ANALYTICAL INVESTIGATION OF THE PERFORMANCE OF GAS TURBINES EMPLOYING AIR-COOLED TURBINE BLADES (PARTS I-III)

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ANALYTICAL INVESTIGATION O F THE FERFGRMAN< E v F tURBINES EM P1 GYI1■!U< A IR

L'UREINE

- /— ■ ,c i X•\T rx’'1 u.Oblm S u b m i t te d to the F acu lty

A u rd u e u n i v e r s i t y

R i c h a r d L e v i Duncan In P a r t i a l F u l f i l l m e n t of the R e q u i r e m e n t s fo r the D e g r e e of D o c t o r of P h ilosop h y June, Li50

ADES

P U R D U E U N IV E R S IT Y

T H IS IS T O C E R T IF Y T H A T T H E T H E S IS P R E P A R E D U N D E R MY S U P E R V IS IO N

i*I_______ _______ ___IiicJiard.Le.v_.D uncan

A n a l y t i c a l In v estig a tio n of the P er fo rm a n c e of Gas

entitled

AirtBnes E m p l o y i n g A i r - C o o l e d Turbine B l a d e s

C O M P L I E S N v r m T H E U N IV E R S IT Y R E G U L A T IO N 'S O X G R A D U A T IO N T H E S E S

A NT) I S A P P R O V E D B Y M E A S F U L F I L L I N G T H I S P A R T O F T H E R E Q U I R E M E N T S

FOR THE D EG REE OF

D o c t o r of AhiIosochy_

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iii

A: b:C "■'as t u r r i i e e n r i c o ,

A sim plifier. m obim,

01

Oohrrnaiw" are

porbu.nw.mce of a you Tub w e a n ' l a e ernpioyinp a i r - c w l m f hollow l a m m e "./lades worse. r a r e consT. a r a b l e lim e aw. e ff o r t f a r tee e e t a w w r w e r aeci a is o a i d am er r e r a n : i w s b u i s r ia. seieet.ii.iy die cewme b e s t s a i le d to a sp e c if ic a i r c r a f t desieai.

The p u r p o s e

01

we s i r dp r e p o r t e d

h e r e i n w a s to a evelo p a. rnethoo of e s t i m a t i o n w.s t u r b in e e.".t i r e p e r ­ f o r m a n c e sa lw r y r a y the a f o r e m e n h o n e a r e c w r r e m e m s ; to p r e s e n t die wow n e c e s s a r y f r o Hie a p p li c a t io n of the m e t e o r to row t u r b i n e en. Inc wi.au n or ro r-:vwre c o n truuwa cnt' ’ t i : n w n u o atm r.O; taaauwi or aue met hoc to a to picul a r o b l e m . T e e m a t e r i a l o.f die t h e s i s is p r e s e n t e r n h r e e punts,

P a r t I is

rote i r e o to p - m s e rt the w w r u i oevoiof mo:o of the iTui. e m e n i o i i e t m o hrac of win l y s i s

to i n d u c e oTw the n n a m n n a ; e t e t w i s of tee

m e t e o r arm Pro r e n s e r i i m involve! to w m o o p o r T rw tiie m a i l r e l a t i o n s in.to it.to Ariel ueso.ii.

The e n t a i l s of tee develo'pmenf of the irmie. relit

l i o n s o r e p on s e n t e a i'i the a. logo ib ic es.

.tppiw-.lion of p.,e

P w h II p r e s e n t s a rcpuwsenie.Tve

- p -S1S | ; r e s w i l e n I ■ P a r t I, lo a e o ' t r o o to ,

n.roo--pr o p a l l e r en /ine.

A s a m p l e c a l c a l a r r m of the a p u h c a t i o o of

the a n a l y s i s is a ls o p r e s e n t e d for P l u s ! r a t i v e p u r p o s e s .

P a r t III is

inte n ded to present, a s u m m a r y aeo d i s c u s s i o n of the c o m p l e t e a n a l y s i s . T e e p r i n c i p l e c o n tr i b u ti o n to the so lutio n of die p r o b l e m of die p e r ­ f o r m a n c e e s t i m a t i o n of "as t u r b in e e a fries e m p loy ino a i r - c o o l e d t u r ­ bine b l a d e s i s comaaned in s e c t i o n 3 and iuie a p p e n d i c e s A, B, C, aao. C, B riefly , bio m a i n c o n t r i b u t i o n s c o n s i s t s of hie folio wh ir t h r e e d e v e l o p -

1. The d e t e r m i n a t i o n oi the e n th a lp y at s ta tio n o independent of the h e a t r e m o v e d f r o m the t u r b in e ; p r e s e n t e d in a pp en d ix A. d,

t h e developm ent p r e s e n t e d in ap pend ix C th at allows the

e ffe ct of a i r - c o o l e d holloa/ b l a d e s upon the p e r f o r m a n c e of the t u r b in e to be e x p r e s s e d i n d e p e n d e n t oi the c o ola n t flow ana to be p r e s e n t e d a s a change in the turbine p r e s s u r e ratio, d. The r e d u c t i o n of the many a sp e c ts of the p r o b l e m to an a n a l y s i s 01

tiie two v a r i a b l e s , (i) die c o o la n t .ilow ratiu v..■,-■•/p., atci yh ine

r a t i o of the h e a t r e m o v e d f r o m the tu r b in e to the w o r k output of the tu r b in e q / h w m aintaining a s u f f i c i e n tl y g e n e r a l m o d e l of a gas tu r b in e c y cle t h a t it is valid f o r m o s t likely d e s i g n s oi c y c l e s em p lo y in g a i r - c o o l e d t u r b i n e b l a d e s . t h e C h a r t s I and II p r e s e n t e d in a p p en d ix G a r e convenient f o r the d e t e r m i n a t i o n of the g e n e r a t o r u s e f u l e n th a lp y f o r v a r i o u s d e g r e e s of cooling and engine o p e r a t i n g c o nditions.

V

Yd 4. A

R i c h a r d d,. Duncan was b o r n in W ingate, In diana on duly 14, Ibid, and a tt e n d e d W in gate G r a m r n e r an d High sch o o l, g r a d u a t in g in IbiG. He e n t e r e d P u r d u e U n i v e r s i t y arid r e c e i v e d the P .S .E .2 . d e g r e e in dune 1937. Yhe following y e a r he e n t e r e d the U.S. Navy F l i g h t T r a i n ­ ing School, P e n s a c o l a , F l o r i d a .

In Ye coder PUT he was a s s i g n e e to

P a t r o l S q u a d r o n Seven a t San Diego, C a l i f o r n i a a s a pilot an d A ssista n t E n gin eering o ff ic e r . In D ecem b er 1140 Mr. Duncan jo in ed P a tro l Yquadron d/wenty lone a t P e a r l H a r b o r a s a gilo t and A s s i s t a n t Madmenarise O fficer, and flew a irp la n es and equipm ent to A u stra lia in F e b r u a r y I t 4:0 He b eca m e a com bat pilot, in F le e t A ir Wing T en, ami in N ovem b­ e r T -w b e c a m e the Wing M a i n te n a n c e Officer of Me Wing H e a d q u a r t e r s Squadron. In Sepdem ber 1043 he w as a s s i g n e d to Naval A i r r e s t C e n te r , P a t u x e n t R i v e r , M a r y l a n d a s a test pilo t and l a t e r as a p ro ject c o ­ ordinator. He r e - e n t e r e d P u r d u e U n i v e r s i t y S e p t e m b e r PTO to take g r a d u a t e w o r k In A i r c r a f t P r o p u l s i o n .

'The U.S. Navy a s s i g n e d h i m to a [Irepu l­

s i o n s u r v e y t o u r of E u r o p e d u r i n g the s u m m e r of 1147 in c o m p a n y with P r o f e s s o r M. J. Zucrow, P u r d u e U n i v e r s i t y .

In June I I'd 3 he

r e c e i v e d the M .S .A e .E . d e g r e e an d w a s g r a n t e d at: a d d itio n al y e a r of s t u d y at P u r d u e U n i v e r s i t y to c o m p l e t e the c o u r s e w ork r e q u i r e m e n t s fo r a P h . D. d e g r e e .

In Julv 1549 he w as a ssig n e d to the Navy L ia iso n

VI

...nice a t the L ew is F ligh t - r e p u l s i o n L a b o r a t o r y , LAC A C’levelar Ohio, d u rin g which a ssig nm ent, he co n du cted the study r e p o r t e d in this t h e s i s . He is a m e m b e r of LA.S., and A .S.M .E .

vii

RAGLE Or CO...'EE' ITS

A C K ,iO W L ED C M E H T S . ,

Page i

. . . , . ____ . . . . . ...........

PirEFACE

ii . . .. ......... . . . . . . . . . . . . ................. . ............... . .

VITA .

.,..

ix

U S E OF T A P L E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST O r ’ F I G U R E S

. . . . ............ . . . . . . . . . . . . .

v

•■■O'fATdOP . . . . . AS. S T R A C T

............... . , . , . ................. . . . . . . . . ........... ..

,....

.

xi xv

P A R E I A c A U f S i S O i CYCLE WliYi A I R - C O O L E D TURHI F E j" LADES Section l ,

I d r e d u c tio n

.............................................. .

Se c tio n A D i s c u s s i o n of F a c t o r s Involved in C o o lie s T u r b i n e s eg the Flow of A i r T h r o u g h Hollow L.Tades . . Se ction 3. P erfo rm a n ce P r e d i c t i o n from G e n e r a l i z e d P aram eters ............

17

S ection 4.

G e n e r a t o r C o d e C alcu lation s

48

Sectio n 5.

Heat T r a n s f e r C o r r e l a t i o n

Se c tio n 6.

V elocity Cycle C a l c u l a t i o n s

..........

48 ..................

51

v iii

PA RT II R E P R E S E N T A T M E A P P . .ICAPIGN OF ANALYSIS Section 7. Metiiod of Application Section S. S a m p le C a lc ula tion o

.............. .....................................

P A P S ' III SUMI vP lRY AND DLICUGSION S ection V Conclu ding Rem arks

........... ......................

