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1/6

E n e r g y C o n v e r s i o n . V o l . 8 p p . 8 5 - 9 0 . F e r g a m o n P re ~ s 1 9 6 8 .

P r i n t e d i n r e a t

B r i t a i n

T h e S u p e r c r i t ic a l T h e r m o d y n a m ic P o w e r C y c le

E G F E H E R t

Received13 January 1968)

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

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

f o r c l o s ed c y c l e e n g in e s t o d a y a r e t h e R a n k in e C y c l e a n d

th e r e c u p e r a t e d B r a y to n C y c l e . B o th a r e c h a r a c t e r i z e d

b y tw o c o n s t a n t p r e s s u r e a n d tw o i s e n t r o p i c p r o c e s s e s .

T h e R a n k in e C y c l e o p e r a t e s ma in ly i n t h e s a tu r a t e d

r e g io n o f i t s w o r k in g f l u id w h e r e a s t h e B r a y to n C y c l e

p r o c e s s e s a r e l o c a t e d e n t i r e ly i n t h e s u p e r h e a t o r g a s

reg ion .

T h e s imp le R a n k in e C y c l e is i n h e r e n t l y e ff ic i en t . H e a t

i s a d d e d a n d r e j ec t e d i s o th e r ma l ly a n d t h e r e f o r e th e

id e a l c y c l e c a n a c h i e v e o v e r 9 0 p e r c e n t o f C a r n o t

e f f ic iency be twe en the sam e tem pera ture s . P ressure r i se

in t h e c y c le is a c c o mp l i s h e d b y p u mp in g a l i q u id , w h ic h

is an e f f ic ien t p rocess r equi r ing sma l l ene rg y input .

T h e r a t i o o f n e t w o r k o u t p u t t o g r o s s w o r k i n t h e c y cl e

i s l a r g e . A mo n g th e l imi t a t i o n s o f t h e c y c l e a r e t h e

f o l l o w in g :

i ) The tem pera ture r ange of the cyc le is seve re ly

l imi t e d b y t h e n a tu r e o f t h e w o r k in g f l u id . A d d in g

s u p e r h e a t i n a n a t t e m p t t o c i r c u m v e n t t h is w i ll d e p a r t t h e

c y c l e f r o m i s o th e r ma l h e a t a d d i t i o n . I n c r e a s in g t h e

t e mp e r a tu r e r a n g e w i th o u t s u p e r h e a t l e a d s t o e x c e s s iv e

mo i s tu r e c o n t e n t i n t h e t u r b in e s , r e s u l t i n g i n b l a d e

e ros ion .

i i) S imp le r e c u p e r a to r c a n n o t b e e mp lo y e d t o r e c o v e r

h e a t f r o m th e t u r b in e e x h a u s t .

i i i ) Expans ion ra t io of the cyc le i s usua l ly la rge ,

r e q u i r i n g i n s o me c a s e s mo r e t h a n 3 0 t u r b in e s t a g es ,

T h e r e c u p e r a te d B r a y t o n C y c l e a d d s h e a t a t c o n s t a n t

p r e s s u re o v e r a t e m p e r a tu r e r a n g e . T h e t e mp e r a tu r e l e v el

i s in d e p e n d e n t o f t h e p r e s su r e l e v el . N o b l a d e e r o s io n

occurs in the turb ine . The pressure r a t io i s low, the re fore

o n e o r tw o t u r b in e s t a g e s a r e a d e q u a t e . A s imp le r e -

c u p e r a to r c a n r e c o v e r mu c h o f t h e t u r b in e e x h a u s t h e a t .

S o me o f t h e l im i t a t i o n s o f t h e c y c l e a r e :

i ) T h e c o mp r e s s io n p r o c e s s r e q u i re s l a r g e e n e r g y

in p u t , t h e r e f o r e t h e n e t w o r k t o g r o s s w o r k r a t i o i s sma l l .

i i) The cyc le i s ve ry sens it ive to c om press or e f f ic iency

a n d p r e s s u r e d r o p .

i ii ) H ea t t r ansfe r sur faces a re la rge for pressure leve ls

t h a t a r e t y p i c a l f o r c u r r e n t B r a y to n e n g in e s .

A th e r m o d y n a m ic p o w e r c y c le h a s b e e n d e v i s e d w h ic h

a v o id s m o s t o f t h e p r o b l e m s o f th e s e c y cl e s a n d y e t re t a in s

ma n y o f t h e i r a d v a n t a g e s . T h i s c y c l e o p e r a t e s e n t i r e ly

above the c r i t ica l p ressure of i t s working f lu id ; i t i s the

Superc r i t ica l Cyc le .

t Astropower Laboratory Missile and Space Systems Division,

Doug las Aircraft Co., Inc., 2121 Cam pus Drive, Newp ort Beach,

California.

