Reference Datums for Available Energy - ACS Symposium Series

5 Reference Datums for Available Energy WILLIAM J. WEPFER Professional Engineering Consultants, Milwaukee, WI 53211

Downloaded by UNIV LAVAL on July 13, 2016 | http://pubs.acs.org Publication Date: May 28, 1980 | doi: 10.1021/bk-1980-0122.ch005

RICHARD A. GAGGIOLI Department of Mechanical Engineering, Marquette University, 1515 W. Wisconsin Ave., Milwaukee, WI 53233 The purpose o f t h i s article is to provide appropriate criteria f o r the p r a c t i c a l s e l e c t i o n o f reference datums f o r the c a l c u l a t i o n o f a v a i l a b l e energy. The s e l e c t i o n o f a reference datum g e n e r a l l y depends on the commodity whose a v a i l a b l e energy is being evaluated, upon the p a r t i c u l a r process (or device) being analyzed, upon the complex o f processes (devices) with which the p a r t i c u l a r process i n t e r a c t s , and upon the ambient environment of the complex. Proper choice o f a reference datum (a dead s t a t e ) is important to e f f i c i e n c y a n a l y s i s and c o s t i n g . I t needs to be recognized that the reference datum f o r a v a i l a b l e energy is an a l t o gether d i f f e r e n t concept than the reference base f o r property tabulations. The s e l e c t i o n o f a reference base f o r property t a b u l a t i o n s is a r b i t r a r y . Whatever base is s e l e c t e d , the base values o f the d i f f e r e n t extensive p r o p e r t i e s will cancel out o f the thermodynamic property c a l c u l a t i o n s - - a s s u m i n g , o f course, that the c a l c u l a t i o n s are c a r r i e d out c o r r e c t l y . (That is, when employing thermochemical property t a b u l a t i o n s t o make property c a l c u l a t i o n s , it is necessary to assure that the base values cancel out; t h i s must be done f o r all extensive p r o p e r t i e s - - e n t h a l p y , entropy,..., availability, etc.) Some c o n t e n d t h a t t h e c h e m i c a l r e f e r e n c e datum f o r a v a i l a b l e e n e r g y can a l s o be s e l e c t e d a r b i t r a r i l y , j u s t l i k e a b a s e f o r thermochemical t a b l e s (while a d m i t t i n g t h a t the thermal r e f e r e n c e d a t u m - - t h e "dead s t a t e t e m p e r a t u r e " - - i s n o t a r b i t r a r y ) . The c o n t e n t i o n is e r r o n e o u s ; c h a n g i n g t h e v a r i o u s v a l u e s o f t h e a v a i l a b l e e n e r g y o f a s p e c i f i c m a t e r i a l by a c o n s t a n t amount ( a s a c o n s e q u e n c e o f c h a n g i n g t h e r e f e r e n c e datum) l e a d s t o m i s c o n c e p t i o n s , t o m i s e v a l u a t i o n s , and t o m i s a l l o c a t i o n s — i n t h e d e t e r m i n a t i o n o f i n e f f i c i e n c i e s and c o s t s . Absolute values of a v a i l a b l e e n e r g y c a n and s h o u l d be e v a l u a t e d . Before proceeding to the c r i t e r i a f o r p r a c t i c a l s e l e c t i o n o f t h e dead s t a t e r e f e r e n c e datum f o r a n a l y z i n g a p a r t i c u l a r p r o c e s s , some b a c k g r o u n d f u n d a m e n t a l s will be p r e s e n t e d .

O-8412-0541-8/80/47-122-O77$05.00/0 © 1980 American Chemical Society Gaggioli; Thermodynamics: Second Law Analysis ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

78

THERMODYNAMICS: SECOND LAW ANALYSIS

Theoretical Preliminaries System A v a i l a b l e Energy. The a v a i l a b l e e n e r g y as commonly d e f i n e d []_) and s y m b o l i z e d by A is a s p e c i a l c a s e o f s y s t e m a v a i l a b l e energy, B. (As will be s e e n , A is c a l l e d t h e s u b s y s t e m a v a i l a b l e e n e r g y in t h i s p a p e r . ) G i v e n a s y s t e m a t an a r b i t r a r y s t a t e , w i t h e n e r g y E(t), e n t r o p y S ( t ) , volume V ( t ) , e t c . , t h e s y s t e m a v a i l a b l e e n e r g y B ( t ) is d e f i n e d a s t h e maximum amount o f e n e r g y t h a t c o u l d be d e l i v e r e d f r o m t h e s y s t e m by p r o c e s s e s w i t h no n e t t r a n s p o r t o f any o t h e r e x t e n s i v e p r o p e r t y . Then


B(t)

= E(t)

-

Ef(t)

(1)

where E 4 ( t ) is t h e e n e r g y a t t h e s t a t e f w h i c h , among all t h e s t a t e s t h a t c o u l d be r e a c h e d w i t h no n e t t r a n s p o r t o f any e x t e n s i v e p r o p e r t y b e s i d e s e n e r g y , has t h e minimum e n e r g y . The e n e r g y , E ( t ) - E*r(t), c o u l d be o b t a i n e d v i a volume t r a n s p o r t s , v i a e n t r o p y t r a n s p o r t s o r v i a any e x t e n s i v e p r o p e r t y transports or combination of extensive property t r a n s p o r t s . T h i s p o i n t is i l l u s t r a t e d by s e v e r a l e x a m p l e s in (2). A v a i l a b l e E n e r g y D e s t r u c t i o n and E n t r o p y P r o d u c t i o n . The S e c o n d Law o f Thermodynamics can be s t a t e d t o d e c r e e t h a t t h e s y s t e m a v a i l a b l e e n e r g y o f an i s o l a t e d s y s t e m d e c r e a s e s in all r e a l p r o c e s s e s . . S i n c e E ( t ) is c o n s t a n t f o r an i s o l a t e d s y s t e m it f o l l o w s t h a t Ê f ( t ) > 0. C o n s i d e r t h e s y s t e m i l l u s t r a t e d by F i g . 1, and d e s c r i b e d in the f i g u r e c a p t i o n . The a v a i l a b l e e n e r g y o f t h e w h o l e s y s t e m is r e p r e s e n t e d by a r e a m f . a ' m on F i g . 2. I f t h e s y s t e m were i s o l a t e d , t h e n f o r any s t a t e 2 t h a t c o u l d be r e a c h e d s p o n t a n e o u s l y f r o m s t a t e 1 o f F i g . 1, t h e dead s t a t e w o u l d have

s

I

f

2 From t h e f i g u r e be g i v e n by

it

= S I I f =Sfi+ T 2

can be s e e n t h a t t h e dead s t a t e e n e r g y w o u l d

=

where

is

SJ2

E

F

L

+

t h e amount o f e n t r o p y

; T

A

produced.