A P P E N D IX A. DERIVATION OF EQUATION F O P h „ .... A P P E N D IX B. ANALYSIS OF SHE MOMENTUM P R E S S U R E PmSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . API-ENDIX C. E F F E C T OF HEA^I REMORA.!... ON OLE P E R F O R M A N C E OF TURBINES . . . . . . . . . . . . . . . . . . . . . . . . . A P P E N D IX D. METHOD OF INCORPOSMi'INCCOM PO NE NT LOSSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .APPENDIX E. H E A P T R A N S F E R CO RRELATION EQUATIONS ............................... . A P P E N D IX F . DERIVATION OF V E L O C IT Y CYCPE EQUATIONS ................................................... A P P E N D IX G. G E N E R A T O R P E R F O R M A N C E CHARTS F O R GAS TURBINE CYCLES ............................. REFERENCES BIBLIOGRAPHY

..................................................... ...... ........ .

ix

L i s t s of T mates a. a F r Iasi, ,1 F... omes

I ’1' mueujronaoe of

■. n.uo...i. a o ^-a:.

of o.

o '.a s e , .

non. t o r t cl or. on a n a . . . . . . . . .

I m r m i s s i u i s ami a c t u a l sp nn w isc 1eciue; al-r re . i s i.ei n i l o m r - e i n . a. t r h o n . rau-nooloo ' F a d e , . . . 3a ."ana..,: j , and c r ite r ia d e s c r i b i n g the

.lamina:;- flow region, a r e s o m e of the fa c to r s ne ed ing i n v e s ti g a t i o n . C u r r e n t l y v a l u e s of h 0 a r e o btain ed f r o m t e s t s on b l a d e s of s i m i l a r d e sig n to those e m p l o y e d in the tu r b in e .

Individual c o r r e la tio n f o r

s e v e r a l d i f f e r e n t blade t y r e s a r e a v a i la b l e (d) (A). D a ta r e g a r d i n g the con vection he at t r a n s f e r c oe f f ic ie n t f o r the i n n e r s u r f a c e appear to be n o n - e x i s t e n t .

G en erally, the e q u atio n that

i s a p p lica b le to the h e a t t r a n s f e r m e c h a n i s m f o r the flow of fluids t hrough t u b e s i s a s s u m e d v a li d f o r the flow of the c o o la n t through the b;.ao.e p a s s a g e s and i s em p lo yed f o r d e t e r m i n i n g the v a l u e s of t r .

It

should be e m p h a s i s e d that the flow of the c oo la n t th r o u g h the cooling p a s s a g e s Is not analogous to the flow of flu id s thro ug h station ary tubes s i n c e the la tter n e g l e c t s the C o r io lis and fr e e con vection f o r c e s ana the e n t r a n c e e ffe cts, both of w hich influ ence the flow c h a r a c t e r i s t i c s of the coola nt.

Since the cooling p a s s a g e r o t a t e s a t the s a m e a n g u la r

v e lo c it y a s the t u r b i n e disc , e n e r g y is added to the c o o la n t a s i t follow: r a d i a l l y o u t w a rd th ro u g h the b l a d e and c a u s e s a f u r t h e r departure of the c o o la n t flow f r o m the flow of a fluid thro u gh a s t a t i o n a r y tube.

It

i s evident that the v a lu es of Ip obtained f r o m heat t r a n s f e r equations v alid for fluid flow through station ary tubes should be applied to a ir c o o led tu r b in e b l a d e s with a d e g r e e of c au tio n .

The v a l u e s of iq f o r

d e sign ca lcu la tio n s should be obtained from, e x p e r i m e n t a l data and t h e r e f o r e t h e r e i s a r e a l ne ed f o r f u r t h e r r e s e a r c h on a i r - c o o l e d b la d e p a s s a g e s .

The h e a t t r a n s f e r is a diffic ult p r o b l e m to a n a l y s e ,

b u t it is a l s o an im portant one, b e c a u s e the c h a n g e in cy cle p e r f o r m ­ ance, when an a i r - c o o l e d turbine is e m p lo yed, depends la r g e ly upon the co o la n t r a t i o Ga/G c ; the r a t i o of the coolin g a i r flow by* to m e to ta l c o m p r e ss o r a ir flew Gc . The coolan t ratio, in turn, is a function of the heat tra n sfe r m echan ism , so that an im p rovem en t in the h e a t tra n sfer a r r a n g e m e n t red u ces the c o o lin g a i r req u irem en t and c o n s e ­ quently the c y c le l o s s e s .

F i n s and other types of in s e r ts p laced in the

coolan t p a s s a g e in c r e a se the heat tra n sfe r su r fa c e , the flow v e lo c ity and the e ffe ctiv e v a lu e of Iq and d e c r e a s e the require;! a m o u n t of c o o l ­ ing a i r . 2.5 h h e P o w e r L o s s Due to Com p r e s s i n g the Coolant F low . The w o r k r e q u i r e d to c o m p r e s s the coo ling a i r m u s t be c on sidere d a s a l o s s , to be d e d u cted f r o m the engine output.

The c o m p re ssio n ma

be a c c o m p l i s h e d p a r t l y in the m ain c o m p r e s s o r and p a r t l y in the flow path fro m the turbine hub to the t u r b i n e blade tip.

I; he e ff ic ie n c y of

that p a r t of the c o m p r e ssio n occu rrin g at the tu r b in e d isc should be high, f o r the co n d itio n s a r e a n alog o us to the flow in a cent; ifugal i m p e l l e r (no d iffusion is r e q u i r e d ) w hich is known to be quite e fficien t. H ence, it is to be exp ected that the c o m p r e ssio n of the coo lin g air should be a c co m p lish ed w ith a s li g h tl y la r g e r e ff ic ie n c y than that for c o m p r e s s i n g the a i r in the m a in c o m pr e s s o r . It a p p e a r s d e s i r a b l e , t h e r e f o r e , to p e r f o r m a s m u c h of the c o o la n t a i r c o m p r e s s i o n a s is p r a c t i c a b l e in an i m p e l l e r on the t u r b i n e d isc.

An i m p e l l e r a t the

tu r b in e d isc would a ls o aid in p r e v e n t i n g l o s s e s due to “ s h o c k ” a t the r o o t of the b l a d e by guiding tne c o o la n t s m o o t h l y into the c o o la nt p a s s ­ a g e s . thereby s o lv in g a p r o b l e m that m ig h t o t h e r w i s e b e c o m e t r o u b l e ­ some. 2.8

The L o ss A s s o c i a t e d With the R e m o v a l of the H e at q F rom the T u r b i n e .

.she Lerms “ c o o ls d t u r b i n e ” and “ un c o o ie d t u r b i n e ” w ill be u s e d to d e n ote the turbine em ploying a ir -c o o le d hollow b la d es and the c u r r e n t h u e of t u r b in e em p lo y in g so lid b l a d e s r e s r e c l i v e l y . T h e h e a t q r e m o v e d from, the m a i n g a s strea m in the c o o le d t u r ­ bine c a u s e s a l o s s in p e r f o r m a n c e b e c a u s e the t h e r m o d y n a m i c p r o c e s s of the turbine m ust then be c o n sid e r e d a s p o l y tr o p i c ra th er than a d ia b a t i c (13). T h e p o ly tr o p ic p r o c e s s above m u s t b e . i s t i n g u i s h e d f r o m the ir r e v e r s ib le adiabatic of the u n c o o le d tu r b in e w ith a e r o ­ d y n a m ic l o s s e s .

The t h e r m o d y n a m i c p r o c e s s of the u n c o o le d t u r b in e

is r e v e r s ib le adiabatic (without a e r o d y n a m i c lo s s e s ) or ir r e v e r s ib le a d ia b a t i c (with a e r o d y n a m i c l o s s e s ) while the a i r - c o o l e d tu r b in e p r o c e s s i s e i t h e r r e v e r s i b l e p o l y tr o p i c (with no a e r o d y n a m i c l o s s e s , but w ith q rem oved from m ain gas stream ) or ir r e v e r s ib le polytropic (with a e r o d y n a m i c ' l o s s e s and n r e m o v e d f r o m the m a i n g a s s t r e a m ) , h h e l o s s a ss o c ia te d with the rem o v a l of the heat q w ill be p r e se n t even though the h e a t i s r e m o v e d f r o m the b o u n d a r y l a y e r and l a t e r r e t u r n e d to the m a i n s t r e a m flowing through the engine a t the tu r b in e

blade tip s .

The r e m o v a l of the h e a t q 'causes a r e d u c t io n in toe a v a i l ­

able tu r b in e w o r k , when the turbine p r e ssu r e r a t i o is held at a given v a lu e (IT). The d e c r e a s e in the a v a i l a b l e work of the t u r b i n e c o n s t it u t e : the m ain in flu en ce that the r e m o v a l of the h e a t q lias upon the cycle perform ance. 1..7 The N e c e s s ity f o r a S im p l if i e d Method of Cycle A n a l y s i s . The p r e c e d i n g d is c u ssio n s have indicated b r i e f l y the d e s i r a b i l i t y of high peak c y c le tem p era tu res, the n e c e s s i t y of cooling the b la d es by enroloying hollow b l a d e s to obtain high t e m p e r a t u r e s , and d e s c r i b e d b r i e f l y die m o r e i m p o r t a n t v a r ia b le s p e r t i n e n t Lc the d e t e r m i n a t i o n of the p e r f o r m a n c e of t u r b o j e t o r t u r b o - p r o p e l l e r e n g in e s e m p loy ing a i r c o o le d hollow t u r b i n e b l a d e s .

'There r e m a i n s to be d e t e r m i n e d , q u a n t i ­

tativ ely, the effe ct of the l o s s e s a s s o c i a t e d with the coolin g p r o c e s s upon the p e r f o r m a n c e , s o that the l i m i t i n g a s w e l l a s the optimum c o n d itio n s f o r the d e s i g n p a r a m e t e r s can be e s t a b l i s h e d .

F o r e x a m p le ,

if the coolin g l o s s e s a r e so la rg e that the l o s s in p erform an ce i s equal to o r g r e a t e r than the gain a s s o c i a t e d with h i g h e r tu r b in e i n l e t t e m p e r ­ a t u r e s , it would be im p r a c tic a l to em p loy cooling to s e c u r e higher c y c l e tem p era tu res. To obtain the d e s i r e d i n f o r m a ti o n it is n e c e s s a r y to c a lc u la te the p erform an ce of the gas turbine engine em ploying a turbine with a ir c o o le d hollow b l a d e s o v e r a wide r a n g e of c y cle t e m p e r a t u r e s and c o m ­ p re s s o r p re ssu re ratios.

The c a l c u l a t i o n s involv ed in d e t e r m i n i n g the

Section d I-erform an ce b red iction F rom G e n e r a l i z e d P a r a m e t e r s id I Gas Cene rater and "Velocity C y c le s . Tn making a n a l y s e s of gas tu r b in e type p o w e r p lants f o r a i r c r a f t , it is d e s i r a b l e to s e p a r a t e the a c t u a l c y c l e into two p a r t s ; 1. a r a m j e t c y c l e ana z . a s t a t i o n a r y g a s tu r b in e c y c le .

