f th e i nv er s e p r o b l e m

A much more f l ex ib le approach app l i cab le to any rad ia l d i s t r ibu t ion o f co is toeva lua te Eqns (5 .18 ) and (5 .20 ) numer ica l ly . F i r s t l e t us de f ine the func t ion

f ( r ) = 2 frh" COrd(r co ) (5 .24)

s o t h a t E q n ( 5 . 1 8 ) b e c o m e s

Cx(r ) = X /K 1 - f ( r ) (5 .25)

F r o m t h e c o n t i n u i t y e q u a t i o n ( 5 . 2 0 ) w e m a y d e f i n e a m a s s f l o w f u n c t i o n

rhS = m

2~rp

Cx (r 2 _ r2 )= - ~

= s r t r ~ /K 1 _ f ( r ) d r

h

a t - ~ u p s t r e a m o f b l ad e ro w

Jt + ~ d o w n s t r e a m o f b l a d e ro w

(5.26)

F o r n u m e r i c a l a n a l y s i s , t h e a n n u l u s m a y b e r e p r e s e n t e d b y m r a d i a l s t e p s b e t w e e n

hub and t ip rad i i r h a n d r t o f th i ck n e ss A r = ( r t - r h ) / m . f ( r ) m a y t h e n b ea p p r o x i m a t e d a t r a d i u s rj by

J

f (r j) = 2 E cOm''''~i ri + 1 Co i+ 1 -- ri c o i) (5 .27)i= 1 r m i

_ . 1w h e r e rm i = 8 9 i + r i+ 1) and Com i ~ (co i + 1 + co i ) . T h e m a s s f u n c t i o n S m a y t h e n a l s o

be eva lu a ted n um er ica l ly i f E qn (5 .26) i s rewr i t t enI n

S 1 -- Ar E r m i ~ / / K 1 - f ( r i ) (5 .28)i= 1



5 . 3 S o l u t i o n o f t h e r a d i a l e q u i l i b r i u m e q u a t i o n 11 7

Since the cons tan t K1 i s in i t i a l ly o f unknown va lue , a method o f success ivea p p r o x im a t io n s w i l l b e r e q u i r e d . T h e t e c h n iq u e a d o p t e d i n t h e P a s c a l p r o g r a mR E - D E S , p r o v id e d o n t h e a c c o m p a n y in g P C d is c , f o ll ow s th a t o f N e w to n . A s a fi rs t

es t imate the va lue o f K1 g iven by the app rox imate ana ly t ica l method o f Eqn (5 .22 )may be used to beg in the p rocess . We may then eva lua te Sa and a l so the nearby va lue

$2 given by

m

$2 = Ar E r m i ~ / / g l -t- A K 1 - f ( r i ) (5 .29)i= 1

w h e r e A K 1 i s a smal l inc remen t in K 1 ( e . g . A K 1 / K 1 = 0 .01 ) . A r ev ised es t imate o f

K 1 then fo l lows by ex t r apo la t ion f rom

- s 1( K1 )re vis e d = K I + A K 1 $ 2 _ S 1 ( 5 . 3 0 )

Ap pl ica t ion o f th i s p roced ure to a so lid body swir l w i th m ean ax ial ve loc i ty Cx = 1.0and t ip swir l veloci ty co = 1 .0 p roduces the so lu t ion shown in Tab le 5 .3 , wh ich showst h e p r e c i s e p r e d i c t i o n f r o m c o m p u t e r p r o g r a m R E - D E S c o m p a r e d w i t h t h eapprox imate ana ly t ica l so lu t ion o f Eqn (5 .23 ) . To ach ieve numer ica l accu racy i t i snecessary to in te rpo la te the in i t i a l (r , co) da ta to p rov ide many more r ad ia l s teps . AL a g r a n g i a n i n t e r p o l a t i o n p r o c e d u r e i s i n c lu d e d i n RE - D E S a n d f o r t h e a b o v ecompu ta t ions m = 400 r ad ia l d iv i s ions o f the annu lus were u sed .

Re s u l t s f o r tw o o th e r v o r t e x f l o w s h a v e a l s o b e e n c a l c u l a t e d u s in g RE - D E S a n d

these a re shown in Tab le 5 .4 . Le t u s now cons ider these in tu rn .

E x a m p le 5 . 3 F r e e - v o r t e x f l o w

Since the func t ion f ( r ) , Eqn (5 .24) , is zero for th is case, the ax ial veloci ty must beu n i f o r m a n d e q u a l t o t h e m e a n v e lo c i t y C x. This i s bo rne ou t by the numer ica lp red ic t ion as can be seen f rom Tab le 5 .4 .

Example 5 .4 Cons tan t swi r l ve loc i ty c o

P r o b l e m

Fo l lowing the l ines o f the ana lys i s g iven in Ex am ple 5 .2 fo r so lid body f low, the r e ade ri s inv i ted to der ive the app rox imate r ad ia l equ i l ib r ium so lu t ion fo r the fo l lowingvor tex specif icat ion :

Co = cons tan t "l

ho = cons tan t

s = cons tan t

(5.31)

S o l u t i o n

M a tc h in g Cx at the r .m.s , rad ius , the analy t ical so lu t ion for th is f low is g iven by

c x J l + 2 ( C x t 2x In (5.32)



118 S i m p l i f i e d m e r i d i o n a l f l o w a n a l ys i s f o r a x i a l t u r b o m a c h i n e s

Table 5.3 Co mp arison of radial equil ibrium axial velocity profi les for solid bod y swirl predic ted

by approx im a te ana lys i s and by num er i ca l m e thod ( com pute r p rogram RE-D ES)

r co Cxoo CxooA pprox . m e thod , N um er i ca l m e thod ,

Eq n (5.52) axial velocity

0.4 0.4 1.356 466 1.379 527

0.5 0.5 1.288 410 1.312 666

0.6 0.6 1.200 000 1.226 007

0.7 0.7 1.086 278 1.114 941

0.8 0.8 0.938 083 0.971 130

0.9 0.9 0.734 847 0.776 591

1.0 1.0 0.400 000 0.472 329

Table 5 .4 Radia l equi l ibr ium prof i les predic ted by program RE-DES for f ree-vor tex andconstant swirl flows

Free-vortex swirl Constant swirl velocity

r c o cx Co cx

0.40 2.500 00 1.000 012 1.0 1.495 487

0.45 2.222 22 1.000 008 1.0 1.414 537

0.50 2.000 00 0.999 999 1.0 1.337 9810.55 1.818 18 1.000 005 1.0 1.264 742

0.60 1.666 67 0.999 995 1.0 1.193 965

0.65 1.538 46 1.000 003 1.0 1.124 930

0.70 1.428 57 1.000 004 1.0 1.057 002

0.75 1.333 33 1.000 005 1.0 0.989 579

0180 1.250 00 1.000 001 1.0 0.922 057

0.85 1.176 47 1.000 001 1.0 0.853 780

0.90 1.111 11 1.000 002 1.0 0.783 979

0.95 1.052 63 1.000 002 1.0 0.711 680

1.00 1.000 00 1.000 001 1.0 0.635 532

T h i s i s f o u n d t o b e i n r e a s o n a b l e a g r e e m e n t w i t h t h e p r e c i s e r e s u l t o b t a i n e d f r o m

t h e n u m e r i c a l p r o c e d u r e .

5.3.2 Solution of the radial equilibrium direct pro blem

W e n o w c o n s i d e r t h e ' d i r e c t ' o r ' a n a l y s i s ' p r o b l e m i n w h i c h t h e b l a d e g e o m e t r y a n d

h e n c e f l u i d d e f l e c t i o n i s s p e c i f i e d a n d w e a r e r e q u i r e d t o c a l c u l a t e t h e r e s u l t i n g a x i a l

v e l o c i t y p r o f i l e C x ( r ) . I n r e a l i t y , a s i l l u s t r a te d p r e v i o u s l y i n F i g . 5 . 3 , ra d i a l e q u i l i b r i u m

d e v e l o p s p r o g r e s s i v e l y a s t h e f l u i d p r o c e e d s f r o m - o o t o + o o . S i m p l e r a d i a l

e q u i l i b r i u m t h e o r y , o n t h e o t h e r h a n d , a s s u m e s t h a t e q u i l i b r i u m i s a c h i e v e d

c o m p l e t e l y b y t h e t i m e t h e f l u i d l e a v e s t h e b l a d e t r a i l i n g e d g e . T h e c o n t o u r a b c d ,



5.3 S o l u t i o n o f t h e r a d ia l e q u i li b r i u m e q u a t i o n 119

Cx l

7 3, , , . . . -

V d

Cx2

F ig . 5 .4 Rad ia l eq u i l i b r i um downs t r eam o f a s ta to r

Cx3"-

. . . . - -

F i g . 5 . 4 , w o u l d t h e n t y p i f y t h e c o n s e q u e n t a p p r o x i m a t i o n t o t h e m e r i d i o n a l

s t r eaml ines , imp ly ing tha t t he swi r l ang le a2 r emains cons t an t a long cd . Th i s i s i n

f a c t a g r o s s a n d u n n e c e s s a r y a s s u m p t i o n w h i c h w e w i l l d r o p l a t e r w h e n m o v i n g o nto more soph i s t i ca t ed ana ly s i s i n Sec t ion 5 .4 . On the o ther hand , i t i s he lp fu l t o

p rog ress ana ly t i ca l ly i n s t ages and to l ook now fo r s imp le r ad ia l equ i l i b r ium so lu t ionsto the f low th rough a s t a to r and a ro to r on the p resen t bas i s .

