Efficiency and Costing - American Chemical Society

15

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on April 18, 2016 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch015

Multiobjective Optimal Synthesis

L. T. FAN and J. H. SHIEH Department of Chemical Engineering, Kansas State University, Manhattan, KS 66506

Conventionally, a food and/or chemical process system has been designed optimally by considering only one objective of economic efficiency (profit or cost), although i t is often d i f f i c u l t or even impossible to express variables and parameters associated with such a system in a monetary unit. For example, the available energy is invaluable to a nation where the energy resources are seriously depleted, because the available energy once lost in a process cannot be recovered by any means; yet the cost of energy resources can be unreasonably low because of the a r t i f i c i a l or manipulated world market condition. Furthermore, the excessive loss of available energy may result in severe pollution, which in turn, may lead to destruction of the environment or of human life. Again it is d i f f i c u l t to assign monetary values to such destruction. It is therefore, natural that the concept of multiobjective analysis should be introduced in synthesizing a chemical or biochemical process system. This paper discusses the basic concepts and terminologies of a multiobjective problem and reviews methods for solving it. The methods are illustrated with an example of the milk evaporation process.

A chemical or biochemical process system has been designed conventionally by considering only one objective function of economic efficiency (profit or cost). However, many of the objectives are often d i f f i c u l t or even impossible to express in a common monetary unit. It i s , thus, natural that the concept of multiobjective optimization be introduced in synthesizing a chemical or biochemical process system.

0097-6156/ 83/ 0235-0307S07.25/ 0 © 1983 American Chemical Society Gaggioli; Efficiency and Costing ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

308

SECOND LAW ANALYSIS OF PROCESSES

A m u l t i o b j e c t i v e o p t i m i z a t i o n problem can be mathematically s t a t e d as: Optimize (minimize or maximize) ^(20

f (x)


2

= [ f ( x ) ] , i = 1, 2, ±

subject

...

(1)

to x e X

gj « to tO U)WOtice that 0)^T.TO,1 ao Q of o>« 0) . m u l t iJp-i p l i e— rL 11 0

?

^

Q

R

»

K

O

T T

"'*

i s the

ratio

fooranoD

For s i m p l i c i t y , suppose that u>^ and u> are such that 2

u>

= 5.972

12

i n the s c a l a r o p t i m i z a t i o n problem corresponding to the example considered i n the preceding s e c t i o n . M i n i m i z a t i o n of J„ over x can be accomplished by any c l a s s i c a l o p t i m i z a t i o n technique. For convenience, the adaptive random search technique (26) i s used here. The optimal s o l u t i o n obtained i s l o c a t e d at x,

= V

= 0.0475 (39)

a

Q 1

=

1.0

where 26.48 9.1

kcal/s

(40)

and f

0

= 2.91

Hi-

Notice that the optimal f- and f ~ are e s s e n t i a l l y i d e n t i c a l to those obtained i n the m u l t i o b j e c t i v e optimizaton problem. In f a c t , i t i s known (9) t h a t , f o r the system with a convex f e a s i b l e region, "12

T

- 12

u

(41

- 12

>

The value of u>^ s e l e c t e d here f o r i l l u s t r a t i o n i s equal to T^ which i s the i n v e r s e of T _ or 0.167435. The f e a s i b l e r e g i o n of the system under c o n s i d e r a t i o n i s not e x a c t l y convex; however, the r e l a t i o n s h i p given i n Equation 41 i s apparently s a t i s f i e d . 2

2

2

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

15.

F A N A N D SHIEH

323


As can be seen from the values o f f ^ and f« i n Equation 40 and F i g u r e 4, the optimal s c a l a r o b j e c t i v e f u n c t i o n i s l o c a t e d on the t r a d e - o f f curve. In other words, as long as Equation 41 i s s a t i s f i e d , the search f o r the s c a l a r o b j e c t i v e f u n c t i o n , 3^* need be c a r r i e d out only along the t r a d e - o f f curve. Thus, we see that under the optimal c o n d i t i o n the appropriate cost o f the d i s s i p a t i o n o f a v a i l a b l e energy, o>^, and that o f the evaporator, u>2 should be such that t h i s r a t i o , w-^* equal to r e v e r s i b l y , the value of be equal to Downloaded by UNIV OF CALIFORNIA SAN DIEGO on April 18, 2016 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch015

