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Pergamon 0196-8904 94)00040-9Energy Convers. M gmt Vol. 36, No. I, pp. 1-5 , 1995Copyrigh t 1995 Elsevier Science LtdPrinted in Great Britain. All rights reserved0196-8904/95 9.50+ 0.00

ENDOREVERSI LE THERMOEC ONOMICS

A L E X I S D E V O SV akgroep voor E l ek t ron ika en In fo rm a t i e sys t em en , U n ive r s i t e i t G en t , S in t P i e t e r sn i euw s t r aa t 41 ,B -9000 G en t , B e lg ium

Received 7 Februa ry 1 994; received for publication 7 September 1994)

A b s t r a c t - -A n endoreve r s ib l e pow er p l a n t i s op t im ized w i th r e spec t t o econom ica l exp lo i t a t i on . T h i s leadsto an op t im um pe r fo rm ance , depend ing on t he r e l a t i ve cos t s o f i nves tm en t and fue l .

E n d o r e v e r s i b le t h e r m o e c o n o m i c s E n d o r e v e r s i b le t h e r m o d y n a m i c s C u r z o n - A h l b o r n e n g i n e

1 . I N T R O D U C T I O NI n 1 9 57 , N o v i k o v [1 ] i n t r o d u c e d a m o d e l f o r a n u c l e a r p o w e r p l a n t , s e e F i g . l ( a ) . I t c o n s i s t s o f t w oh e a t r e s e r v o i r s : o n e a t t h e h i g h t e m p e r a t u r e T~ a n d o n e a t t h e l o w t e m p e r a t u r e T 2. B o t ht e m p e r a t u r e s a r e a s s u m e d c o n s t a n t a n d o f k n o w n v a l u e . B e t w e e n th e t w o r e s e r v o i rs a r e tw oc o m p o n e n t s :

o n e r ev e r s ib l e c o m p o n e n t : a C a r n o t e n g i n e o n e i r re v e r s ib l e c o m p o n e n t : a t h e r m a l c o n d u c t o r g .

T h e h o t t e r s id e o f t h e C a r n o t e n g i n e ( o r , e q u i v a l e n tl y , t h e c o l d e r si de o f t h e c o n d u c t o r ) i s a s s u m e dt o b e a t s o m e ( v a r i a b l e ) t e m p e r a t u r e T 3 .

I f T3 e q u a l s T , , th e e n g i n e c o n v e r t s h e a t i n t o w o r k a t i t s m a x i m u m e f fi c ie n c y , i .e . a t t h e C a r n o tef f ic iency , r /c = 1 - Tz/T~ H o w e v e r , t h e n t h e e n e r g y c o n v e r s i o n h a p p e n s i n fi n it e ly s lo w l y a n d t h u s ,n o f in i te p o w e r W is p r o d u c e d . T h e r e f o r e , N o v i k o v p r o p o s e d n o t t o m a x i m i z e t h e e ff ic ie n c y r/,b u t r a t h e r t h e p o w e r p r o d u c t i o n r a t e W . H e d e m o n s t r a t e d t h a t t h is h a p p e n s w h e n T3 e q u a l s~ . T h e co n v e rs i o n e ff ic ie nc y r / is l o w er t h a n th e C a r n o t v a lu e a n d e qu a ls 1 -

I n 1 9 75 , C u r z o n a n d A h l b o r n [2 ] ( u n a w a r e o f th e N o v i k o v r e s u lt ) i n t r o d u c e d a m o d e l f o r ap o w e r p l a n t , w i t h t w o t h e r m a l c o n d u c t o r s g t a n d g 2, s ee F ig . l (b ) . W e h a v e n o w t w o i n t e r m e d i a t et e m p e r a t u r e s , i .e . T 3 a n d / 4 . T h e C u r z o n - A h l b o r n s c h e m e r e d u c e s to t h e N o v i k o v s c h e m e i f t h et h e r m a l c o n d u c t a n c e g 2 t e n d s t o i n f i n i t y , s u c h t h a t T 4 b e c o m e s i d e n t i c a l ly e q u a l t o T 2 . I t i sr e m a r k a b l e t h a t t h e C u r z o n - A h l b o r n g e n e r a li z a ti o n o f th e N o v i k o v m o d e l l e ad s t o t h e s a m e r e su l tf o r t h e m a x i m u m - p o w e r e f f ic ie n c y, n a m e l y