85

2 . D e s c r i p t io n o f t h e C y c l e

F o r t h e t h e r mo d y n a mic a n a ly s i s o f t h e S u p e r c r i t i c a l

C y c l e , t h e p r o p e r t i e s o f i t s w o r k in g f l u id a r e r e p r e s e n t e d

in F ig s. 1 a n d 2 . A p u r e s u b s t a n c e s u c h a s w a t e r o r

c a r b o n d i o x i d e ) i s s h o w n o n a t e m p e r a t u r e - e n t r o p y

d i a g r a m a n d o n a n e n t h a l p y - e n t r o p y d i a g r a m . A l so

E n t r o p y

F i g . 1 . T e m p e r a t u r e e n t r o p y d i a g r a m f o r t h e s u p e r c r it ie a l

cycle.

- i e

. . . . . . . , ~ m V ' f C ~ P o i n J

I i

Entropy

F i g . 2 . E n t h a l p y e n t r o p y d i a g r a m f o r t h e s u p e r cr i ti c al c y c l e .

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8 6 E.~ G ; F E l l E R . . . .

s h o w n a r e l i n e s o f c o n s t a n t p r e s s u r e , c o n s t a n t e n th a lp y ,

c o n s t a n t t e mp e r a tu r e , a n d t h e s a tu r a t i o n l i n e a n d c r i t i c a l

p o in t .

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

a b , b d , d e , a n d e a . S e g m e n t a b r e p r e s e n t s a n i s e n t r o p i c

c o m p r e s s i o n o f t h e s u b c o o l ed l i qu i d f r o m p r e ss u r e P t t o

p2 . Segment

b d

r e p r e s e n ts h e a t a d d i t i o n a t c o n s t a n t p r e s-

s u r e p 2 t o t h e h ig h e s t t e mp e r a tu r e o f t h e c y c le a t p o in t d .

F r o m d t o e , i s e n t r o p i c e x p a n s io n o c c u r s f r o m p r e s s u r e

p z t o p l , w i t h a c c o m p a n y i n g w o r k o u t p u t . H e a t i s ex -

t r a c t e d f r o m e t o a a l o n g c o n s t a n t p r e s s u r e l i n e p l . A

p o r t i o n o f t hi s h e a t, r e p r e se n t e d b y e n t h a l p y d r o p f r o m

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

f l u id , r a i s i n g i ts e n th a lp y f r o m b t o c a t c o n s t a n t p r e s s u r e

p 2 . N e t h e a t r e j e c t e d i s i n d i c a t e d b y t h e e n th a lp y d r o p

f r o m f t o a a t c o n s t a n t p r e s s u r e p l . P o i n t a i s a t t h e l o w e st

t e m p e r a t u r e o f t h e c y cl e a n d a b o v e t h e t e m p e r a t u r e o f

t h e h e a t - r e c e iv in g r e s e r v o i r . N e t h e a t i n p u t t o t h e c y c l e

o c c u r s f r o m c t o d a t c o n s t a n t p r es s u r e p 2 . N e t w o r k

o u t p u t i s t h e d i f fe r e n c e b e tw e e n w o r k o u tp u t f r o m d

t o e a n d p u m p w o r k i n p u t f r o m a t o b .

A p s e u d o - S u p e r c r i t i c a l C y c l e h a s b e e n e mp lo y e d f o r

s t e a m p o w e r p l a n t s p r e v io u s ly . I n t h is c o mb in a t i o n o f th e

R a n k i n e a n d B r a y t o n C y c l es th e w o r k i n g f l ui d i s p u m p e d

f rom the sa tura ted s ta te to a supe rc r i t ica l p ressure . (See

F ig . 3 .) H e a t i s a d d e d i n a c o n s t a n t p r e s s u r e p r o c e s s o v e r

E n t r o p y

F i g . 3. Temperature entropy diagram for a p s e u d o s u p e r c r i t i c a l

cycle,

a t e mp e r a tu r e r a n g e . T h e s a me c y c l e , u s in g c a r b o n d i -

o x id e a s t h e w o r k in g f l u id h a s b e e n : d i s c u s s e d b y t h e

R u s s i a n , V . L . D e k t i a r e v [ 1] , a n d t h e I t a l i a n , . G . A n g e l i n o

[ 2] . T h i s C O 2 c y c le w a s d i s c u s se d b y t h e a u th o r , w i th o u t

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

in i t i al d i s c lo s u re o f t h e S u p e r c r i t i ca l C y c l e i n a n E n g in -

ee r ing Repor t [3] .

3 . C y c l e P e r f o r m a n c e A n a l y s i s

Refe r r ing to F ig . 2 , the idea l the rm a l e f f ic iency of the

Superc r i t ica l Cyc le i s de f ined a s

Wout

_ ( h a - - h e ) - - ( h o - - h a )

~ t - Q i n h a - h c ' (1)

w h e r e h i s t h e v a lu e o f t h e e n th a lp y a t t h e p o in t i n d i c a t e d

by the subsc r ip t .

Th e e f f ic iency of the cyc le i s h igh because , ( i ) the pum p

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

( i i ) mo s t o f t h e h e a t e x t r a c t e d f r o m th e c y c l e c a n b e t r a n s -

f e r r e d b a c k t o l h e w o r k in g f l u id b y r e g e n e r a t i o n , t h u s

r e d u c in g t h e t o t a l e n e r g y r e je c t e d f r o m th e c y c l e .

