Efficiency and Costing - American Chemical Society

11 StrategicUseofThermoeconomics forSystemImprovement

Downloaded by UCSF LIB CKM RSCS MGMT on December 3, 2014 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch011

Y. EL-SAYED and M. TRIBUS Center for Advanced Engineering Study, Massachusetts Institute of Technology, Cambridge, MA 02139

Second Law analysis combined with a cost consideration helps to understand the economics of lost work i n an energy intensive system. This understanding may guide to innovative conceptual system designs. Familiar examples i n the steady state are used for i l l u s t r a t i v e purposes.

A process engineer creating a conceptual design for an energy intensive system is expected to consider the extremization of two leading c r i t e r i a : an energy efficiency c r i t e r i o n and a cost criterion which includes competing material demands. Both are usually subject to constraints imposed for technical, environmental, economic or legal reasons. The options open to meet these c r i t e r i a include: • Choose another operating point (different pressure, flow rate, . . . ) • Redesign the flow chart (introduce new device, add or remove items of equipment, provide by-pass flows, ...) • Redefine the purpose (consider a dual purpose, find a use for a wasted stream, . . . ) • Modify environment (relax or control a boundary constraint, negotiate a legal constraint, . . . ) • Identify areas for R&D implementations With the help of computer programs and suitable strategy for their use, i t i s possible to examine a great number of alternative solutions energy-wise and cost-wise and to enhance the evolution to competitive alternatives. A good strategy takes into account the evolutionary nature of design concepts, the appropriate level of detail of describing the system and the degree of accuracy of available and assumed data for properties, performance and cost. The purpose of this study is to develop such a strategy

0097-6156/83/0235-0215S06.75/0 © 1983 American Chemical Society In Efficiency and Costing; Gaggioli, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

216

SECOND LAW ANALYSIS OF PROCESSES

from a s o l i d t h e o r e t i c a l base and to f o r m a l i z e what many good engineers appear to do i n f o r m a l l y . The b a s i c i d e a i s to t r e a t a system or a p l a n t as i f i t i s embedded i n two environments: 1) A p h y s i c a l environment described by a r e f e r e n c e pressure Po» r e f e r e n c e temperature T , and a s e t of reference chemical p o t e n t i a l s [ y ] (or a l t e r n a t i v e l y reference compositions [ x ] ) 2) An economic environment described by a s e t of r e f e r e n c e p r i c e s [c^] i n c l u d i n g that of c a p i t a l With the system embedded i n the p h y s i c a l environment a l l m a t e r i a l s and energies of i n t e r e s t are evaluated according to t h e i r work p o t e n t i a l s ( e x e r g i e s ) • The c r i t e r i o n f o r the system i s a work measure. A t y p i c a l f u n c t i o n r a t e d per u n i t time i s 0

c


c

•

0

f

Q

Q

- EEI - EEP

e

(1)

where the f i r s t term i s the sum of a l l input e x e r g i e s , the second term i s the sum of a l l u s e f u l product e x e r g i e s , and the f u n c t i o n i s t o be minimized. The c r i t e r i o n may a l s o take an e f f i c i e n c y form to be maximized. With the system embedded i n the economic environment, a l l the energies and m a t e r i a l s of i n t e r e s t are evaluated according to t h e i r economic p o t e n t i a l s ( c o s t s ) . The c r i t e r i o n f u n c t i o n i s a cost measure. A t y p i c a l f u n c t i o n r a t e d per u n i t time i s o,c

=

E

c

f

F

+

l

z

'

2 c

p

p

2

where the f i r s t term i s the cost of major feed s t o c k s , the second i s the cost of p r o c e s s i n g devices of competing m a t e r i a l content and the t h i r d i s the value of u s e f u l products. The f u n c t i o n i s to be minimized. Thermoeconomics makes the connection between these two e v a l u a t i o n s v i s i b l e throughout the system;. The i d e a of c o u p l i n g p h y s i c a l and cost streams i s not new. The e a r l i e s t example of which we are aware i s due to Manson Benedict i n an unpublished set of notes i n 1949 reported by G y f t o p o u l i s (1). Other examples have been reported (10, 11, 12). P o t e n t i a l Work and Lost Work Functions The i r r e v e r s i b i l i t y accompanying a steady flow process may be computed f o r any zone by n o t i n g the f l u x e s of entropy i n t o and out of the zone. Leaving a s i d e the problems posed by semipermeable membranes, which introduce ambiguities i n t o the meanings of heat and work, (2), equation (3) provides such a balance: c r

S = gmeSe-EmiSi-EQb/Tb (3) I D As de-Nevers has pointed out (3), equation (3) s u f f i c e s to compute the " l o s t work", T S , of the process i n the zone, c r

0

In Efficiency and Costing; Gaggioli, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

11.