r/om = 1 - ~ T ~ ,

a v a l u e o f t e n r e f e r r e d to i n t h e l i t e r a tu r e a s t h e C u r z o n - A h l b o r n e ff ic ie n c y.E n g i n e s su c h a s th e N o v i k o v e n g in e a n d t h e C u r z o n - A h l b o r n e n g i n e a re k n o w n a s en d o r e -

v e r s i b l e e n g in e s , a f t e r t h e n a m e p r o p o s e d b y R u b i n [3 ] f o r i r r e v e r s i b l e e n g i n e s , w h e r e a l l t h ei r re v e r s ib i l it y i s r e s tr i c te d t o p r o c e s s e s h a p p e n i n g i n t h e c o m m u n i c a t i o n b e t w e e n t h e c o n v e r t i n ge n g i n e a n d t h e e x t e r n a l w o r l d a n d w h e r e , t h u s , n o i rr e v e rs i b il it ie s h a p p e n i n t h e e n e r g y c o n v e r t e ri ts el f. F o r a n o v e r v i e w o f e n d o r e v e r s i b l e t h e r m o d y n a m i c s , t h e r e a d e r i s r e fe r r e d t o R e f s [ 4- 7 ].

I n t h e p r e s e n t p a p e r , w e w i ll s h o r t l y r e v ie w t h e t h e r m o d y n a m i c s ( s ee S e c t io n 2 ) a n d t h e ni n t r o d u c e t h e t h e r m o e c o n o m i c s ( se e S e c ti o n 3 ) o f th e N o v i k o v p l a n t . A l l re s u lt s a re e q u a l l y v a l idf o r t h e m o r e g e n e r a l C u r z o n - A h l b o r n p l a n t . B u t , i n o r d e r t o m a k e t h e m a t h e m a t i c s m o r e

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2 DE VOS: ENDOREVERSIBLETHERMOECONOMICSt ra nspa re n t , e xp l i c i t c a l c u l a t i ons r e fe r t o t he mode l i n F i g . I ( a ) . The re a de r wi l l e a s i l y ge ne ra l i z ef o r t h e m o d e l i n F i g . l ( b ) . W h e r e a s , i n t h e r m o d y n a m i c s , w e t r e a t h e a t a n d w o r k f l u x e s ( in J /s = W ) ,i n t h e r m o e c o n o m i c s , w e w i l l a l s o t r e a t m o n e y f l u x e s ( i n e c u / s ) .

2 E N D O R E V E R S I B L E T H E R M O D Y N A M I C ST h e h e a t f lu x Q i s g o v e rn e d b y t h e c o n s t it u t iv e l a w o f t h e t h e r m a l c o n d u c t o r . L i k e N o v i k o v a n d

C u r z o n a n d A h l b o r n , w e w i l l a s s u m e a l i n e a r ( i . e . N e w t o n i a n ) t r a n s p o r t e q u a t i o n :Q = g ( T ~ - T3). (1)

T h e h e a t - t o - w o r k c o n v e r s i o n i s g o v e r n e d b y t h e C a r n o t l a w :

T3)T o g e t h e r , t h e a b o v e f o r m u l a e l e a d t o

W = g ( 1 - ~ ) ( T t - T 3), (2 )w h e r e W a p p e a r s a s a f u n c t i o n o f t h e v a r ia b l e t e m p e r a t u r e T 3. F i g u r e 2 ( b ) s h o w s t h e f u n c t i o nW ( T 3) . T h e c u r v e h a s t w o z e r o s : o n e a t /'3 = / ' 2 a n d o n e a t T 3 = T j. B e t w e e n t h e t w o z e r o s , t h ec u r v e d i s p l a y s a m a x i m u m . T h e r e a d e r w i l l e a s il y v e ri f y t h a t d W / d T 3 = 0 leads to T3 = ~ an d,s u b s e q u e n t l y , t o W , , ,x= g ( v / ~ - x / / ~ ) 2. W i t h th e e ff ic ie nc y e q u a t io n r / = W / Q = I - T 2 / T 3 ,th is le a d s to qopt = 1 - ~ .N o t e t h a t , i n th e r a n g e T2

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D E V OS : E N D O R E V E R S I B L E T H E R M O E C O N O M I C S 3

F i g . 1 . E n d o r e v e r s i b l e e n g i n e s . ( a ) N o v i k o v e n g i n e ;( b ) C u r z o n - A h l b o r n e n g i n e .

max

o[W~

Wmox

m

r p x -L

F i g . 2 . H e a t a n d p o w e r a s a f u n c t i o n o f t e m p e r a t u r e T 3.( a) H e a t c o n s u m p t i o n ; ( b ) p o w e r p r o d u c t i o n .