The r a m j e t cy cle o p e r a t e s

in f r o n t of the gas t u r b i n e c y cle; f i r s t su g g e s t e d by w. L u ts in F e b r u a r y io4Q (Li). D r . I. E. D r i g g s h a s develop ed f r o m t h is i d e a a m o s t c o n ­ v e n ie n t g e n e r a l i z e d m ethod of a n a ly sis, which w ill be applied h e r e to the p e r f o r m a n c e of a gas t u r b in e eq uip ped with a i r - c o o l e d hollow b l a d e s (vG) (34). In this m a n n e r it is p o s s i b l e to apply the s a m e method of a n a l y s i s to the t u r b o j e t and the t u r b o - p r o p e l l e r engine. The d iv is io n of the e ngin e is i l l u s t r a t e d sc h e m a tic a lly in f i g u r e 4. The unit c o m p r i s i n g Lie c o m p r e s s o r , the c o m b u s t o r , the turb in e , and the enthalpy availab le f o r propulsion a t s t a ti c c on d itio ns is c a l l e d the gas generator.

The e n th a lp y a v a i la b l e f r o m the gas v e n e r a t o r c y c le is

c a l l e d the u s e f u l generator enthalpy ahU;:r, s e e f i g u r e 5. The ramjet: c y c l e i l l u s t r a t e d in f i g u r e 5 by the c y c l e cycle.

0171 is c a l l e d the v e lo c ity

'The v e lo c it y c y c l e c o n t r i b u t e s to the o v e r a l l p r o p u l s i o n power

the enth alp y Lhu v . The total u sefu l enthalpy f r o m the a c tu a l c y c le is equal to nhUD- plus Ahuv, and m a y be converted to t h r u s t by e ith er a jet or a p r o p e lle r, or both. By m ean s of the aforem en tion ed approach the perform an ce of both th e t u r b o j e t and t u r b o - p r o p e l l e r e n gin e s

reT„Gr-;';,r.or

° J-

e m plo yin g a i r - c o o l e d ivdlov/ b l a d e s in the t u r b i n e c a n be obtained f r o an a n a l y s i s of the g a s g e n e r a t o r p o rtion of the c o m p l e t e cycle. h

..Generaliz e d P a r a m e t e r s to C o r r e c t f o r Altitude and F l i g h t S p e e d Conditions,

h o r e d u c e the n e c e s s a r y c a l c u l a t i o n s f u r t h e r it. i s co nv en ien t to avoid the n e c e s s i t y of c o v e r i n g a range of flight s p e e d s and a l t i t u d e s by g e n e r a l i z i n g the c y c l e .

The g a s g e n e r a t o r c y c l e and the v e lo c ity

c y c l e can be g e n e r a l i z e d by apply ing the following p a r a m e t e r s (23).

p _ Am bient s i a l i c tem p eratu re at altitude A m b i e n t s t a t i c t e m p e r a t u r e at std . S.L .

A m bient tota l tem p eratu re a t a l titude _ T'to A m bient sta tic tem p eratu re at altitude Tr,

A m b i e n t s t a t i c p r e s s u r e a t a l t i t u d e_ Am bient sta tic p r e s s u r e at std. S.L.

A m b ie n t t o ta l p r e s s u r e a t a ltitu d e A m b i e n t s ta tic p r e s s u r e a t altitud e

_

The above p a r a m e t e r s a r e e m p l o y e d to c o r r e c t the v a r i a b l e s to s t a n d a r d s e a l e v e l s t a t i c conditions. The p a r a m e t e r s r a nd 5 c o r r e c t for the in flu e n ce of a lt i tu d e , and - and w f o r f lig h t s p e e d (23) A pplication of the c o r r e c t i o n s y s te m r e s u l t s in the follow in g

C orrected en th a lp y C orrected pressure

....... .............

C o r r e c t e d air weight, flow

C orrected fu el w eight flow

.. hA p/h^

.........

OTT

................ dll. Gu A

-

t

Corr ected t h r u s t .. . .. . d r r e c t e d te rr: pe r a tu r

3T

The Sch e m a t i c D i a g r a m of a C y c le E m p lo y in g A i r - C o o led I-TqII ow T l a d e s In the T u r b i n e R o to r.

T h e s c h e m a t i c d i a g r a m of the gas g e n e r a t o r m odel to be em p loyed for the a n a l y s i s of the gas t u r b i n e with a i r - c o o l e d hollow b l a d e s i s i l l u s t r a t e d in fig u r e G. The assu m p tion s in volved in c o n s t r u c t i o n of the m o d e l a r e : a.

The c o r r e c t total p r e s s u r e at the turbine blade r o o t is

o b ta in e d by b l e e d in g the c o o la n t f r o m the c o m p r e s s o r ou tle t (s t a ti o n 1). b.

The t e m p e r a t u r e s and p r e s s u r e s at the c o m p r e s s o r d i s c h a r g e

a r e equal to the t e m p e r a t u r e and p r e s s u r e at the r o o t of the tu r b in e blade. c.

E qu al v a l u e s of heat, a r e r e m o v e d f r o m e ac h b l a d e and equal

v a lu e s of the c o o la nt flows th r o u g h e a c h blade. d. No h e a t is lost.

All of the h e a t q i s r e t u r n e d to the m ain gas

flow d o w n s t r e a m of the t u r b i n e r o t o r . e. No l o s s in total p r e s s u r e o c c u r s whe n the coolant flow Gr. m ix es with the gas flow G* d o w n s t r e a m of the t u r b i n e r o t o r . f. The coolan t Gc and the g a s flow Gt a re co m p letely m ixed at s t a t i o n 6. if the s t a t i c p r e s s u r e of the g a s s t r e a m and the st a ti c p r e s s u r e of the c o ola nt flow a r e m a t c h e d at the tip of the t u r b in e blade, the d i s c h a r g e of the c o o l a n t flow into the m a i n g a s s t r e a m will be a c c o m p l i s h e d a t a high v e lo c ity of the coolan t flow and e ffic ie n t sc av e n g in g of the cooling p a s s a g e s w ill b e obtained. The s t a t i c p r e ssu r e in the r o t o r of an im ­ p u lse tu r b in e , when the turbine n o z z l e b l a d e s a r e choked, w ill be a p p r o x i m a t e l y o n e - h a l f that of the t o ta l p r e s s u r e in the n o z z le b l a d e s . The tota l p r e s s u r e of the m a i n g a s s t r e a m at the tu r b in e noz zle b l a d e s i s e q u al to the t o ta l p r e s s u r e a t th e c o m p r e s s o r d i s c h a r g e ; n e g l e c t i n g the c o m b u s t o r to ta l p r e s s u r e l o s s .

If, as in a s s u m p t i o n (a),

the to ta l p r e s s u r e of the c oo la n t flow at the b l a d e r o o t i s m a d e e q u a l to the to ta l p r e s s u r e a t the c o m p r e s s o r d i s c h a r g e , the c o o la n t flow t o ta l p r e s s u r e at the b l a d e r o o t w ill a l s o oe eq ual to the to ta l p r e s s u r e of the m a i n gas s t r e a m a t the n o z z l e b l a d e s .

The s t a t i c p r e s s u r e of the

c o o la n t a t the b l a d e tip when the cooling p a s s a g e s a r e c h o k e d will then be e q u a l to the s t a t i c p r e s s u r e of the g a s s t r e a m a t the b l a d e tip and

s u f f i c i e n t p r e s s u r e f o r the c o o la n t flow i s thus o b tain ed .

A ssum ption

(b) is v a lid if the t h e r m a l and p r e ssu r e l o s s e s in the ducting fo r the flow of the c o o la n t f r o m the c o m p r e s s o r to the b l a d e r o o t a r e s m a l l enough to be n e g l e c t e d .

A s s u m p t i o n (e) r e q u i r e s a co n d itio n of r o t a ­

tio na l s y m m e t r y f o r the r e m o v a l of the h e a t q a nd the flow of the c o o l a n t Gc . T h e d i s t r i b u t i o n of the coolan t flow and the h e a t r e m o v e d w ill not be e x a c t l y s y m m e t r i c a l around the c ir c u m fe r e n c e of the rotor, b u t f o r p u r p o s e s of p e r f o r m a n c e c a l c u l a t i o n s the a v era g e v a l u e s a r e a ccep ta b le, and the assu m p tion is valid .

In u n cooled tu rb in es the

a s s u m p t i o n th a t the therm odynam ic p r o c e s s is a d i a b a t i c is p e r m i s s i b l e . A s s u m p t i o n (d) is the corresp on d in g con d ition fo r the co o led t u r b i n e an d t h e r e f o r e i s a l s o p e r m i s s i b l e .

G’he m ix in g l o s s e s a s s o c i a t e d with

the m ix ing of the coolan t flow and the g a s s tr e a m behind the turbine r o t o r a r e n e g l e c t e d in a c c o r d a n c e with a s s u m p t i o n (e).

The p r e s s u r e

of the m a i n g a s stre am , i s a s s u m e d to be u n a f f e c t e d by the m ix in g process.

Sin ce the p r e s s u r e of the coolan t flow is slig h tly h igh er than

that of the gas s t r e a m the l o s s e s due to m ix in g w ill ten d to be com perts a t 0(1, ./to s u m p t io n (f) m e a n s th a t s t a t i o n 6 m u s t be taken s u f f i c i e n tl y f a dow nstream of the t u r b i n e r o t o r t h a t th e flow i s o n e - d i m e n s i o n a l f o r all p ra c tic a l pu rp o ses. The g e n e r a l i t y of the a n a l y s i s i s not g r e a t l y r e d u c e d by the above assum ptions,

't h e followin g t h r e e m e t h o d s of o b tain in g the r e q u i r e d

p r e s s u r e of the c o o la n t a t th e b l a d e r o o t a r e s p e c i f i c a l l y p e r m i s s i b l e ,

p rovid in g the e f f i c i e n c i e s of the v a rio u s c o m p r e ssio n p r o c e s s e s are a p p r o x i m a t e l y e q u a l when two or m ore c o m p r e s s o r s a r e em ployed in the sa m e engine: (a)

T he c o m p r e s s i o n is p e r f o r m e d e n t i r e l y in the m a i n c o m ­

pressor. (b) T he c o m p r e s s i o n i s a cco m p lish ed partly in the m ain c o m ­ p r e s s o r and p a r t l y in an a u x i l i a r y c o m p r e s s o r (for e x a m p l e , f r o m the t u r b i n e r o t o r hub to the t u r b i n e b l a d e r o o t ) . (o) T h e c o m p r e s s i o n i s p e r f o r m e d c o m p l e t e l y by an a u x i l i a r y com pressor. The m o d e l is a lso a p p licab le to coo lin g of the com b u stor and the tu r ­ bine n o z z l e b l a d e s . 3.4

The A dditional P a r a m e te r s R e q u i r e d fo r E n gin es with C ooled Turbines.