Gaggioli; Thermodynamics: Second Law Analysis ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


WEPFER AND GAGGIOLI

τ

Reference

Datums

τ

I P

I

V

Figure 1.

II P

II

I

II

m I

II

A system having T 7 7 > T 7 , S77 > S7, p 7 7 = p 7 and m 7 7 = m 7




Figure 2. T-S diagram illustrating dead state it corresponding to State 1 in Figure 1. State f£ is the dead state for some State 2 that could be reached spontaneously from State 1.


5.

WEPFER AND GAGGIOLI

Reference

81

Datums

Where t h e i n t e g r a l is t w i c e t h e a r e a f i f 2 J (Since

ï Ef

k f

ι

shown on F i g . 2.

, t h e i n t e g r a l is e q u a l t o o r l e s s t h a n

area

mf4a'm. The s t a t e f 2 w h i c h y i e l d s t h e maximum i n t e g r a l , e q u a l t o a r e a mf4a'm is t h e e q u i l i b r i u m s t a t e t h a t t h e s y s t e m w o u l d r e a c h f r o m s t a t e f i , were i s o l a t i o n m a i n t a i n e d . ) Since Β = Ε Ε 4 , it f o l l o w s t h a t


*i»i.«

( t )

• • M "

• -Tf

( t

>S„

( t

>

( 2 )

Subsystem A v a i l a b l e E n e r g i e s . The p u r p o s e o f t h i s s e c t i o n is t o show t h a t s u b s y s t e m a v a i l a b l e e n e r g i e s , A , c a n be d e f i n e d s u c h t h a t t h e s y s t e m a v a i l a b l e e n e r g y , B , o f any s y s t e m is e q u a l t o t h e sum o f t h e s u b s y s t e m a v a i l a b l e e n e r g i e s . T h a t is, f o r any breakdown o f t h e s y s t e m i n t o d i s t i n c t s u b s y s t e m s Β = ΣΑΊ.

(3)

where Α,· is an e x t e n s i v e p r o p e r t y Β is n o t an e x t e n s i v e p r o p e r t y . ) t h a t f o r any s u b s y s t e m A = E

p

+

f

V - T

f

S - *

o f s u b s y s t e m i ( Β Φ ΣΒ Ί · ; i . e . , F u r t h e r m o r e , it will be shown

i

f

N .

( 4 )

where p * , T f , u j f , . . . , a r e t h e p r e s s u r e , t e m p e r a t u r e , and c h e m i c a l p o t e n t i a l s . . . o f t h e s u b s y s t e m a t t h e dead s t a t e o f t h e composite (whole) system. Before proceeding t o t h e above-mentioned developments, it needs t o be n o t e d t h a t t h e s y s t e m a v a i l a b l e e n e r g y B j y j j o f t h e c o m p o s i t e I U I I o f two s y s t e m s I and I I e q u a l s t h e sum o f t h e i r i n d i v i d u a l s y s t e m a v a i l a b l e e n e r g i e s , Βτ + B J J , p l u s t h e s y s t e m a v a i l a b l e e n e r g y B j 4 u i i f o f t h e c o m p o s i t e when I is in its dead s t a t e and I I ' i s in its dead s t a t e : B

IUII=

B

I

+

B

II

+

B

IfUIIf

(5)

C o n s i d e r t h e c o m p o s i t e s y s t e m shown in F i g . 3 where s u b s y s t e m s A and Β c a n exchange e n t r o p y , v o l u m e , and components i = 1, The a v a i l a b l e e n e r g y o f A is t h e same a s t h a t o f AUC, where C is an i n f i n i t e e n v i r o n m e n t a t Τ 4 , ρ 4 , μ.4..., t h e dead s t a t e t e m p e r a t u r e , p r e s s u r e - a n d c h e m i c a l p o t e n t i a l s o f AUB. S i m i l a r l y f o r Β a n d BUD. The a v a i l a b l e e n e r g y o f t h e c o m p o s i t e , BAND, is t h e same a s t h e a v a i l a b l e e n e r g y o f t h e c o m p o s i t e RjUR2:




Figure 3. A composite system consisting of Subsystems A and B, at a state of the composite which has dead-state properties equal to T , p , and μ . . . . Subsystem Β is surrounded by an infinite environment D having the same T , p , and /x. . . . Simihrly for C, which surrounds A. Note that Rj = AUC and R« = BUD. f

f

ι{

f

f


fJ

5.

WEPFER AND GAGGioLi

B

AUB "

B

Reference

83

Datums

R!UR2

= BRL

+ BR 2 + %

f

U

R

2

f

(

6

)

However, because both and R2 a r e i n f i n i t e in e x t e n t and have t h e same d e a d - s t a t e i n t e n s i v e p r o p e r t i e s Τ 4 , ρ 4 , μ 4 . . . , it f o l l o w s t h a t B R i U R 2 is e q u a l t o z e r o . Furthermore, the a v a i l a b l e e n e r g y o f R4 is e q u a l t o t h e maximum e n e r g y t h a t c a n be e x t r a c t e d f r o m t h e c o m p o s i t e AUC f

f

= {E-Ef}

BRL

+ {E-Ef}c

S i n c e t h e maximum e n e r g y is e x t r a c t e d w i t h p r o c e s s e s h a v i n g S4=0 t h e t e r m { E - E f } r c a n be r e w r i t t e n by u s i n g a f o r m o f t h e G i b b s E q u a t i o n ( 3 - 7 ) , dE = Τ dS - pdV + z y i d N - | :


9

{E-Ef}c = where BRL

P f

{Vf-V}c

- Tf{Sf-S}c

•mif{N1-N1f}c

r e p r e s e n t s t h e components (.2,4.) o f C . Thus = {E-Ef}A + pf{Vf - V}c - T f { S

S } c + Σμ-jf{N-j - N f } c

r

H o w e v e r , n o t e t h a t a volume b a l a n c e on s y s t e m and C) g i v e s {Vf - V}c = - { V f - V}A

( 7 )

(subsystems A

S i m i l a r b a l a n c e s c a n be w r i t t e n f o r e n t r o p y and components s o t h a t E q . 7 c a n be w r i t t e n in t h e f o l l o w i n g f o r m BRl