EL-SAYED AND TRIBUS

Thermoeconomics for System Improvement

217

r e q u i r i n g only knowledge o f T , the r e f e r e n c e temperature and of absolute e n t r o p i e s , which a r e u s u a l l y computed anyway as part o f c y c l e c a l c u l a t i o n s . Q


I t i s easy t o apply equation (3) t o a l l the system o r t o a subsystem. However, i t i s a l s o d e s i r a b l e t o evaluate the l o s t work which accompanies the discharge of a stream i n t o a r e f e r e n c e environment or a l t e r n a t i v e l y the maximum work which can be obtained i f the stream i s brought r e v e r s i b l y t o the s t a t e of the r e f e r e n c e environment. The f u n c t i o n "Essergy" equation (4), d e r i v e d and named by Evans (4) leads t o such e v a l u a t i o n s E = D +

P o

V - T S - Ey Q

c > c

N

(4)

c

(see nomenclature). Equation (4) r e q u i r e s i n a d d i t i o n t o the knowledge o f T , the knowledge o f a r e f e r e n c e pressure p and reference compositions [ x ] . When equation (4) i s a p p l i e d to a flowing stream equation (5) f o r the "Flow Essergy" i s obtained Q

Q

cQ

E

f

= H - T S - Sy N c * G

(5)

c

There seems t o be an agreement t o name equation (5) "Exergy". The l o s t work ( c a l l e d a l s o d i s s i p a t i o n ) , i n a system o r a subsystem may a l s o be computed from an "exergy balance" according to equation (6). c r

T S =EE 0Q

f

f

W

- E E +EEQ -EEQ +EE* - E E -dE _ ,/dt in out in out i n out stored

(6)

In u s i n g equation (6) work f l u x i s taken as equal t o exergie f l u x . Heat f l u x i s m u l t i p l i e d by the Carnot r a t i o t o o b t a i n i t s exergie flux, i . e . Q

E =Q(T-T )/T

(6a)

Q

The d i s s i p a t i o n i n a p l a n t i s equal t o the sum o f t h e d i s s i p a t i o n s i n the separate zones of the p l a n t . Therefore, i f the t o t a l d i s s i p a t i o n i n the p l a n t i s , T S , i t may be decomposed i n t o : c r

Q

TS 0

c r

=Ea k

k

(6b)

k

where, 0*^, represents t h e d i s s i p a t i o n i n zone k. The C o s t i n g of Exergies and D i s s i p a t i o n s There a r e mainly two reasons f o r a t t a c h i n g a p r i c e t a g t o an exergy or to a d i s s i p a t i o n somewhere i n the system whether the c r i t e r i o n i s an energy e f f i c i e n c y c r i t e r i o n such as equation (1) or a cost c r i t e r i o n such as equation (2). The reasons a r e : • To o b t a i n a d i s t r i b u t i o n throughout the system o f the items i n v o l v e d i n the c r i t e r i o n f u n c t i o n


218



•

To change the d i s t r i b u t i o n i n a d i r e c t i o n extremizing the c r i t e r i o n f u n c t i o n The f i r s t reason r e s u l t s i n a s e t of d i r e c t p r i c e s [ y ] . The second reason r e s u l t s i n a s e t of d i f f e r e n t i a l p r i c e s [6]. The two kinds of p r i c e s permit two d i f f e r e n t modes of a n a l y s i s : Thermoeconomic accounting using a l g e b r a i c a l l y determined p r i c e s p e r m i t t i n g comparisons of subsystems and t h e i r c o s t i n g items as though they were r e l a t i v e l y independent. Thermoeconomic o p t i m i z a t i o n using d i f f e r e n t i a l l y derived p r i c e s , p e r m i t t i n g the a n a l y s i s of the system's l o c a l and g l o b a l responses to w e l l s p e c i f i e d s m a l l changes i n the s t a t e of the system, and l e a d i n g to s e n s i t i v i t y a n a l y s i s and o p t i m i z a t i o n techniques. Two subsets of the d i f f e r e n t i a l p r i c e s [6] are most conveni e n t i n d i r e c t i n g to improved values of the c r i t e r i o n f u n c t i o n : Marginal p r i c e s [0] and shadow p r i c e s [X]. A marginal p r i c e 0 i s the change of the c r i t e r i o n f u n c t i o n per u n i t change of a f r e e v a r i a b l e . A shadow p r i c e X i s the change per u n i t change of a dependent v a r i a b l e . Exergy and d i s s i p a t i o n p r i c e s f o l l o w by the chain r u l e . For example c o n s i d e r i n g the energy c r i t e r i o n , equation ( 1 ) , f o r a s i n g l e exergy input EI and a s i n g l e f i x e d produce EP, a l l d i r e c t p r i c e s are u n i t y , i . e . (7a) and the d i f f e r e n t i a l p r i c e of a d i s s i p a t i o n o*j due of a f r e e v a r i a b l e y, i s ej^-OEI/ay^/Oaj/By^

to a change

(7b)

Performance-Cost Modeling The computation of exergies and d i s s i p a t i o n s and t h e i r c o s t i n g r e q u i r e s a mathematical d e s c r i p t i o n of the system's performance and c o s t . A s u i t a b l e l e v e l of d e t a i l f o r the purpose of comparing a l t e r n a t i v e design concepts i s the d e s c r i p t i o n of the performance and the cost of each subsystem by o v e r a l l d e s c r i p t o r s . A more d e t a i l e d d e s c r i p t i o n may be achieved by s u b d i v i d i n g the subsystems i n t o s m a l l e r subsystems. E s s e n t i a l r e l a t i o n s d e s c r i b i n g each subsystem are o v e r a l l equations f o r mass balance, f o r energy balance, f o r performance, and f o r c o s t i n g i n terms of performance. P r e s e n t l y a v a i l a b l e cost trends (17) i n terms of c a p a c i t y parameters (e.g. area, mass r a t e , power, ...) are s u i t a b l e c o s t i n g equations to s t a r t w i t h . They may be implemented to i n c l u d e the i n f l u e n c e of v a r i a b l e s such as pressure, temperature or e f f i c i e n c y whenever s u f f i c i e n t data are a v a i l a b l e .



