13
Thermoeconomic Optimization
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 5, 2016 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch013
of a
Rankine
Cycle
Cogeneration
System
ROBERT M. GARCEAU1 and WILLIAM J. WEPFER School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332
This paper presents the application of thermoeconomics to the optimization of a Rankine cycle cogeneration system which supplies process heat i n the form of hot water. The method described in this paper i s well-suited for application to thermal systems. This technique does not require that available-energy (exergy) be used e x p l i c i t l y i n the optimization although the effects of available-energy dissipations are evaluated v i a the shadow and marginal prices. The design space for this cogeneration system consists of five independent variables and three parameters that reflect "market" conditions; fuel cost, e l e c t r i c i t y cost, and the required hot water supply temperature. Results (suboptimizations) are presented for several given sets of market conditions. Thermal systems can be completely described using balance equations for mass, energy, and entropy i n conjunction with thermophysical property relations and/or equations of state, equipment performance characteristics, thermokinetic or rate equations, and boundary/initial conditions. With the thermal system adequately described, i t can be optimized by any current technique. Although the approach presented i n this paper is not explicit i n Second Law terms, i t never-the-less w i l l yield the optimal design and with the appropriate transformations, w i l l yield any desired Second Law quantity. The variables which are used to describe the system usually are not a l l independent since there exist many equations of constraint. It may be possible to substitute the constraint equations into the objective function, leaving only independent 1
Current address: Aerospace Systems Division, Harris Corporation, Box 37, Melbourne, FL 32901 0097-6156/83/0235-0263S07.25/0 © 1983 American Chemical Society In Efficiency and Costing; Gaggioli, Richard A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 5, 2016 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch013
264
SECOND LAW ANALYSIS OF PROCESSES
v a r i a b l e s to be optimized. Unfortunately, i t i s not always mathematically d e s i r a b l e to e l i m i n a t e a l l the dependent v a r i a b l e s from the o b j e c t i v e f u n c t i o n . One a l t e r n a t i v e i s the use o f Lagrangian m u l t i p l i e r s . The a p p l i c a t i o n o f Lagrange's Method to l a r g e s c a l e thermal systems i s well-known and wide-spread. Brosilow and Lasdon (1-2) and Gembicki (3) a p p l i e d t h i s technique to complex chemical proc e s s i n g p l a n t s . El-Sayed and Evans (4) developed a method whereby a complex thermal system i s decomposed i n t o i t s component p a r t s , each component buying and s e l l i n g available-energy w i t h other components. T h i s approach requires that the problem be cast i n terms o f available-energy coordinates, that i s , the c o n s t r a i n t equations r e p r e s e n t i n g the i n t e r n a l economic t r a n s a c t i o n s (supply and demand equations) must be e x p l i c i t i n available-energy flows. The b e n e f i t o f such a transformation i s that the Lagrange m u l t i p l i e r s represent p r i c e s that d e s c r i b e the i n t e r n a l s a l e s and purchases o f a v a i l a b l e - e n e r g y . I n t u r n , these p r i c e s can be used to show the economic t r a d e - o f f s between c a p i t a l investment and available-energy d e s t r u c t i o n f o r each component o f the system. In order to o b t a i n the available-energy balance equations (the intercomponent available-energy s a l e s and purchases), i t i s necessary to incorporate a l l the r e l e v a n t thermodynamic cons t r a i n t s . I n general no g u i d e l i n e s e x i s t concerning the s e l e c t i o n of the form f o r these c o n s t r a i n t equations. Furthermore, the r e s t r i c t i o n that a l l s t a t e v a r i a b l e s (temperatures, pressures, mole f r a c t i o n s , mass flows ...) be transformed to available-energy functions i s d i f f i c u l t and q u i t e t e d i o u s . As a r u l e , t h i s t r a n s formation from s t a t e v a r i a b l e s to available-energy flows r e s u l t s i n a set o f h i g h l y non-linear a l g e b r a i c equations which must be solved i n order to o b t a i n the set o f optimum d e c i s i o n v a r i a b l e s . The technique to be described i n t h i s paper follows that o f T r i b u s and El-Sayed (5) i n that no transformation i s made to available-energy coordinates. Rather the problem i s optimized u s i n g thermodynamic and equipment coordinates. I n t h i s way any d e s i r e d Second Law q u a n t i t i e s can be obtained upon c o n c l u s i o n o f the o p t i m i z a t i o n using an appropriate transformation. D e s c r i p t i o n o f the O p t i m i z a t i o n
Technique
T h i s method r e q u i r e s the o b j e c t i v e f u n c t i o n to be expressed i n terms o f the dependent (or s t a t e ) v a r i a b l e s , {x^}, and independent (or d e c i s i o n ) v a r i a b l e s , {y^}.