={E-Ef}A -

P f

{Vf-V}A + Tf{Sf-S}A - Jii1f{NrN1f}

A

( 8 )

B

(9)

Likewise BR2

= {E-Ef}B -

P f

{Vf-V}B + Tf{Sf-S}B - 4

i f

{NrN

i f

}

S u b s t i t u t i o n o f E q s . (8) and (9) i n t o E q . (6) y i e l d s B

AUB

=

E

-

À

{ E

PfVA -

+

A

f

+

P f

T

V

fSA "

A f "

T

4ifNTA

f

S

A f - 4 i f

N

i A f >

+


84


+ EB

+ pfVB - TfSB -

~{EBf

I

U i f

+ PfVBf " TfSBf -

N

i B

5uifNBf}

(10)

From c l a s s i c a l t h e r m o d y n a m i c s ( 4 , 6 ) it is known t h a t a f o r m o f t h e G i b b s e q u a t i o n can be i n t e g r a t e d t o g i v e Ε = TS - pV + Σμ-,-Nj. T h u s , t h e t e r m s in t h e b r a c k e t s in E q . 10 a r e i d e n t i c a l l y equal to z e r o . T h a t is B

AUB =

E

A

PfvA "

+


+ EB

T

fsA

"

4ifNiA

+ pfVB- TfSB - l u . f N

The s u b s y s t e m a v a i l a b l e e n e r g y Α

Ε

Ξ

+ pfV

- TfS

-

T h i s d e f i n i t i o n a l l o w s E q . 11 subsystem a v a i l a b l e energies B

AUB

=

A

A

+

A

(11)

i B

is

d e f i n e d as f o l l o w s

mIFN1

(

t o be e x p r e s s e d in t e r m s

1

2

)

of

B

(13)

F i n a l l y , it will be shown t h a t s u b s y s t e m a v a i l a b l e e n e r g y an e x t e n s i v e p r o p e r t y ; i . e . , t h a t Α / 4 β = A A + Α Β · Α

Α·,,η

Ε

Ξ

AUB

=

E

B

+

=

A

E

A

+pV

AUB

- T S

f AUB

+

PfV

B

+

+

A

T

fSB '

PfVB -

f AUB

Σμ

if

Ν

iAUB

*1f"lA

fSB -

T

-

is

* i f

N

i B

B

(14)

T h u s , f o r any o b j e c t , its s u b s y s t e m a v a i l a b l e e n e r g y e q u a l s t h e sum o f t h e s u b s y s t e m a v a i l a b l e e n e r g i e s o f its p a r t s , p r o v i n g t h a t A is e x t e n s i v e . I t can a l s o be shown (2) t h a t t h e s u b s y s t e m a v a i l a b l e e n e r g y changes as a r e s u l t o f t r a n s p o r t s a n d / o r d e s t r u c t i o n s o f s u b s y s t e m a v a i l a b l e energy; i . e . Subsystem

+

Âx + Απ


(15)

5.

WEPFER AND GAGGiOLi

where t h e t r a n s p o r t s Ατ -Â.


'IT

= Ετ

Reference

and d e s t r u c t i o n s

+ pfVT - TfST

85

Datums

-

*

are 1 F

N

1 F

TfS4

(16) (17)

I t is i m p o r t a n t t o r e c o g n i z e t h a t E q s . 12-17 a r e all v a l i d a t any i n s t a n t , even i f p 4 , T f , μ - j f , . . . a r e c h a n g i n g . The e q u a t i o n s d e v e l o p e d in t h i s s e c t i o n f o r s u b s y s t e m a v a i l a b l e e n e r g y a r e g e n e r a l l y f o u n d in t e x t b o o k s on t h e r m o d y n a m i c s where t h e y a r e p r e s e n t e d as t h e a v a i l a b l e e n e r g y o f an object or system. Furthermore, the textbooks express the t r a n s p o r t c o e f f i c i e n t s - - T f , p f , y i f . . . - - as TQ, PQ> y - j o - - - t o r e p r e s e n t the i n t e n s i v e p r o p e r t i e s of the ambient s u r r o u n d i n g ( s ) - - a s s u m e d t o be in s t a b l e e q u i l i b r i u m . H o w e v e r , as shown in t h i s a r t i c l e these equations r e a l l y represent the c o n t r i b u t i o n of a subsystem to the o v e r a l l a v a i l a b l e energy o f a composite system. F o r c a s e s in w h i c h a s u b s y s t e m is s u r r o u n d e d by a s t a b l e a m b i e n t s u b s y s t e m t h a t is i n f i n i t e in e x t e n t t h e n T f , p f , andyjf...are i d e n t i c a l t o t h e t e m p e r a t u r e , p r e s s u r e , and c h e m i c a l p o t e n t i a l s . . . o f t h e a m b i e n t s u r r o u n d i n g s ; in many s u c h c a s e s , t h e s e T 4 , u - j f c a n be assumed t o be c o n s t a n t — p r o v i d e d t h a t t h e a m b i e n t is u n p e r t u r b e d by o t h e r s y s t e m s ( n o t s u b s y s t e m s ) . If the ( l o c a l ) a m b i e n t is s t a b l e and l a r g e b u t a f f e c t e d by o t h e r s y s t e m s , t h e n T f , p.f and μ Ί · 4 a r e f u n c t i o n s o f t i m e . If a l a r g e a m b i e n t s u r r o u n d i n g s is n o t s t a b l e , t h e n T f , p f a n d y - j f a r e e q u a l t o t h e v a l u e s o f T , p , and μ a t t h e dead s t a t e o f t h e a m b i e n t as a s y s t e m a l o n e ; t h e s e T f , p f and μ - j f a r e f u n c t i o n s of time, a l s o . The r e m a i n d e r o f t h i s a r t i c l e a d d r e s s e s t h e s e l e c t i o n o f a v a i l a b l e e n e r g y s y s t e m s and s u b s y s t e m s as w e l l as t h e c h o i c e o f dead s t a t e s f o r a n a l y s e s o f p r a c t i c a l p r o b l e m s . The S e l e c t i o n o f R e f e r e n c e

Datums f o r S u b s y s t e m A v a i l a b l e

Energy.