11.

EL-SAYED AND TRIBUS


219

Thermoeconomic Accounting Thermoeconomic accounting u s u a l l y i n v o l v e s a cost c r i t e r i o n . I n t h i s mode o f a n a l y s i s the e f f e c t o f the economic environment i s t r a c e d by f o l l o w i n g each input stream and a s s i g n i n g monetary values i n accordance w i t h the exergie content. F o r example, t h e pressure p o r t i o n o f t h e exergy may be followed from a pump t o the f r i c t i o n a l d i s s i p a t i o n i n a heat exchanger. The cost t o produce the f l u i d mechanical exergy a t the pump can be assigned t o the pressure drop process. T h i s assignment makes i t evident at what cost per u n i t o f pressure l o s s i t pays t o change the heat exchanger. The exergy i n f l u x w i t h stream " i " may be apportioned t o e i t h e r an e x i t i n g u s e f u l stream ( " j " ) , an e x i t i n g wasted stream ("£") o r a d i s s i p a t i o n process ("k"). I n the accounting mode the engineer simply s t u d i e s the process diagram and assigns values t o the x's i n equation (8):

The p r i c e s o f exergies [EP] and [ER] and o f d i s s i p a t i o n s [o] a r e obtained by m u l t i p l y i n g each input E I ^ by i t s boundary p r i c e and summing over a l l i n p u t s . Consider f o r s i m p l i c i t y a p l a n t w i t h only one input and one product stream, w i t h the o b j e c t i v e t o reduce the cost as much as p o s s i b l e . F o r such a system the objective function i s : $ =aEI+Ez -gEP k 0

(9)

k

a and 3 are t h e boundary p r i c e s f o r input and product e x e r g i e s . I f the exergie output i s f i x e d , EI, may be e l i m i n a t e d between (8) and (9) t o g i v e : $ = E (aa +z )+EaER£+(a-g) EP k % 0

k

k

(10)

On the other hand, i f the input were f i x e d , as i n r a t i o n i n g , the value o f EP would have been e l i m i n a t e d t o g i v e : Z

+

*o" (^+z ) ^BER +(a-B)EI (11) k % The f i r s t terms i n equations (10) and (11) a r e the cost o f a m o r t i z a t i o n and d i s s i p a t i o n i n each zone. I n (10) t h e d i s s i p a t i o n i s " p r i c e d " at the cost o f the e x t r a f u e l r e q u i r e d t o make up f o r t h e l o s s e s . I n (11) the p r i c e r e f l e c t s l o s t revenue through l o s s i n production a t f i x e d i n p u t . When there are s e v e r a l inputs and outputs, the assignment of the x's i n equation (8) may become a r b i t r a r y . When t h e p r i c e s per u n i t o f exergy i n the input streams are not too d i f f e r e n t , i t i s u s e f u l t o d e f i n e an average p r i c e a * and use t h i s average i n t h e c a l c u l a t i o n s . T h i s k i n d o f accounting allows the comparison o f t h e cost o f d i s s i p a t o r s w i t h the cost o f t h e i r dissipations* k

il



220


Another accounting approach i s proposed by G a g g i o l i (11) by which the cost of input energy and the cost of p r o c e s s i n g devices are a l l o c a t e d to a set of u s e f u l product exergies w i t h i n the system. The p r i c e of a d i s s i p a t i o n i s the p r i c e of i t s immediate exergy input. T h i s k i n d of accounting permits the comparison of a l t e r n a t i v e subsystems producing the same product exergy. The accounting approach i s the simplest to use and o f t e n r e v e a l s o p p o r t u n i t i e s f o r improvement which are not otherwise obvious. The accounting method, however, does not use knowledge of system behaviour. I t does not r e v e a l how a c a p i t a l investment i n one part of the p l a n t may a f f e c t the entropy generation i n another. In the accounting method, entropy generation i s a number not a f u n c t i o n . Thermoeconomic accounting i s demonstrated i n example 1. Thermoeconomic

Optimization

D i r e c t p r i c e s do not take i n t o account the e f f e c t a d e c i s i o n i n one part of a p l a n t may have on the i r r e v e r s i b i l i t i e s i n another. Marginal and shadow p r i c e s do t h i s but are more complicated to compute. They depend upon the system o f equations (and t h e i r f i r s t d e r i v a t i v e s w i t h respect to the v a r i a b l e s of i n t e r e s t ) rather than upon only the s t a t e s of v a r i o u s zones. The mathematic a l d e s c r i p t i o n of a thermodynamic process r e q u i r e s the s p e c i f i c a t i o n of a s e t of "equations of c o n s t r a i n t " , represented here by the s e t , [ f c ^ O ] . The thermodynamic performance and stream v a r i a b l e s are d i v i d e d i n t o two s e t s , s t a t e and d e c i s i o n v a r i a b l e s , represented by [x.] and [y^]* and each of the d e f i n i n g f u n c t i o n s , [ $ . ] , i s expressed i n terms of these v a r i a b l e s . I f the o b j e c t i v e f u n c t i o n , $ , (whether i t i s an energy o b j e c t i v e or a cost o b j e c t i v e ) i s s i m i l a r l y expressed, a Lagrangian may be d e f i n e d according t o : L= +EA.$. (12) j The way Lagrange's method of undetermined m u l t i p l i e r s i s i n t e r preted here i s not conventional. The approach i s described i n Appendix A. To guarantee that L i s independent o f the s e t [ X J ] , set: 3

3

3L/3x =EA b ..+b j=0 j

i

i

o

(13)

The s o l u t i o n of the s e t of l i n e a r equations represented by (13) determines the shadow p r i c e s . In terms of these p r i c e s the marginal p r i c e s are given by: e =3L/3y =EX k

k

i a i k +

When the f u n c t i o n s $ d e f i n e d by 6 =0.