Ra d ia l eq u i l ib r iu m d i rec t a n a lys i s f o r a s in g le s ta to rLet u s cons ider t he s imp le case i l l u s t r a t ed in F ig . 5 .4 where ho and s a re bo th cons t an t

and the swi r l ang le a2 downs t ream o f t he s t a to r i s spec i f i ed as a genera l func t ion

o f r a d i u s t h r o u g h

C02t a n a 2 = ~ = f ( r ) ( 5 .3 3 )

Cx2

T h e r a d i a l e q u i l i b r i u m e q u a t i o n ( 5 . 1 6 ) t h e n b e c o m e s

d c x 2Cx2 --d-~-r+

Cx2 tan O~ dr d--; (rC x2 t an a2 ) = 0

w h i c h m a y b e r e w r i t t e n

dcx2{ 1 + tan 2 r162 -~r

tan ot2 d )+ (r tan a2) Cx2 -- 0 (5 .34)

r d r

T o s u m m a r i s e , t h i s m a y b e e x p r e s s e d a s t h e l i n e a r f i r s t - o r d e r d i f f e r e n t i a l e q u a t i o n

dcx_____g2_dr ~- R(r) Cx e = 0 (5 .35)



120 S i m p l i f ie d m e r i d i o n a l f lo w a n a ly s is f o r a x ia l t u r b o m a c h i n e s

w h e re R ( r ) is a function of radius given by

R ( r ) =

t a n o t 2 d (r t a n o r 2 )

r d r

1 + tan 2 a 2 (5.36)

The general so lu t ion of Eqn (5 .35) is g iven by

Cx2 = K exp ( - f R ( r ) dr) (5.37)

where the cons tan t K mus t be de te rmined f rom the con t inu i ty equa t ion (5 .20) .

Exa mple 5 .5 Cons tan t a z s t a to r

P r o b l e m

Derive an express ion fo r CxE/Cx downstream of a s ta tor g iven that tan Ot2 - - c o n s t a n t .

S o l u t i o nThe func t ion R(r ) , Eqn (5 .36 ) , now reduces to

1 ( t an2a2 ) = s in2a2 _ p

R ( r ) = 1 + tan 20~ r - -

w he re p = sin 20~ . Th u s

f R ( r ) dr = In (re)

and

exp ( - f R ( r ) dr) = r -p

Equat ion (5 .37) thus y ields the so lu t ion

C x 2 = K r - P

Applicat ion of the cont inui ty equat ion (5 .20) then resul ts in

Irt

Cx 7r(r2 - r 2 ) = 27rK r I -p dr

h

27rK

2 - p~ [ r 2 - p _ r2 -p ]

and hence the cons tan t K i s de te rmined th rough

K = C x ( r 2 - r2)(1 - p / 2 )

r -P



5 .3 S o l u t i o n o f t h e r a d ia l e q u i l i b r i u m e q u a t i o n 121

L

O3( -

Cx2

O3v

f

I _ _ . w02 I C02

F i g . 5 . 5 Radia l equ i l i b r ium downs t ream o f a fan ro to r

J J ~ T cl= . --

u

J . ~ uf r

w 2 ~ ~ 2

C x l

cz2

Finally, putting h = rh/rt , we have the solution

Cx2 __ ( 1 - h 2 ) ( sin2 a2 ) (~ tCx

1 - T

s i n 2 a 2 ( 5 . 3 8 )

R a d i a l e q u i l i b r i u m d i r e c t a n a l y s i s f o r a s i n g le r o t o r a x i a l f a n

Let us consider next the flow through the axial fan rotor shown in Fig. 5.5. In thiscase we will assume that the relative outflow angle/32 is specified as a function ofradius. For incompressible flow and zero inlet swirl co l , the Euler pump equat ionmay be written

1. 2 . ( P o2 - - Pol) = r12co2 (5.39)P

If the inlet stagnation pressure Pol is constant we may differentiate the above toobtain

1 d p o 2 = ~[~ d(rc02)

p dr dr

which may be introduced into the radial equilibrium equation (5.14) to yield

dcx2 d ( rco2 ) co 2d ( r [ l - c 0 2 ) dC x 2 T : ['~ dr r dr ( rco2) = r -dr (rc~ (5.40)

But from velocity triangles, Fig. 5.5,

c ~ = r ~ - - w ~ }

Wo2 = Cx2 tan/32(5.41)



122 Simpl i f ied merid ional f low analys is for axia l turbomachines

In t roduc t ion o f these equa t ions in to Eqn (5 .40 ) l eads f ina l ly to the fo l lowingf i r s t -o rde r l inea r d i f fe ren t i a l equa t ion"

dcx2 ( t a n 1 2 d{ 1 + t an 2 f i E } ~ "+ " r d r (r t a n f 1 2 ) ) r - " 21~ ta n f 1 2 (5 .42a)

w h i c h m a y b e s u m m a r i s e d a s

dcx2d----~+ fl(r)Cx2 = fz(r) (5 .42b)

where the two func t ions o f rad ius a re g iven by

f1(r) = ( tan [32 d (r tan f12) ) 1r d r 1 + tan 2/32

21~ ta n f i E

f2(r ) = 1 + ta n 2 fiE

(5.43)

T h e s t a n d a r d p r o c e d u r e f o r s o l u t io n o f E q n ( 5 .4 2 ) is t o m u l t ip l y t h r o u g h o u t b y

e x p ( f f l ( r ) d r ) an t i then in tegra te wi th re spec t to rad ius , re su l t ing in

f f 2 ( r ) ( f f l ( r ) dr) dr + K1x p

Cx2 = (5 .44)

e x p ( f f l ( r ) d r )

A s f o r t h e p r e v i o u s e x a m p l e o f t h e s t a t o r b l a d e r o w , t h e c o n s t a n t K 1 m u s t b ed e t e r m i n e d f r o m t h e c o n t i n u i t y e q u a t i o n ( 5 . 2 0 ) . W e o b s e r v e t h a t t h e s t a t o r s o l u t i o n ,E qn (5 .37 ) , i s s imply a subse t o f the above fo r the case wh en ~ = 0 and henc ef~(r) = 0 . For the f ree -vor tex s t a to r , on the o the r hand , s ince t an /32 = K1/r, f l ( r ) isa l so ze ro and Eqn (5 .44) reduces a s expec ted to Cx2 = K1 = Cx.

Num erical so lu t ion o f the d irect prob lem fo r a fan ro torAlthough a so lu t ion has been ob ta ined above in c losed ana ly t i c fo rm, the in tegra l s

in Eqn (5 .44) would s t i l l i n mos t cases need to be eva lua ted numer ica l ly . In v iewof th i s a be t t e r s t ra tegy i s to so lve Eqn (5 .42) numer ica l ly . Thus a f t e r in tegra t ion theso lu t ion may be expres sed a s

Cx2 = L ( r , r + K1 ( 5 . 4 5 )w h e r e

Ir ]

L(F j, Cx2 -- ( -- Cx 2fl (r j) d- f2 (t]) } d r

h

i~- Ar 2 { - Cx 2fl(ri) + f2(ri) }i = 1

(5 .46)

w h e r e f 1 2 a n d t h u s f l (r i ) a n d f2(ri) a re spec i f i ed a t m equa l ly spaced rad i i f rom rh



Evaluate f l ( r ) and f2(r )

1s t es t imate of L ( r , C x 2 )

assuming Cx2 = C x

1st es t imate of K1

Calculate Cx2

Calculate L(r,Cx2)

!Calcula te K 1

n o

yes

F ig . 5 .6 Flow diagram for numerical solution of radial equil ibrium dow nstream of a rotor



124 S i m p l i f ie d m e r i d i o n a l f l o w a n a ly s is f o r a x ia l t u r b o m a c h i n e s

t o r t w i t h A r = ( r t - r h ) / m . T h e c o n s t a n t o f i n t e g r a t i o n f o ll o w s f r o m t h e c o n t i n u i t yequa t ion (5 .20 ) wh ich l eads to

2 I s tK1 = Cx + (~ _ r2 ) rL(r , Cx2)d r

2A r m

Cx + (r 2 _ r2 ) ~ ri L(r i, Cx2)i= 1

(5 .47)

H o w e v e r , w e n o t e t h a t Cx2 a l so appear s i n t he exp ress ion fo r L ( r , Cx2) , E qn (5 .46) ,and an i t e ra t ive app roach i s r equ i red as shown in F ig . 5 .6 .

T h e c o m p u t e r p r o g r a m R E - A N A L , t h e s o ur c e c o de o f w h i c h is g iv e n o n t h ea c c o m p a n y i n g P C d is c , e x e c u t e s t h is c o m p u t a t i o n a l s e q u e n c e f o r w h i c h s a m p l e o u t p u t

i s g iven in Table 5 .5 .