I n t h e a c tu a l c y c l e , t h e g r e a t e s t l o s s w i l l o c c u r i n t h e

e x p a n s io n p r o c e s s , w h ic h w i l l p r o c e e d a lo n g t h e p o ly -

t r o p i c l i n e

d e '

i n s t e a d o f t h e i s e n t r o p i c l i n e

d e .

T h e

e f fi c ie n c y o f t h e t u r b in e m a y b e d e f in e d a s

h a - he A H ~

e t - - - - (2)

h a - - h e A H t

T h e t o t a l w o r k o u t p u t i s r e d u c e d b e c a u s e o f th e l o w e r e d

turb ine e f f ic iency . Th e the rm a l e f f ic iency of the cyc le i s

a l s o l o W e re d . H o w e v e r , s o m e o f t h e e n e r g y n o t a v a i l a b l e

a s w o r k , i s a v a i l a b l e a s h e a t a n d c a n b e t r a n s f e r r e d t o

th e h e a t i n p u t p r o c e s s o f t h e c y c le . I n o th e r w o r d s , t h e

e n t h a l p y d r o p h e - - h e i s u t i l i z e d i n r a i s i n g t h e e n th a lp y

f r o m h e t o h e ' , t h u s r e d u c i n g t h e n e t h e a t i n p u t f r o m

h a - - h e to h a - - h e .

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

in c r e a s e i n p u mp w o r k . T h e l i q u id p u mp h a s h ig h e f f i -

C i e n c y , a n d t h e ma g n i tu d e o f t h e e n th a lp y d i f f e r e n c e

b e tw e e n i d e a l a n d a c tu a l p u mp w o r k i s s ma l l . T h e

e f fi c ie n c y o f t h e p u m p m a y b e d e f in e d a s

h o - - h a A H v

- 3 )

e v - - hb ' - - h a A H r

T h e a c tu a l t h e r m a l e f f i ci e n cy o f a r e a l c y c le b e c o me s

( h a - - h e ') - - ( hb - - h a )

(4)

V l at : h a - - h e

4 . R e g e n e r a t i o n

T h e r e g e n e r a t i o n p r o c e s s i s e s s e n t i a l t o t h e a c h i e v e -

m e n t o f h ig h t h e r m a l e f f i c ie n c y i n t h e S u p e r c ri t ic a l

C y c l e . T h e p r o c e s s i s mo r e c o mp l i c a t e d t h a n t h a t f o r

t h e B r a y to n C y c l e d u e t o t h e l a r g e d e v i a t i o n o f fl u id

p r o p e r t i e s f r o m th e i d e a l i n t h e v i c in i t y o f t h e c r i t ic a l

p o in t . F o r t h e p u r e s u b s t a n c e , t h e s p e ci fi c h e a t a t c o n s t a n t

p r e s s u r e

i n c r e a s e s w i th o u t b o u n d s a t t h e c r i t i c a l p o in t .

B y d e f in i t i o n , t h e s p e c i f i c h e a t a t c o n s t a n t p r e s s u r e i s

t h e r a t i o o f i n c re a s e i n e n th a lp y i n a c o n s t a n t p r e s s u re

p r o c e s s t o t h e c o r r e s p o n d in g i n c r e a s e i n t e mp e r a tu r e ,

expressed a s a de r iva t ive . Symbol ica l ly ,

C , = ( O 0 ~ I , ) ~ . (5)

I f t h e p r o p e r t i e s o f a p u r e s u b s t a n c e a r e r e p r e s e n t e d

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

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T h e S u p e r c r it i ca l T h e r m o d y n a m i c P o w e r y c l e

8 7

5 ' ] / ~, ~ 2OO

)

- * - 100

< : ] ' ' , ~ - - - - - - o

J " o C , ) 7L,,3 8 0 0 900 l OOO H OO 1200

I u r~ ?c l a t : , r c Io [ j

Fig. 4 . Enthaipy-temperaturediagram--high A H r .

o r d in a t e , t e mp e r a tu r e a s t h e a b s c i s s a , a n d l i n e s o f

c o n s t a n t p r e s s u r e a r e d r a w n , t h e n t h e s l o p e a t a n y p o in t

i s p ro po r t ion a l to the v a lue Of the Spec if ic hea t . F igu re 4

shows such a se t o f curves for wa te r . On the c r i t ica l

p r e s s u r e l i n e t h e ma x imu m s lo p e i s i n f i n i t e a n d o c c u r s

a t t h e c r i t i c a l p o in t . T h e ma x imu m s lo p e i s f i n i t e a t

h ighe r pressures and i t s va lue dec reases wi th inc reas ing

pressure .