Q

a

o k

and

(14) [$.] are a n a l y t i c , the optimum i s

k



11. EL-SAYED AND TRIBUS

221

As proposed by Fax and M i l l s ( 6 ) , i t i s e a s i e r t o keep t r a c k of the v a r i o u s d e r i v a t i v e s i f they a r e put i n t o a m a t r i x as shown i n Table ( 1 ) . I n t h e general case, the shadow and marginal p r i c e s may be found from:


^ k ^ ^ o k i - ^ J ^ i j i "

1

^ ^

(

1

5

)

I f the b a s i s matrix can be d i a g o n a l i z e d , the marginal and shadow p r i c e s may be found i n a d i r e c t way. The matrices i n d i c a t e d i n equation (15) a r e known as Jacobians. The p r o p e r t i e s of Jacobians have been o f i n t e r e s t to thermodynamicists f o r many years and may be used t o f i n d an i n t e r e s t i n g i n t e r p r e t a t i o n o f the meanings of t h e shadow p r i c e s . Using the methods described i n (7., 8), d e f i n e :

J[A]=J

L l -- Vl x

#

, X

i

i+l -'- nJ

, x

f

x

*n]

J[B]=J

X-....

. J[C]=J

,

1

x n

1'

jv

11

I

JJ

$ ,$ | n o I V kJ

(16)

!

y

With these d e f i n i t i o n s , t h e shadow and marginal p r i c e s a r e found to be:

v-JM/JIBW

{ftl"!;i;r.lj=-< w w ^ - o ] *~T^J=< V V[yo] 3

3

(

i

7

)

(18)

When t h e s t a t e v a r i a b l e s s a t i s f y t h e equations o f c o n s t r a i n t , the Lagrangian i s equal t o t h e o b j e c t i v e f u n c t i o n . Since the Lagrangian i s then independent of the s t a t e v a r i a b l e s , there a r e only t h r e e ways i n which t h e o b j e c t i v e f u n c t i o n can be changed: 1. Change a d e c i s i o n v a r i a b l e ( i . e . , change t h e o p e r a t i n g point) 2. A l t e r a c o n s t r a i n t 3. Change something i n the r e f e r e n c e environments ( p h y s i c a l or economic)


222


Table 1.

System M a t r i x (n+1) (n+f) Derivatives 3$i/3y =a • • Xj • • • • • Yk


k

*1

b .. •

•• b y • • •

*i

b ..

•• ij

n

± 1

*n

D

•• ln

b

b

b

b

n l "

• * nj

b

o l "

•

in

b

a

ll--

a

io--

nn

a .. l k

a

i k v

f

. .a

l f

a

•• i k - *

'* if

•• ^ k ' *

*

# a

nf

Objective *0

,

b

b

j

0

a

o n

o l "

a

•• ok*• •

# a

of

For f i x e d environments, we have: S$ =£ 0 $ / 3 y ) d y + i : ( 3 $ /3$ ) 6$ k j But i n view of the previous equations,

(19)

6$ =Ee,dy.-SX.6$. ° k j Not ing that $± =$ . -x., we J 3

(20)

0

Q

k

k

3

3

1

have:

6$ =i:e,dy,+i:x,6x, (21) k j We have found t h a t shadow p r i c e s are e a s i e r to i n t e r p r e t i f they are expressed as m u l t i p l e s of exergy p r i c e s by u s i n g the equation: fc

A

j

=

V

x

j

/

3

8

E

J

( 2 2 )

j

Marginal p r i c e s are more i n f o r m a t i v e i f expressed i n non-dimens i o n a l form, 0^, u s i n g : < W k

/

$

( 2 3 )

o

Once the s e t s of p r i c e s , [ 6 ] , [X^], are kitown, the p r i c e s assoc i a t e d w i t h a change i n an exergy or a d i s s i p a t i o n may be obtained from the corresponding 6 or X i by the c h a i n r u l e . I f , f o r example, an exergy stream E , has y and x-^ as v a r i a b l e s , k

k

g

k


11.