The d e f i n i t i o n o f s u b s y s t e m a v a i l a b l e e n e r g y , A , w h i c h is an e x t e n s i v e p r o p e r t y , is c r u c i a l t o p r a c t i c a l Second Law efficiency analysis. B e f o r e a p r o c e s s , d e v i c e , o r system can be a n a l y z e d , it is n e c e s s a r y t o a s c e r t a i n ( o r assume o r a p p r o x i mate) t h e dead s t a t e s o f all r e l e v a n t m a t e r i a l s and e q u i p m e n t . More p r e c i s e l y , p f , T f , and μ - j f must be known f o r e a c h s u b s y s t e m b e f o r e E q . 12 c a n be e m p l o y e d t o e v a l u a t e A o f any s u b s y s t e m . In t h e o r y , once t h e r e l e v a n t s y s t e m is d e f i n e d , t h e c o n v e n t i o n a l t h e r m o d y n a m i c p r i n c i p l e s o f e q u i l i b r i u m c a n be u s e d t o f i n d t h e P f , T f , μ i f . . . , o n c e t h e p o s s i b l e v a r i a t i o n s (344) a r e (assumed and) p r e s c r i b e d . In p r a c t i c e , t h e e s t a b l i s h m e n t " o f t h e s e properties with equilibrium principles involves elaborate s e a r c h methods ( 8 , 9 ) .




86

In most i n s t a n c e s t h e d e t e r m i n a t i o n o f p f , T f , u j f . . . i s more s t r a i g h t f o r w a r d , so t h a t e q u i l i b r i u m c a l c u l a t i o n s need n o t be made. I n any c a s e , t h o u g h , t h e f i r s t s t e p in t h e a n a l y s i s is t o e s t a b l i s h "the r e l e v a n t composite system". T h a t is, a p r o c e s s o r d e v i c e o r p l a n t ( w i t h its l o a d ) c a n n o t be a n a l y z e d in i s o l a t i o n from the r e s t o f the " u n i v e r s e " . The Second Law a n a l y s i s r e q u i r e s c o n s i d e r a t i o n o f t h e c o m p o s i t e o f all s u b s y s t e m s w i t h w h i c h t h e p r o c e s s s y b s y s t e m must i n t e r a c t in o r d e r to accomplish the d e s i r e d purpose of the p l a n t . Thus, f o r e x a m p l e , t h e r e l e v a n t c o m p o s i t e s y s t e m f o r an a u t o m o t i v e e n g i n e p l a n t would c o n s i s t of (at l e a s t ) the engine hardware, the f u e l , t h e c o n f i n e d c o o l a n t , and a m b i e n t a i r ( u s e d f o r b o t h c o o l i n g and f o r c o m b u s t i o n ) . Once t h e r e l e v a n t c o m p o s i t e s y s t e m is d e f i n e d , t h e d e t e r m i n a t i o n o f T f , p f , y i f , . . . c a n be t a c k l e d . Stable Reference Environments. When one o f t h e r e l e v a n t s u b s y s t e m s is a t c o m p l e t e s t a b l e e q u i l i b r i u m and is v e r y l a r g e compared t o all o f t h e o t h e r s u b s y s t e m s t o g e t h e r , it can be c a l l e d a s t a b l e ambient environment. Denote its ρ , Τ , μ by p Q , TQ, vy). Then T f = TQ f o r all s u b s y s t e m s . F u r t h e r m o r e , f o r all m a t e r i a l s w h i c h a r e n o t p r e c l u d e d f r o m p r e s s u r e e q u i l i b r i u m w i t h t h e l a r g e s y s t e m ( t h a t is, volume may be f r e e l y e x c h a n g e d ) , P f = PQ. When a m a t e r i a l is p r e v e n t e d from a t t a i n i n g p r e s s u r e e q u i l i b r i u m w i t h the e n v i r o n m e n t , because it is c o n f i n e d in an e n v e l o p e ( f l e x i b l e o r i n f l e x i b l e ) , t h e n P f e q u a l s P(TQ,V') where VQ is t h e s p e c i f i c volume o f t h e m a t e r i a l when it and its c o n f i n i n g e n v e l o p e a r e in t h e i r dead s t a t e with the environment. S i m i l a r comments h o l d f o r μ,-f. If s u b s t a n c e i o f a m a t e r i a l c a n i n t e r a c t w i t h its components in the ambient environment, then = μ · . Otherwise μ . = U j ( T 0 , P f , x - f ) where t h e x - - f a r e t h e mole f r a c t i o n a t t h e dead s t a t e of tne m a t e r i a l . η

0

f

An example o f an i n s t a n c e where P f e q u a l s P(TQVQ) w o u l d be t h e c a s e o f t h e H20 " s e a l e d " w i t h i n t h e e q u i p m e n t o f a power c y c l e ; a n o t h e r example w o u l d be r e f r i g e r a n t ( s a y NH3) c o n f i n e d i n s i d e t h e components o f a r e f r i g e r a t i o n c y c l e . The r e f r i g e r a n t w o u l d a l s o be an e x a m p l e o f a c a s e where μ 4 d i f f e r s f r o m t h e v a l u e f o r t h e components in t h e s u r r o u n d i n g e n v i r o n m e n t ; f o r t h e refrigerant, μ = g(TQ,pf) = g(T0,p(TQ,v')). A t an i n s t a n t o f t i m e a l a r g e a m b i e n t s u b s y s t e m may be a t a s t a b l e e q u i l i b r i u m s t a t e , b u t t h e s t a t e may change w i t h t i m e . F o r e x a m p l e , t h e c o o l i n g w a t e r f o r a power c y c l e may be s u p p l i e d f r o m a l a r g e l a k e w h i c h , a t a g i v e n i n s t a n t is (more o r l e s s ) u n i f o r m in t e m p e r a t u r e and c o m p o s i t i o n . However, the temperature may v a r y s i g n i f i c a n t l y f r o m s e a s o n t o s e a s o n ( a s a c o n s e q u e n c e o f u n c o n t r o l l a b l e i n f l u e n c e s o f o t h e r s y s t e m s , from o u t s i d e t h e c o m p o s i t e o f l a k e and power p l a n t ) . T h e o r e t i c a l l y , the ι Ύ



5.

