~ Pergamon Energy Convers. Mgmt Vol, 38, No. 15-17, pp. 1535-1542, 1997

/ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII: S0196-8904(96)00215-4 0196-8904/97 $17.00 + 0.00

EXERGY-A IDED COST MIN IMIZAT ION

GEORGE TSATSARONIS ~ and MICHAEL J. MORAN 2 'Institute for Energy Engineering, Technical University of Berlin, Marchstr. 18, 10587 Berlin, Germany 2Department of Mechanical Engineering, The Ohio State University, 206 W. 18th Avenue, Columbus, OH.

43210-1154, U.S.A.

Abstract--This paper shows how exergy-related variables can be used to minimize the cost of a thermal system. These variables include the exergetic efficiency, the rates of exergy destruction and exergy loss, an exergy destruction ratio, the cost rates associated with exergy destruction, capital investment and operating and maintenance, a relative cost difference of unit costs and an exergoeconomic factor. A simple cogeneration system is used as an example to demonstrate the application of an iterative exergy-aided cost minimization method. 1997 Elsevier Science Ltd.

NOMENCLATURE

c - cost per unit of exergy C' = cost rate associated with exergy

C'D = cost rate of exergy destruction L" -- exergy flow rate

/~O = time rate of exergy destruction EF = exergy rate of fuel L'L ----- time rate of exergy loss L'p ---= exergy rate of product f = exergoeconomic factor

HRSG -- heat-recovery steam generator m -- mass flow rate p --= pressure

PEC = purchased-equipment cost r -- relative cost difference T = temperature

yD ---- exergy destruction ratio yc = exergy loss ratio

ZCi ~ cost rate associated with capital investment ~#OM __= cost rate associated with operating and maintenance expenses

2 = 2c'+ ~oM

Greek symbols

E --- exergetic efficiency ~/sc = isentropic efficiency of compressor q,, = isentropic efficiency of turbine

Subscripts

D --- destruction F = fuel L = loss P = product k = kth component

tot = overall system.

INTRODUCTION

Thermoeconomics combines exergy analysis and economic principles to provide the system designer or operator with information not available through conventional energy analysis and economic evaluations, but crucial to the design and operation of a cost-effective system. Thermoeconomics can be considered as exergy-aided cost minimization.

The objective of a thermoeconomic analysis might be: (a) to calculate separately the cost of each product generated by a system having more than one product; (b) to understand the cost formation

1535

1536 TSATSARONIS and MORAN: EXERGY-AIDED COST MINIMIZATION

Feedwater Saturated Vapor, @ ~ 20 bars, 14 kg/s Air Preheater

L i . @ .. - . ~ Natural Gas

Steam Generator 1 . ~ I~

Combust ion~ ,~ Chamber [ / , . , /

J Power to Air Compressor

(9

iNet Power 30 MW

Air

Fig. 1. Base-case design of the cogeneration system.

process and the flow of costs in the system; (c) to optimize specific variables in a single component; or (d) to optimize the overall system. A thorough discussion of thermoeconomics is presented in Ref. [1]. In this paper we discuss how certain exergy-related variables can be used to minimize the cost of a thermal system. The iterative optimization technique presented here requires a minimum of available data and provides effective assistance in optimizing thermal systems, particularly in dealing with complex systems and/or in cases where conventional optimization techniques cannot be applied to the system optimization.

CASE STUDY

Figure 1 presents the base-case design of a cogeneration system that develops a net power output of 30 MW and provides 14 kg/s of saturated water vapor at 20 bar. The first five columns of Table 1 show relevant thermodynamic and economic data. The second column of Table 2 shows the purchased-equipment costs (PEC) for each component in the base-case design.

Table 1. Mass flow rate, temperature, pressure, exergy rate, and cost data for the streams of the cogeneration system

State Stream

Mass flow Exergy flow Cost flow Cost per rate Temperature Pressure rate rate exergy unit m T p ~ ~' c

(kg/s) (K) (bar) (MW) (S/h) (S/G J) 1 Air 2 Air 3 Air 4 Combustion products 5 Combustion products 6 Combustion products 7 Combustion products 8 Water 9 Water

10 Methane 11 Power to air compressor 12 Net power

91.28 298.1 1.01 0.000 0 0 91.28 603.7 10.13 27.538 2756 27.80 91.28 850.0 9.62 41.938 3835 25.40 92.92 1520.0 9.14 101.454 5301 14.51 92.92 1006.2 1.10 38.782 2026 14.51 92.92 779.8 1.07 21.752 1137 14.51 92.92 426.9 1.01 2.773 145 14.51 14.00 298.1 20.00 0.062 0 0.00 14.00 485.6 20.00 12.810 1256 27.23 1.64 298.1 12.00 84.994 1398 4.57 -- - - - - 29.662 2003 18.76 - - - - - - 30.000 2026 18.76