T ab le 5.5 'Back to back' test using output from design radial equil ibrium program RE-DES

as input to analysis radial equil ibrium program RE-ANAL

Initial input data to

R E - D E S

Output predicted by

R E - D E SFinal output from

R E - A N A L u sin g f 1 2 o o

values f rom RE-DES

(column 4)

r ca2 Cxoo [3200 C x~ C02

0.4 0.4 1.379 529 16.169 731 1.379 893 0.400 1060.5 0.5 1.312 669 20.851 998 1.313 066 0.500 151

0.6 0.6 1.226 010 26.076 781 1.226 347 0.600 1650.7 0.7 1.114 944 32.122 009 1.115 265 0.700 2020.8 0.8 0.971 134 39.481 002 0 .9 7 1 01 0.800 2200.9 0.9 0.776 596 49.209 612 0 .7 7 6 30 0.900 272120 1.0 0.472 335 64.716 982 0 . 4 7 2 13 1.000 165

E xam ple 5 .2 o f a 'so l id body swi r l ' s t a to r i s r ec ons idere d he re w he re co2 i sp ro po r t iona l t o r ad iu s r , co lumn s 1 and 2 . Th e ax ia l ve loc i ty Cxooa n d c o n s e q u e n t e x i t

s w i r l a n g l e / 3 2 0 0 p r e d i c t e d b y p r o g r a m R E - D E S a r e r e c o r d e d i n c o l u m n s 3 a n d 4 . T oc h e c k t h e a c c u r a c y o f t h e t w o c o m p u t e r p r o g r a m s a ' b a c k t o b a c k ' t e s t h a s b e e n

u n d e r t a k e n h e r e i n w h ic h th e/ 32 0 0 v a l u e s o u t p u t f r o m R E - D E S w e r e u s e d a s i n p u tt o th e d i r e c t a n a ly s i s p r o g r a m R E - A N A L , s e t t i n g t h e r o t a t i o n a l s p e e d f ~ = 0 . T h e

o u t c o m e is ta b u l a t e d i n c o l u m n s 5 a n d 6 w h e r e Cxoos h o w s v e r y c l o s e a g r e e m e n t w i t h

t h e r e s u l t s p r e d i c t e d b y R E - D E S , c o l u m n 3 . T h e u l t i m a t e t e s t i s t h e f i n a l p r e d i c t i o n

o f t he ca2 va lues , c o lum n 6 , wh ich a re i n very c lo se ag ree m en t w i th the o r ig ina l des ign

d a t a .

5 . 4 A c t u a t o r d i s c t h e o r y a p p l i e d to a n a x ia l t u r b o m a c h i n e

b l a d e r o wA c t u a t o r d i s c t h e o r y p r o v i d e s a s i m p l e m e a n s f o r i m p r o v e m e n t t o r a d i a l e q u i l i b r i u m

analys i s t o a l low fo r t he p rog ress ive deve lopmen t o f t he ax ia l ve loc i ty p ro f i l e t h rought h e b l a d e r o w a s i l lu s t r a te d b y Fi g. 5 .7 . T h e m e t h o d h a s b e e n e x t e n s i v e ly d o c u m e n t e d



5.4 Actu ato r disc theory applied to an axial turb omac hine blade row 125

D

Cx

l r

l rh

/ /c x Cxo,

F i g . 5 . 7 Actuator disc model of a blade row

o~

~ 2 _ _shed

A

Cl ~ ~ ~2w. . -- , ,

III

by Hor lock (1978) and more de ta i led ana lys i s w i l l be g iven in Chap te r 6 . In th i ssec t ion the bas ic p r inc ip les and f ina l r esu l t s w i l l be p resen ted and app l ied to a s ing leb lade row. In la te r sec t ions the method wi l l be ex tended to a se r ies o f increas ing ly

complex des ign and ana lys i s p rob lems .The concep t o f the ac tua to r d i sc , bo r rowed f rom p rope l le r theo ry , i s i l lu s t r a ted

in F ig . 5 .7 . The mer id iona l d i s tu rbances wh ich p roduce the r ad ia l sh i f t o f thes t r eaml ines a re in f ac t caused by the shedd ing o f vo r tex shee ts y f rom the b ladet r a i l ing edges . The mechan isms under ly ing th i s w i l l be d i scussed in more de ta i l inChap te r 6 . In f ac t the vo r tex shedd ing bu i ld s up p rog ress ive ly f rom the lead ing edgeto the t r a i l ing edge due to any var ia t ion o f the b lade c i r cu la t ion wi th r ad ius . Fo rs imp l ic i ty , however , we wi l l a s sume in s tead tha t the t r a i l ing vo r t ic i ty i s shed

d i s c o n t in u o u s ly i n t h e p l a n e A D o f t h e s o - c a l l e d actuator disc. An ac tua to r d i sc i sthus a s imp le mathemat ica l model o f a b lade row cons is t ing o f a p lane d i scon t inu i tya t wh ich the f lu id def lec t ion and assoc ia ted vo r tex shedd ing a re assumed to occu r

in s t a n t a n e o u s ly .Al te rna t ive ly we cou ld th ink o f an ac tua to r d i sc as a r ea l b lade row wi th the same

cascade sha pe bu t w i th an in fin i te num be r o f b lades Z o f in f in i tes imal cho rd l (i . e .Z ~ ~ as l---~0) . S ince the ac tua to r d i sc r ep resen ts the cen t r e o f vo r tex shedd ing , i tw o u ld s e e m r e a s o n a b l e t o l o ca t e A D in t h e p l a n e o f t h e c e n t r e o f b o u n d c i r c u l a t io nF o f the b lade p ro f i les , i . e . a t the cen t r e o f l i f t . The usua l p rac t ice i s to loca te ADat the one- th i rd b lade cho rd pos i t ion fo r a s ta to r as i l lu s t r a ted in F ig . 5 .7 and a t theha l f cho rd pos i t ion fo r a h igh s tagger ro to r . A l te rna t ive ly the cen t r e o f l i f t cou ld be

c a l c u l a t e d f r o m th e p r e s s u r e d i s t r i b u t i o n s p r e d i c t e d b y t h e p r o g r a m CA S CA D E . I fwe now express the ax ia l ve loc i ty in the fo rm

Cx = Cx + Cx (5..48)



126 Simp li fied m eridional f low analysis for axial turbomachines

ICx

e ( x ) = - ; -Cx=,

1

0 .9

0 .8

0 .7

0.6

0.5

0 .4

0.3

0 .2

0.1

0- 2 -1'.5 -'1 "-0'.5 0 o l s i , ; i s 2

r t - r h

Table 5.6 Actuator disc

coefficients k

rh / r k

0.3 3.2935

0.4 3.23300.5 3.1967

0.6 3.1731

0.7 3.1567

0.8 3.1480

0.9 3.1435

1.0 3.1416

Fig. 5.8 Growth of ax ial veloc i ty perturbat ions through an actuator disc

w he re Cx i s a smal l pe r tu rb a t io n o f the m ean ax ia l ve loc i ty Cx, ac tua to r d i sc ana ly s i ss h o w s t h a t t h e p e r t u r b a t i o n s t h r o u g h o u t t h e a n n u l u s a r e g i v e n b y

c;tC X =

1= ~exp = F ( x ) f o r x < X A Dr t - r h

1= 1 - ~ e xp = F(x) for x > XAD

r t - r h

(5 .49)

Cx= i s t he r ad ia l equ i l i b r ium per tu rb a t io n wh ich ob ta in s as x- -~ oo, F ig . 5 . 7 , and kis a c o n s t a n t , t h e v a l u e o f w h ic h d e p e n d s u p o n t h e h u b / t i p r a t i o o f t h e a n n u l u s ,Tab le 5 .6 .

T h e f u n c t i o n F( x ) h a s b e e n e v a l u a t e d i n F i g . 5 . 8 f o r a n a n n u l u s w i t h rh/rt = 0 .5 ,s h o w i n g h o w t h e a x i a l v e l o c i t y p e r t u r b a t i o n s g r o w e x p o n e n t i a l l y f r o m - o o t o + ~ .

I t is o f par t i cu l a r i n t e re s t t o no te t ha t t he per tu rb a t io ns Cx reach exac t ly ha l f o f t herad ia l eq u i l i b r ium va lue Cx= a t t he p l ane o f t he ac tua to r d i sc XAD = 0 . A t a n y o t h e r

loca t ion ( x , r ) t he ax ia l ve loc i ty may thus be exp ressed in t e rms o f t he r ad ia l

equ i l i b r ium ax ia l ve loc i ty a t t he same rad iu s th rough

C = C x Jr C x o o F ( x - X A D ) = C x -4- ( C x o o - C x ) F ( x - X A D ) (5.5o)

5 . 5 A c t u a t o r d i s c a n a l y s i s f or a s i n g l e r o to r a x i a l f a n

W e a r e n o w i n a p o s i t i o n t o i m p r o v e o n t h e n u m e r i c a l s c h e m e f o r d i r e c t a n a l y s i sc o n s i d e r e d i n t h e l a s t s e c ti o n . L e t u s m a k e t h e f o l l o w i ng a s s u m p t i o n s f o r o u r a c t u a t o r

d i sc model o f t he s ing le ro to r f an , F ig . 5 . 9 "(1 ) The ac tua to r d i sc i s l oca t ed a t t he mid -cho rd pos i t i on XAD.

(2 ) The b l ade re l a t ive ou t l e t f l ow ang le ~ 2 i s d e t e r m i n e d a t t h e t r a i l i n g e d g e

p lane x t e .



5 .5 A c t u a t o r d i s c a n a ly s i s f o r a si n gl e r o t o r a x ia l f a n 127

A

XADD

u , . . - -

r h

Xte

' - V xFig , 5, 9 Location of actuator disc plane (AD ) and trailing ed ge plane (te) for axial fan

Equation (5.41) may then be rewritten

W 0 2 : C x t e tan f12 (5.51)

Thus the radial equilibrium Eqn (5.40) may be modified to read

d cx ~ Wo 2 d

= ~ - - ( r 2 f l - r w o 2 )Cx= d r r d r

tan/32 (2ftr - d )- " C x t e r -~r ( C x t e r tan f12)

(5.52)

If/32 is specified as a function of radius, the above may be written in simplifiedform

dcx~d r = p ( r , C x~ (5.53)

where

p ( r , C x = ) = C x t e ta n f12 ( 2fir -- dCx~ r d'r'r(Cxter tan/32) ) (5.54)

We note that Cxte is a function of Cx= throu gh E qn (5.50), na mely

C x t e = C x + c x ~ F ( x t e - XAD ) = C x J r ( C x ~ - C x ) F(x te - XAD ) ( 5 . 5 5 )

Equation (5.53) may now be integrated with respect to radius to provide a form ofsolution analogous to Eqn (5.45) suitable for iterative numerical analysis, namely

Cx= = L ' ( r , Cx=) + g l (5.56)



12 8 S i m p l i f i e d m e r i d i o n a l f l o w a n a ly s is f o r a x ia l t u r b o m a c h i n e s

w h e r e a t r a d iu s rj

i

rj

L' ( r j , Cx ~) = p ( r , Cx ~)dr

h

J

~ . A r E p ii=1

(5.57)

The p rev ious i t e r a t ive scheme, F ig . 5 .6 , may then be u sed wi th s l igh t mod i f ica t ionto ach ieve a numer ica l so lu t ion fo r Cx~ b y s u c c e s s iv e a p p r o x im a t io n s a n d h e n c e f o rCx a t any o ther ax ia l loca t ion in the annu lus mak ing use o f Eqn (5 .50) .