EL-SAYED AND TRIBUS


then the p r i c e s a s s o c i a t e d w i t h the changes of E

s

223

through the

(24)


(25) Both modes o f thermoeconomic a n a l y s i s , accounting and o p t i m i z a t i o n , are i l l u s t r a t e d i n example 2, c o n s i d e r i n g a cost c r i t e r i o n f u n c t i o n . The case o f an energy c r i t e r i o n f u n c t i o n may be i n c l u d e d as a s p e c i a l case. The D e r i v a t i o n o f c o s t i n g equations When the cost of a device i s best evaluated i n terms o f i t s geometrical v a r i a b l e s , c o s t i n g equations can be d e r i v e d i n terms of e s s e n t i a l performance v a r i a b l e s determining the performance of t h e device i n a system. Geometrical and performance v a r i a b l e s are g e n e r a l l y r e l a t e d by design r e l a t i o n s . The v a r i a b l e s and the design r e l a t i o n s determine together t h e design degrees o f freedom. Three cases a r i s e 1) E s s e n t i a l performance v a r i a b l e s determining the p e r f o r mance of the device i n a system a r e equal t o the design degrees of freedom. In t h i s case the problem i s a problem of exchange of v a r i a b l e s . 2) E s s e n t i a l performance v a r i a b l e s a r e l e s s than the design degrees of freedom. In t h i s case, t h e cost i s minimized w i t h respect t o the excess degrees o f freedom and the envelope o f minimum c o s t s i s the c o s t i n g equation. 3) E s s e n t i a l performance v a r i a b l e s a r e more than the design degrees o f freedom. I n t h i s case a d d i t i o n a l c o n s t r a i n t s a r e imposed on the e s s e n t i a l performance v a r i a b l e s . Since the second case a r i s e s o f t e n i n p r a c t i c e , i t i s t r e a t e d i n the t h i r d of t h e example problems. A Thermoeconomic Strategy In conceptual designs the concern i s as much w i t h c r e a t i v i t y as i t i s w i t h a n a l y s i s . C r e a t i v e approaches a r e apt t o introduce s i g n i f i c a n t changes i n flow c h a r t s , thereby a l t e r i n g the analyses. As options a r e discovered o r created, changes i n plans occur. Therefore a s t r a t e g y , r a t h e r than a p l a n i s r e q u i r e d . F i g u r e 1 presents the flow chart f o r a s t r a t e g y which conforms t o the general o u t l i n e o f a design process as described by Rosenstein (9). To s t a r t t h e process, t h e engineer d e f i n e s as a c c u r a t e l y as p o s s i b l e the i n p u t s , outputs and energy and cost o b j e c t i v e f u n c t i o n s . Then the r e f e r e n c e environments, both p h y s i c a l and economical a r e d e f i n e d . These a r e not n e c e s s a r i l y s t r a i g h t forward s t e p s , as f o r example when d e a l i n g w i t h high r a t e s of



Conceptual

Design

Detailed

Define purpose

Design

Optimum s p e c i f i c a t i o n s

1 Define Environments Physical ( P , T {u ,o}) Economic ({cf},{c },{c }) 0

0

p


Design hardware

c

z

Define system Energy c r i t e r i o n *o,e < >y) Cost c r i t e r i o n *o,c ( »y) Mathematical model j (x,y)=o x

x

Detailed cost estimations P r o f i t a b i l i t y analysis

Find a solution {y} — » {x} — » * o>e, o.c Accounting Exergies {E}, Dissipations {o} Direct prices {v} Processing devices {Z}

No^-Go to start

Optimizing Shadow prices {X} Marginal prices {6} Accounting d i s t r i b u t i o n s

Yes-t-Go to start

Figure 1.

System matrix (n+1) (n+f).



II.

EL-SAYED AND TRIBUS


225

i n f l a t i o n o r d e f i n i n g t h e chemical p o t e n t i a l o f a r a r e substance i n t h e environment. The next step i s t o develop a mathematical d e s c r i p t i o n o f the system. From the p e r s p e c t i v e o f mathematical a n a l y s i s , t h e set o f equations d e s c r i b i n g t h e c o n s t r a i n t s i s the system. An a l g o r i t h m which permits the computation o f the performance o f the system ( i . e . , s o l v e s the equations of c o n s t r a i n t simultaneously) i s a means t o s a t i s f a c t o r y s o l u t i o n s . If the d e s c r i p t i o n of the system requires m v a r i a b l e s , and there are n equations o f c o n s t r a i n t , there w i l l be (m-n) degrees of freedom. The designer may choose these as " d e c i s i o n v a r i a b l e s " . The remaining n v a r i a b l e s a r e c a l l e d " s t a t e v a r i a b l e s " and a r e found u s i n g t h e algorithm. As i n d i c a t e d i n t h e s t r a t e g y , a f t e r the system has been d e f i n e d , i t i s necessary t o f i n d a t l e a s t one f e a s i b l e s o l u t i o n to the s e t of equations, i . e . , t o demonstrate that the p l a n t has a t l e a s t one o p e r a t i n g p o i n t . F o r that o p e r a t i n g p o i n t the energy and cost o b j e c t i v e f u n c t i o n s may be computed. According t o t h e s t r a t e g y , accounting methods a r e f i r s t employed i n t r a c i n g the exergie flows from the v a r i o u s i n l e t s t o the d i s s i p a t i v e processes. T h i s step leads t o a comparison between t h e c a p i t a l costs and exergy d e s t r u c t i o n costs i n each zone. While such a p i c t u r e does not r e v e a l t h e system's behaviour, and t h e r e f o r e i s not always accurate (as f o r example when a d i s s i p a t i v e process merely gets d i s p l a c e d from one part o f the system to another), n e v e r t h e l e s s , because t h i s i s the l e a s t c o s t l y form of a n a l y s i s and i t o f t e n suggests f r u i t f u l avenues t o pursue, from a s t r a t e g i c p e r s p e c t i v e , i t i s the most a t t r a c t i v e f i r s t step. I t may a l s o be the only p o s s i b l e a n a l y s i s i n the absence of s u f f i c i e n t data. See the f i r s t example a t the end o f t h i s paper f o r a case i n which t h e a n a l y s i s r e v e a l s a u s e f u l set o f o p p o r t u n i t i e s . Other examples are a v a i l a b l e . Based upon t h i s a n a l y s i s the designer may decide e i t h e r t o change t h e c o n f i g u r a t i o n o r t o change t h e operating p o i n t . As i n d i c a t e d i n the s t r a t e g y , i f the flow chart i s not t o be changed, the next step i s t o compute the shadow, [X^], and marginal, [9^]* p r i c e s . The marginal p r i c e s form a v e c t o r which points t o d i r e c t i o n of an improvement i n the e x i s t i n g design by changing the d e c i s i o n v a r i a b l e s . (This step i s e s s e n t i a l l y the same as conventional optimization.) A l t e r n a t i v e l y , the designer may use t h e shadow p r i c e s t o i n v e s t i g a t e , approximately, the b e n e f i t o f a sought c o n f i g u r a t i o n m o d i f i c a t i o n . In each i t e r a t i o n w i t h a new marginal p r i c e v e c t o r , t h e d i s t r i b u t i o n s by thermoeconomic accounting are always a c c e s s i b l e i f needed. These elements o f t h e s t r a t e g y and t h e technique o f developing t h e c o s t i n g equations a r e i l l u s t r a t e d i n t h e f o l l o w i n g three examples.