TSATSARONIS and MORAN: EXERGY-AIDED COST MINIMIZATION 1537

Table 2. Values of the purchased-equiment costs (PEC) and the thermoeconomic variables for the base-case design (T~ = 850 K; T4 = 1520 K; pz/p~ = 10; ~/~ = r/~, = 0.86)*

PEC E IZt~ yD CV Cv CD 2 Or~ + 2 r f Component [1065] [%] [MW] [%] [$/GJ] [S/G J] [S/h] [S/h] [S/h] [%] [%] Combustion 0.34 80.37 25.48 29.98 11.45 14.51 1050 68 1118 26.7 6.1 chamber Gas turbine 3.74 95.20 3.01 3.54 14.51 18.76 157 753 910 29.2 82.7 Air compressor 3.73 92.84 2.12 2.50 18.76 27.80 143 753 896 48.2 84.0 HRSG 1.31 67.17 6.23 7.33 14.51 27.36 326 264 590 88.5 44.8 Air preheater 0.94 84.55 2.63 3.09 14.51 20.81 137 189 326 43.4 57.9 *For the overall plant we have CP.,o, = $3617/h and CL.~o, = (77 = $145/h.

These costs are obtained from the cost equations given in Appendix B of Ref. [1]. These cost functions are used for illustrative purposes only and do not provide current costs for purchasing the respective equipment items. In the design of thermal systems cost equations such as those in Ref. [1] are not usually available. Thus, in the discussions of thermoeconomic evaluation and iterative optimization, we recognize that after each design modification the new purchased-equip- ment costs would be calculated by a cost engineer. For simplicity of presentation, however, we assume that the cost values provided by the cost engineer are in full agreement with the corresponding values calculated from these cost equations. The remaining direct costs, as well as the indirect costs, are estimated using average factors. The total capital investment of the cogeneration system in the base case is estimated at approximately 46 million (mid-1994) dollars. Table 7.9 in Ref. [1] summarizes the parameters and assumptions used in the economic analysis, which is based on the revenue-requirement method discussed in Ref. [2].

The year-by-year economic analysis results in the levelized annual costs for fuel ($10.4 106), operating and maintenance ($5.9 x 106) and carrying charges ($10.5 x 106), for a levelization time period of 10yrs. The values given in the previous sentence are the corresponding levelized current-dollar costs obtained for the base case. The levelized costs are used as input data for the thermoeconomic analysis and optimization. The cost flow rates in the system are obtained by dividing the levelized annual costs by the number of hours of system operation per year.

In this paper principles of thermoeconomics are used to determine changes in the design variables of the cogeneration system that result in an improvement of the cost effectiveness of the overall system. The methodology used provides a plausible exploratory approach for improving the cost effectiveness of thermal systems generally and the case study of Fig. 1 in particular.

THERMOECONOMIC VARIABLES

A detailed thermoeconomic evaluation of a thermal system is based on a set of variables calculated for each component of the system as presented in the Appendix. For the kth component these include the following:

Exergetic efficiency E~ (Section 3.5.3). Rates of exergy destruction /~D,k and exergy loss EL.k (Section 3.5.2). Exergy destruction ratio yo.k and exergy loss ratio yL,k (Section 3.5.2). Cost rates associated with capital investment 2",' c~ , operating and maintenance expenses 2~ TM , and

their sum Zk (Section 8.1). Cost rate of exergy destruction (~o,~ (Section 8.2.1). Relative cost difference r, (Section 8.2.2). Exergoeconomic factor j~ (Section 8.2.3).

The section references given in parentheses locate in Ref. [1] where the corresponding variable is introduced.

1538 TSATSARONIS and MORAN: EXERGY-AIDED COST MINIMIZATION

THERMOECONOMIC EVALUATION AND OPTIMIZATION

The following methodology can be used in an exploratory approach aimed at improving the cost effectiveness of a thermal system:

(1) Rank the components in descending order of cost importance using the sum Zk + (~D,k. (2) Consider design changes initially for the components for which the value of this sum is high. (3) Pay particular attention to components with a high relative cost difference rk, especially when

the cost rates Zk and Co,k are high. (4) Use the exergoeconomic factorf~ to identify the major cost source (capital investment or cost

of exergy destruction): (a) if the f~ value is high, investigate whether it is cost effective to reduce the capital investment for the kth component at the expense of the component efficiency; (b) if the j~ value is low, try to improve the component efficiency by increasing the capital investment.