T w o c o m p u te r p r o g r a m s a r e g iv e n o n t h e a c c o m p a n y in g P C d i s c w h ic h u n d e r t a k eac tua to r d i sc ana lyses fo r s ing le b lade rows . P rog ram AD-ANAL so lves the ' ana lys i s 'p rob lem, p red ic t ing the f l0w th rough a b lade row o f p rescr ibed e f f lux ang le /32 .P r o g r a m A D - D E S s o lv e s t h e o p p o s i t e ' d e s ig n ' p r o b l e m , p r e d i c t i n g t h e e f f l u x a n g l e/32 req uire d in order to gene rate a prescr ib ed swir l veloci ty d is tr ibut io n c02 . Studieswi l l be under taken in the nex t two sec t ions to i l lu s t r a te these des ign and ana lys i sp r o b l e m s .

5.5 .1 Ac tuator disc design of a sol id body swirl s tator

To b r ing ou t the def ic ienc ies o f r ad ia l equ i l ib r ium ana lys i s , the r - c02 da ta g iven inT a b l e 5 . 5 h a v e b e e n u s e d a s i n p u t i n to t h e a c tu a to r d i s c d e s ig n p r o g r a m A D - D E Sa n d t h e r e s u l t s a r e g iv e n i n T a b l e 5 . 7 . A l th o u g h A D - D E S h a s b e e n w r i t t e n t o d e a l

wi th f an ro to r des ign , a s ta to r may a l so be des igned by s imp ly spec i fy ing ze ro speedof ro ta t ion , f l = 0 .Two so lu t ions are i l lus tra ted here as fo l lows:

(1) The rad ia l e qu i l ib r ium so lu t ion , ob ta ined by p lac ing the ac tua to r d i scar ti fic ia l ly a very long way ups t r ea m o f the b lade row (XAD =- -1 0 0 0 wasused here ) .

(2) The ac tua tor d i sc so lu t ion with the fo l lowing locat ions:Lea d ing edge loca ted a t XLE = 0 .0A ct ua tor d isc located at XAD = 0.1Trai l in g e dge locate d at XTE = 0.2

Our des ign a im here i s to p red ic t the b lade e f f lux ang le /32 d is t r ibu t ion wh ich wou ldgen erat e t he specif ied swir l veloci ty c02 g iven in co lu mn 2 with a me an axial veloci tyC x = 1 .0 . Two observa t ions may be made :

( a ) So lu t ion (1 ) i s in c lo se ag reemen t wi th the p rev ious r ad ia l equ i l ib r iumso lu t ion shown in Tab le 5 .5 .

(b ) The t rue b lade t r a i l ing edge e f f lux ang les /32 acco rd ing to ac tua to r d i scanalysis , so lu t ion (2) , d if fer s ignif icant ly f rom the/32~ values a long waydowns t r eam de l ivered by r ad ia l equ i l ib r ium ana lys i s .

5.5.2 A ctuator disc design and analysis of a single stage rotor axial fan

For ou r second s tudy le t u s r econs ider the ax ia l f an i l lu s t r a ted in F ig . 5 .5 wh ichcompr ises a s ing le ro to r on ly . Our a im wi l l be to genera te ve loc i ty t r iang le des igndat a f ro m an in i t ia l specif icat ion of c02 versus rad ius , includ ing also me r id io nal f lowa n a ly s is b y a c tu a to r d i sc th e o r y . P r o g r a m A D - D E S w i ll t h e n b e u s e d t o d e m o n s t r a t e



5.5 Actuator disc analysis fo r a single rotor axial fan 129

Table 5.7 Design of a solid rotation sw irl stator blade row : com parison of radial equilibrium

and a ctuator disc methods

I n i t i a l i n p u td a t a

S o l u t i o n 1Radial equilibrium method

XAD ~ --

Solution 2Actuator d isc method

XLE = 0 .0 , XAD = 0 .1 , XTE = 0 .2

r c02 CxTE fl2oo CxTE [32

0.4 0 .4 1 .379 52 16.169 82 1 .268 81 17.497 78

0.5 0.5 1.312 66 20.852 20 1.221 45 22.261 73

0.6 0.6 1.225 99 26.077 20 1.160 06 27.348 60

0.7 0.7 1.114 91 32.122 79 1.081 39 32.915 60

0.8 0.8 0.971 09 39.482 26 0.979 52 39.239 32

0.9 0.9 0.776 57 49.210 37 0.841 75 46.915 401.0 1.0 0.472 86 64.692 23 0.626 64 57.927 32

h o w c o m p e t i t i v e d e s i g n s m a y b e p o s t u l a t e d f o r d i f f e r e n t t y p e s o f v o r t e x f l o w co2(r)a n d i n p a r t i c u l a r w e w i l l c o m p a r e t h e f r e e - v o r t e x d e s i g n m e t h o d e x p o u n d e d f u l l y i nSec t ion 5 .1 wi th non - f ree-vo r t ex swi r l d i s t r i bu t ions . A su i t ab l e app roach towards the

l a t t e r w o u l d b e t o a d o p t a m i x e d - v o r te x w h i c h c o m b i n e s t h e t w o v o r t e x t y p e s a l r e a d yc o n s i d e r e d , n a m e l y t h e f r e e - v o r t e x a n d t h e f o r c e d - v o r t e x o r s o l i d - b o d y s w i r l . T h u s

let us speci fy

aco2 = - + br (5 .58)r

C o n t r o l o v e r t h e v o r t e x m i x a n d t h u s t h e r a d i a l d i s t r i b u t i o n o f l o a d i n g m a y b e

e x e r c i s e d b y c a r e f u l s e l e c ti o n o f t h e c o n s t a n t s a a n d b . H o w e v e r , a m u c h b e t t e rdes ign s t r a t egy wou ld be to cons ider i n s t ead the work coef f i c i en t ~ , wh ich may be

e x p r e s s e d a s

l ( a )= ~ = U = ~ ~ + b (5 .59 )

By spec i fy ing the w ork coef f i c i en t qJm a t t he m ean rad iu s r m and ~o a t any o the r r ad iu sr0 , Eqn (5 .59 ) may be so lved fo r t he coef f i c i en t s a and b to y i e ld the fo l lowing

e q u a t i o n f o r ~ ( r ) :

( l / r 2 - 1 / r 2 )= ~ m q- ( ~ 0 - ~ m ) 1/rE _ 1/r2 (5 .60)

I f we e l ec t t o spec i fy the f an du ty ( t~m , ~m) a t t he mean rad iu s , t he swi r l ve loc i ty

d i s t r i b u t i o n , f r o m E q n ( 5 . 5 9 ) , b e c o m e s

c02 ~, U ~ rm

C x - t ~ m U m ~ m r m

1 r {t~m rm I/ tm

+ (q Jo - qhn ) ( 1/r2- 1/r2l l r E - 1 / r 2 ) }

(5 .61)



130 S i m p l i f i e d m e r i d i o n a l f l o w a n a ly s is f o r a x ia l t u r b o m a c h i n e s

w h e r e C x i s t he mean ax ia l ve loc i ty . C lose in spec t ion con f i rms tha t t h i s equa t ionc o n f o r m s w i t h t h e o r ig i n a l m i x t u r e o f f r e e -v o r t e x a n d f o r c e d - v o r t e x , E q n ( 5 .5 8 ) , b u t

in s t ead m ake s u se o f muc h m ore he lp fu l i n i ti a l des ign da t a . Thu s Co2 i s now

de te rm ine d in t e rms o f t he des ign du ty (~bm, ~ 'm) a t rm , and fo r a p res cr ibe d w orkcoef f i c i en t q '0 a t any ch osen re fe re nce rad iu s r0.

A t t h i s po in t a t t en t ion migh t u sefu l ly be d i rec t ed to the fo l lowing two spec ia l vo r t excases .

(1) F r e e - v o r t e x s w i r l ( b = 0 ) . A f ree -vo r t ex swi r l d i s t r i bu t ion i s ob ta ine d i f wespec i fy tha t

Co 2 0 rm

C02m ro

and hen ce t/,0 mu s t b e g iven the va lue

= (5 .62)

As shown in Sec t ion 4 .7 fo r ze ro in l e t swi r l , t he r eac t ion o f such a ro to r -on lyf a n b e c o m e s

~ 0R = I

2

fo r t he f r ee -vo r t ex f an

[4.41]

Thu s th e r ad ia l var i a t ion o f bo th wo rk coef f i c i en t t/, and reac t ion wi l l be ve rycons iderab le fo r a f r ee -vo r t ex f an , as was shown in Tab le 5 .1 .

(2) F o r c e d - v o r t e x o r c o n s t a n t r e a c t io n f a n r o to r . A pure so l id -body swi r l o rforc ed- vor tex w i ll be de l ivere d b y E qn (5 .61) by spec i fy ing q 'o = q'm at there fe rence rad iu s r0 . In t h i s case Eqns (5 .60 ) and (5 .61 ) r educe to

~' = ~'m (5. 60 a)

c02 qJm r= (5 .61a)

Cx q~m rm

Thu s a f an ro to r d es igned to gener a t e a fo rced -vo r t ex ex i t swi rl c0 2 wi ll havethe same work coef f i c i en t a t a l l r ad i i . F rom Eqn (4 .41 ) we see tha t t he

reac t ion R wi l l a l so be cons t an t a t a l l r ad i i and equa l t o R = 1 - ~m/2 .