The o p e r a t i o n of the m o d e l c o n s t r u c t e d u n d e r a s s u m p t i o n s s e t forth in s e c t i o n 3.1 m ay b e r e p r e se n te d by a m odified T ray ton c y c le . The co o lin g flow Gc i s c o m p r e ss e d w ith the m ain flow in the c o m p r e s s ­ o r , c o n d u c te d f r o m s t a t i o n 2 to the r o o t of the t u r b i n e r o t o r b la d e , arid t h r o u g h the hollow b l a d e w h e r e i t a b s o r b s the h e a t q f r o m the m a i n g a s s t r e a m by c o n d u c t io n and c o n v e c ti o n t h r o u g h the b la d e w a l l s .

The

c o o la n t flow Gc i s th en d i s c h a r g e d a t the b l a d e tip and m i x e d with the m a i n g a s flow d o w n s t r e a m of the t u r b i n e .

The c y c l e p r o c e s s of the

c o o l e d e ngin e d i f f e r s f r o m the B r a v t o n c y c l e only by the e ff e c t of the

two v a r i a b l e s Gy ana q.

T h erefore the in flu en ce of a i r - c o o l e d hollow

b l a d e s upon the c y c l e p e r f o r m a n c e can b e r e d u c e d to the a n a l y s i s of the b e h a v i o r of hie two v a r i a b l e s Gc anti q. It i s s t i l l n e c e ss a r y , of c o u r s e , to d e t e r m i n e the in flu ence of I T and q upon the c o m p o n e n t c a r t s of the ermine. .he T h e r m a l C ycle and b r e s s u r e Cycle. 1.0 an a ly se the Influence of G,-. and a upon, the p e r f o r m a n c e of the c o o le d engine c y cle, it i s c o n venien t to s u b - d i v i d e the gas g e n e r a t o r c y c l e into two p a r t s .

P a r t one c o n s i s t s of a c o n s i d e r a t i o n of enth a lp y

o r e n e r g y l e v e l s th ro u g h the gas g e n e r a t o r f r o m sta tio n 1 to and i n ­ clu ding s t a t i o n 7.

F o r c o n v e n i e n c e p a r t one i s denoted a s the ‘T h e r m a l

c y c l e ” . P a r t two d e note d a s the ^ p r e s s u r e c y c l e ” i s b a s e d upon a c o n s i d e r a t i o n of the to ta l p r e s s u r e l e v e l s f r o m s t a t i o n 1 up to and including s t a t i o n 7. 3.1 The Inf l u e n c e of A i r - C o o l e d Hollow T u r b i n e B l a d e s on the T h erm al C ycle. The enthalpy r i s e In the t h e r m a l c y c l e f r o m s t a t i o n 1 to sta tio n 1, Lh c , c a n be c o m p u t e d from, m e th o d s d e v e l o p e d f o r u n c o o ie d e n g in e s (s e e , for exam p le, r e fe r e n c e 10). The c h a n g e in enthalpy le v e ls in m e com bustor,

Blip, can b e obtained by m ethods s im ila r to the m e t h o d s

em p olyed f o r u nc oo le d c y c l e s , p r o v i d e d that the d e c r e a s e in w eight flow th ro u g h the com b u stor due to r e m o v i n g the coolan t flow at the c o m p r e s s o r d i s c h a r g e is tak e n into c o n s i d e r a t i o n (se e r e f e r e n c e 30).

The w o r k done in the t u r b i n e I q oar. a ls o be obtained f r o m m e t h o d s employed, on u n c o o le d t u r b i n e s .

A m eth o d Including the effe ct of

c o m b u s t i o n g a s e s is p r e s e n t e d in r e f e r e n c e CO. The t u r b in e c y cle d i a g r a m on the h-S p la n e is i l l u s t r a t e d in f i g u r e 7. The tu r b in e wort: L g i n d ic a t e d by the e n th a lp y d e c r e a se fro m station 1 to station 4 can b e o btain ed by em ploying m e t h o d s v a lid f o r uncoo le d t u r b i n e s .

Em ­

ploying the m e t h o d s fo r uncooled engines a knowledge of the value of q is n e c e s s a r y to ob tain h 4 and a knowledge of the v a lue of h5 is n e c e s s a r y to obtain lw.

it Is d e s i r a b l e to avoid the above d iffic u lties

and d e r i v e an e qu atio n f o r d e t e r m i n i n g hg independent; of the v a lu e s of h 4, hr,, and q.

The d e sir e d e x p r e s s i o n is d erived in Appendix A and the

r e su lt in the f o r m of a g e n e ra liz ed equation is

h’ ■ T - T k n l i - i ' b0b_ q ...

va.

7T

b ■ m O P T ------------

Al:

A The term f /b i s the f u e l - a i r r a t i o b a s e d upon the flow of a: th ro u g h the com b u stor and is defined by the equation

and i s to be d i s t i n g u i s h e d f r o m the f u e l - a i r r a t i o b a s e d upon the flow of a i r t h r o u g h the c o m p r e s s o r T

on

E q u a d - A i !7 sho w s that h c r a n be d e t e r m i n e d in d ep e n d en t of the v a r i e s of h 4, i n , and q.

calculations

The e n th a lp y l e v e l s n e c e s s a r y f o r the c y c le

of the c o o le d enyine can. be o b tain ed w itho u t the . ;no

••

ledge of q; h o w e v e r , the v a r i a b l e Gr, do es e n t e r into the t h e r m a l c y c le

as

a

non-dimensional

w e ig h t flow.

p a r a m e t e r in c o n ju n c tio n with the

compressor

It i s not n e c e s s a r y then to know the valu e of C>, but only

the valu e of the c o o la n t r a t i o Gc /G a . T h e r e f o r e the influ e nce of a i r c o o le d

hollow

t u r b i n e b l a d e s upon the

performance

of the t h e r m a l cy cle

can be e x p r e s s e d a s a function of the m vdaru r a t i o . 1.7 The in flu e n ce of C o m b u s t o r i r r e s s u r e T o s s e s on the

Pressure

C ycle ,

The c o m p r e s s o r p r e s s u r e l e v e l s of the p r e s s u r e c y c le can be o b ta in e d by m e t h o d s e m p l o y e d f o r u n c c o i e d en g in e s,

Che c h a n g e in

total p r e s s u r e f r o m s t a t i o n 2 to s t a t i o n f a r i s e s L o r n die l o s s e s in hie c o m b u s t o r due to fl u id friction, an d r a t e of c h a n g e of the flow.

of

Both t y p e s of p r e s s u r e l o s s i n c r e a s e with an i n c r e a s e in the

temperature (77).

momenturn

r i s e in the c o m b u s t o r , a s s u m i n g c o n s t a n t in le t c o nd ition s

The l o s s e s m a y b e r e d u c e d by

increasing

s e c t i o n a l a r e a to the w e ig h t flow A/Gq.

die r a t i o of the c r o s s -

Glie r a t i o A/Gp d e c r e a s e s f o r a

c o o le d engine b e c a u s e the c o olin g a i r d o e s not e n t e r the c o m b u s t o r . Since of

the temperature

A/Gt

rise

is

larger

in a cooled

engine and

i s s m a l l e r the l o s s e s tend to r e m a i n u nchanged.

A

the ratio one -

d i m e n s i o n a l flow a n a l y s i s of the t o ta l p r e s s u r e l o s s due to the i n c r e a s e

in m.o men turn i s p r e s e r v e d in appendix P.. s h e r e s u l t s i n d ic a t e that with a c o n s t a n t outle t Maori n u m b e r the c h a r g e it', total p r e ssu r e l e s s cue to m e change in m o m e n t u m f o r en gin es with cooled and u n c o c l e d t u r b i n e s will dif f e r by l e s s t h a n two p e r c en t, and can t n e r e f o r e be n e g le c te d ,

'hhe c o m b u s t o r will have to b e r e d e s i g n e d to w i t h sta n d the

h i g h e r tem p era tu res, and em p h a sis should be upon s e c u r i n g a low in le t Mach n u m b e r to m a i n t a i n low v a l u e s of the p r e s s u r e l o s s e s . 8.8 T h e r m o d y n a m i c P r o c e s s e s of a T u r b i n e with A ir -C o o le d Hollow .-H a d e s . As p r e v i o u s l y s t a t e d in s e c t i o n Mb, the tu r b in e w o r d output, d e c r e a s e s when the h e a t q is r e m o v e d by c o o l i n g the b l a d e s ,

hhe

d e c r e a s e in turbine w ork can be illu str a te d with the aid of the a d ia ­ batic effic ie n cy and f i g u r e s 8 and 8. h h e two ex trem e m e t h o d s of r e m o v i n g the h e a t f r o m the t h e r m o d r m a m i c p r o c e s s of the t u r b i n e a r e : :. At c o n s t a n t p r e s s u r e b e f o r e the w o r n i s p e r f o r m e d and 2. c o n s t a n t p r e s s u r e after the tu r b in e -work h a s b e e n c o n s u m e d .

at Figure 8

i l l u s t r a t e s the th er r n o d y n a m io p r o c e s s e s of c a s e 1 w h e r e the h e a t q is a s s u m e d to be r e m o v e d a t constant, p r e s s u r e f r o m s t a t i o n

Applying

the definition of the adiabatic efficie ncy r e s u lts in

a - ” - O n = i - r_ I.V

Lt

w

F i g u r e h i l l u s t r a t e s the therm odynam ic p r o c e s s e s of c a s e 2 w h e r e the

Jr

C

Ci

h e a t q i s a s s u m e d to be

removed

at

constant

p r e s s u r e f r o m station

3-1iU

C om parison of the two c a s e s in d ica tes that r e m o v i n g the heat q a t the be ginning of the p r o c e s s , s t a t i o n 3, r e s u l t s in a l o w e r efficiency, and c o n s e q u e n t ly l e s s wort:

than

in the c a s e w h e r e the h e a t is r e m o v e d

at s t a t i o n 4. In fa c t , r e m o v i n g th e h e a t at s ta tio n 4 h a s no influ ence upon the tu rb ine w o r k ,

The t h e r m o d y n a m i c p r o c e s s e s of hie a c t u a l

tu r b in e em p lo y in g a i r - c o o l e d hollow b l a d e s w ill nave the n e a t r e m o v e d co n tin u o u sly from, s t a ti o n 3 to s t a t i o n 4 and t h e r e f o r e sh o u ld lie b e tw e e n c a s e 1 and c a s e 3. She The Influence of Cooled T u r b i n e Upon the - m e a s u r e Cycle. R a t h e r then em p lo y the change in t u r b i n e w o r k a s s o c ia te d with the r e m o v a l of a given a m o u n t of h e a t , it is m o r e c o n v e n i e n t f o r p u r p o s e s of c a l c u l a t i n g p e r f o r m a n c e of an engine with an a i r - c o o l e d t u r b i n e to c o n s i d e r the t u r b i n e w o r k c o n s t a n t and d e t e r m i n e the c h a n g e in tu r b in e p r e ssu r e ratio req u ired to s a t i s f y that condition. The change in the tu r b in e p r e s s u r e ratio ca u sed by the rem oval of a giver am ount of heat q from the turbine c a n be d eterm in ed f r o m the a n a l y s i s p r e s e n t e d in ap p en d ix C. The a n a l y s i s in a p p en d ix C of the in flu e n ce of u upon the p r e s s u r e c y c l e i s a n alo go u s to a n a l y s i s of a c o o le d c o m p r e sso r b y C. F . W islicenu:

the

ur.cooled

olytromlc

m

eff i c i e r

defined

Hj

:n e a n a ly s is is Das?