To i l lus t r a te the e f fec t o f the pressure level and p ressure

r a t i o o n t h e r e g e n e r a t i o n p r o ce s s l e t t h e l o w e r a n d u p p e r

pressures Of an idea l supe rc r i t ica l cyc le be 3500 ps ia an d

4500 ps ia , r e spec t ive ly . In F ig . 4 the en tha lpy d i f f e rence

a t c o n s t a n t t e m p e r a tu r e f o r t h e s e p re s s u r e s is p lo t t e d a s a

f u n c t io n o f t e m p e r a t u r e a n d l a b el e d A H . T h e c o n s t a n t

t e m p e r a t u r e l in e a t t h e m a x i m u m p o i n t o f t h e A H c u rv e ,

A H r ,

i s d rawn. This l ine in te r sec ts the 3500 ps ia and

4 5 0 0 p s ia c o n s t a n t p r e s s u r e l i n e s a t

a o

and b0 , re spec t ive ly .

N o w , e q u a l i n c re m e n t s o f e n t h a l p y a re t a k e n o n t h e t w o

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

d e c r e a s in g e n th a lp y . T h e c o r r e s p o n d in g p o in t s a r e

labeled a~, bl , a l l , bl l , a2, b2, a22, b22, e tc . I t ca n be seen

th a t f o r c o n s t a n t e n th a lp y i n c r e me n t s i n b o th d i r e c t i o n s

th e t e mp e r a tu r e i n c r e me n t i s i n c r e a s in g . T h e i n c r e a s e i s

t h e g r e a t e s t o n t h e l o w e r p r e s s u r e l i n e i n t h e i n c r e a s in g

e n th a lp y d i r e c t i o n a n d o n t h e h ig h e r p re s s u r e l i ne i n t h e

d e c r e a s in g e n th a lp y d i r e c t i o n . T h e e x i s t e n c e o f t h i s

p o s i t i v e t e mp e r a tu r e g r a d i e n t a l l o w s h e a t t r a n s f e r t o

t a k e p l a c e i n t h e s u p e r c r i t i c a l r e g io n f r o m th e l o w e r t o

the h ighe r pressure .

O f t h e tw o a s p e c t s o f t h e e n th a lp y ~ - t e mp e r a tu r e

r e la t io n s , n a m e l y t h e e x i st e nc e o f a m a x i m u m A H

b e tw e e n t h e c o n s t a n t p r e s s u r e r e l a t io n a n d t h e i n c r e a s in g

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

t h e f o r m e r i s a d e t r ime n t t o c y c l e e f fi c ie n c y a n d t h e

l a t t e r i s a n a id i n r e d u c in g r e c u p e r a to r s i z e . T h e f o r me r

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

c o n v e r s io n t o w o r k . A s s u c h , i t l e a d s t o a n a l t e r n a t e b u t

equiva len t de f in i t ion of cyc le the rma l e f f ic iency ,

( h a - - h e ) - - ( h b - - h a )

i t ( h d - - h e ) + ( h e - - h e )

_ ( h a - - h e ) - - ( h o - - h a )

- - ( h a - - h e ) + ( h i - - h b ) 6 )

w h e r e

h e - - h e = h - h b = A H r .

E x p r e s s e d i n a n o th e r

f o r m ,

1 - - A H v / A H t

~ . = 7 )

1 + A H r / A H t "

F r o m th i s r e l a t i o n i t i s e v id e n t t h a t ~ , t c a n b e i n c r e a s e d

b y r e d u c in g

A H r .

I f A H f o r a h ig h e r s e t o f p r e s su r e s , s a y 4 50 0 a n d 5 50 0

ps ia i s p lo t ted ( see F ig . 5 ) a lower

A H r

i s ob ta ined . I t i s

ev ident tha t a s the pressure leve l ( above the c r i t ica l

1 5 0 0

5 0 0

C r i t i c a l

Point

- - ' i ' ] ~t } ~ 3 " ~ - " -U 0 ] l I B I I . I i b

a l l ;

; - " / - - i b i - 0

500 600 700 800 900 1000 I I 00 1200

Ternperatu re l F }

Fig. 5 . E nthalpy-temperaturediagram--low

AHr.

p r e s s u r e ) i s i n c r e a s e d ( w i th in u n d e t e r min e d l imi t s ) a n d

as the pressure r a t io i s dec reased , the

A H r

i s dec reased .

T h e v a lu e o f t h e u n t r a n s f e r a b l e e n th a lp y i s

A H r

i n a n

id e a l re c u p e r a to r . A t t h a t p o in t t h e A T a c ro s s t h e r e -

c u p e r a to r i s z e r o a n d t h e o r e t i c a l ly a n i n f in i t e su r f a c e

w o u ld b e r e q u i r e d f o r h e a t t r a n s f e r . I n a r e a l re c u p e r a to r

t h e A T h a s a p o s i t i v e v a lu e A T p ~ a n d i s c a l le d t h e p in c h

t e m p e r a tu r e . T h e s u r f a ce r e q u i r e d f o r h e a t t r a n s f e r

b e c o me s s ma l l e r a s A T i s i n c r ea s e d . A n i n c re a s e i n A T

resul t s in an inc rease in A H r t o s o me v a lu e A H ~ ,

a c c o r d in g t o t h e r e l a t i o n

A H ' - - A H r

: Cp -avg >(

A T p i .