T w o f a n d e s i g n s h a v e b e e n c o m p l e t e d o n t h i s b a s i s u s i n g t h e p r o g r a m A D - D E S

f o r t h e c o m m o n d a t a s p e c i f ic a t io n g i v e n a t t h e h e a d o f T a b l e 5 .8 . F o r t h e s e d e s i g n s

a hub / t i p r a t io h = 0 .4 was chose n and rm was se t a t t he r .m . s , r ad iu s . A t r m these l ec t ed du ty coef f i c i en t s w e r e t ~m = 0 .5 and ~tm = 0 . 3 . T h e h u b s e c t i o n w a s c h o s e n

fo r t he r e p res en ta t ive r ad iu s r0 a t wh ich the w ork coef f i c i en t was se t a t q'0 = 1 .0 8 7 5fo r t he f r ee -vo r t ex f an ( see Eqn (5 .62 ) ) and a t 0 .5 fo r t he mixed -vo r t ex f an .



5 .5 A c t u a t o r d i s c a n a l y s is f o r a s i n gl e r o t o r a x ia l f a n 131

Table 5.8 Vortex and loading specificat ions for two al ternative fan designs

C o m m o n d e s ig n d a ta :

Hub/ t ip ra t io h = rh/rtrm/ r (r .m.s. radius)

Duty coefficients at rm:

= 0 . 4= 0.761 577

t~m "- 0 .5

q,m --- 0 .3

Leading edge locat ion X l e / r - - 0 . 0Trail ing edge location X t e / r t - - 0 . 2

Ac tuato r disc location X AD/rt = 0.1

Refere nce radius r0 = 0.4

Free-vor tex des ign Mixed-vortex design

r/ r Co2 /C ~b R Co2/Cx ~b R

0.4 0 1.142 37 qJ0 = 1.087 5 0.45 6 25 0.630 27 qJ0 = 0.60 0 00 0.70 0 00

0.45 1.015 44 0.859 259 0.570 37 0.606 30 0.513 05 0.743 48

0.50 0.913 89 0.696 000 0.652 00 0.592 00 0.450 86 0.774 570.55 0.830 81 0.575 207 0.712 40 0.584 74 0.404 84 0.797 58

0.60 0.761 58 0.483 333 0.758 33 0.582 75 0.369 84 0.815 08

0.65 0.702 99 0.411 834 0.794 08 0.584 82 0.342 60 0.828 70

0.70 0.652 78 0.355 102 0.822 45 0.590 08 0.320 99 0.839 50

0.75 0.609 26 0.309 333 0.845 33 0.597 88 0.303 56 0.848 22

0.80 0.571 18 0.271 875 0.864 06 0.607 76 0.289 29 0.855 36

0.85 0.537 58 0.240 830 0.879 59 0.619 35 0.277 46 0.861 27

0.90 0.507 72 0.214 815 0.892 59 0.632 35 0.267 55 0.866 23

0.95 0.481 00 0.192 798 0.903 60 0.646 56 0.259 16 0.870 42

1.00 0.456 95 0.174 000 0.913 00 0.661 78 0.252 00 0.874 00

F r o m T a b l e 5 . 8 it c an b e s e e n t h a t t h e r e is c o n s i d e r a b l e r a d i a l v a r i a t i o n o f q , a n d

r e a c t i o n R f o r t h e f r e e - v o r t e x d e s ig n a s e x p e c t e d . O n t h e o t h e r h a n d t h e m i x e d - v o r t e x

d e s i g n e x h i b it s o n l y m o d e s t r a d i a l v a r i a t i o n o f R a n d r e d u c e d s p r e a d o f t h e w o r k

c o e f f i c i e n t q J . I n d e e d , t h e o b j e c t i v e o f t h e m i x e d - v o r t e x f a n d e s i g n h e r e i s t o s h i f t

a e r o d y n a m i c l o a d i n g t o w a r d s t h e b l a d e t i p s b y i m p o s i n g g r e a t e r w o r k c o e f f i c i e n t s ,

a t t h e s a m e t i m e u n l o a d i n g t h e b l a d e r o o t r e g i o n . T h e s e c o m p e t i n g f a n d e s i g n s m a k e

a n i n t e r e s t i n g c o m p a r i s o n , a s i l l u s t r a t e d b y T a b l e 5 . 9 , g i v i n g a n i n s i g h t i n t o t h e

d e s i g n e r ' s a r t .

T h e p r e d i c t e d v e l o c i t y t r i a n g l e d a t a f o r t h e s e t w o d e s i g n s a r e p r e s e n t e d i n T a b l e

5 . 9 . F r o m t h e s e d a t a t h e p i t c h / c h o r d r a t i o t / l w a s a l s o c a l c u l a t e d a s s u m i n g a d i f f u s i o n

f a c t o r o f D F = 0 .5 a n d u s i n g E q n ( 2 .2 8 ) . A b l a d e c h o r d s c al e w a s t h e n o b t a i n e d

a c c o r d i n g t o t h e f o l l o w i n g d e f i n i t i o n : c h o r d s c a l e = ( c h o r d a t r ) / ( c h o r d a t rh ) . T h e

f o l lo w i n g i t e m s o f c o m p a r i s o n b e t w e e n t h e t w o d es i g n s m a y b e d r a w n o u t f r o m t h e

d e t a i l e d d e s i g n d a t a i n T a b l e 5 . 9 "

( 1 ) T h e a x i a l v e l o c i t y p r o f i le C xte a t t h e t r a i l i n g e d g e p l a n e f o r t h e f r e e - v o r t e x

f a n i s c o n s t a n t a s w e w o u l d e x p e c t f o r t h i s c o n s t a n t w o r k i n p u t d e s i g n .

F o r t h e m i x e d - v o r t e x f a n , o n t h e o t h e r h a n d , Cxt e v a r i e s e n o r m o u s l y f r o m

o n l y 0 . 7 2 3 1 8 a t t h e h u b t o 1 . 2 4 5 0 7 a t t h e t i p . T h i s i s t h e c o n s e q u e n c e o f

t h e i n c r e a s e o f sp e c if ic w o r k i n p u t f r o m h u b t o ti p f o r t h e m i x e d - v o r t e x

d e s i g n i n t r o d u c e d b y th e f o r c e d v o r t e x c o m p o n e n t o f t h e s p e ci fi e d sw i rlc 0 2, T a b l e 5 . 8 .

( 2) W e n o t i c e c o n s i d e r a b l e v a r i a t i o n o f t h e r e l a ti v e o u t l e t a n g l e / 3 2 f o r t h e

f r e e - v o r t e x d e s i g n , w h i c h h a s a d r a m a t i c e f f e c t u p o n t h e r o t o r d e f l e c t i o n e R .



132 S i m p l i f i e d m e r i d i o n a l f l o w a n a l y s i s f o r a x i a l t u r b o m a c h i n e s

Table 5.9 Comparison of free-vortex and mixed-vortex fan designs

Free-vortex design

r/r Cxte]Cx c x J Cx [31 [32 ~2

0.4 1.0 1.0 46.410

0.5 1.0 1.0 52.708

0.6 1.0 1.0 57.599

0.7 1.0 1.0 61.455

0.8 1.0 1.0 64.546

0.9 1.0 1.0 67.067

1.0 1.0 1.0 69.154

--5.252

21.761

39.149

49.852

56.827

61.682

65.250

48.802

42.424

37.292

33.136

29.734

26.918

24.558

r/rt eR q~te ~ t/l Chord scale

0.4 51.661 0.951 97 1.087 50 0.4886 1.0000

0.5 30.947 0.761 58 0.696 00 0.5503 1.1097

0.6 18.450 0.634 65 0.483 33 0.9359 0.7830

0.7 11.603 0.543 98 0.355 10 1.5460 0.5530

0.8 7.719 0.475 99 0.271 87 2.3257 0.4201

0.9 5.385 0.423 10 0.214 81 3.2494 0.3383

1.0 3.904 0.380 79 0.174 00 4.3049 0.2837

Mixed-vortex design

r/r Cxte[Cx cx J C x 1 3 1 1 3 2 ~2

0.4 0.723 18 0.609 17 49.854 30.157 45.975

0.5 0.783 91 0.694 91 55.247 42.609 40.428

0.6 0.862 89 0.806 42 59.086 49.008 35.853

0.7 0.952 01 0.932 24 61.932 52.667 32.332

0.8 1. 046 82 1. 066 11 64.119 54. 996 29. 686

0.9 1.144 95 1.204 65 65.851 56.520 27.6961.0 1.245 07 1.346 01 67.255 57.632 26.182

r/rt eR q~te ~b t/ l Chord scale

0.4 19. 697 0. 688 45 0. 600 00 1. 2604 1.0000

0.5 12.638 0.597 01 0.450 86 1.8468 0.8531

0.6 10.078 0.547 63 0.369 84 2.1231 0.8905

0.7 9.265 0.517 88 0.320 99 2.0783 1.0613

0.8 9.153 0.498 27 0.289 29 1.8794 1.3413

0.9 9.330 0.484 43 0.267 55 1.6444 1.7246

1.0 9.623 0.474 11 0.252 00 1.4230 2.2142



5 . 6 A c t u a t o r d is c t h e o r y a p p l i e d t o m u l t ip l e b l a d e r o w s - t h e d e s ig n p r o b l e m 133

( 3 )

(4)

Thus the cos t o f demand ing cons t an t spec i f i c work a t a l l r ad i i i s t ha t eR mus tvary f r om a m ere 3 .9 0 4 ~ a t t he t i p sec t ion to an un rea l i s t i ca l ly h igh va lue o f