CiO)

turbi

vap - n

n ht

where:

me

m roine word.

/ v d p = the m e c h a n i c a l w ork a v a i l a b l e from, me diibd flowing th r o u g h the t u r b i n e r o t o r in

reversible

process.

'“ he d i a g r a m s of tine t h e r m o d y n a m i c p r o c e s s e s f o r an ideal

turbine (h-S

em p lo y in g a i r - c o o l e d hollow b l a d e s a r e p r e s e n t e d in f i g u r e s 1C

plane) a n a 11 ( P - v plane). F r o m

consideration

of the p r o c e s s e s in

f i g u r e 10

and a n a lo g o u s to e q u a tio n C5

it tig -

0

h vdp = q

t h e r e f o r e f o r the c o o le d tu r b in e

Co

or m me ideal c a s e

Li = ,/Vdp

L a e con d itio n that the t u r b i n e w o r k ^ r e m a i n c o n s t a n t i n d e p e n d ­ e nt of the v a lu e of q d i c t a t e s th a t J vdo a l s o rem ain c o n sta n t.

F i g u r e 11

show s that to s a t i s f y the a f o r e m e n t i o n e d r e q u i r e m e n t the a r e a ado b m u s t be e q u a l to the a r e a a34e. F o r n > k i t i s s e e n a l s o th a t R v L- , and c o n s e q u e n t ly . ip / H h < R , / i h . Lhe m a g n itu d e of the c h a n g e in p r e s s u r e ra tio for a s p e c i f i c v a lu e of turbine w o r d L T and a s p e c i f i c v a lu e of n e a t q rem o v ed f r o m the turbine c a n b e determ in ed f r o m the e q u a t io n s d e r i v e d in a p p e n d ix C. A s s u m i n g v a l u e s f o r L :, and Id /Lp , the tu r b in e w o r k Lq i s d e t e r m i n e d f r o m

d -

cd

S e l e c ti n g a v a lu e of the r a t i o q / L p the v a lu e of n / n - 1 i s o b ta in e d i r o n

c

f

in

- w

t h e p r e s s u r e ra tio f o r the c o o le d t u r b i n e i s then d eterm in ed fro m

A

w

.

015

Lhe c h a n g e in the t u r b i n e p r e s s u r e r a t i o due to coo lin g the b l a d e s i s e x p r e s s e d a s a p e r cen t of the u n o o c le d turbine p r e s s u r e r a t i o a n a d e n ote d a s a c o r r e c t i o n f a c t o r •

correction factor = o

„

/i // T-. * / ^4 '

ie c o r r e c tio n f a c t o r s w e r e d e t e r m i n e d f o r a r a n g e of v a l u e s of / P p , and q / I q .

The r e s u l t s a r e p r e s e n t e d in f i g u r e 12.

3.10 S u m m a r y of the Influence of a C ooled T u r b i n e Upon

The in flu en ce of a i r - c o o l e d hollow tu r b in e b l a d e s upon the p r e s s u r e c y c l e p a r t of the gas g e n e r a t o r i s r e d u c e d to the effe ct of the co o lin g upo n the t u r b i n e p r e s s u r e r a t i o and c a n b e e x p r e s s e d in t e r m s of the n o n -d im en sio n a l p a r a m e t e r c/'Lq by the c o r r e c t i o n , fa cto r p r e s e n t e d in f i g u r e ib.

2 'he in flu en ce of the c o o l e d b l a d e s upon the

therm al c y c l e

p a r t of the g a s g e n e r a t o r i s e x p r e s s e d b y e q u a tio n A l l and is a function of the n o n - d i m e n s i o n a l p a r a m e t e r Gc / G a . The a p p li c a t io n of the two p a r a m e t e r s Ga / G a and q/Iq-, t h a t e x p r e ss the in f lu e n c e of a i r c o o le d t u r b i n e b l a d e s upon the g e n e r a t o r p e r f o r m a n c e , to the c y c l e c a lc u la tio n s fo r the r a s g en erator is p r e sen ted in s e c t i o n 4.

2000E ■ 25 00R 3000 R 3 500R

'

2000R 2500R

3000R I

■

3S 00R

200 OR 2 5 0 0 J? q _ 3000Ri t

P000 H op, 00}? 3 0 0 OR 3500R

12,

Ccrreo ticn chert for turbine pressure retie when beet i s r e mo v e d f r o m t h e q e s s t r e a m .

tO

beeiior 4 G e n e r a t o r Cycle C alculation s 4.1 M ethod of D eterm in ing ChUr:- of Id eal C y c le s . By applying the a n a l y s e s d e s c r i b e d in the p r e c e d i n g s e c t i o n the e ffe c t of e m p lo y in g a i r - c o o l e d holloa/ t u r b i n e b l a d e s upon the p e r fo r m ­ ance of the g a s g e n e r a t o r c a n be d eterm in ed . F ig u re 12 togeth er with equatio n AM m a y be e m p l o y e d in c o njun c tion with the c y c l e c a l c u l a t i o n / of en '-j gin es w ith uncooled turbin es to obtain the u s e f u l enthalpyj of the j.

g a s g e n e r a t o r c y cle e m p lo y in g a i r - c o o l e d hollow tu r b in e b l a d e s .

Cite

follow ing a s s u m p t i o n s w e r e m a d e to p e r f o r m Use c a l c u l a t i o n s : a.

The c o m p r e sso r p olytropic e ffic ie n c y

b.

..he c o m b u s t i o n e ff ic ie n c y

c.

Che turbine polytropic e ffic ie n c y .............................. 0.38

c.

s h e h y d r o g e n - c a r b o n r a t i o of the fuel

e.

The com b u stor p r e ssu r e lo s s

.......... ..........

.................................... .

..... .

..................

OTd 0.475

0.167 0

'The v a l u e s of Cc / G a , q / L p i 0 /T, , and T'q a r e s e l e c t e d and the s t a n d a u s e a le v e l v a l u e s of p r e s s u r e and t e m p e r a t u r e a r e assu m ed for T dp, r e s p e c t i v e l y .

c i e s of unity.

and

The c a l c u l a t i o n s w e r e m ade f o r c o m p o n e n t e f f i c i e n ­

C h a r t I of r e f e r e n c e 30 w as e m p l o y e d to obtain £ h c .

C h a r t s I and IV of r e f e r e n c e 50 w e r e em p loyed to obtain f / h and L h.m I p i s o b t a i n e a f r o m the following r e l a t i o n

v_-ia

_ (:m. _ Gc + Of) j-t

4.1

The t u r b i n e p r e s s u r e ratio corresp on d in g to Iq for an u n cocled t u r b i n e i s obtained from C h a r t IT of r e fe r e n c e 30. To c o r r e c t the turbine p r e s s u r e ratio obtained above f o r the effect of r e m o v i n g q/Lq, the p r e s s u r e ra tio i s m u ltip lied b y the c o r r e c tio n fa cto r obtained fro m f i g u r e lb.

The a v a i l a b l e p r e s s u r e r a t i o f r o m s t a ti o n 0 to s t a t i o n 7 i s

o b ta in e d f r o m

/h

x T /P„

- T /P„

'Ih.e v a lu e of h fiis o btain ed f r o m eq u atio n A17 ( e r = 1) and h7 i s o btain ed f r o m C h a r t h i of r e f e r e n c e 30.

The id e a l u s e f u l g e n e r a t o r

e n th a lp y is the d iffe re n c e b e tw e e n h G and h 7 . b.he c y c le c a lc u la tio n s above w e r e m a d e f o r c o m p r e ss o r p r e s s u r e r a t i o s f r o m 2 to 64, f o r t u r b i n e i n le t t e m p e r a t u r e s from. 2000T to 3500R. f o r c o o la n t r a t i o s f r o m 0 to 30 p e r cent, and f o r q/Lq v a l u e s f r o m 0 to 20 p e r cent. F ig u re 13 p r e se n ts the r e s u lts of the c a lc u la tio n s in the c o n v e n t io n a l m ari ner b y p lo ttin g rdiUg v e r s u s h, /'b> with tb5 a s a p a r a m e t e r .

In

fig u r e 13 the coolan t ratio Cc /G a is a lso plotted a s a p a ra m eter. In o r d e r to r e d u c e the n u m b e r of c u r v e s in f i g u r e 13 a c o n s t a n t v a lu e of q/Lq of 10 p e r c e n t w a s p lo tte d .

Gc/Ga

F i g u r e 13 show s that the c o o la n t r a t i o

h a s a m a r k e d influ e nce upon the u s e f u l e n th alp y f o r a given

42

■Sf?

m

soo

i'sts:

m

;01

5i

o p tim u m p r e ssu r e r a t i o f o r a given valu e of T... d e c r e a s e s a s Gr. / G 0 increases.

To f a c i l i t a t e i n t e r p o l a t i o n betw een v a l u e s of d s , Tie r e ­

s u l t s of the ca lcu la tio n s a r e p r e s e n t e d a s c a r p e t plots in C h a r i 1 of ap pen d ix G.