D

e

o

r

Nomenclature 2 A

contact area o f the evaporator, m

BR1

a v a i l a b l e energy to the evaporation compressor 1, k c a l

system through

BR2

a v a i l a b l e energy to the evaporation compressor 2, k c a l

system through

E

b o i l i n g point elevation, K

e

allowable

_f

objective function vector

1?

o b j e c t i v e f u n c t i o n v e c t o r defined as i^2

f^

s c a l a r f u n c t i o n a s s o c i a t e d with the i - t h subsystem

£

constraint function vector

g

l e v e l v e c t o r f o r the o b j e c t i v e f u n c t i o n v e c t o r , f^

y

i

••••»

v e c t o r f u n c t i o n a s s o c i a t e d with the i - t h subsystem

J

scalar objective function

vl

vector objective function j - t h objective function

Mj. . mass flow r a t e o f the aqueous milk s o l u t i o n a t the i n l e t , kg/s t , X

Mf , e

M

mass flow r a t e o f the aqueous milk s o l u t i o n at the e x i t , kg/s

- mass flow r a t e o f the vapor, kg/s s, J. Continued on next page


324


M. M


M

. marginal r a t e of s u b s t i t u t i o n of f . f o r f .

. mass flow r a t e of steam at the i n l e t , s, 1 s ,e

mass flow r a t e of condensate at the e x i t ,

P

pressure i n the evaporator,

T

temperature exit, K

T

. temperature r, 1

f

T.

kg/s

atm

of the aqueous m i l k s o l u t i o n and vapor at the

of the aqueous milk s o l u t i o n at the i n l e t ,

K

. t r a d e - o f f r a t i o between the i - t h and j - t h o b j e c t i v e s

>3

1

T

kg/s

. temperature s ,1

T

temperature

of steam at the i n l e t ,

K

of steam at the e x i t , K

s ,e U

2 heat t r a n s f e r c o e f f i c i e n t , k c a l / s , m , K

i< w

weighting c o e f f i c i e n t v e c t o r d e f i n e d as [w^, w^,

w^

i - t h component of the weighting c o e f f i c i e n t v e c t o r , w

w^.

surrogate worth f u n c t i o n a s s o c i a t e d with the i - t h and j - t h objectives

x

decision vector

w^]

T

x^ ^ c o n c e n t r a t i o n of the m i l k i n the aqueous s o l u t i o n at the ' i n l e t i n mole f r a c t i o n x^ '

c o n c e n t r a t i o n of the m i l k i n the aqueous s o l u t i o n at the e x i t i n mole f r a c t i o n

Greek Symbols 3

s p e c i f i c enthalpy measured i n r e f e r e n c e to the dead s t a t e , kcal/kg

e 8

,

1

- s p e c i f i c exergy f o r the steam generated i n the evaporator, kcal/kg

e

s p e c i f i c exergy, k c a l / k g

X

Lagrangian

_X

Lagrangian m u l t i p l i e r v e c t o r

multiplier


15. OL^

FAN AND SHIEH


325

structural parameter from unit 1 to unit 2 structural parameter from unit 1 to unit 1

a

n i

structural parameter from unit 1 to unit 0

Literature Cited


1.

2.

3.

4. 5.

6. 7. 8. 9.

10.

11. 12.

13. 14.