51 .661 ~ a t t he hub . To ach ieve such h igh def l ec t ion f rom a d i f fu s ing cascade

wou ld in f ac t be very d i f f i cu l t and wou ld be assoc ia t ed wi th h igh lo sses . Them i x e d - v o r t e x f a n , o n t h e o t h e r h a n d , e x h i b i t s m u c h m o r e m o d e s ta e r o d y n a m i c r e q u i r e m e n t s w i t h a m u c h m o r e u n i f o r m d e f l e c t io n e R i n t h e

rang e o f 10 ~ to 20 ~T h e d e f l e c t i o n l e v e l s a r e r e f l e c t e d t o s o m e e x t e n t i n t h e r e c o m m e n d e d t /lv a l u e s f o r t h e t w o f a n s a n d i n t h e c o n s e q u e n t v a l u e s o f c h o r d s c a l e . T h el a t t e r s h o w t h a t t h e f r e e - v o r t e x f a n b l a d e w i l l t a p e r c o n s i d e r a b l y f r o m h u b t ot ip wh ich i s obv ious ly advan tageous fo r ca r ry ing cen t r i fuga l s t r esses . Fo r t hem i x e d - v o r t e x f a n , o n t h e o t h e r h a n d , t h e b l a d e c h o r d s a c t u a l l y i n c r e a s e w i t hr a d i u s i n o r d e r t o a c c o m m o d a t e t h e g r e a t e r s p e c i f i c w o r k a n d i t s a s s o c i a t e da e r o d y n a m i c l o a d i n g .

B e c a u s e t h e s p e c i f i c w o r k i n p u t i n c r e a s e s t o w a r d s t h e o u t e r r a d i i o f t h em i x e d - v o r t e x f a n , t h e g r e a t e r o u t l e t s t a g n a t i o n p r e s s u r e p r o d u c e s t w o e f f e c t s .

F i r s t l y t he ax ia l ve loc i ty p ro f i l e i ncreases i n t he t i p r eg ion . Second ly , and

c o n s e q u e n t l y , t h e m a s s w e i g h t e d p o w e r i n p u t i s g r e a t e r f o r th e m i x e d - v o r t e xdes ign . Thus in t eg ra t ing f rom hub to t i p , t he average spec i f i c work inpu t s fo r

the two des igns a re as fo l lows :

Spec i fi c w ork inpu t , f r e e -vo r t ex des ign = 1.20 W kg -1

Spec i fi c wor k inpu t , m ixed -vo r t ex des ign = 1 .247 3 W kg -1

T h e m i x e d - v o r t e x f a n i s t h u s c a p a b l e o f t r a n s m i t t i n g m o r e p u m p i n g p o w e rin to the f lu id , a l t hough un l ike the f r ee -vo r t ex i t wi l l no t be un i fo rmlyd i s t r i b u t e d b u t w i l l b e c o n c e n t r a t e d m o r e t o w a r d s t h e o u t e r r a d i i .

T h e b l a d e p r o fi le s a t h u b , a r i t h m e t i c m e a n a n d t i p s e ct i on s f o r th e s e t w o f a n s h a v eb e e n d e s i g n e d by m e a n s o f t h e p r o g r a m C A S C A D E , r e s u lt in g in t h e c a m b e r a n d

s t a g g e r a n g l e s s h o w n i n T a b l e 5 . 1 0 w h i c h a r e r e q u i r e d t o a c h i e v e t h e c o r r e c t o u t l e t

ang le /32 wi th shock - f ree in f low.The resu l t i ng b l ade p ro f i l es a re shown in F ig . 5 .10 , wh ich revea l s t he marked

d i f f e r e n c e i n a e r o d y n a m i c d e s i g n r e q u i r e m e n t s f o r t h e s e t w o t y p e s o f f a n v o r t e xd e s i g n . I n p a r t i c u l a r t h e f r e e - v o r t e x f a n b l a d e i s b o t h s t r o n g l y t w i s t e d a n d t a p e r e d ,w h i l e t h e m i x e d - v o r t e x f a n b l a d e h a s m i n i m a l t w i st a n d i n fa c t in c r e a s e d b l a d e c h o r dat the t ip sect ion .

5 ,6 A c t u a t o r d is c th e o r y a p p l ie d t o m u l t i p le b l a d e r o w s -t h e d e s i g n p r o b l e m

S o f a r w e h a v e c o n s i d e r e d t h e m e r i d i o n a l f l o w i n d u c e d b y a s i n g l e b l a d e r o w o n l y ,

such as t he in l e t gu ide vanes shown in F ig . 5 . 7 o r t he ax ia l f an ro to r , F ig . 5 . 9 . Fo r

these s imp le con f igu ra t ions the vo r t ex f low crea t ed by the b l ade row i s convec ted

u n h i n d e r e d d o w n s t r e a m a n d w i l l g r o w p r o g r e s s i v e l y t o w a r d s t h e r a d i a l e q u i l i b r i u m

sta t e as x ~ oo in the m an ne r i l l u s t r a t ed by F ig s 5.7 and 5 .8 . M ore f r equ en t ly an ax ia l

t u r b o m a c h i n e w i ll c o m p r i s e s e v e r a l b l a d e r o w s e a c h d e s i g n e d t o d e v e l o p a n e w v o r t e x

swir l co = f ( r ) i n o r d e r t o c o n t r o l t h e e n e r g y t r a n s f e r b e t w e e n b l a d e s a n d f l u i d .

H o r l o c k ( 1 9 5 8 , 1 9 7 8 ) d e m o n s t r a t e d t h e u s e o f m u l t i p l e a c t u a t o r d i s c s t o m o d e l t h ec o n s e q u e n t i a l b l a d e r o w i n t e r f e r e n c e a n d t o p r e d i c t t h e c o m p l e x m e r i d i o n a l f l o w f o r

t h e w h o l e a s s e m b l y .A su i t ab l e ac tu a to r d i sc mode l t o s imu la t e a two-s t age ax ia l f an is show n in F ig .



13 4 S i m p l i f ie d m e r i d i o n a l f lo w a n a ly s is fo r a x ia l t u r b o m a c h i n e s

I

, >I )i re c ti o n o f m o t i ~

( a ) ( b )

F ig . 5 . 1 0 C omparison of fan rotor blade geometries for (a) free-vortex and (b) mixed-vortex

designs

T ab le 5.10 Cascade design parameters for free-vortex and mixed-vortex fan rotor design

selected to achieve shock-free inflow , using the C4 profile an d circular arc ca mb er 0

Design Section r/r t /l A 0

Free-vortex

Mixed-vortex

H ub 0.4 0 .4886 18.44 58

M ean 0.7 1.5460 54.2 32Tip 1.0 4.3049 65.0 40

H ub 0.4 1.2604 38.5 38M ean 0.7 2.0783 55.62 34

Tip 1.0 1.4230 61.4 30

5 .11 and we wi l l cons ider here the d e s i g n p r o b l e m , w h i c h m a y b e s t a t e d a sfo l lows:

(1 ) T he swi r l d i s t r i bu t ions Co l , c0 2 e t c . gene ra t ed by each b l ade row a rep r e s c r i b e d a s f u n c t io n s o f ra d i u s a n d a r e a s s u m e d t o b e c r e a t e d a t t h e

a c t u a t o r d i s c p l a n e s A D 1 , A D 2 e t c .

(2 ) The resu l t i ng ax ia l ve loc i ty p ro f i l es and swi r l ang les a re t hen to be ca l cu la t ed

fo r t he l ead ing a nd t r a i l i ng edge p l anes o f the b l ade rows , X le and X te.

W e o b s e r v e f r o m F i g . 5 . 1 1 t h a t t h e v o r t e x f i e l d e m a n a t i n g f r o m e a c h a c t u a t o r d i s c

i s i n e f fec t t e rmina ted by the nex t ac tua to r d i sc and rep laced by a new vo r t ex f i e ld .

To s imp l i fy mat t e r s a t t h i s s t age l e t u s cons ider f i r s t t he s ing le vo r t ex f i e ld bounded

by just the f i rs t two actuator d iscs , F ig . 5 .12 .

A s i l l u s t r a t e d a b o v e , t h e v o r t e x f i e l d b o u n d e d b y a c t u a t o r d i s c s A D 1 a n d A D 2

m ay be t r ea t e d as t he superp os i t i on o f vo r t ex f ie ld s fo r two i so l a t ed ac tua to r d i scs

b o t h e x t e n d i n g t o x = ~ . T h e v o r t e x f ie ld e m a n a t i n g f r o m A D 2 h e r e is t h e n e g a t i v eo f t h e v o r t e x f i e l d e m a n a t i n g f r o m A D 1 . T h u s t h e f i r s t t a s k r e q u i r e d f o r s o l u t i o n

o f t he mer id iona l f l ow i s ca l cu la t ion o f t he r ad ia l equ i l i b r ium so lu t ion fo r t he vo r t ex

f i e ld c rea t ed by AD1 , y i e ld ing the ax ia l ve loc i ty C x ~ l = C x + C x ~ l . The ax ia l ve loc i ty




T h e l a s t t e r m a c c o u n t s f o r t h e v o r t e x f i e l d c r e a t e d a t t h e l a s t a c t u a t o r d i s c A D 4w h i c h is a s s u m e d t o e x t e n d t o x = oo. F r o m t h is d i s c u s si o n w e m a y s e t o u t t h e s i m p l e

f l o w d i a g r a m g i v e n i n F i g . 5 . 1 3 t o s u m m a r i s e t h e v a r i o u s s t a g e s r e q u i r e d o f t h e

d e s i g n p r o c e s s . T h e c o m p u t e r p r o g r a m M U L T I h a s b e e n w r i t t e n t o p e r f o r m t h i sd e s i g n s e q u e n c e w h i c h w i ll n o w b e i l l u s t ra t e d b y c o n s i d e r i n g t h e d e s i g n o f a t w o - s t a g e

axia l fan .