226

EXAMPLE 1; Manufacture of an I n d u s t r i a l Chemical At the time the a n a l y s i s was undertaken, i n f o r m a t i o n on the system performance was a v a i l a b l e f o r only one o p e r a t i n g p o i n t and cost i n f o r m a t i o n c o n s i s t e d of i n i t i a l i n s t a l l a t i o n costs plus estimates of l i f e times. Equipment costs were not r e l a t e d t o o p e r a t i n g c o n d i t i o n s . Under these circumstances only the accounting approach could be used. The a v a i l a b l e i n f o r m a t i o n i s summarized i n Table I I . (For more d e t a i l s , see r e f e r e n c e 10) The exergy flow diagram, Figure 2, shows the r e s u l t s of the exergy balances on the v a r i o u s zones of the system. The r e f e r e n c e s t a t e of zero exergy was taken as P =1 atm, T =528 R, mass f r a c t i o n water=0.3, mass f r a c t i o n sodium palmitate=0.7. The feed c o n d i t i o n i s a convenient r e f e r e n c e s t a t e to take when the a n a l y s t i s not i n t e r e s t e d i n p o t e n t i a l uses of the product i n other environments. The e v a l u a t i o n of the x's i n equation (8) i s r e l a t i v e l y simple s i n c e a l l exergy inputs by heat may be assumed t o be converted to f l u i d thermal exergies or to be d i s s i p a t e d i n entropy generation. There i s , however, some ambiguity i n the f l a s h tank where the input mechanical and thermal exergies are converted t o chemical exergy, thermal exergy and d i s s i p a t i o n . Since the costs o f the input exergies are not too d i f f e r e n t from one another, i t i s convenient and reasonable t o a s s i g n an average p r i c e to the d i s s i p a t i o n and consider that the input exergies c o n t r i b u t e i n p r o p o r t i o n t o t h e i r presence. The r e s u l t i n g d i r e c t p r i c e s , [ y ] , d i s s i p a t i o n v a l u e s , [y.O] and a m o r t i z a t i o n s , [ z ] , are given i n Table III. The data suggests that improvements should be made as follows: S u b s t i t u t e a b o i l i n g type heat exchanger (new zone 3) i n p l a c e of the separate heater and f l a s h tank (zones 3 and 4). Use the l a t e n t heat of the exhaust steam i n the preheater (zone 1 ) . S u b s t i t u t e o r d i n a r y c o o l i n g water at 68 F f o r the b r i n e i n zone 7, u s i n g a l a r g e r heat exchanger w i t h much greater a g i t a t i o n . Table I I I presents the r e s u l t s o f the a n a l y s i s and a comparison of three cases: The p e r f e c t l y r e v e r s i b l e process The o r i g i n a l system The modified system Q

Q

9

EXAMPLE 2: A Simple Open C y c l e Gas Turbine T h i s example t r e a t s a simple open-cycle gas which the cost o b j e c t i v e f u n c t i o n , equations of c o s t i n g equations are a l l a v a i l a b l e i n a n a l y t i c 3 shows these f u n c t i o n s along w i t h the f i x e d d e c i s i o n v a r i a b l e s . Since the s e t of equations

turbine for c o n s t r a i n t and form. F i g u r e and v a r i a b l e i s diagonalized,


11.

EL-SAYED AND TRIBUS


Available

Table I I .

2


F

l

F

2

P

l

R

l Internal S

S

3

S

4 5

s

68

0

180

4500

0

158

0.22

520

0

217

1.0

30

-

0.3

180 182

5000 570

2

S

S

35

20

5000

l

0.3

5000

0.3

5000 350

353

0.3

4480

0

212

0.22

4480

90

212

0.22

4500

50

212

0.22

w

2

3

w

4

Q

l

Q

2

J

J

3

J

4

6

Pump

Preheater

14/18F Brine E l e c t .

p

l -

7 Chiller

9.0

Z

0.2

20

2

15

3.5

14

46

0.2

25

12

0.0

20

13

0.3

14

45

7.5

14

18

0.5

14

Z

Z

Z

3 4 5 6 7

4.27

212

0.34

212

5.9

450 p s i g s t .