Kuhn, H.W.; Tucker, A.W., "Nonlinear Programming"; Proc. of the Second Berkeley Symposium on Mathematics and Probabili t y : Univ. of California Press, Berkeley, Ca, 1951; pp. 481-692. Cohon, J.L.; Marks, D . H . , "A Review and Evaluation of Multi-objective Programming Techniques"; Water Resources Research 1975, 11, 521. Huang, S . C . , "Note on the Mean-Square Strategy for Vector Valued Objective Functions"; Journal of Optimization Theory Applications 1972, 9, 364. Zadeh, L.A., "Optimality and Non-Scalar Valued Performance Criteria"; IEEE Transactions 1963, AC8, 59. Reid, R.W.; Vemure, V., "On the Non-Inferior Index Approach to Large Scale M u l t i - c r i t e r i a Systems"; Journal of the Franklin Institute 1971, 291, 4, 241. Charnes, A; Cooper, W.W., "Management Models and Industrial Application of Linear Programming"; John Wiley: NY, 961. Major, D . C . , "Benefit-Cost Ratios for Projects in Multiple Objectives Investment Programs"; Water Resources Research 1969, 5, 1174. Lee, S.M., "Goal Programming for Decision Analysis"; Auervach: Philadelphia, 1972. Haimes, Y . Y . ; H a l l , W.A., "Multiobjectives in Water Resources Systems Analysis: The Surrogate Worth Trade-Off Method"; Water Resources Research 1974, 10, 615. Cohon, J.L.; Marks, D . H . , "Multiobjective Screening Models and Water Resource Investment"; Water Resources Research 1973, 9, 521. Roy, B . , "Problems and Methods with Multiple Objective Functions"; Mathematical Programming 1971, 1, 50. Haimes, Y . Y . , et al., "Multiobjective Optimization in Water Resources Systems: The Surrogate Worth Trade-Off Method"; Elsevier Scientific Publishing: Amsterdam, 1975. Nakayama, H . ; Sawaragi, Y., "Decision Making with Multiple Objectives and its Applications (in Japanese)"; Systems and Control 1976, 20, 511-520. Fan, L.T.; Shieh, J . H . "Thermodynamically Based Analysis and Synthesis of Chemical Process Systems"; Energy 1980, 5, 955.


326 15. 16. 17. 18.


19. 20.

21. 22. 23. 24. 25.

26.

SECOND LAW ANALYSIS OF PROCESSES Denbigh, K . C . , "The Second-Law Efficiency of Chemical Processes"; Chem. Engr. S c i . 1956, 6, 1. Bruges, E . A . , "Available Energy and the Second Law Analysis"; Academic Press: London, 1959. Gaggioli, R . A . , "Thermodynamics and the Non-Equilibrium System"; Ph.D. Dissertation: University of Wisconsin, 1961. Baehr, von H.D.; Schmidt, E.F., "Definition und Berechnung von Breenstaffexergien"; BWK 1963, 15, 375. Szargut, J ; Petela, R . , "Egzergia"; Wydawnictwa NaukowoTechniczne: Warsawa, Poland, 1965 (in Polish). Evans, R . B . ; Tribus, M . , "Thermo-Economics of Saline Water Conversion"; I & EC Process Design and Development 1965, 4, 195. Reistad, G . , "Availability: Concepts and Application"; Ph.D. Dissertation: University of Wisconsin, 1970. Riekert, L., "The Efficiency of Energy-Utilization in Chemical Processes"; Chem. Engr. S c i . 1974, 29, 1613. Gaggioli, R . A . ; P e t i t , P.J., "Use the Second Law First"; Chemtech 1977, August, 496. Rodriguez, L . in "Calculation of Available Energy Quantities "; Gaggioli, R.A, E d . : ACS SYMPOSIUM SERIES NO. 122, American Chemical Society: Washington, D . C . , 1980; p. 39. Wepfer, W . J . ; Gaggioli, R.A. in "Reference Datumo for Available Energy"; Gaggioli, R . A . , E d . : ACS SYMPOSIUM SERIES NO. 122, American Chemical Society: Washington, D . C . , 1980; p. 77. Chen, H . T . ; Fan, L.T., "Multiple Minima in a Fluidized Reactor-Heater System"; AICLE J. 1976, 22, 680.


15.