W e n o w a s s u m e t h a t Q m ,x i s a n a p p r o p r i a t e m e a s u r e f o r t h e s i ze o f t h e p l a n t. T h u s , t h e r u n n i n gc o s t s C o f t h e p l a n t e x p l o i t a t i o n a r e g i v e n b y

C = a Q m a x + bQ,w h e r e t h e c o e f fi c ie n t s a a n d b b o t h h a v e t h e d i m e n s i o n e c u / J .

T h e p r o f i t s a r eWq = - - C

a q u a n t i t y e x p r e s s e d i n J / e c u . I t i s t h i s q u a n t i t y w e w i l l m a x i m i z e . T h i s a p p r o a c h i s s o m e w h a td i f f er e n t f r o m

B e j a n [1 0 ] w h o m i n i m i z e s t h e s i m p l e c o s t f u n c t i o n g T t / H I , t h a t l e a d s t o n e g l e c t i n g t h e f u e l c o s t ,a n d

I b r a h i m et al. [1 1] w h o m i n i m i z e a v e r y c o m p l i c a t e d c o s t f u n c t i o n t h a t l e a d s t o c o m p l i c a t e dc o m p u t a t i o n s i n v o l v i n g L a g r a n g e m u l t i p l i e r s .

T h u s , w e w i l l , i n t h e p r e s e n t p a p e r , m a x i m i z eW( T 3)q( T 3) = aQm,x + b O T~)

a q

9.5

T a b l e 1. R e l a t iv e f u e l c o s t f o r v a r i o u s e n e r g y s o u r c e sT3 Fue l f ( )oT d T T I~ R e n e w a b l e 0

U r a n i u m 2 5C o a l 3 5G a s 5 0F i g . 3 . P r o f i t a s a f u n c t i o n o f t e m p e r a t u r e T ~.

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DE VOS: ENDOREVERSIBLETHERMOECONOMICSt ~ o p t

I -

0 . 51 2

O0

q o p tI

- 1 = 0 . 5 - . . - . I - T 2 0 . 5 -f

i I I I I I I r I ~ O0 . 5 OFig. 4. O ptima l efficiency s a function of relative fuel costf, calculated numerically for the cas e T 2 / T ~ = 0.5.

I I i [ I I i f [0.5 Ire/'rl

Fig. 5. O ptima l efficiencyas a function of the tem peratureratio T 2 / T I .

w i t h r e s p e c t t o / ' 3. W i t h e q u a t i o n s ( 1 ) a n d ( 2 ), th e e x p r e s s i o n b e c o m e s1 T 3 - T z ) T , - T3)q ( T 3 ) = a T 3 [ ( T , - T 2 ) + f l ( T , - T3)] '

w h e r e f l i s t h e d i m e n s i o n l e s s r a t i o b / a .F i g u r e 3 s h o w s t h e f u n c t i o n q ( T 3 ) f o r v a r i o u s v a l u e s o f t h e f l- p a r a m e t e r . F o r f l = 0 , t h is c u r v e

i s o f t h e s a m e f o r m a s t h e W ( T 3 ) - c u r v e o f F i g . 2 ( b ) . I n d e e d , i f f l = 0 , t h e n1q( T 3) = ~ W( T 3) ,

w i t h a m a x i m u m a t T 3 = T w / ~ ~ . F o r f l- -, + o o , t h e m a x i m u m o f th e c u r v e co m e s c l o s e r a n d c l o s e rto i t s r e ve r s ib l e po in t a t T 3 = T , .