The in d iv id u a l c a r n e t p lo ts a r e con stru cted at a given

vaiue of E; / E

with L hus;/^T a s the o r d i n a t e , and Gc / G a , q / L p and

df/tet a s p a r a m e t e r s .

The influence of a change in q/Lq upon d h ua f o r

c h a n g e s in the v a l u e s of Gc / G a is neglig ib le , and t h e r e f o r e the i n f l u ­ en ce of q / L | i s p reserved on die c a r p e t i lots onlp fo r the value of Gc / G a e q u a l to z e r o .

F o r v a l u e s of Gc / G a otrier than z e r o , only the

10 p e r c e n t v a lu e of q /L q i s p r e s e n t e d on the c a r p e t p lo ts . O'

E /E

F o r values

l e s s than B the d if f e r e n c e in v a lu e s of L h1J0, f o r v a l u e s of ojL^

f r o m 0 to 20 p e r c e n t w e r e too s m a l l to be included. 4.2 M ethod of D e t e r m i n i n g &iiuc for

A ctual C y c le s.

The l o s s e s o c c u r r i n g in the c o m p o n e n ts of actual e n g in e s c a n be taken into a c c o u n t by introducing the polytropic e ffic ie n c ie s of the c o m p r e ss o r and tu r b in e (24).

F i g u r e 14 illu s tr a te s sc h e m a tic a lly the

c y c le w ith lo s s e s and the c o r r e s p o n d i n g i d e a l c y c le . 4 n r. for the id eal and actu al c y c le s are m ade equal; consequ en tly, the value of Lt for the i d e a l and a c t u a l c y c l e s a r e a ls o equal.

How ever, the c o m p r e s s o r p r e s ­

s u r e r a tio s corresp on d in g to the i d e a l and actu al c y c le s w ill not be equal, an d the va lu e of E l / l G will b e d i f f e r e n t f o r the i d e a l and the a c tu a l c y c l e s .

The l o s s in Ahup- due to the i n e f f ic ie n c y of the com p on-

■LI ■

A

~ ,L> . .

%~

e a t s can oe obtained by e x p r e s s i n g the c o m p o n e n t l o s s e s by trie d i f f e r ­ e n c e in e n t r o p y PS-; b e tw e e n s ta tio n fy and s t a t i c r i (se e f i g u r e Id). r he l o s s in the c o m p r e s s o r can be e x p re s s e d , b y the e n lr e p y d i f f e r e n c e dSc b e tw e e n s t a t i o n hi and s ta tio n P.

r h e r e l a t i o n d e r i v e d in a pp en d ix

D, f o r obtaining h S (, i s given by

P,SC - c ( i -

c ) in p y -

DIO

S im ila rly f o r the tu r b in e

a S | = c r) (1 - 1/"j) in

p—

jjll

p’he to ta l e n t r o p y d i f f e r e n c e I S 1 ■is c.vual to i.hc s u m of the e n tr o p y d i f f e r e n c e s of the indiv idual c o m p o n e n ts , uS.,, f Si-,, and h S+. (2-). :dbe c y c l e c a l c u l a t i o n s d e s c r i b e d in s e c t i o n d,l w e r e r e p e a t e d em p lo y in g Ohio f o r the v a lu e of m- a n d 'd j.

g h e r e s u l t s of the above

c y c le c a l c u l a t i o n s a r e p r e s e n t e d a s c a r p e t p lo ts in Chart II of a p p e n d i x G by a m eth o d i d e n t i c a l to th at e m p lo y e d f o r oresen tin g the d a t a of the c y c l e c a l c u l a t i o n s d e s c r i b e d in s e c t i o n -hi. 4.3 G e n e r a l i z e d Method of u b t a i n i n g the G enerator P e r f o r m a n c e . A m ore g e n e r a l , though som ew h at m ore involved, m e t h o d of p r e ­ d ic tin g the p e r f o r m a n c e of a gas g e n e r a t o r em p lo yin g a i r - c o o l e d hollow t u r b i n e b l a d e s can be developed.

D r i g g s ~ show s th a t the

O c L ‘1

. dp

C 0 0 0 r O x i U ' . V Cl

[ d o .

0 :X_.j

-

W rO V .S

" :'C

d / h e r e 'he n ob n of hie eii uaJp a i r flow.

Jh is show

---

w ro n s a r e FhhU p e r pound of c o m p r e s s o r

in tip o r e In ha* ; he j. e r f o r r n a n c e c h a r a e i e r i e : ice

ni bicin oe.iure. o r c -cies eLiiolo imx a i r c oole d hollow u rrb h w olad.es can a l s o be c o r r e l a t e d if 'he p a r a r i t u e r dlib is W-.biiplied b

of c o i n p r e s n o r a i r flow.

(1. •

b'ignre lb can no h x . n o e a direcop

■lie ijeuex*a..or* nSefiil e .n hu ip

.o o' ai .

vinei when me -/alLies on .vuf», dm-./ ■> ,3

p x./-3 ci.. J .-hip a r e Known xor an. u l . r wbe a ■.. i _i Injii . e p e e s , : i r . o'alon. s of hh-.(, obtained fro.:.. hipnre Id a r e v alid fo r c o s ' iI d r o v e n - c a r o o n ioeln.

i iiereiorej

lox

c o r o r o o.i i n u r e In a r e m o r e

g e n e r a l .hen h o c h a i n s ; m x . i c J in ap p en dix oh

47

is. 3tt

o

c>

section o H e at T r a n s f e r C o r r e l a t i o n 5.1 H eat T r a n s f e r C o r r e l a t i o n R e l a tions. The h e a t t r a n s f e r p r o c e s s f o r the flow of f l u id s thro u gh and a r o u n d tu bes h a s r e c e i v e d c o n s i d e r a b l e a tte n tio n and th e r e s u l t s of e x p e r i m e n t s h ave been s a t i s f a c t o r i l y c o r r e la te d by equations of the form

Nu =

(Re, Hr)

5.1

E x p e r i m e n t a l e v id e n c e i n d i c a t e s the above r e l a t i o n is not valid f o r high v a l u e s of the a verage flu id v e lo c ity and the r a t i o of the a v e r a g e fluid t e m p e r a t u r e to the tube w a ll t e m p e r a t u r e (14) (28) (29). H owever, a c o r r e l a t i o n of the e x p e r i m e n t a l d a t a can be obt ained f o r the above c o n d itio n s by em p lo yin g an e qu atio n of the f o r m (la)

Nu =

(Re, P r , Tc / T b)

5.2

where T r, ^ the mixing cup t e m p e r a t u r e of the fluid ‘"'E Tp = the t e m p e r a t u r e of the tube w a l l ° E 5.2 R e la tio n b e tw e e n Gc / G a and q/Ly. Gc /G a an d q /Ly a r e not i n d e p e n d e n t v a r i a b l e s and t h e r e f o r e , Gfl/ G a m a y b e e x p r e s s e d a s a fu nction of q/Ly. The f o r m of the fu nction r e l a t ­ ing G c / G a and q/Ly will depend upon the design of the h e a t t r a n s f e r

can b e a p p r o x i m a t e d by em p lo y in g the e n e r g y e q u atio n and the c o r r e l ­ ation. r e l a t i o n d i s c u s s e d in s e c t i o n 5.1.

The d e r i v a t i o n of h e a t t r a n s f e r

c o r r e l a t i o n e q u atio n s f o r a ty p ic al t u r b i n e i s p r e s e n t e d in a p p e n d i x E. The r a t i o of the c o r r e l a t i o n equatio n f o r the c o o la n t flow t h r o u g h the hollow b l a d e to the c o r r e l a t i o n equation f o r the g a s flow a r o u n d the b l a d e i s given by

When the c o m p r e s s i o n w o r k add ed to the c o o la nt iron:, the blade r o o t to the tip i s n e g l e c t e d a nd c n is a s s u m e d c o n s t a n t , the o n e - d i m e n s i o n a l e n e r g y e q u a tio n a p p lie d to the floe-; of the c o o la n t thro u gh the hollow blade becom es

q

I -LtLT_m S

E ld

CI

The d e s i r e d r e l a t i o n b e tw e e n Gc / G a and q/Ly i s d e t e r m i n e d by the s i m u l t a n e o u s so lu tio n of e qu ation s E9 and Eld.

The a f o r e m e n t i o n e d

so lu tio n is p r e s e n t e d in a g r a p h i c a l f o r m in f i g u r e 1G. R e s u l t s of the above a n a ly sis a re approxim ate and only r e p re sen ta tiv e or p relim in a ry d esig n v a lu es a re o b t a i n e d by i t s a p p lic a tio n .

The a c t u a l v a l u e s of the

r e q u i r e d c o o la n t r a t i o sh o u ld be o b ta in e d f r o m t e s t data.

20CK

Lg 2500 R T'b = I800 R

Tg rr. 3 0 0 0 R

"'b ~ l^OOR"

T,r ~ 3000R Tv - 2 0 0 OR

O c c la n t R a t i o , Pig,

16.

C c c ln n t re t i c

- 3500R Tb = 2000R

- v 100

requirement

from. h e a t

t r a n s f e r oechaniam

Section o V elocity Cycle Calculations 6.1

v e lo c ity C yc le Useful Enthalpy.

C h a r t I and II of a p p e n d i x G in clu d e only a p o r t io n of the effect of the flig ht v e lo c it y upon the p e r f o r m a n c e of the c y c l e .

It i s n e c e s s a r y

to add, to the v a lu e of a h UP, o btained f r o m C h a r t I and II, the va lue of the u s e f u l e n th a lp y of the v e lo c ity c y c l e Ghuv b e f o r e the c o r r e c t i o n foi the eff e ct of flig ht v e lo c it y i s c o m p l e t e .

Che e n th a lp y ch an ge L b nv of

the v e lo c it y c y c le c o r r e s p o n d i n g to -ohy of m e gene r a t e r c y c l e is d e ­ r i v e d in a p p e n d ix F . Che r e s u l t i s

,-hbv = (1 - - c / G a ) Lhh - L f , „

Ft

Having the valu e of n h p v , the v a l u e s of the v e lo c it y c y c l e u se f u l e n th a lp y h h uv (see f i g u r e o) c a n b e o btain ed by m e a n s of the c o n v e n ­ tio n a l m e t h o d s f o r a n a l y s i n g gas t u r b i n e c y c l e s .