Appendix A:



FAN AND SHIEH

327

Thermodynamic Background

In c a r r y i n g out the thermodynamic f i r s t - and second-law analyses of a process system, the c a l o r i f i c thermodynamic f u n c t i o n s of m a t e r i a l species i n v o l v e d i n the process are necessary. Conventionally the enthalpy, the entropy and the Gibbs f r e e energy are evaluated on the b a s i s of a reference s t a t e where a l l are known elements are i n t h e i r pure s t a t e s , the temperature i s 25°C (298.15 K) and the pressure i s 1 atm. An adoption of such a r e f e r e n c e s t a t e o f t e n y i e l d s negative energy and Gibbs f r e e energy, and these, i n t u r n , render the a n a l y s i s of process systems d i f f i c u l t , i f not impossible. To circumvent such a d i f f i c u l t y a great d e a l of e f f o r t has been spent f o r determining the thermodynamically meaningful reference s t a t e that tends to y i e l d a p o s i t i v e c a l o r i f i c f u n c t i o n . One of the f r e q u e n t l y employed reference s t a t e s i s the s o - c a l l e d dead s t a t e where the reference m a t e r i a l species (datum l e v e l m a t e r i a l s ) are e s s e n t i a l l y the products of complete combustion, the reference concentrations (datum l e v e l concentrations) are the environmental concentrations of these products, the r e f e r e n c e temperature (datum l e v e l temperature) i s the environmental temperature, and the reference pressure (datum l e v e l pressure) i s the e n v i r o n mental pressure (14,18,19,22,23,25). T h i s reference s t a t e has been adopted i n t h i s work f o r e v a l u a t i n g the energy and a v a i l a b l e energy contents f o r c a r r y i n g out the thermodynamic f i r s t - and second-law analyses. The s p e c i f i c enthalpy r e l a t i v e to the dead s t a t e , 3, i s d e f i n e d as (14 19,24) 9

6

h - h

5

(A-l)

Q

where h i s the s p e c i f i c enthalpy at any s t a t e and h^ that at^the dead s t a t e . By adding the enthalpy at the standard s t a t e , h , i n t o and s u b t r a c t i n g i t from Equation A - l , we have (h - h°) + (h° -

h ) Q

(A-2)

Since h Equation A-2

(A-3)

becomes = (h° -

e

h ) 0

(A-4)

By d e f i n i n g (14) (A-5) T B

T

E

T°'

C

P

d T


(A-6)

328

SECOND LAW ANALYSIS OF PROCESSES P B

P

E

J

I

v

T

" © p l

d

P

(

'

A

_

7

)

Equation A-3 can a l s o be expressed as 3 = 3° + 3

+ 3

T

(A-8)

p

Here, 3^ i s termed the s p e c i f i c chemical enthalpy, 3^ the s p e c i f i c thermal enthalpy and 3 the s p e c i f i c pressure enthalpy. The combination o f the s p e c i f i c thermal enthalpy, (? , and the s p e c i f i c pressure enthalpy, 3p, may be named the s p e c i f i c p h y s i c a l enthalpy. When the m a t e r i a l species i s one o f the components i n a s o l u t i o n , Equations A - l through A-7 are v a l i d , provided that the s p e c i f i c q u a n t i t i e s a r e changed to the p a r t i a l molar q u a n t i t i e s . Note that s u p e r s c r i p t 0 r e f e r s to the standard s t a t e , and s u b s c r i p t 0 r e f e r s to the dead s t a t e ; Cp i s the s p e c i f i c heat, and v i s the s p e c i f i c volume. The s p e c i f i c exergy, , i defined as (14,16,18-23) p


T

e

e

s

5

(h - h ) - T ( s - s )

E

(h° - h ) - T ( s ° - s ) + (h - h°) - T ( s - s°)

Q

0

Q

Q

0

Q

(A-9)

Q

S u b s t i t u t i n g Equation A-3 and

s

"

s

°•

T

d

< / ?

T

"

D

c / © P

p

P

(

A

1

"

0

)

i n t o Equation A-8 g i v e s r i s e to T T J c (l- ^)dt

e - (h° - h ) - T ( s ° - s ) + Q

0

+

Q

p

T )(|g ldP

[v - (T -

Q

p

(A-ll)

where s i s the s p e c i f i c entropy a t any s t a t e , the s p e c i f i c entropy a t the dead s t a t e , and T the datum l e v e l temperature. By d e f i n i n g (14,19,22,23) n

e° = (h° - h ) - T ( s ° - s ) Q

0

(A-12)

Q

T £

T

„/ T

c (l-^.)dP

(A-13)

p

and P £

p%o/

I v

"

(

T

-V©p

J

d

P

'


( A

"

1 4 )

15.