Specify ann ulus geo m etry rh, rt , l ' l, C x and for each blade row X l e , X t e , X A D

ISpecify Co as a funct ion of radius downstream of each blade row

I

Solve the radial equi l ibrium equat ion for region downstream of each blade row

ICalculate the axial velocity profiles at le and te locations, Eqn (5.64)

ICalculate velocity triangle data and also ~b, q~, t /l etc. versus radius for each blade row

Fig . 5 . 1 3 F low diagram for mer idional analysis and design of multi -stage fan by actuator disc

theory

5.6.1 Theory for constant specific w ork multi-stage axial fans

I n S e c t i o n 5 . 5 . 2 a n a l y s i s w a s d e v e l o p e d f o r a s i n g l e r o t o r a x i a l f a n f o r w h i c h t h ed o w n s t r e a m v o r t e x f i e l d w a s f o r m e d o f a m i x t u r e o f f l e e - v o r t e x a n d f o r c e d - v o r t e x

s w i r l , E q n ( 5 . 5 8 ) . T h i s s t r a t e g y m a y b e e x t e n d e d t o m u l t i - s t a g e a x i a l f a n s o r

c o m p r e s s o r s b y s p e c i f y i n g t h e s w i r l d o w n s t r e a m o f t h e s t a t o r s a n d r o t o r s t h r o u g h

a

= - - + b ro 1r

a

C o2 = - + b rr

d o w n s t r e a m o f a s t a t o r

d o w n s t r e a m o f a r o t o r

(5 . 65)

Appl i ca t i on o f t h i s t o t he t wo- s t age f an i l l u s t r a t ed i n F i g . 5 . 11 w i l l r e su l t i n i den t i ca l

s t a g e s , e a c h a b s o r b i n g c o n s t a n t s p e c i f i c w o r k a t a l l r a d i i . T h u s f r o m t h e E u l e r p u m p

e q u a t i o n f o r c o m p r e s s i b l e f l o w , E q n ( 4 . 3 ) , t h e s p e c i f i c w o r k i n p u t o f o n e s t a g e a t

r ad i us r i s g i ven by

lYC(J kg -1 ) = Aho = U ( c o 2 - c o l )

= 2 a l ) = c o n s t a n t (5 . 66)

T h u s t h e f a n w i ll d e l i v e r t h e s a m e s t a g n a t i o n e n t h a l p y r i s e A h o f o r a ll m e r i d i o n a l

s t r e a m l i n e s f r o m h u b t o c a s in g , t h e r e b y p r e v e n t i n g t h e p o s s i b l e a c c u m u l a t i o n o f r a d i a l



5 .6 A c t u a t o r d i sc t h e o ry a p p l ie d to m u l t ip l e b la d e r o w s - t h e d e s ig n p r o b l e m 137

s t r o n g g r a d i e n t s d h o / d r and thus s t rong var i a t ions in t he ax ia l ve loc i ty p ro f i l e a t ex i tf rom the f an . Th i s s ty l e o f vo r t ex des ign c l ea r ly o f fe r s g rea t a t t r ac t ions a l though there

a r e o t h e r l i m i t a t i o n s a s w e s h a l l s e e . T h e c o n s t a n t a m a y b e e v a l u a t e d i n t e r m s o f

a speci f ied du ty (~bm, ~ 'm) at th e m ea n or r .m .s , rad iu s rm s ince

Aho 2a l l 2a t~ m

~ t m - - " ~ m = U2 -" Cx rm

a n d t h u s

a rm ~m= (5 .67)

C x 2 ~m

The cons t an t b may a l so be exp ressed in t e rms o f u sefu l i n i t i a l des ign inpu t var i ab lesb y r e f e r e n c e t o t h e v e l o c i t y t r i a n g l e s a t t h e m e a n r a d i u s , F i g . 4 . 6 . T h u s a d d i n g E q n s

( 5 . 6 5 ) w e o b t a i n f o r t h e m e a n r a d i u s r m ,

b Col § co2 1 - R m= = (5 .68)

C x 2 C x r m q~mrm 9

The d imens ion less swi r l ve loc i t i es c o / C x a r e n o w p r e s c r i b e d a t a l l o t h e r r a d i i b yin t roduc ing these r esu l t s i n to Eqn (5 .65 ) , r esu l t i ng in

CO 2 -~ m ( ~ - ~ ) 1 - - R m ( r ~ )Cx - - ( 4 - ) § t~ m .... ( 5 . 6 9 )

w i t h ( - ) f o r s t a to r s a n d ( + ) f o r r o t o r s . T h e v o r t e x f ie ld is t h u s d e t e r m i n e d e n t i r e l y

by the se l ec t ion o f t he ke y overa ll des ign du ty var i ab les a t t he r .m . s , r ad iu s rm , nam ely

~bm, ~ 'm and the re ac tio n Rm.

5.6 .2 Sample design of a two-stage constant spec if ic wor k axia l fan

To i l l u s t r a t e t he above ana ly s i s , a two-s t age f an wi l l be des igned fo r t he fo l lowing

overal l speci f icat ion :

H u b r a d i u s r h = 0 .6Tip rad iu s r t = 1 .0

r .m .s , rad iu s rm = W'(~h + ~) / 2 = 0 .824 62

A t rm, ~bm = 0.5~ tm - 0 . 2 5R m = 0 .6

The ax ia l l oca t ions o f l ead ing edge , t r a i l i ng edge and ac tua to r d i scs fo r t he fou r b l ade

rows a re spec i f i ed as i n Tab le 5 .11 fo r a f a i r ly t i gh t ly packed mach ine wi th a good

d e a l o f m e r i d i o n a l i n t e r a c t i o n b e t w e e n t h e b l a d e r o w s .

The resu l t i ng des ign swi r l d i s t r i bu t ions fo r s t a to r s and ro to r s as ca l cu la t ed wi th theP a s c a l p r o g r a m C O N S T W K , g i v e n o n t h e a c c o m p a n y i n g P C d i s c , a r e s h o w n i n F i g .

5 .14 .

Thus a f a i r ly modes t swi r l c o / C x i s i n t roduced in t he d i r ec t ion o f ro t a t ion by the



138

co

c~

S i m p l i f ie d m e r i d i o n a l f l o w a n a ly s is f o r a x ia l t u r b o m a c h i n e s

1 . 2

0 . 8

0 . 6 :

0 . 4

0 . 2

, . ., d k' . . . . " d r " ~

. ... de"'" ...de~ ....

..,...--"

. , . . , ,-", , . , , - ",..--

o0 . 6 0 : 7 0 : 8 0 '. 9 1

r a d i u s r

I s t ~ t o ~ n o . 1 1 I S t a = n o .2 1

Xle~

x . . _ _ . ~ , _ ~ ~--- --~ ~ [ [_J ] ~R o t o roJ ,RotorNo'.2,

I~ Stator -~-' rotor !

Fig . 5 .14 Des ign swi r l d is t r ibu t ions downst ream of exam ple two-stage axial fan stator and rotor

blade rows

Table 5.11 A xial location of tw o-stage fan blade ro ws and equivalent actua tor discs

Item Stator No. 1 Ro tor No. 1 Stator No. 2 Ro tor No. 2

Leading edge Xle 0.0 0.15 0.3 0.45Trailing edge xte 0.1 0.25 0.4 0.55

Ac tuator disc XAD 0. 05 0.2 0.35 0.5

f i r s t s ta to r to p recond i t ion the en t ry f low to the f i r s t ro to r . On the o ther hand , f a i r ly

subs tan t ia l swi r l ve loc i t i es o f the o rder c o / C x ~ 1 .0 e m a n a t e f r o m R o to r N o . 1 a n d

the pa t te rn i s r epea ted fo r the second s tage . The ax ia l ve loc i ty p ro f i les p red ic ted by

a c tu a to r d i s c t h e o r y , u s in g c o m p u te r p r o g r a m M U L T I , a r e s h o w n in F ig . 5 . 1 5 f o r

the lead ing and t r a i l ing edge p lanes , toge ther wi th the r ad ia l equ i l ib r ium p ro f i les .The fo l lowing observa t ions may be made f rom these r esu l t s .

(1 ) The r ad ia l equ i l ib r ium ax ia l ve loc i ty p ro f i les a re iden t ica l fo r the r eg imesd o w n s t r e a m o f s t a to r s 1 a n d 2 a n d d o w n s t r e a m o f r o to r s 1 a n d 2 a s o n ewou ld expec t fo r iden t ica l p rescr ibed swir l d i s t r ibu t ions .

(2) Th e r ad ia l equ i l ib r ium p ro f i les s lope muc h more hea v i ly dow ns t r e am o f thero to r s due to the s t ronger vo r tex f lows .

(3 ) The ac tua to r d i sc smoo th ing e f f ec t t ends to r educe the lead ing and t r a i l ingedge p ro f i le s lopes fo r the ro to r s wel l be low the r ad ia l equ i l ib r ium va lues .

(4 ) The r ever se i s t rue fo r s ta to r 2 . Be ing sandwiched be tween two ro to r s , i t sax ial veloci ty prof i le s lope is greater even than that of i ts own radialequ i l ib r ium p ro f i le .

(5 ) S ta to r 1 , be ing sub jec t to les s mu tua l b lade row in te r f e rence , exh ib i t s on ly

modest prof i le s lopes at Xle and Xte.(6 ) The r ad ia l equ i l ib r ium so lu t ions a lone wou ld g ive a qu i te inaccu ra te

p red ic t ion o f the mer id iona l f low wh ich i s dear ly s t rong ly in f luenced bym u tu a l i n t e r f e r e n c e b e tw e e n t h e b l a d e r o w s .