E m p lo y in g the a f o r e ­

m e n t i o n e d c o n v e n tio n a l m e t h o d s , c y c l e c a l c u l a t i o n s m e r e m a d e f o r a n u m b e r of d if f e r e n t flig h t s p e e d s and g e n e r a t o r u s e f u l enth alp y v a lu e s, Ghe r e s u l t s of the above c a l c u l a t i o n s a r e p r e s e n t e d in f i g u r e 17. a huv i s o btain ed f r o m a k nowledge of Gc / G a , E ly , -Gi n u m b e r M.

and the i li g n t Iviacn

The t o t a l u s e f u l en th a lp y :,hu -P i s the s u m of Lh.aP. and

- hu v 6• ^ R e la ti o n b e tw e e n U s ef u l E n th a lp y and T h r u s t . F o r the t u r b o j e t engine the t o ta l u s e f u l enth alp y i s r e l a t e d to the

1 40

1 120 1.1

)Q.

80' lU V /a Z

80

40

0. 5

20

Q ufr-0

_

____________ _________________________ 200 300 400 500

100 “

Fig,

17,

0 c / ' 0 p ) h h h /@z

The v e l o c i t y

-

h h U (5 / g r

cy c le performance in v o lv in g a ir - c o o l e d b la d es in the t u rb in e,

t h r u s t by the m om entum r e l a t i o n and is given

F = 223.7 G , , - n , , h u r;. - h a m 4 g

s.-

F o r the t u r b o - p r o p e l l e r engine the m a x i m u m t h r u s t is o b t a i n e d by dividing the use fu l enth alp y jet nozzle.

hu ,- betw een the p r o p e l l e r and the

Toe o p iirnum d iv isio n of the u s e f u l enthalpy

tained a p p r o x i m a t e l y ,

o h u -is cre­

b y c o n v e r t i n g the v e l o c i t y c y c le u s e f u l enthalp,

;_.huv into t h r u s t in the n o z z l e and c o n v e n i n g the g e n e r a t o r u se f u l enth alp y '-*hUfy into t h r u s t by e m p lo y in g the p r o p e l l e r (24). The t h r u s d e v elo p ed by the j e t n o z z l e by e m p lo y in g the above divisior:. of r fy- h given by

Fj

g . ' / o n ,-.nuv o

o

and the th ru st de v elop ed by the p r o p e l l e r is given b T?

_p

_ 778 U D Gp- L q O' 5 5 ffv y -

6.3 A i r Intake Ductin g L o s s e s . The l o s s e s in the a i r i n ta k e and d i f f u s e r s y s t e m a r e tak e n into a c c o u n t b y d e t e r m i n i n g the v a lu e of

t

f r o m the a c t u a l p r e s s u r e r a t i o

k-

m e e n e o tiv e M a ch num oer lvie corresp on d in g to t ootam ea w orn equation G.5 is given by k -f

'2 Ig M e b k b n p ;;

I

k _ l;

Em ploying the e ffective M ach number Me i n s t e a d of the actual m g : i t M ach num oer Ivl to ootain tne u s e t u : enthalpy

01

tne velocity

c' yv c le bin-v thus tne w b from fig u r e 17 r e s u lts in a s m a l l e r value, of Ah,.,,.and Li. v d i f f u s e r l o s s e s a r e aceounted f o r by a d e c r e a s e in t h r u s t , ow D e t e r m i n a t i o n of u h y / l ‘r and i / b f T. To f a c i l i t a t e the d eterm in ation of the t e r m

i

i

u

■’

“

w

'

v w

l

ST

-p -y

""

-

m

~ T T ^

r e o u i r e d to obtain. b h iry f r o m f i g u r e 17, the v a l u e s of tne c o r r e c t e d en th a lp y change — L occu ring in the c o m b u s t o r f o r d i f f e r e n t v a lu e s of the c o r r e c te d tem p eratu re T? Am and the id e a l c o m p r e sso r p r e s s u r e ratio Ft i / P T

a r e p r e s e n t e d in f i g u r e 18,

The da ta f o r the c u r v e s

1. The i d e a l and a c t u a l c o m p r e s s o r r a t i o s a r e r e l a t e d by py j. imu i

\p j \ - 1/

G G CO n G G

■V

G Jr-4 _P

CO XC r*"* G 0 -~s

*X r~i G

o 6 S i to

.c'C!

co CO

CO

■ ti

G .r-i +-■' — . CC O G CO r~i • Q) G 3

G

t-q r>

o CO oO G CQ G u

^•'Ll

G o 40 CO

L\: ■' Gr

CC O —,

9 9 q _ 'j

CC

M 0)

f-'i ..0

V.

G XT' G X

q

r-G

G 00

H-t G

£ O O cO cx

r.G G ,? C“

>„

H

■j 4=

CO X 4-j

X) r vjg n6 -

a Gy - A a ) A . + -...q nhp +

A8

h,

T he r a t e of gas flow th rou gh the turbine 3A d if f e r s f r o m the r a r e of a i r flow through the c o m p r e s s o r Ga by the addition of the r a t e of fuel flow Gf and the s u b t r a c t io n of the r a t e of coo ling flow Gr>. E ence

'a “ u c ■ "'i die rate of flow dow nstream of the tu rb in e i s given

A10

T OS

S u b stitu tin g e q u a tio n s Ad and A 10 into equ atio n AS y ield s the following re la tio n

(Ga + Gf) llr = Gf n

+ Ga h , + G, Ghb

A ll

D ividing by (Ga + Gf) equation A l l b e c o m e s h :, - -■—S iL~— ~~ Ini, + s_~* 2 a.'m _ . h ,i + Ga^ - Gc + Gf 'w Ta -*■ - f u-a + wf vva + vjf

Lh

A12

By d e fin itio n O? / A.

•'"h

J-, O

—- '■ ' ■' ] I

/ / ./-i

/--n

3,

~

\

- ■ i

S u b s titu tin g e q u a tio n A id into equ atio n A l l and dividing by d a / ( G a - Gr.), one o b tain s

h

= f/biv i

^

c / ' Jla

+ f/b

1 - 0 7 0 ,,! i

~ ~

i/o

;i + i/b) drib

T •

...

i

Ale

l" - G e / w a ”+~f7o

•e a rra n g in s /A aJ i/b y; _ n..» /ni y/«'n iJ" A- '-'0./ ‘-‘a/T s-■

'

t~~r-ri a { .

/ --la) (i /A"'™'"b + i / DV)i"” + (-L - w c/w a) d/o / -i

,• v

i g,

•■'--aail--.

fA I r ;

S h e f i r s t t e r m on the r i g h t hand s id e of e q u a tio n A id is n e g lig ib le in c o m p a r i s o n to the s e c o n d and t h ir d t e r m s , b e c a u s e (1 - Gr>/Ga)f/b -:u f o r a ll p r a c tic a l v a lu e s of i / b . u

111 +

H ence

(-1 ~ b c / G a ) ( l + i / b )

ohp

1 + 0 -Gc/Ga) f/b ■

S u b s titu tin g the e x p r e ssio n fo r c o r r e c t e d enthalpy h / 9 c into equation A lo r e s u lts in the follow ing g e n e r a l i z e d fo rm i i i + (1 - Gc / G a) (l - f / b) -Adlb n6 _ oT •; gt i r a ~ ;'j c/dma) i / i

Ail /

E q u a tio n s A16 and A17 a r e in d e p e n d e n t of q and t h e r e f o r e , the

e n th a lp y a t the d if f e r e n t s t a ti o n s f r o m i to h of ihe c y c l e of a p-as turbine engine em p loyin g a ir -c o o le d hollow tu rb in e b la d e s c an be c o m p u te d w ith out the know ledge of the a m o u n t of h e at t r a n s f e r r e d to hie e w n a n t.

A P I ENDIX B ANALYSIS I P I LIE MOMENTUM PR ESSU R E SPSS Due to the chang e in m o m e n tu m a cc o m p a n y in g an addition of h e a t to a c o m b u stio n c h a m b e r , a lo s s in to ta l p r e s s u r e o c c u r s ; u s u a lly d enoted a s the m o m e n tu m to ta l p r e s s u r e lo s s .

The m o m e n tu m to ta l

p r e s s u r e lo s s c a n be d e te r m i n e d f r o m a o n e -d im e n s io n a l a n a l y s i s in the follow ing m a n n e r: The c o m b u s tio n c h a m b e r is a s s u m e d to be c ir c u la r with constan t c r o s s - s e c t i o n a l a r e a , and the c o m b u s tio n c h a m b e r o u tle t M ach num ber M„ is a ssu m ed to be c o n s ta n t.

P R m ay be held c o n s ta n t fo r an i n c r e a s e

in the t e m p e r a t u r e r i s e by e i t h e r i n c r e a s i n g the c r o s s - s e c t i o n a l a r e a o r d e c r e a s i n g the w eight r a t e of flow Gp

T he r e l a ti o n s f o r h e a t a d d i­

tion to a c o m p r e s s i b l e , in v isc id flu id flow ing in a c o n s ta n t a r e a d u c t a r e the e q u a tio n s fo r a R a y lie g h line, given by (hi) M"

"i" 1

art M' UP

M

1 + k PI }

B2

w h e re ( )* - r e f e r s to c o n d itio n s w h e re M = 1 If c o m b u s tio n c h a m b e r s f o r an engine without c o o lin g and an engine w ith a i r - c o o l e d tu r b in e b la d e s c o n s u m e the s a m e w eight rate

of fu el, the r e l a t i o n of the enthalpy c h a n g e s in a cco rd a n ce with the f ir s t law of th e r m o d y n a m ic s is given by

pc - !1u - '- 't --m o r a s s u m in g h h = c D h P w ith c n c o n s ta n t

(1 + Gc / G t ) Ag

= cm

E-

w n e re ( )., - r e f e r s to uncooled engine v a lu e s ( )n - r e f e r s to a ir -c o o le d engine v a lu es A s s u m in g v a lu e s of the r a t i o of the tem p eratu re r i s e to hie to ta l in le t / rp tem p eratu re -pi-A- fo r the unc.ooled engine the v a lu e s of the ra tio fo r 11 the c o r r e s p o n d i n g a i r - c o o l e d eng ine f o r d if f e r e n t v a lu e s of Gr /3G- c an be obtained fro m equation B3. t /a lu e s of P j / p R a rid h l/T f* in a g r e e m e n t w ith e q u a tio n s E l and B2 r e s p e c t i v e l y , a r e g iven a s fu n c tio n s of the Ivlach n u m b e r in the ta b le s of r e f e r e n c e 31. Hence, f o r a v a lu e of M;: the c o r r e s p o n d i n g v a lu e s of P r / P A and TRi/iLg* c an b e o b ta in e d .