(8)

T h i s r e l a t i o n h o ld s w h e r e A T p , i s le ss than 15F , which

is a prac t ica l va lue for r ea l appl ica t ions . Over th is

r a n g e t h e d i f fe r e n ce s b e tw e e n C p _ a vg a n d t h e ma x im u m

a n d m i n i m u m v a l u es o f

C p

a re n egl ig ib ly sma l l .

T h e e f fi c ie n c y o f t h e r e c u p e r a to r m a y b e d e f in e d a s

th e r a t i o o f t h e i d e a l a n d a c tu a l e n th a lp y d i f f e re n c e s,

A H r

e r : AH ~ (9)

Usin g Eq ua t io ns (2), (3) , (7) and (9) , the ac tua l the rm a l

e f f iciency of a r ea l cyc le becom es

1 - - A H p / A H t e t e v

~ a t i - ~ - A - H r - / A H t e t e r " (10)

5 . W o r k i n g F l u id s

In pr inc ip le , the Superc r i t ica l Cyc le can be ope ra ted

w i th a n y f l u id , j u s t a s a B r a y to n C y c l e c a n b e o p e r a t e d

w i th a n y g a s . I n p r a c t i c e , t h e c h o i c e o f w o r k in g f l u id

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

t e mp e r a tu r e s . T a b l e 1 l is t s c r i ti c a l p r o p e r ti e s o f s o me

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8 8 E . G . F E H ER

Table 1. Critical constants of working fluids

Name Formula

C r i t i c a l C r i t i c a l

temperature pressure

(F) (psia)

Am mo nia NH3 271 2 1636

Carbon dioxide CO2 87.8 1072

Hexa fluorobenzen e C6F6 460 402

Perfluoropropane CaFa 161 4 388

Sulfu r diox ide SO2 315 5 1143

Sulfur hexafluoride SF6 114 546

Water H20 705 3206

Xenon Xe 61.9 853

o f t h e w o r k in g f l u id s w h ic h c a n b e u s e d i n p r a c t i c a l

a p p l i c a t i o n s .

F o r i n i t i a l i n v e s t i g a t i o n s c a r b o n d io x id e w a s s e l e c t e d

as the working f lu id for seve ra l r easons . F i r s t , i t s c r i t ica l

p r e s s u r e i s o n e t h i r d t h a t o f w a t e r , a l l o w in g l o w e r

o p e r a t i n g p r e s s u r e s . S e c o n d , i t i s k n o w n to b e a s t a b l e

a n d i n e r t m a t e r i a l t h r o u g h o u t t h e t e m p e r a t u r e r a n g e o f

i n t er e s t. T h i r d , t h e r e i s a c o n s id e r a b l e b o d y o f l i t e ra tu r e

o n t h e p r o p e r t i e s o f c a r b o n d io x id e , h e n c e c y cl e a n a ly s i s

i s b a s e d o n r e a s o n a b ly f i r m d a t a . A n d f i n a l l y , c a r b o n

d io x id e i s a b u n d a n t , n o n - to x i c a n d r e l a ti v e ly i n e x p e n siv e .

T h e t h e r m o d y n a m i c a n d t r a n s p o r t p r o p e r t ie s o f c a r b o n

d io x id e w e r e a s s e mb le d f r o m s e v e r a l s o u r c e s , n o t a b ly

f r o m R . L . S w e ig e r t et al [4] , G . C . Kennedy [5] , D .

Pr ice [6] , D . M. Newi t t et aL [ 7 ] , a n d L . H . C h e n [ 8 ].

T h i s d a t a c o v e r s t h e t e m p e r a t u r e r a n g e f r o m 3 2 F t o

1 8 0 0 F a n d t h e p r e s s u r e r a n g e f r o m 0 t o 5 0 0 a tm.

6 . E f f e c t o f C o m p o n e n t P a r a m e t e r s

I n s p e c t i o n o f E q u a t i o n ( 7 ) r e v e a l s t h e i n f l u e n c e o n

id e a l c y c le ef f ic i e nc y o f t u r b in e w o r k , p u m p w o r k a n d

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

F ig u r e 6 i s a p lo t o f c y c l e e f fi c ie n c y a s a f u n c t i o n o f t h e

in d e p e n d e n t v a r i a b l e s o f E q u a t i o n ( 7 ) ,

AH~/AHt

a n d

AHr/AHt

S u p e r imp o s e d o n t h i s p lo t a r e s o me ty p i c a l

i d e a l c a r b o n d io x id e c y c l e s o p e r a t i n g b e tw e e n p r a c t i c a l

t e mp e r a tu r e a n d p r e s s u r e l imi t s . I t i s e v id e n t t h a t t y p i c a l

v a lu e s o f AHp/AHt r a n g e f r o m 0 .0 7 t o 0 . 2 0 a n d t h a t

t h e v a lu e s o f AHr/AHt usu a l ly lie be tw een 1 /3 and 1 .