5 .6 A c t u a t o r d i sc t h e o r y a p p l i e d t o m u l t i p le b l ad e r o w s - t h e d e s ig n p r o b l e m 13 9

F ig . 5 . 1 5 Axial velocity prof iles at various locat ions in a two-stage fan

Now we wou ld expec t the two s ta to r des igns to be qu i te d i f f e ren t s ince s ta to r 1

receives zero swir l a t in let while s ta tor 2 has to absorb the s t rong swir l ing f lowemerg ing f rom ro to r 1 . On the o ther hand the two ro to r s r ece ive and e jec t iden t ica lswir l veloci t ies Col and c02 and we wo u ld hope there fo re to be ab le to adop t ide n t ica lb lade p ro f i le geomet ry . Unfo r tuna te ly , however , as shown by F ig . 5 .15 , the ax ia lveloci ty prof i les for the two ro tors do in fact d if fer , resu l t ing in s l ight ly d if ferentve loc i ty t riang les . Th is i s bo rn e ou t by the ta bu la t ion o f p red ic ted r e la t ive in f low andoutf low angles g iven in Table 5 .12 .

5.6 .3 Meridional f low reversals due to excess ive vortex swir l

As a l r eady exp la ined wi th r e fe rence to the r ad ia l equ i l ib r ium equa t ion (5 .14 ) andas i l lu s t r a ted in Example 5 .3 , Sec t ion 5 .3 .1 , the ax ia l ve loc i ty Cx i s cons tan t fo r af r ee -vo r tex f low whatev er the vo r tex s t r eng th , w h ich mak es i t a very a t tr ac t ive des ignop t ion , espec ia l ly fo r tu rb ines . Fo r non - f r ee-vo r tex f lows , on the o ther hand ,excess ive ly h igh swir l d i s t r ibu t ions may p roduce such s t rong mer id iona l d i s tu rbances




Table 5.12 Relative flow angles predicted for the two-stage fan rotors

Rotor No. 1 Rotor No. 2

r / r t fl l ~ f12 ~ fl l ~ f12 ~

0.6 45.69 22.82 43.77 21.84

0.7 49.44 31.83 48.18 31.02

0.8 53.88 40.82 53.50 40.55

0.9 59.06 50.35 59.95 51.111.0 65.49 61.69 68.44 64.41

t h a t t h e a x i a l v e l o c i t y c o u l d b e c o m e n e g a t i v e a t t h e h u b o r t h e c a s i n g d e p e n d i n g o nt h e v o r t e x t y p e . I n s u c h s i t u a t i o n s t h e r e a l f l o w w o u l d b r e a k d o w n a n d r e v e r s e .

C o n s e q u e n t l y n o s o l u t i o n t o t h e r a d i a l e q u i l i b r i u m e q u a t i o n w o u l d b e p o s s i b l e . F o r

e x a m p l e , t h e a p p r o x i m a t e s o l u t i o n f o r s o l i d b o d y s w i r l , E q n ( 5 . 2 3 ) , i n d i c a t e s t h a t

C x t / C x = 0 at the t ip rad ius r t i f the t ip sw ir l veloci ty i s set a t C o t / C x = 1 .091 089 wi tha hub/ t ip ra t io h = 0 .4 .

Th i s p rob lem rep resen t s a r ea l phys i ca l l im i t on p rac t i ca l des ign wh ich can be

d e t e c t e d d u r i n g n u m e r i c a l a n a l y si s b u t is q u i te d i f fi cu lt to p r e d e t e r m i n e . F o r e x a m p l e ,

fo r t he f an du ty spec i f i ed in Sec t ion 5 .6 .2 a des ign i s imposs ib l e fo r a hub t i p r a t ioh < 0 .5 and the p r og ram M U LT I has d i f f icu l ty cop ing wi th such a spec i f i ca t ion and

c a n n o t c o m p u t e C x va lues in r ever sed f low reg ions . I t i s t hus essen t i a l t o avo id suchf l o w r e g i m e s a n d t h e r e a r e t w o o p t i o n s a v a i l a b l e t o t h e d e s i g n e r :

(1 ) P rescr ibe a l ess power fu l vo r t ex type .

(2 ) Increase the hub / t i p r a t io .

We wi l l now pu rsue the f i r s t o f t hese two op t ions .

5.6 .4 Pow er law vortex flows for low hub/ t ip ratio ax ial fans and compr essors

A l t h o u g h t h e r e a r e m a n y p o s s i b l e t y p e s o f v o r t e x f l o w a v a i l a b l e , a w i d e r r a n g e o fcons t an t spec i f ic wo rk f lows ma y be cons idered i f Eq ns (5 .65 ) a re m od i f i ed asfo l lows:

a

= _ _ + b r p0 1r

a

C o2 = - + b r pt

d o w n s t r e a m o f a s t a t o r

d o w n s t r e a m o f a r o t o r

(5.70)

Fo l lowing the same s t r a t egy as t ha t ou t l i ned in Sec t ion 5 .6 .1 fo r mu l t i - s t age f ans and

compresso r s , t he coef f i c i en t s a and b may be exp ressed in t e rms o f overa l l des ign

par am ete r s ~bm, qJm and Rm spec i f i ed a t t he r .m . s , r ad iu s rm , r esu l t i ng in t he vo r t ex

spec i f i ca t ion

c e X R m ( r )< - ( -+ ) - - + 7 2m ( 5. 7a )



5 .6 A c t u a t o r d i sc t h e o r y a p p l i e d to m u l ti p l e b l a d e r o w s - t h e d e s ig n p r o b l e m 14 1

1.6

casing

U .._ Cx

hu b

1.5

Cxh

Cx- 1.4

1.3

1.2

1.1

o o15 i ~15 2 215 3 3.5x

F i g . 5 .1 6 Ax ia l ve loc ity Cxh/Cxat the hub radius of a ten-stage axial compressor p redicted by actuator

disc theory assuming incompressible f low

Fig . 5. 1 7 Axial v elocity profi les compared with rad ial equil ibrium profi les for ten-stage axial

compressor

w i t h ( - ) f o r s t a to r s a n d ( + ) f o r r o t o r s, p = 1 .0 o b v i o u s l y c o r r e s p o n d s t o t h e sp e c i alc a s e o f th e f o r c e d v o r t e x f o r t h e s e c o n d t e r m . M o r e m o d e s t v a l u e s o f p < 1 . 0 w il l

t h u s r e s u l t i n r e d u c e d m e r i d i o n a l d i s t u r b a n c e s a n d a x i a l v e l o c i t y p r o f i l e s l o p e s a n dper m i t the d es ign e r to s e lec t a sma l le r va lue o f hub / t ip ra t io i f so des i red . To i l lus t ra tet h is a n d t o c o n c l u d e t h i s c h a p t e r t h e a c t u a t o r d i sc so l u t io n h a s b e e n u n d e r t a k e n u s i n g

p r o g r a m M U L T I f o r a t e n - s t a g e c o m p r e s s o r w i t h t h e f o l l o w i n g o v e r a l l d e s i g n



142 S imp l i f i ed m er id iona l f low ana lys is fo r ax ia l tu rbomach ines

specif ica t ion wi th p = 0 .25"

H u b / t i p r a t i o r h / r t = 0 .4

r .m.s , rad ius rm/r = 0 .812 40

A t r m , t ~ m - - 0 . 5

~tm " - 0 . 2 5

R m = 0 . 5

Vo rtex po w er coe f f i cien t p = 0 .25

Figure 5 .16 i l lus t ra te s how the mer id iona l ve loc i ty a t the hub rad ius bu i lds up

rap id ly dur ing the f i r s t two s tages and se t t l e s down in to a sma l l pe r iod ic va r ia t ionf r o m s t a t o r to r o t o r a r o u n d a v a l u e i n t h e r e g i o n o f c x J C x = 1 .52 . The p red ic ted ax ia lve loc i ty p ro f i l e s a t the s t a to r and ro to r t ra i l ing edge p lanes fo r s t age 5 a re comparedin F ig . 5 .17 wi th the re la ted rad ia l equ i l ib r ium so lu t ions .

F r o m t h e s e s t u d i e s t w o c o n c l u s i o n s m a y b e d r a w n . F i r s t l y , t h e m e r i d i o n a l f l o w

tends to s e t t l e down fa i r ly qu ick ly to a regu la r pa t t e rn such tha t iden t i ca l b ladegeomet ry cou ld be adop ted fo r a l l s t ages excep t the f i r s t and l a s t . Second ly , thet ra i l ing edge ax ia l ve loc i ty p ro f i l e s fo r s t a to r and ro to r a re a lmos t iden t i ca l and l i er o u g h l y h a l f - w a y b e t w e e n t h e t w o r a d i a l e q u i l i b r i u m s o l u t i o n s f o r s t a t o r a n d r o t o r .I t s h o u l d b e p o i n t e d o u t t h a t a n e x t r a s t a t o r h a s b e e n p r o v i d e d h e r e d o w n s t r e a mof the l a s t s t age to remove the ex i t swi r l .

A f in a l a n d m o s t i m p o r t a n t o b s e r v a t i o n t o m a k e i s t h a t i n p r a c t ic e f o r a g a s

c o m p r e s s o r t h e a r e a s h o u l d b e r e d u c e d p r o g r e s s i v e l y p r o c e e d i n g t h r o u g h t h e s t a g e sto ma in ta in cons tan t ax ia l ve loc i ty Cx a s t h e d e n s i t y i n c r e a s e s . I n t r o d u c t i o n o fcompres s ib i l i ty in to ac tua to r d i s c ana lys i s to hand le th i s p rob lem wi l l be dea l t wi thin the nex t chap te r .

axial turbines

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