329


FAN AND SHIEH

Equation A - l l reduces to e - e° + e

+ e

T

(A-15)

p

where i s termed the s p e c i f i c chemical exergy, e the s p e c i f i c thermal exergy, and e the s p e c i f i c pressure exergy. Again, the combination of the s p e c i f i c thermal exergy, e , and the s p e c i f i c pressure exergy, e , may be named the s p e c i f i c p h y s i c a l exergy 114,19,22,23). Equations A - l through A-15 are the working formulas f o r e v a l u a t i n g the energy and a v a i l a b l e energy contents of a m a t e r i a l s p e c i e s which i s i n v o l v e d i n a chemical process. T

p

T


p

Appendix B:

L i s t of the Performance Equations f o r Determining the Non-Inferior Set

The o b j e c t i v e f u n c t i o n to be minimized i s J

- f

2

= A

2

(B-l)

T h i s i s Equation 13 i n the t e x t . The e q u a l i t y c o n s t r a i n t s i n c l u d e d i n Equation 12 i n the text are M

f.1

x

M

x

M

f , i " f,e - s , l

M

f .A

f , i " f,e f , . -

M

,i

M

+

M

< s,i

=

+

M

M

" < s,i

a

01

V

+

a

=

M

l l

+

"21

e

+

f,i f,i M

=

+

" ( s,i

( B

0

( B

a

s,l 01 +

M

) e

a

s,i

s,l 01

) f J

M

3

- f,e f,e "

M

A

_ "

(6

2 )

3 )

"

6

s,l s,l (B

s,e = °

e

s,i s,i a

s,l 01

) e

+

B

R

+

l

s,e - M

B R

4

~ >

2

f > e

e

f > e

- M

s > 1

a

2 1

e , s

2

U - 0.0474 - 0.0339 x, f »e

A

~

0

M

M

0

(B-6)

. - B )(M . + M .a.,) s,i s,e s,i s , l 01 U(T . - T - E) s,i / v

(B-5)

7

v

7

)

/


(

B

_

330


3 . = 55.5556(6.9756 + 0.013023T .+ 1.557 x 10" s,i s,i - 1.547 x 1 0 "

8

T

3

.)

(B-8)

= T - 298.15 s,e

(B-9)

7


3 i = 55.5556(6.9756 + 0.013023T + 1.557 x 10"* T s,l 1.547 x 1 0 "

3

f

±

B-

e

= (T

f ±

2

T s,i

y1

S

3 s,e

7

8

2

3

T )

- 298.15) + x

f

±

= (T - 298.15) + x-

(B-10)

(5978.918 - 0.642 T

f

±

)

(B-ll)

(5978.918 - 0.642 T)

(B-12)

T . tof °'*) 298.15'

. = 55.5556C-0.153674 - 2.29605 s,i

v

7

+ 0.007564T

2

. - 1.387475 x 1 0 " T . s,i s,i + 1.38838 x 1 0 " T . - 5.648159 x 1 0 " T . s,i s,i P . + 0.59242 £ n [ > ]} (B-13) 9

3

1 2

s

4

1

( p

S , i

e

s,e

sat.

- 55.5556(0.56207 - 0.003433T s,e 8

2

+ 8.296 x 1 0 " T ) s,e e

f

±

- 5900.506 x

f

(B-14)

+ (1 - 0.742 x

±

f

±

)(T

f ±

- 298.15

T - 298.15 £n ^ f f f j ) + (1 - x

f

= 5900.506 x, ~

2 9 8

'

1 5

+ (1 - x

-

+

x

2

2 9 6

'

f

2905> e

)

+

f,±

£ n x

(B-15)

) ( T - 298.15

2-296[x itn x

Jtn(l - x

f,i

)]

f

+ (1 - 0.742 x. to

[ x

f e

f

e

) ]

f

e

(B-16)


15.


FAN AND SHIEH

. - 55.55561-0.153674-2.29605 £n I

e

Z 7 0 . 7

- 1.387475 x K f T

2

1 2

- 5.648159 x 1 0 ~ T

e

Q


Efficiency and Costing - American Chemical Society

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