F ig u r e 7 s h o w s t h e c h a n g e i n a l l v a r i a b le s o f E q u a t i o n

' 0 . 6

~ 0 4

1 -

0 . 8

o 2 . ~

O

Turbine nlet Temp.

1600% - u

1200F - o

800F =~ --

P~IPI = 2 2 ~

~ ~P21PI *.I'25

0.2 0 4 0.6 O.B

Z~Hpz~H

Fig. 6. Cycle efficiency vs. pum p work to turbine work ratio.

0 " 65

-

0 . 6 -

~, 0.55.

0 " 5 -

0 . 4 5 -

f ~ ,

f

T u r b i n e In ldTemp, ) . 300OF

- Pump In let Temp, 6 ~'F

Pump Inle t Press, L~O00 s i a

Cycle Press D rop - 0

q - % - e r . l

Cnrn0t Efficleney 0.70

-W0rklng Fluid: C02

0 . 4 - - O

1 5 2 2 . 5 3 3.5

Pump Discharge P r e s s u r e

Pump in let Pressure .

Fig. 7. Cycle efficiency vs. pressure ratio.

- 0 " 8

-0"5

- 0 . 4

- 0 - 2

( 7) a s a f u n c t i o n o f p r e s s u r e r a ti o , f o r a p u m p in l e t

pressu re o f 2000 ps ia . Id ea l cyc le e f f ic iency i s a l so

c o m p a r e d t o C a r n o t e f f ic i en c y . T h e h ig h e s t e f fi c ie n c y

o c c u r s f o r a p r e s s u r e r a t i o o f mo r e t h a n 3 . 5 , b u t i t i s

s i g n i f ic a n t t h a t t h e e f f i ci e n cy r e ma in s a lm o s t c o n s t a n t

d o w n to a p r e s s u r e r a t i o o f 2 .

T h e d i f fe r e n c e b e tw e e n i d e a l a n d a c tu a l t h e r m a l

e f f ic ienc ies a s r eprese nted by E qu a t io ns (7) an d (10),

r e spec t ive ly , i s due to the e f f ic ienc ies of the th ree ma jor

c o m p o n e n t s o f a n e n g in e o p e r a t i n g o n t h e S u p e r c r i t i ca l

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

r e c u p e r a to r .

F ig u r e 8 s h o w s t h e r e c u p e r a to r e f fi c ie n c y a n d c y c le

e f fi c ie n c y a s a f u n c t i o n o f p r e s s u r e r a t i o a n d p in c h

t e m p e r a tu r e A T tn . H e r e t h e d e g r a d in g i n f l u e n ce o f t h e

g

1 1.5 2 2 5 3 3 5

Pump Discharge Pressure

Pump In l e t Pr essu r e

F i g . 8 . C y c l e e f f i c i e n c y a n d recuper tor e f f i c i e n c y vs p r e s su r e r a t i o

and pinch temperature.

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T h e S u p e r c r it i c a l T h e r m o d y n a m i c P o w e r C y c l e 8 9

0 , 8 -

Tu rbine I nl e t Temp, - ~ 300 F . ~

Pump Inlet T emp, - 58 F

0.7 - Turbine Inlet Press, 40(X)psia -

Tu rbine Ou tlet Press, - 20(]0

s i a

e rCyclelPress" Drop

-

0 _ ~

0.6- WorkingFluid: CO

0 . 4 -

G

- rbi n (ep = 1

0 . 3 -

(3 .2 ~ - - ~ _ _

0 . 1

0-

0"2 0,4 0,6 0,8 1

Turb ine and Pump

f f i c i e n c y

F ig 9 . C yc le e f f i c ie nc y vs . tur b ine and pum p e f f ic ie nc y .

recup era tor e f f ic iency on cyc le e f fic iency i s show n

q u a n t i t a t i v e ly . I n c r e a s in g A T ~ r e p r e se n t s d e c r e a s in g s iz e

r e c u p e r a to r s . I t i s t o b e n o t e d t h a t t h e h ig h e r t h e p in c h

t e mp e r a tu r e , t h e mo r e ma r k e d t h e e f f e c t o f p r e s su r e

ratio on cycle eff ic iency.

T h e e f fe c t o f p u m p a n d t u r b in e e ff ic ie n c ie s i s sh o w n

in F ig . 9 . As would be expec ted , the pump e f f ic iency

h a s

the sma l le r e f fec t , p roduc ing le ss than 10 pe r cen t

d e g r a d a t i o n i n c y c l e e f fi c ie n c y f o r a d r o p f r o m 1 00 p e r

c e n t t o 6 0 p e r c e n t p u mp e f fi ci e nc y . T h e e f fe c t o f t u r b in e

e f fi c ie n c y i s s imi l ar t o t h a t f o r a r e g e n e r a t e d B r a y to n

Cyc le .