S

3

2 -

Flash Tank

Heater

Elect.

Elect.

iw,

4 —

-

20

2.25

212-217

Elect.

150 p s i g s t .

Z

-92.4

180

J

6.0

183.0

14-18

Q

3 Reject i

Z

151.0

460

2

Variables

i

z

0.67

365

Cost

C a p i t a l Maintenance L i f e k$ k$/yr yrs

4.5

-

l

w W

Zones

Rate kw

T F

X

External

data, example 1

Thermo. Var.

Thermodynamic Variables Energy Streams P T m H 0 F lb/hr psig (mass)

Matter Streams

s

k- 6-l

6 Mixer

hi J

"5 Pump

3i

Input Energy P r i c e s 150 p s i g steam 450 p s i g steam E l e c t r i c a l energy 14/18 F Brine

2.7 $/I000 lb 3.05 $ / l 0 0 0 lb 4.37 C/kwhr 1.9 $/kwhr c o o l i n g

227

load



228

t

i

54.5 4. 5+

-0.735 2.245 16.91 0.0 2.38 , j 0.0 ; Pump * 0 .39

1

28.41 0.92 0.0 ! Heater!

66 45 1 47 0 0

Components

S 10.1 6.pJ_

0V67

34.5

4.86

25.3 9.19 0.0

6.26 0.0 0.26

21.47 0.19 0.36 T

[Chiller-

-1.29 6.15 0.0

21.47 0.34 0.26

Mixer ^ 1.285

Figure 2.

Table I I I .

W

l

W

2

W

3

W

4

Q

l

Q

2

Q

3

F

l

F

2

4.5 .67

kw

C/kw hr

4.37

4.5

4.37

4.37

0.67 4.37

6.0

4.37

6.0

4.37

4.37

9.0

4.37

54.5

3.0

1.26 3.00

78

3.2

55.86 3.00

^ 0 ^ 0

-

0

-

\

-0.144 0.33 0.0

i J Pump |

21.73 21.47 0.0 0.26 0.928

0

-

Modified System

Current System

@ {a]

17.4

kw

Exergy flow diagram, example 1.

Modified

9.0

10.1

Thermal Mechanical Chemical

D i s s i p a t i o n vs. a m o r t i z a t i o n , example 1

{EI}

{EI}

Stream kw C/kwhr

34.33 0.0 1.25

i JFl.Tank!

,Q

.W

Current System

9.74 1.47 -1.51

0.186

22.07

.6.52

35.57

67 93 ,

19.29

16.56 0.12 0.0

i ; Pre- | heater"

9.7

29.33

16.68^ 37.9 0.025 0.0

0.07 0.15 0.0

V 78

1.5

0.15


•

;

37.93

{a, E P

,

kw

a

l

°2 0

3

°4 °5 °6 °7 ER

X

E J

1

E J

2 3

E J

EJ

4

EJ

5

C/kw hr

37. 93

vs{z}

EJ}@{Y]

C/hr

3.00 114

{a,... {ya,. . }vs{z}

20

1. 5

4.37

6.56 21.42

29. 33

3.24

94.94 36.8

9. 7

3.39

32.88 12

«/hr

kw

For example, a heat exchanger w i t h n=0.6, A P U Q ^ O . 0 5 atm, A cold * atm, should not cost more than 211 k$/yr. Examination o f c o s t i n g equation r e v e a l s t h a t exchangers i n t h i s range should cost about 3.6 k$/yr., so that there i s obviously an opportunity t o improve t h e system through a d d i t i o n of a heat exchanger. The r e d i s t r i b u t i o n of cost given i n Table V I shows t h a t t h e pressure l o s s penalty i s high on the hot p

= 0

0 5


230


i»f (Feed) Fixed decision variables {y^} = {W , c , e , P ^ Q

f

f

3

comb.

hr,Y , R, c } Variable decision variables {y } = {r^, r , n , P , m } p

R

2

3

3

z

2

a

Basis state variables {x.} = {P„, T„, W , VL 3 i 3 c t

a tC

"O

n


Thermodynamic Constraints P

2

= P /r

T

2

=T (l+ ((P /P )

3

=

2

x

2

( y _ 1 ) / y

-

1

l)/

n]L

> = *

f 2

4 Exhaust (Reject)

A i r at P , T

W = W + W t 0 c P = P

1

1

(Feed)

1

V VW! '-"^!)""""' EI = n . c ( T a

p

Costing

3

T j )

Constraint

z,+z +z. = 2.0 m 0

1

a

+ 14 m

a

Objective

(0.9 - n ) " P / P J l n P / P + 1.4 m 1

2

1

2

1

1

(.92 - n )' (£nP /P )(l 3

3

(0.995 - r )

a

+ 5exp(.02T

l4

_ 1

2

3

(l

+ 5exp(.01T - 28)) 3

- 56))

= *

g

Function

Figure 3. Direct Prices iy)