WEPFER AND GAGGiOLi

Reference

87

Datums

e f f i c i e n c y a n a l y s i s o f t h e power c y c l e w o u l d u t i l i z e t h e i n s t a n t a neous l a k e t e m p e r a t u r e f o r T f . Then, t o analyze the annual p e r f o r m a n c e , it w o u l d be n e c e s s a r y t o i n t e g r a t e t h e i n s t a n t a n e o u s r e s u l t s over the y e a r . H o w e v e r , in p r a c t i c e good r e s u l t s c a n be o b t a i n e d by making j u s t a few a n a l y s e s , s a y one f o r e a c h s e a s o n , and t h e n w e i g h i n g t h e r e s u l t s — o r even w i t h j u s t one a n a l y s i s , e m p l o y i n g an a p p r o p r i a t e a n n u a l a v e r a t e TQ. Thus, t h e i n t e g r a t i o n o f i n s t a n t a n e o u s r e s u l t s ( o r t h e summing o f i n c r e m e n t a l r e s u l t s ) c a n be a v o i d e d when ( i ) t h e v a r i a t i o n s o f t h e Τ 4 , ρ 4 a n d u . j f a r e r e l a t i v e l y s l o w and s m a l l , and ( i i ) i f t h e l o a d s on t h e s y s t e m a r e u n c o r r e c t e d w i t h T f , P f and μ . f . A n o t a b l e c o u n t e r - e x a m p l e , where t h e a n a l y s i s o f t h e s y s t e m must be i n s t a n t a n e o u s , is t h e c a s e o f an a i r - c o n d i t i o n i n g s y s t e m (10,11). The r e l e v a n t a m b i e n t s u r r o u n d i n g s is t h e o u t d o o r a i r . B o t h T i ) t h e t e m p e r a t u r e TQ a n d t h e h u m i d i t y ( a n d hence t h e μ Ί · — f o r t h e N 2 and 0 2 as w e l l as f o r t h e H ? 0 ! ) change r a p i d l y and s u b s t a n t i a l l y . And ( i i ) t h e l o a d on t h e a i r - c o n d i t i o n i n g s y s t e m is s t r o n g l y d e p e n d e n t upon o u t d o o r t e m p e r a t u r e and humidity. On t h e o t h e r h a n d , f o r t h e a n a l y s i s o f a c h e m i c a l p l a n t ( e . g . , 1 2 ) , t h e v a r i a t i o n s o f a m b i e n t c o n d i t i o n s may be n e g l i gible. I f the l a r g e s t c o n t r i b u t i o n s to the a v a i l a b l e energies of m a t e r i a l s a r e t h e i r chemical a v a i l a b i l i t i e s ( 1 2 , 1 3 ) , then v a r i a t i o n s o f TQ may be i n c o n s e q u e n t i a l . I n f a c t , t h e usage o f TQ = 7 7 ° F = 2 5 ° C is o f t e n j u s t i f i a b l e — f o r t h e s a k e o f t h e convenience o f thermochemical property c a l c u l a t i o n s — e v e n i f the a v e r a g e o u t d o o r t e m p e r a t u r e is somewhat d i f f e r e n t . M e t a s t a b l e and U n s t a b l e A m b i e n t E n v i r o n m e n t s . I n some i n s t a n c e s , an a m b i e n t e n v i r o n m e n t m i g h t be a t a s t e a d y s t a t e b u t not a t a completely s t a b l e e q u i l i b r i u m s t a t e . F o r example ( 8 ) , n i t r o g e n , N 2 , is n o t in s t a b l e e q u i l i b r i u m w i t h t h e c r u s t o f t h e e a r t h and t h e s e a s . A more s t a b l e c o n f i g u r a t i o n o f n i t r o g e n is in n i t r a t e s . ( C o n c e i v a b l y , t h e s t a n d a r d c o m p o s i t i o n o f a i r in t h e e n v i r o n m e n t is m a i n t a i n e d in a s t e a d y m e t a s t a b l e s t a t e by i n t r u s i o n f r o m an e v e r l a r g e r s y s t e m , s u c h as t h e s u n a n d / o r the e a r t h ' s magnetic f i e l d ; i . e . , seemingly p o s s i b l e v a r i a t i o n s (3) a r e p r e v e n t e d by u n r e c o g n i z e d i n f l u e n c e s . ) Then t h e a p p r o p r i a t e p r a c t i c a l c h o i c e f o r T f , p f , and μ 4 a r e To, p 0 , and μ Ί ·Q— t h e s t e a d y v a l u e s in t h e l o c a l a m b i e n t e n v i r o n m e n t . W i t h t h i s s e l e c t i o n o f T f , P f , and μ - f , t h e t o t a l z A j , summed o v e r all t h e s u b s y s t e m s j b e s i d e s t h e a m b i e n t e n v i r o n m e n t , does n o t r e p r e s e n t t h e a b s o l u t e s y s t e m a v a i l a b l e e n e r g y . Rather Β = zAj + B Q

(18)

where B Q is t h e s y s t e m a v a i l a b l e e n e r g y , o f t h e c o m p o s i t e o f t h e a m b i e n t e n v i r o n m e n t and all o t h e r s u b s y s t e m s , when t h e o t h e r s u b s y s t e m s have t h e i r T , p , and μ Ί · r e d u c e d t o t h e a m b i e n t v a l u e s


88


( o r t o P(TQ,V'), e t c . ) . I n s o f a r as t h e o t h e r s u b s y s t e m s a r e v e r y s m a l l compared t o t h e a m b i e n t , BQ is e s s e n t i a l l y t h e a b s o l u t e a v a i l a b l e energy o f the environment a l o n e . It follows t h a t f o r a subsystem


E

i

+

Po v i "

T

0Si

" 4iO =

E

i

+

PfVi

"

T

fSi

" 4if

N

i

(]9)