The com bin ed e f fec t on cyc le e f f iciency of the th ree

c o m p o n e n t e f fi ci e nc ie s a s a f u n c t i o n o f p r e s su r e r a t i o i s

b

z 1:5 z 2;5 3 3;3

Pump DischargePressure

Pump Inlet Press ure

F ig . 10 . Cyc le e f f i c ie nc y vs . pr e s s ur e r a t io and c ompone nt

e f f ic ie nc y .

sho wn in F ig . 10 . Peak cyc le e f f iciency occurs a t p ro-

gress ive ly lower pressure ra t ios a s component e f f ic ienc ies

decrease .

I n a p r a c t i c a l e n g in e , t h e t o t a l p r e s su r e d r o p a r o u n d

the cyc le a lso has a degrading e f fec t on cyc le e f f ic iency .

T o ta l p r e s su r e d r o p c a n b e r e f e r r e d t o t h e p u mp a s

in c re a se i n p u m p p r e s su r e r a t i o o v e r t h e t u r b in e p r e s su r e

ra t io .

S y mb o l i c a l l y

APc~

=

APv - - APt 11)

and expressed a s a r a t io ,

A P e y A P p I . 1 2)

Pt Pt

The e f fec t o f inc reas ing pressure drop on cyc le e f f ic iency

a t seve ra l pu m p a nd turb in e e f f ic iencies i s sho wn in

Fig.

1 1 .

Tu rbine Inle t Temp. 1300OF

Pump Inlet Temp. - 68F

Turbine Inlet Press. - 4000psia

Turbine Outlet Press, - 2000psia

0 ,6 - - - O r . ,9 0

Pincll Temp. - 8OF Approx.) i

Carnot Efficiency 0,70

Working Fluid: CO ~ e t - ep 1 ]

0.55

/- - e o e = 0.9

~ 0,5 i ~

0.45

i

~ , ~ j F e t = e p : 0 " 7

0.4 ~

r

0 . 3 5 ~ I J

0.025 0.05 0.075 0,1

~P

Cycle Pressure Drop, ~- I

F ig . 11 . Cyc le e f f i c ie nc y vs . c yc le pr e s s ur e dr op .

o.6-~ 7 - ~

I I

App i ?

[ C y c l e P r e s s ' D r p " ~ t " 1 ; [

. 3 1

-

F ~ nlet Press. = 3000psie i

0 '35 ~ - - - - ~ Turbine Out let Press, = 2000psia

r Pump Inlet Temp. - 68OF

[ e ~ ep = e r - 0 . 8 i

[ ~,'orkmgluid: CO j

o . 3 ~ i I l

I000 1100 1200 1300 1400 1500 1600

Tu rbine Inlet Temperature,OF

F ig . 12 . C yc le e f f i c ie nc y vs . tur b ine in le t t e mpe r atur e.

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9 0 E . G . F E H E R

o.~ -- ~- __

0 "45 I

~-, ,~Pp , /

_ z I / / '

0.35- - - ~ / ~ 1 ~ / ~

o - / , 1 /

I l-- ,t i

0.3- Turbine InletTemp. -

Tu rb i ne I n l e t P re ss . - 3 0 ~O ps i a I

Tu r b i ne O u t l e t P re ss . - 2 0 0 0 ps i a j

0 . 2 ~ - [ ~ ~ e - e . - e r - 0 - 8 - - $ . . . . . d

Worki~n~j Fluid . CO-~

Cr i t ic a l Te mpe ra t u r e i

. 2 . . . . . i I ]

40 60 80 100 120 140 160 180

Pu mp I n l e t Te mpe ra t u r e , O F

F i g . 1 3 . C y c l e e f f ic i e n c y v s . p u m p i n l e t t em p e r a t u r e .

A m o n g t h e s e v e r a l c y c l e p a r a m e t e r s t h a t i n fl u en c e

cyc le e f fi ci ency there remains the tu rb ine and pu m p

in le t tem pera tu res to cons ider . The e f fec t o f tu rb ine in l e t

t em pera tu re on cyc le ef fi ci ency a t s evera l cyc le p res su re

drops i s shown in F ig . 12 . The var i a t ion i s much the

same as fo r the regenera ted Bray ton Cycle . F igure 13

s h o w s t h e e f f ec t o f p u m p i n le t t e mp e r a t u r e o n c y c l e

efficiency at severa l cycle press ure d rops . Th e cyc le

ef f i c i ency var i a t ion here i s qu i t e d i f fe ren t f rom tha t o f

a Bray ton Cycle . At t empera tu res lower than the c r i t i ca l

t em pera tu re the e f fec t i s small . A change o f 40F in

p u m p i n le t t e mp e r a t u r e c a u s e s a 1 p e r c e n t ag e p o i n t

change in cyc le e f fi ci ency. A t t em pera tu re s h igher than

ab ou t 15F a bov e the c r i t ica l t em pera tu re the e f fec t i s

sudden ly l a rge . A cha nge o f ab ou t 8F c auses a 1 per -

c e n t a ge p o i n t c ha n g e . A t t e mp e r a t u r e s f a r t h e r a b o v e t h e

cr i ti ca l t em pera tu r e the cyc le behave s much as a h igh

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

i s a h igh den s i ty compress ib le gas .