Problem d e f i n i t i o n , example 2, Marginal Prices

Shadow Prices 1

= a

h \ -

a 3/3T

3

X 3/3P •5 4 = 3/3V^ l4

(z (

z X * ')

2 +

X

3 +

7

7

6 * W

X

A

3

X

2

X

l

8

1

9

2

e

3

6

4

9

5

l

=

3/3^ ( z + X * )

=

3/3r

=

3/3n

1

3/3m

a

3/3W

=

3/3Tc 2

(X

3

3

+ x

=

2

3/3P =

2

(z +X * «)

2

(z X « ' X * ') 3 +

(z

i +

1

z

1

2 +

z

+

3 +

6

6

X * ' X « 3

3

+

6

6

)

,)

7

2

{y }- Wo=2500 Btu/sec. c = $36/Barrel, e =16000 Btu/lb, P =latm, T =520R, hr= hrs/yr =.243, R=0.07 Btu/lbR, y=l.H f

F

{0 } k

+ 0.28

f

-0.584

.6 -502 < k>/VJo 25 lb/sec 0.8 e

+7

1

1

-0.375

-0.168

-1.72

-274

-11.6

-1392

0.94

10 atm

0.85

?1

{y } F

200 $/yr(Btu/sec) 2.37 )

.

13, 14, 15,

1

0

0

0

XlSOD - ^ -^ -

7 5

0

^) -

9 7

1

*!;- !!^

3'r,-design, geometric, rate c,m,a=cost, manufacture, amortization i = l , 2 , . . . I feeds j = l , 2 , . . . n constraints, basis variables j = l , 2 , . . . J products k = l , 2 , . . . K zones k = l , 2 , . . . F degrees of freedom=number of decision variables

Acknowle dgmen t s The work described i n this report was funded by a grant from the Department of Energy, whose support is gratefully acknowledged. Literature Cited 1.

2. 3.

Gyftopoulos, E l i a s , Lazaridis, Lozaros J. and Widner, Thomas, Potential Fuel Effectiveness i n Industry, Ballinger Publishing Company, Cambridge, MA 1974 Tribus, M, Thermostatistics and Thermodynamics, D. van Nostrand C o . , Inc. 1961. pp 605, 607 De-Nevers, Joel and Seader, J. D. "Lost Work: A Measure of Thermodynamic Efficiency", Energy, V o l . 5, No. 8-9, AugustSept. 1980, pp 7859-769


11. EL-SAYHD AND TRIBUS 4.

5.

6.


7. 8.

9. 10.

11.

12. 13. 14. 15. 16. 17.


Evans, R. B . , "A Proof that Essergy is the Only Consistent Measure of Potential Work for Chemical Systems", PhD thesis, Dartmouth College, NH, 1969 Tribus, M . , El-Sayed, Y. "Thermoeconomic of Energy Systems", Final Report, DoE Agreement, DE-AC0-79ER10518.A000, Center for Advanced Engineering Study, MIT, Cambridge, MA 1982 Fax, D. H. and M i l l s , R. R . , "Generalized Optimal Heat Exchanger Design", Trans. ASME, A p r i l 1957, p 653 reference 2, pp 456-460 Tribus, M. and El-Sayed, Y., Progress Report, DoE Grant, DoE Agreement, DE-AC02-79ER10518.A000, June 1980, MIT, Cambridge, MA Rosenstein, A. B . , Rathbone, R. R . , Schneerer, W. F., "Engineering Communications", Prentice H a l l , 1964 Tribus, M. and El-Sayed, Y., "Thermoeconomic Analysis of an Industrial Process", report, Center for Advanced Engineering Study, MIT, Cambridge, MA Gaggioli, R., "A Thermodynamic-Economic Analysis of the Synthane Process", Final Report for US DoE, Marquette University, Milwaukee, Wisconsin, Nov. 1978 Evans, R. B . , Tribus, M . , Thermoeconomics of Saline Water Conversion", I&EC Process Design and Development, V o l . 4, A p r i l , 1965, pp 195-206 Peters, M . , Timmerhaous, K . , "Plant Design and Economics for Chemical Engineers", McGraw Hill, 1968, Chapter 14 Welty, J . R . , Engineering Heat Transfer, Wiley International, 1974 Popper, H., "Modern Cost-Engineering Techniques", McGraw Hill, 1970 Garrison, C. M. and Steinmetz, F. J., "Energy Optimization of Interchanges", Heat Transfer--Orlando, 1980, pp 301-309, AICHE Symposium Series. V o l . 76, No. 199 Humphreys, Kenneth and K a t e l l , Sidney, "Basic Cost Engineering", Marcel Dekker, 1981

Appendix A Lagrange's Method Revisited The mathematical description of a system is determined by a set of equations of constraint, [$j=0], j = l , 2 , 3 , . . . n The expressions, $j, are functions of two sets of variables: Decision variables, y^, which, within l i m i t s , may be set by the designer, and State variables, X j , which are determined by the equations of constraint, once the decision variables have been chosen. If the total number of variables is m, the number of equations of constraint is n, the degrees of freedom w i l l be f=m-n, equal to the number of decision variables. If the performance of the system is defined by an "objective function", denoted by $ , the task of the designer may be stated as follows: Q


237

238

SECOND LAW ANALYSIS OF PROCESSES Extremize $

Q


Subject to

-*

Efficiency and Costing - American Chemical Society

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