where p 4 , T4., a n d y 4 a r e t h e v a l u e s a t t h e s t a b l e dead s t a t e o f t h e c o m p o s i t e s y s t e m a n d PQ, TQ, and y . Q a r e t h o s e a t t h e pseudo dead s t a t e — a t t h e s t e a d y s t a t e u l t i m a t e l y a c h i e v e d by c o m m u n i c a t i n g w i t h t h e a m b i e n t e n v i r o n m e n t , w h i c h is n o t a t stable equilibrium. The r e a s o n t h a t t h e a p p r o p r i a t e p r a c t i c a l c h o i c e f o r T4r, p f , y . f a r e t h e s t e a d y v a l u e s in t h e m e t a s t a b l e l o c a l a m b i e n t e n v i r o n m e n t is t h a t t h e a v a i l a b l e e n e r g y BQ is n o t a c c e s s i b l e p r a c t i c a l l y - - a s l o n g as t h e m e t a s t a b i l i t y c a n n o t be overcome w i t h p r a c t i c a l means. I f t h e m e t a s t a b i l i t y c a n be o v e r c o m e , t h e n , o f c o u r s e , t h e p r o p e r s e l e c t i o n f o r T f , ρ * , y n - f w o u l d be t h e f i n a l v a l u e s in t h e u l t i m a t e dead s t a t e o r t h e a m b i e n t s u r r o u n d ings. In some i n s t a n c e s an a m b i e n t e n v i r o n m e n t m i g h t be u n s t a b l e . F o r e x a m p l e , a power p l a n t may i n t e r a c t w i t h two l a r g e e n v i r o n m e n t s , s u c h as t h e s u r r o u n d i n g a t m o s p h e r i c a i r ( u s e d f o r c o m b u s t i o n and f o r d i s p e r s i o n o f e x h a u s t g a s e s ) a n d a l a r g e body o f w a t e r ( u s e d f o r c o o l i n g ) . T o g e t h e r t h e two a m b i e n t s m i g h t n o t be a t e q u i l i b r i u m — b e c a u s e t h e a i r is a t a d i f f e r e n t t e m p e r a t u r e f r o m t h e w a t e r a n d / o r is n o t s a t u r a t e d w i t h w a t e r . T h e o r e t i c a l l y , t h e c o m p o s i t e o f t h e two e n v i r o n m e n t s has n o n z e r o s y s t e m a v a i l a b l e e n e r g y BQ. I f t h e r e a r e no r e a l i s t i c a l l y p r a c t i c a l means f o r o b t a i n i n g BQ, t h e n a r e a s o n a b l e s e l e c t i o n o f T f , p f , y 4 f w o u l d be t o make d i f f é r e n t c h o i c e s d e p e n d i n g upon the process o r device being a n a l y z e d . For example, the a n a l y s i s o f t h e power c y c l e s h o u l d u s e T w a t e r f o r T f in t h e c a l c u l a t i o n of A j ' s . Whereas, the a n a l y s i s o f t h e combustion process s h o u l d u s e T a i r f o r T f and y H 2 Q I A I R f o r y . H0

T h i s p r o c e d u r e is a p p r o p r i a t e Β

=

system

A

cy

+ A

co

+ Β

>

f

inasmuch as 0,(cyUw)U(coUa)

where A c y = Β 4 is t h e s u b s y s t e m a v a i l a b l e e n e r g y o f t h e c y c l e (cy) r e l a t i v e to the water ( w ) , A = B n is t h e subsystem a v a i l a b l e energy o f t h e combustion subsystem ( c o ) r e l a t i v e t o t h e a i r ( a ) , a n d BQ,(cyUw)U(coUa) is t h e s y s t e m a v a i l a b l e energy o f the o v e r a l l ' c o m p o s i t e , the crucial p o i n t is t h a t B 0 ( c „ y w ) u ( c o U a ) i S P r a c t i c a 1 4 e ( 4 u a l t 0 B w U a . I f B w y a is n o t r e a l i s t i c a l l y a v a i l a b l e in p r a c t i c e , t h e n it is r e a s o n a b l e t o u s e B s y s t e m = A c y + A C Q . c o

a



5.

WEPFER AND GAGGiOLi

Reference

Daturas

89

T h e r e is y e t a n o t h e r t y p e o f u n s t a b l e e n v i r o n m e n t w h i c h needs t o be c o n s i d e r e d , n a m e l y , an e n v i r o n m e n t w h i c h is made u n s t a b l e by i n f l u e n c e s f r o m t h e p l a n t . The g u i d i n g p r i n c i p l e s f o r s e l e c t i n g a dead s t a t e u n d e r t h e s e c i r c u m s t a n c e s will be i l l u s t r a t e d v i a an e x a m p l e . C o n s i d e r a power p l a n t w h i c h d i s p e r s e s S 0 2 i n t o the environment. The e n v i r o n m e n t c o n t a i n i n g SO2 is u n s t a b l e ; t h e SO2 c o u l d r e a c t s p o n t a n e o u s l y w i t h H2O in t h e e n v i r o n m e n t t o a c h i e v e a more s t a b l e c o n f i g u r a t i o n o f t h e s u l f u r — i n s u l f u r o u s and/or s u l f u r i c a c i d , f o r example. Still, n e i t h e r o f t h e s e a c i d s l a c k s p o t e n t i a l t o c a u s e change ( i . e . , n e i t h e r is i n e r t ) . T h e o r e t i c a l l y , t h e d e t e r m i n a t i o n o f t h e dead s t a t e w o u l d r e q u i r e t h e s e a r c h methods r e f e r r e d t o e a r l i e r , a p p l y i n g t h e thermodynamic p r i n c i p l e s o f c h e m i c a l e q u i l i b r i u m mentioned earlier. The a p p l i c a t i o n o f t h e s e p r i n c i p l e s r e q u i r e s p r e c i s e s p e c i f i c a t i o n o f the r e l e v a n t subsystems, of t h e i r i n i t i a l c h e m i c a l c o n s t i t u t i o n (and o f t h e p o s s i b l e v a r i a t i o n s ) . In p r a c t i c e , t h i s i n f o r m a t i o n may n o t be a c c e s s i b l e . Suppose, f o r example, t h a t CaC03 ( i . e . , l i m e s t o n e — r e l a t i v e l y a b u n d a n t ) is a v a i l a b l e in t h e e n v i r o n m e n t , t h e n t h e SO2 can combine w i t h t h e C a C 0 3 ( a n d o t h e r components o f t h e e n v i r o n m e n t ) t o p r o d u c e g y p s u m , C a S O 4 h 4 O , w h i c h is v e r y i n e r t . T h e r e is a c a l c u l a b l e (12,13) a v a i l a b l e e n e r g y a t t a i n a b l e .from this reaction. Two q u e s t i o n s a r i s e : (1) I s t h e r e a n o t h e r c a l c i u m - b e a r i n g compound a v a i l a b l e in t h e e n v i r o n m e n t w h i c h c o u l d c o m b i n e w i t h t h e SO2 (and o t h e r components) t o y i e l d g y p s u m , and w h i c h c o u l d y i e l d more a v a i l a b l e e n e r g y in t h e p r o c e s s ? (2) Is t h e r e a n o t h e r s u l f u r - b e a r i n g compound b e s i d e s gypsum t h a t w o u l d be even more i n e r t ( w h i c h means t h a t more a v a i l a b l e e n e r g y c o u l d be o b t a i n e d , s a y upon r e d u c i n g t h e gypsum t h e r e t o ) ? These q u e s t i o n s c o u l d n e v e r be a n s w e r e d d e f i n i t i v e l y ; t h e s e a r c h f o r t h e s e o t h e r compounds c o u l d c o n t i n u e i n d e f i n i t e l y . In p r a c t i c e , a r e a s o n a b l e p r o c e d u r e f o r e s t a b l i s h i n g an a p p r o p r i a t e s t a b l e c o n f i g u r a t i o n o f an e l e m e n t is by ( a ) m a k i n g a more o r l e s s q u i c k s t u d y o f c h e m i s t r y t e x t b o o k s a n d / o r r e f e r e n c e books t o a s c e r t a i n what compounds, b e a r i n g t h e e l e m e n t , a r e i n e r t , and (b) what compounds a r e c o n c e i v a b l e e n v i r o n m e n t a l components which c o u l d r e a c t w i t h the element to produce the v a r i o u s i n e r t compounds. Then, c o n s i d e r i n g the d i f f e r e n t combinations of r e a c t a n t s and p r o d u c t s , d e t e r m i n e t h o s e w h i c h y i e l d t h e maximum a v a i l a b l e energy. A h r e n d t s ( 8 ) , and Fan and S h i e h ( 9 ) , have made e x h a u s t i v e a n a l y s e s in t h i s m a n n e r . For example, they f i n d t h a t more a v a i l a b l e e n e r g y is o b t a i n e d f r o m s u l f u r b e a r i n g compounds i f t h e y a r e r e d u c e d t o gypsum w i t h C a ( N 0 3 ) 2 t h a n w i t h CaC03. H o w e v e r , C a ( N 0 ) o is r a r e in t h e e n v i r o n m e n t , w h e r e a s C a C 0 3 is r e l a t i v e l y a b u n d a n t . Which is t h e b e t t e r c h o i c e ? The a n s w e r l i e s in t h e d e f i n i t i o n o f " t h e r e l e v a n t s y s t e m " I f it i n c l u d e s b o t h t h e C a l c i u m N i t r a t e and t h e C a l c i u m C a r b o n a t e , t h e n c l e a r l y t h e n i t r a t e is t h e a p p r o p r i a t e c h o i c e . If it 3