T he valu e of b l£ /Gq x is given by

ana no, i i. i _ -l r

m / rp, _ fMh42_L_ r7 h o xin/ -ia

die v a lu e of P r / P i * c o r r e s p o n d i n g to the v a lu e of T r / T

o b ta in e d f r o m e q u a tio n B3 c an be d e te r m i n e d f r o m the ta b le s of r e f e r e n c e 31.

The r a t i o of the in le t and outlet, to ta l p r e s s u r e s of the c o m ­

b u stio n c h a m b e r is given by

P h L = P t l / P th p t2 The c o m p a r i s o n of the m om entum p r e s s u r e lo s s in c o m b u s tio n c h a m b e r s is a c c o m p lis h e d by d e te r m in in g the v alue of the r e l a ti o n

(PT. / P i r ) . . ( V lT P t su by e m p lo y in g the ta b le s of r e f e r e n c e 31 and e q u a tio n s E4, Bo, and B7 a t d if f e r e n t v a lu e s of Ma , Gc / G g and t e m p e r a t u r e r i s e d d u . d'he d a ta o b ta in e d by a p p ly in g the c o m p a r is o n m ethod ou tlin ed above a r e p r e ­ s e n te d in f ig u r e 24.

F i g u r e 24 show s th a t the c h an g e in the m o m e n tu m

to ta l p r e s s u r e lo s s a s s o c i a t e d w ith a i r - c o o l e d g as t u r b in e type en gin es w ill not be m u c h d i f f e r e n t th an the v a lu e f o r the c o r r e s p o n d i n g uncooled e n g in e ; t h e r e f o r e , the e ffe c t of m om entum to ta l p r e s s u r e lo s s m ay be n e g le c te d in an a n a l y s i s of the l o s s e s a s s o c i a t e d with the a i r - c o o l i n g of hollow tu rb in e b la d e s .

oDOr‘

oa

+

c i

11 I—! •* ■’

t "■

C J. I r ~i —

|

t - |E-

AT P E NT) IX C E F FE C T CF REMoTTNG THE HEAT

0

FROM T E E 'TURBINE

As d i s c u s s e d in s e c tio n 2 .T the th e rm o d y n a m ic p r o c e s s e s of a cooled tu rb in e w ithou t a e r o d y n a m ic lo s s e s (pj. = 1.0) should be c o n s id ­ e red a s r e v e r s i b l e polytropic p r o c e s s e s and the p r o c e s s e s w ith a e r o ­ dynam ic l o s s e s a s i r r e v e r s i b l e p o ly tro p ic p r o c e s s e s . F or r e v e r s ib le p clytrop ic expansion p r o c e s s e s the shaft w o rk i s equal to the a v a ila b le h y d ro d y n a m ic w o rk of the flu id /v d p , o r

The p h y s ic a l s ig n ific a n c e of equation Cl and the in terp retation of the p r o c e s s e s f o r a cooled tu rb in e a r e p resen ted in s e c tio n s 3.C and

F o r the r e v e r s i b l e a d ia b a tic ex p an sio n L:.ru - j

L

in = n vdp

F r o m f ig u re 10 the r e l a t i o n of hh and ...j. fo r the r e v e r s i b l e p o ly ­ tro p ic exp an sio n is s e e n to be q - ant Hence, equation C l b e c o m e s

L h t - /v d p = q By definition the p o ly tro p ic e ffic ie n c y of the u n co o led tu rb in e is given by rp = T h y / / v d p

C3a

qin c e

Lhf = L+ f o r the u n cooled tu rb in e e q u atio n C’4 m a y be w ritte n a = j vdp

- Of

or L t - Ut /v d p = 0

Co

S i m i la r ly f o r the c o o le d tu rb in e

or Co

h h t - T'1| / v d p - q S u b stitu tin g fo r y h *■f r o m e q u atio n C3, equ atio n CS b e c o m e s 1.11 -

p t/v d n = 0

07

F r o m e q u a tio n s C7 and C5 it is e v id e n t th at th e p o ly tro p ic efficiency c a n be a p p lie d to the co oled t u r b in e in a m a n n e r a n alog ous to the m eth o d em p loyed f o r uncooled t u r b in e s . The a n a ly sis of m e e ffe ct of q upon the perform an ce of the t u r ­ b in e is d e v elo p ed f r o m e q u atio n C l . 7 h e s h a f t w ork T t can be ex p r e sse s b y (see f i g u r e 10) T,, = c I T - T . ^ P L ^^ ""T ;

PR

o r in t e r m s of the t u r b in e p r e s s u r e r a tio k -1

Lt = c pw

. 1 - (P4 */P~

~ ) 1c~ i

Co

T he r e l a ti o n b e tw e e n p r e s s u r e s and t e m p e r a t u r e s of a p o ly tro p ic

p r o c e s s is P v lL = c o n s ta n t

CIO

E m p lo y in g e q u atio n Gr 0, the a v a ila b le hydrodynam ic w o rk is given by ■4 vdp = P

n v

3 d-o TO

^ 11

I n te g ra tin g fpT-) l"

p v n J' c n m T

i _ fli - ~ \ P -

nn

or in t e r m s of tne l e m o e r a t u r e

vdp = — -

‘

p

i

' L

t

- ;

,

“ 4 J

F r o m e q u a tio n C7 f o r a r e v e r s i b l e p o ly tro p ic p r o c e s s the sh a ft w ork and availab le flu id w ork a re equal. Equating e q u atio n s C lc and Co and so lv in g f o r n /n -1 y ie ld s

n n -1

R (T

- T.: 1 - T, )

. .

H aving th e v a lu e of n / n - 1 , the tu rb in e p r e s s u r e r a t i o P.. / l c a n be o b tain ed f r o m n fh _ tR n h P:, " O T , /

The c a lc u la tio n s a r e p e r f o r m e d by: (a) a s s u m in g v a lu e s f o r T„ and P3 / P ,

; (b) em ploying T:; and P. / P , in e q u atio n C9 to obtain

l t , and the valu e of Ty

f r o m equation C8; (c) a s s u m in g a v alu e of

q / .dp rI 4 is determ in ed fro m the r e s u l ti n g value of L h given by equ atio n C3; (d) n /n -1 is o b tain ed f r o m e q u atio n 0 1 4 ; a n a (e) P, /P_ is o b tain ed by em ploying e q u atio n 0 1 5 . A c o r r e c t i o n f a c t o r defined by the r a t i o of E, / p .

to ly / l y : is o b tain ed f r o m the r e s u l t s of

ca lc u la tio n s above and p resen ted in fig u r e 12 a s a function of T, , P~ / P i

and q /L t . s h e p r e s s u r e r a t i o B / F g

i s m u ltip lie d by the

c o r r e c t i o n f a c t o r c o r r e s p o n d in g to the g iven tu rb in e o p e r a tin g conditions to obtain the actual p r e s s u r e ratio req u ired by the tu rb in e to p erform the s h a f t v/ork L|-

A P P E N D IX D METHOD OF INCORPORATING THE COM PO NENT LOSSES It Is c o n v en ien t to a c c o u n t f o r the l o s s e s o c c u rin g in the com p o n en t; of actual c y c le s by in tro d u c in g polytropic e ffic ie n c ie s . F ig u r e 14 i l l u s ­ tr a te s by the aid of a h -S diagram the c y c le with lo s s e s and the c o r r e ­ sponding id e a l c y c le .

The value of the id e a l c o m p r e sso r work is m ade

eq u al to the a c tu a l c o m p r e s s o r w o rk ; t h e r e f o r e , the id e a l and a c tu a l tu rb in e w ork s a r e a lso equ al. The l o s s e s a r e acc o u n te d f o r by c o n s i d e r ­ ation of the change In e n tro p y f o r the actual p r o c e s s e s in the follow ing way F o r a r e v e r s i b l e p r o c e s s (19) dT vdp d S = Cr, —- + ~ ~

u.

F or a p e r fe ct gas D1 m a y a lso be e x p r e s s e d by

d S = cp N + R —■

D2

f o r a c o n s ta n t t e m p e r a t u r e p r o c e s s tv dP P

D:

I n te g r a tin g

L S - R In gr1 o

d
,, a t the b la d e r o o t e q u al to the total tem p eratu re at the c o m p r e s s o r d i s c h a r g e df and dividing by the tu rb in e o u tp ut e q u a tio n E12 becom .es

F o r the c a s e w h e re cr . is c o n s ta n t e q u atio n E I 4 b e c o m e s

n-1//T = rLm£L_m_c — L cm

i__i \

rp

p :■,

J— ; J- -J

The s im u lta n e o u s s o lu tio n of e q u a tio n s Ed and E15 is the d e s i r e d r e la tio n b etw een Gc /G a and

q /h q .

A g r a p h i c a l m ethod of so lv in g

equations EO and E15 sim u lta n eo u sly is p r e sen ted in figu re 16. The in te r se c tio n of the curve of equation EC w ith the cu rve of equation E l d i s the d e sir e d sim u ltan eou s solu tion . A fa m ily of c u r v e s of e q u a tio n s E l and E l d a r e included in fig u r e lb , so th a t the r e l a ti o n of Tr. / T p and q /L -r

c an b e ob tain ed f o r a w ide r a n g e of the d e s ig n p a r a m e t e r s e m ­

p lo y ed for the cooling of hollow turbine b la d e s .

lOO

A P P E N D IX F DERIVATION OF THE V E L O C ITY CY CLE EQUATIONS The v e lo c ity c y c le is s im ila r to a Drayton c y c le and th erefore th m ethods of determ in in g the p erform an ce for a gas turbine c y c le can a lso be em ployed to obtain the p erform an ce of the v e lo c ity c y c le . F ig u re 5 show s that the e n th a lp y change in the v e lo c ity c y c le c o r r e spending to Thy in the g en erator c y c le is given by

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Dm " Uiiug “ n i

T he in flu en ce of f / b in e q u atio n F2 Is n e g lig ib le , h e n c e ^ nbv = v1 “ vJc / vJa) L 'nb ~ a nug

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Since the value of n h g v can be d eterm in ed , the p erform an ce of the v elo cizy c y c le f o r a given v a lu e of flig h t v e lo c ity c a n b e o b tain ed f r o m conventional m ethods em p loyed for c a lcu la tin g the p e r f o r m a n c e of B rayton c y c le s .

T he p e r f o r m a n c e of the v e lo c ity c y c le f o r d iff e re n t

v a lu e s of flig h t M ach n u m b e r M w a s c o m p u te d ; the r e s u l t s a r e p r e ­ s e n te d in f ig u r e 17.

A PPE N D IX G rENERATOR PE R F O R M A N C E CHARTS F D R GAS TURBINE CYCLES

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