7 . C o n c l u s i o n

The Supercr i t i ca l Cycle o f fe rs the fo l lowfng charac-

teris t ics , which are desi rable in a pract ical appl icat ion.

High thermal e f f i c i ency , l ow vo lume to power ra t io , no

b lade e ros ion in the tu rb ine , no cav i t a t ion in the pump,

s ing le s t age tu rb ine and pump, s ing le phase f lu id in the

hea t re j ec t ion p rocess , and insens i t iv i ty to compress ion

efficiency.

Some app l i ca t ions fo r a supercr i t i ca l eng ine a re :

e l ec t r i c power genera t ion fo r space ,

t e r res t r i a l e l ec t r i c power genera t ion ( s t a t ionary and

por t ab le ) ,

s h a f t p o w e r f o r ma r i n e p r o p u l s i o n ( s u r f a c e a n d

sub-su r face) .

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

nuclear reac to r as the hea t source , t he supercr i t i ca l

e n g i n e c a n b e a c o mp a c t , p o r t a b l e e l e c t r i c p o w e r

genera to r .

The reasons fo r the neg lec t o f the Supercr i t ica l Cy cle

u n t i l n o w a r e n o t k n o w n . I t c a n b e c o n j e c t u r e d t h a t i t s

in t roduct ion to use has been de layed by the eng ineer ing

r e q u i r e me n t s o f s o me o f t h e c o mp o n e n t s . I t i s c l e a r ,

howeve r , t ha t t he l eve l o f toda y s t echno log y i s adeq uate

fo r the success fu l p rac t i ca l u t i li za t ion o f the S upercr i ti ca l

Cycle.

C p

Cp-avg

e

h

P

Q i n

T

Wout

~Tat

r/ey

~Tit

A H

A H

A T

A T ~

N o m e n c l a t u r e

spec i f i c hea t a t cons tan t p res su re

average spec i f i c hea t a t cons tan t p res su re

componen t thermal e f f i c i ency

e n t h a l p y

pressu re

h e a t i n p u t

t e mp e r a t u r e

w o r k o u t p u t

actual thermal eff iciency

cycle therm al eff iciency

ideal thermal eff iciency

idea l en tha lpy d i f ference

ac tua l en tha lpy d i f fe rence

tempera tu re d i f fe rence

p inch t empera tu re (d i f fe rence)

u b s c r i p t s

a to f s t a t e po in t s

p p u m p

r r e c u p e r a t o r

t t u rb ine

u p e r s c r i p t s

ac tua l po in t o r p rocess (as opposed to idea l )

Acknowledgments W o r k p r e s e n t e d h e r e i n w a s c o n d u c t e d a t t h e

A s t r o p o w e r L a b o r a t o r y , M i s s i le a n d S p a c e S y s t e m s D i v i s io n ,

D o u g l a s A i r c r a f t C o m p a n y , I n c . u n d e r c o m p a n y - s p o n s o r e d

R e s e a r c h

a n d D e v e l o p m e n t

f u n d s .

R e f e r e n c e s

[ 1] V . L . D e k h t i a r e v , O n d e s i g n i n g a l a r g e , h i g h l y e c o n o m i c a l

c a r b o n d i o x i d e p o w e r i n s t a l l a t i o n . Elecrtichenskie Stantskii ,

5 : 1-6 , May 1962 .

[ 2] G . A n g e l i n o , Perspectives for the Liquid Phase Compression Gas

Turbine. A S M E P a p e r N o . 6 6 - - G T - 11 1 , 1 3 -1 7 M a r c h 1 96 6.

[3 ] E . G. Feher , Supercritical Thermodynam ic Cycles fo r Exte rna l

and Internal Combustion Engines. A s t r o p o w e r , I n c . E n g i n e e r i n g

R e p o r t M a y 1 9 6 2 .

[ 4 ] R . L . S w e i g e r t , P . W e b e r a n d R . L . A l l e n , Ind. Engng. Chem.

3 8 185 1946).

[ 5 l G . C . K e n n e d y , P - V - T r e l a t i o n s i n C O 2 a t e l e v a t e d t e m p e r a -

t u r e s a n d p r e s s u r e s . Am. J . Sci . 2 2 , 225-241 1954).

[6 ] D. P r ice , The Thermodynamic Properties o f Carbon Dioxid e up

to 1000C and 1400 bars . N a v o r d R e p o r t 3 8 4 6 , N o v . 1 9 5 4 .

[ 7 ] D . M . N e w i t t , N . V . P a l , N . R . K u l o o r a n d J . A . W . H u g g i l l ,

Thermodynamic Functions of Gases,

V o l . 1 t E d . F . D i n ) . B u t t e r -

w o r t h , L o n d o n 1 9 56 ).

[ 8 ] L . C h e l a , T h e r m o d y n a m i c a n d t r a n s p o r t p r o p e r t i e s o f g a s e o u s

c a r b o n d i o x i d e , i n t h e A . S . M . E . b o o k

Thermodynamic and

Transport Properties of Gases, Liquids and Solids. M c G r a w - H i l l

(1959).