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c o n t a i n s o n l y t h e c a r b o n a t e , t h e n a c h o i c e must be made: Either (1) s e l e c t t h e c a r b o n a t e as t h e a p p r o p r i a t e e n v i r o n m e n t a l compon e n t f o r r e d u c i n g s u l f u r , o r (2) R e - d e f i n e " t h e r e l e v a n t s y s t e m " , e n l a r g i n g it t o i n c l u d e t h e n i t r a t e . I f n i t r a t e is a v a i l a b l e n e a r b y t h e n (2) is t h e p r o p e r p r a c t i c a l c h o i c e ; i f it e x i s t s o n l y in g e o g r a p h i c a l l y r e m o t e l o c a t i o n s , t h e n (1) is a p p r o p r i a t e . G e n e r a l l y , t h e c h o i c e is n o t c l e a r c u t . What is " n e a r b y " and what is " r e m o t e " ? T h a t is, when w o u l d it be w o r t h e x p a n d i n g "the r e l e v a n t system"? Only i f the a v a i l a b l e energy o b t a i n e d w i t h t h e n i t r a t e is s u b s t a n t i a l l y l a r g e r t h a n w i t h t h e c a r b o n a t e . A c t u a l l y , it is o n l y s l i g h t l y l a r g e r , and in t h i s c a s e t h e c a r b o n a t e , w h i c h is much more a b u n d a n t is c l e a r l y t h e b e t t e r c h o i c e t h a n t h e n i t r a t e ( e x c e p t in p e c u l i a r l o c a l e s where t h e n i t r a t e is r e a d i l y a v a i l a b l e in t h e e n v i r o n m e n t ) . S t i l l , t h e r e may be c i r c u m s t a n c e s t h e r e t h e c a r b o n a t e may n o t be an a p p r o p r i a t e p r a c t i c a l c h o i c e , b e c a u s e it may be scarce and/or expensive. I f t h i s is t h e c a s e , t h e n t h e s e a r c h must be c o n t i n u e d . O f t e n , H2O c o u l d be u s e d . However, c o n s i d e r the p r o s p e c t o f d i s p e r s i o n o f S 0 i n t o a d e s e r t e n v i r o n m e n t ; it is c o n c e i v a b l e t h a t t h e dead s t a t e o f t h e s u l f u r w o u l d need t o be t a k e n as t h e SO4, a t its p a r t i a l p r e s s u r e in t h e a i r in t h e i m m e d i a t e v i c i n i t y o f t h e power p l a n t — o r even a t its p a r t i a l p r e s s u r e in t h e e x h a u s t g a s e s - - a n d a t a m b i e n t t e m p e r a t u r e . From t h e p r a c t i c a l s t a n d p o i n t , s u c h a c h o i c e w o u l d be j u s t i f i e d , a s s u m i n g t h a t t h e r e w o u l d be no r e a l i s t i c means f o r u t i l i z i n g any a v a i l a b l e e n e r g y t h e S 0 w o u l d have r e l a t i v e t o d e s e r t environment. On t h e o t h e r h a n d , i f HpO is a v a i l a b l e in t h e e n v i r o n m e n t , t h e n t h e S 0 c o u l d f o r e x a m p l e , be c o n v e r t e d t o a c i d . But t h e a c i d w o u l d n o t be i n e r t , u n l e s s d i l u t e . The e x t e n t t o w h i c h it c o u l d be d i l u t e d w o u l d depend upon t h e amount o f H2O in " t h e r e l e v a n t s y s t e m " . I f t h e p l a n t is n e a r t o t h e s e a , t h e n t h e c o n c e n t r a t i o n o f s u l f a t e i o n s in t h e s e a w o u l d d i c t a t e t h e dead s t a t e c o n f i g u r a t i o n o f t h e s u l f u r ( i . e . , t h e e x t e n t o f d i l u t i o n o f SO4); s e e (14) f o r t h e l i s t i n g o f a v a i l a b l e e n e r g y values o f v a r i o u s elements r e l a t i v e to standard sea w a t e r . 2

2

2

Closure. T h i s a r t i c l e has p r o v i d e d t h e g u i d e l i n e s f o r p r a c t i c a l s e l e c t i o n o f r e f e r e n c e daturns f o r a v a i l a b l e e n e r g y . The a p p l i c a t i o n o f t h e s e g u i d e l i n e s is i l l u s t r a t e d by t h e v a r i o u s S e c o n d Law a n a l y s e s p r e s e n t e d ( o r r e f e r r e d t o ) in t h i s v o l u m e . S e v e r a l a u t h o r s have p r o p o s e d c h e m i c a l r e f e r e n c e datums f o r s e v e r a l e l e m e n t s and compounds, in v a r i o u s e n v i r o n m e n t a l c i r c u m s t a n c e s ; e . g . , se ( 8 , 9 , 1 2 , 1 3 , 1 4 ) . A l t h o u g h it is o f t e n i n t i m a t e d t o t h e c o n t r a r y , a c r u c i a l p o i n t is t h i s : In e n g i n e e r i n g p r a c t i c e t h e s e l e c t i o n o f an a p p r o p r i a t e r e f e r e n c e datum must t a k e i n t o a c c o u n t t h e p l a n t



5.