E x p l i c it m a n i p u l a t i o n o f d q / d T 3 = 0 f o r a r b i t r a r y f l l e ad s t o t h e q u a d r a t i c e q u a t i o n[T, -- (1 + f l ) 7 ' 2] T23 + 2 f lT~ T 2 T 3 - [(1 + fl) T l - T2] T, T 2 = 0

w i t h s o l u t i o n

a n d , t h e r e f o r e , t oT ~ - I + ,O ) T 2

t/opt = 1 _ ~ T T , - ( I + f l ) T 2, / 1I n s t e a d o f e x p r e s s i n g t h e r e s u l t i n t e r m s o f fl, a n u m b e r d i f f ic u l t t o r e c o v e r f r o m t h e l i t e r a t u r e ,

w e c a n a l s o e x p r e s s i t in t e r m s o f t h e f r a c t i o n a l f u e l c o s tb Q f l ( T , - T3)f = a Q m a x + b Q = ( T , - ~ ) + [3 (7 1 - T 3)

F o r v a r i o u s t e c h n o l o g i e s, t h is n u m b e r i s a p p r o x i m a t e l y g i v e n b y T a b l e 1. T h e v a l u e s in t h e t a b l eh a v e b e e n d e d u c e d f r o m t h e B e l g i a n s i t u a t i o n [ 1 2 - 1 5 ], b u t h a v e , n e v e r t h e l e ss , a n i n d i c a t i v e g e n e r a lva lue .

F o r v a r i o u s v a l u e s o f fl, w e c a n n o w c a l c u l at e , n u m e r ic a l ly , b o t h f a n d rlopt, giv ing r i se t o F ig . 4 .W e s ee h o w t h e o p t i m u m r / v a ri e s s m o o t h l y f r o m t h e N o v i k o v v a l u e f o r f = 0 , i. e. fo r e n e r g y s o u r c e sw h e r e t h e i n v e s t m e n t i s t h e p r e p o n d e r a n t c o s t , t o t h e C a r n o t v a l u e f o r f = 1 , i. e. f o r e n e r g y s o u r c e sw h e r e t h e f u e l i s t h e p r e d o m i n a n t c o s t .

I f w e r e p e a t t h i s c a l c u l a t io n f o r v a r i o u s r a t io s I 2/7'1, w e f i n a l l y g e t c u r v e s o f F i g . 5 . A s a l l f - v a l u e sin T a b le 1 a r e i n t he r a n ge 0 ~< f ~< 0 .5 , we se e f r om F ig . 5 t ha t , i n a l l c a se s , t he op t im um i s c lose rt o 1 - ~ t h a n t o 1 - - I 2 / T l .

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DE VOS: ENDOREVERSIBLE THERMOECONOMICS 54 . CONCLUSION

We have calculated the optimum economic management of a power plant. We treated variousenergy conversion technologies, the relative fuel cost ranging from 0 (renewable energy sources)to 50% (natural gas). It turns out that the optimal working point lies between the Novikov orCu rz on- Ahl bor n working point and the Carnot or revers ible working point, somewhat c loser tothe former than to the latter.Ac k now l e dge m e n t s - - The present research is supported by the Commission of European Communities in the frame-workof the Copernicus programme (pan-European network on thermodynamics and thermoeconomics). Professor J. Parrott(University of Wales---College of Cardiff), Professor E. Selleslagh (Universiteit Gent and Electrabel and Engineer G.Goossens (E.S.P.) are greatly acknowledged for valuable discussions.

R E F E R E N E S1. 1. Novikov, Atomnaya Energiya 3, 409 (1957); in English translation: J. Nucl . Energy H 7, 125 (1958).2. F. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975).3. M. Rubin, Phys. Rev. A 19, 1272 (1979).4. A. Bejan, Advanced Engineering Thermodynamics, pp. 404-527. Wiley, New York (1988).5. A. De Vos, Her lngenieursblad 59, 62 (September 1990).6. A. De Vos, Endoreversible Thermodynamics of Solar Energy Conversion. Oxford Univ. Press, Oxford (1992).7. A. De Vos, Sol. Energy Mater. Sol . Cells 31, 75 (1993).8. J. Gordon and M. Huleihil, J. Appl . Phys. 69, 1 (1991).9. J. Gordon, Am. J. Phys. 59, 551 (1991).10. A. Bejan, Trans . A.S .M .E .- -J . Energy Resour . Technol. 115, 148 (1993).11. O. Ibrahim, S. Klein and J, Mitchell, AES--Vol. 24, Thermodynamics and Energy Systems: Fundamentals, Education,and Computer-aided Analysis, A. S.M ,E ., pp. 15-20 (1991).12. G. Goossens, Private communications (1992).13. E. Selleslagh, Private communications (1992).14. G. Frederick, J. Gillon and J. Delvoye, Electricity generation costs in Belgium. Conf. Spanish Nucl . Sot . , Puerto deSanta Maria (30 October 1992),15, Jaarverslag 1991 van her Controlecomit~ voor de elektriciteit en het gas, Brussels (1992).