The equations o f c o n s t r a i n t are d i v i d e d i n t o two groups. One set o f c o n s t r a i n t s , r e f e r r e d to as s u b s t i t u t i o n c o n s t r a i n t s , are used to e l i m i n a t e s e l e c t i v e dependent v a r i a b l e s from the o b j e c t i v e f u n c t i o n , , and from the set o f Lagrange c o n s t r a i n t s , j . The other s e t , c a l l e d Lagrange c o n s t r a i n t s are used d i r e c t l y i n the Q
In Efficiency and Costing; Gaggioli, Richard A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Rankine Cycle Cogeneration System
13. GARCEAU AND WEPFER
265
o p t i m i z a t i o n scheme, each having an associated Lagrange m u l t i p l i e r , A j . The c o n s t r a i n t s are defined by equations o f the form
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 5, 2016 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch013
h
=
• j
-
X
(
J
2
)
where UJ cr iUJ
0.20 0.30 0.40 WORK/HEAT RATIO, W/Q Figure 10. Net revenue generated as a f u n c t i o n o f work/ heat r a t i o f o r v a r i o u s market p r i c e s o f e l e c t r i c i t y , a hot water supply temperature o f 250°F, and a f u e l cost o f $3 per m i l l i o n Btu.
In Efficiency and Costing; Gaggioli, Richard A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 5, 2016 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch013
13. GARCEAU AND WEPFER
Rankine Cycle Cogeneration System
0.20
0.30
287
0.40
WORK/HEAT RATIO, W/Q Figure 11. Net revenue generated as a f u n c t i o n o f work/ heat r a t i o f o r v a r i o u s market p r i c e s o f e l e c t r i c i t y , a hot water supply temperature o f 250°F, and a f u e l cost o f $4 per m i l l i o n Btu.
In Efficiency and Costing; Gaggioli, Richard A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
288
SECOND LAW ANALYSIS OF PROCESSES
Acknowledgments The authors would like to acknowledge the financial support provided by the School of Mechanical Engineering, Georgia Institute of Technology and the Celanese Corporation. Literature Cited
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 5, 2016 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch013
1.
2. 3. 4. 5.
6. 7.
8.
9. 10. 11. 12.
13. 14.
15.
Brosilow, C. B . ; Lasdon, L., "An Optimization Technique for Recycle Processes" paper presented at the IChE-AIChE joint meeting, London, England, 1964. Lasdon, L., Systems Research Center Report 50-C-64-19, Case Institute of Technology, Cleveland, 1964. Gembicki, S. Ph.D. Thesis, Dartmouth College, Hanover, NH, 1968. Evans, R. B . ; El-Sayed, Y. M . , Trans. ASME, Journal of Engineering Power, 1970, 92, 27-35. El-Sayed, Y. M . ; Tribus, M . , "The Strategic Use of Thermoeconomic Analysis for Process Improvement," paper presented at the AIChE Meeting, Detroit, August, 1981. Gaggioli, R. A . ; et al., Proceedings American Power Conference, 1975, 37, 671-679. P e t i t , P. J.; Gaggioli, R. A . , i n "Thermodynamics: Second Law Analysis;" Gaggioli, R. A . , E d . ; ACS SYMPOSIUM SERIES No. 122, American Chemical Society: Washington, D . C . , 1980; Chap. 2. Gaggioli, R. A . ; Wepfer, W. J., "Available-Energy Accounting-A Cogeneration Case Study," paper presented at the 85th AIChE Meeting, Philadelphia, June, 1978. Wepfer, W. J. and Gaggioli, R. A . , International Journal of Mechanical Engineering Education, 1981, 9, 283-295. Clark, F . D. and Lorenzoni, A. B . , "Applied Cost Engineering," Marcel Dekker, Inc.: New York, 1978. Humphreys, K. K. and K a t e l l , S., "Basic Cost Enginering," Marcel Dekker, Inc.: New York, 1981. Peters, M. S. and Timmerhaus, K. D . , "Plant Design and Economics for Chemical Engineers," 2nd E d . , McGraw-Hill: New York, 1968. Garceau, R. M . , M. S. Thesis, Georgia Institute of Technology, Atlanta, Georgia, August, 1982. "Steam," Computer Subroutine, Georgia Institute of Technology, Atlanta, Georgia, based on ASME Steam Tables, 3rd ed., ASME, New York, 1977. Wepfer, W. J.; Crutcher, B. G . , Proc. American Power Conference, 1981, 43, 1070-1082.
RECEIVED July 7, 1983
In Efficiency and Costing; Gaggioli, Richard A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