WEPFER AND GAGGiOLi

Reference

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b e i n g a n a l y z e d and its s u r r o u n d i n g s . T h e r e is no one " t h e o r e t i c a l l y c o r r e c t " r e f e r e n c e e n v i r o n m e n t . The p r a c t i c a l a n a l y s i s o f a p l a n t s h o u l d r e f l e c t its r e a l c i r c u m s t a n c e s . Nevertheless, t h e v a r i o u s r e f e r e n c e e n v i r o n m e n t s f o r c h e m i c a l s w h i c h have been p r o p o s e d a r e h e l p f u l t o t h e p r a c t i t i o n e r , as p r o s p e c t i v e c h o i c e s o f r e f e r e n c e d a t u m s - - o r as g u i d e s t o t h e f o r m u l a t i o n o f an a p p r o p r i a t e r e f e r e n c e d a t u m . Two f i n a l p o i n t s need t o be made: (1) D i f f i c u l t i e s w i t h t h e s e l e c t i o n o f c h e m i c a l r e f e r e n c e datum s h o u l d n o t be a l l o w e d t o be an o b s t a c l e t o t h e a p p l i c a t i o n o f S e c o n d Law a n a l y s i s . The p r a c t i t i o n e r is e n c o u r a g e d t o r e l y on j u d g m e n t , and t h e n p r o c e e d . S u b s t a n t i a l e r r o r s will r e s u l t o n l y i f t h e j u d g m e n t is very poor. ( F u r t h e r m o r e , j u d g m e n t is i m p r o v e d by experience.) (2) I f t h e p r a c t i t i o n e r is w i l l i n g t o f o r e g o t h e i n f o r m a t i o n g i v e n by t h e a b s o l u t e v a l u e s o f a v a i l a b l e e n e r g y f 1 o w s , b u t w o u l d be s a t i s f i e d w i t h t h e e v a l u a t i o n o f a v a i l a b l e energy consumptions o n l y , then (a) A v a i l a b l e e n e r g y b a l a n c e s c a n be u s e d t o e v a l u a t e t h e c o n s u m p t i o n s w i t h o u t even s e l e c t i n g a dead s t a t e f o r chemical a v a i l a b l e energy. Because, t h e c o n s u m p t i o n will a l w a y s e q u a l d i f f e r e n c e s in a v a i l a b l e e n e r g i e s , so t h a t t h e dead s t a t e v a l u e s will cancel. I n any c a s e , t h o u g h , a dead t e m p e r a t u r e , T f , is n e e d e d . (b) In f a c t , i f o n l y c o n s u m p t i o n s , A , a r e d e s i r e d , t h e n it is u n n e c e s s a r y t o u s e a v a i l a b l e e n e r g y balances. I n s t e a d , e n t r o p y b a l a n c e s c a n be u s e d , since A = T 4 . G e n e r a l l y , entropy balances a r e s i m p l e r to u s e , s i n c e t h e y i n v o l v e f e w e r thermochemical property c a l c u l a t i o n s . However, the e v a l u a t i o n of the e f f i c i e n c y of a process r e q u i r e s a b s o l u t e values of the a v a i l a b l e energy— f o r example, the a v a i l a b l e energy content of the fuel. S i m i l a r l y , S e c o n d Law c o s t i n g depends upon evaluation of absolute values of a v a i l a b l e e n e r g i e s . T h u s , in o r d e r t o make e f f i c i e n c y a n a l y s e s , o r t o do c o s t i n g , a t l e a s t an a p p r o x i m a t i o n t o t h e r e f e r e n c e datum must be made. Literature Cited 1.

G a g g i o l i , R.A., volume.

" P r i n c i p l e s o f Thermodynamics,"

2.

Wepfer, W.J., " A p p l i c a t i o n o f the Second Law to the A n a l y s i s and Design o f Energy Systems," Ph.D. D i s s e r t a t i o n , U. o f Wisconsin, Madison, 1979.


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Keenan, J.H. Thermodynamics.

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4. Hatsopolous, G.N., and Keenan, J.H. Principles of General Thermodynamics. Wiley: New York, 1965. 5.

Obert, E.F. 1948.

Thermodynamics.

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Obert, E.F. Concepts of Thermodynamics. New York, 1960.

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Obert, E.F., and Gaggioli, R.A. Thermodynamics. 2nd ed., McGraw-Hill: New York, 1963.

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8. Ahrendts, J., "Die Exergie Chemisch Reaktionsfahiger Systeme," VDI-Forschungsheft 579, VDI-Verlag, Dusseldorf, 1977. 9.

Fan, L.T. and Shieh, J.H., "Thermodynamically-based Analysis and Synthesis of Chemical Process Systems," presented at the U.S.D.O.E. Workship on the Second Law of Thermodynamics, George Washington U., Aug. 15-17, 1979; to be published in Energy: The Inter national Journal.

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Wepfer, W.J., Gaggioli, R.A., Obert, E.F. "Proper Evaluation of Available Energy for HVAC." Trans. A.S.H.R.A.E., 85, 1 (1979)

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Gaggioli, R.A., Wepfer, W.J., and Elkouh, A.F. "Avail able Energy Analysis for HVAC, I. Inefficiencies in a Dual-Duct System." Energy Conservation and Building Heating and Air-conditioning Systems, A.S.M.E. Symposium Volume, Η00116, 1978, 1-20.

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Gaggioli, R.A., and Petit, P.J. "Second Law Analysis for Pinpointing the True Inefficiencies in Fuel Conversion Systems" or "Use the Second Law F i r s t . " Chemtech, 7, 8 (1977), 496-506.

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Rodriguez, L., "Calculation of Available Energy Quantities," this volume.

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Szargut, J. and Dziedziniewicz, C., "Energie utilisable des substances chimiques inorganiques," Entropie, 40, 14-23, 1971.

RECEIVED October 17, 1979.


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