(5) Eliminate any subprocesses that increase the exergy destruction or exergy loss without contributing to the reduction of capital investment or of fuel costs for other components.

(6) Consider improving the exergetic efficiency of a component if it has a relatively low exergetic efficiency or relatively large values of the rate of exergy destruction, the exergy destruction ratio, or the exergy loss ratio.

When applying this methodology, it is important to recognize that the values of all thermoeconomic variables depend on the component types: heat exchanger, compressor, turbine, pump, chemical reactor and so forth. Accordingly, whether a particular value is judged to be high or low can be determined only with reference to a particular class of components. It is also important to consider the effects of contemplated design changes in one component on the performance of the remaining components. These effects may be determined either by inspection of the system flow sheets or by using a simulation program.

The methodology introduced above will now be applied to the case study cogeneration system. The objectives are to identify the effects of the design variables on the costs and suggest values of the design variables that would make the system more cost effective. The key design variables, the decision variables, for the cogeneration system are the compressor pressure ratio pz/pl, the isentropic compressor efficiency r/so, the isentropic turbine efficiency q~, the temperature of the air entering the combustion chamber T3, and the temperature of the combustion products entering the gas turbine T4.

First iteration The following nominal values of the decision variables correspond to the first workable design

(base-case design) developed for the cogeneration system of Fig. I, Tables 1 and 2:

p2/p~ = 10, r/~ = qst = 0.86, T3 = 850 K, T4 = 1520 K.

The last two columns of Table 1 and the last 10 columns of Table 2 summarize the values of the thermoeconomic variables calculated for each component of the cogeneration system for the base-case design. In accord with the presented methodology, the components are listed in order of descending value of the sum ~'D + 2.

The combustion chamber, the gas turbine and the air compressor have the highest values of the sum Z + ~'D and are, therefore, the most important components from the thermoeconomic viewpoint. The low value of the variable f for the combustion chamber shows that the costs associated with the combustion chamber are almost exclusively due to exergy destruction. A part of the exergy destruction in a combustion chamber can be avoided by preheating the reactants and by reducing the heat loss and the excess air, but this usually leads only to a small reduction in the exergy destruction. For simplicity, we assume here that the heat loss cannot be further reduced. The excess air is determined by the desired temperature T4 at the inlet to the gas turbine. The temperature T4 is a key design variable, for it affects both the performance of the entire system (exergy destruction in the combustion chamber, gas turbine, air preheater and HRSG, and exergy loss associated with stream 7), and the investment costs of the components.

An increase in the heat transfer rate in the air preheater achieved through an increase in temperature T3 also results in a decrease of the exergy destruction in the combustion chamber.

TSATSARONIS and MORAN: EXERGY-AIDED COST MINIMIZATION 1539

Thus, the temperature T3 is also a key design variable because, in addition to the combustion chamber, it affects the exergy loss associated with stream 7 as well as the performance and investment costs of the air preheater and the heat-recovery steam generator. Holding all other decision variables constant, the higher the temperature T3, the smaller the average temperature difference in the air preheater and the heat-recovery steam generator. A decrease in the average temperature difference in these heat exchangers results in an increase in both the exergetic efficiency and the capital investment for each heat exchanger.

Summarizing, by considering measures for reducing the high cost rate associated with the exergy destruction in the combustion chamber of the cogeneration system, two key design variables have been identified, the temperatures T3 and T4. An increase in these temperatures reduces the CD value for the combustion chamber and other components, but increases their capital investment costs.

Turning next to the gas turbine, which has the second highest value of the sum 2 + CD, the relatively large value of factor f suggests that the capital investment and O&M costs dominate. The capital investment costs of the gas turbine depend on temperature/4, pressure ratio p2/p~ and isentropic efficiency r/st. To reduce the high ,~ value associated with the gas turbine, we should consider a reduction in the value of at least one of these variables.

The air compressor has the highestfvalue and the second highest relative cost difference r among all components. Thus, we would expect the cost effectiveness of the entire system to improve if the Z value for the air compressor is reduced. This may be achieved by reducing the pressure ratio p2/p~ and/or the isentropic compressor efficiency t/~.

The heat-recovery steam generator has the lowest exergetic efficiency and the highest r value among all the components. As thefvalue indicates, almost 45% of the relative cost difference is caused by the Z value in this component, with the remaining 55% caused by exergy destruction. Thus, we might conclude that a decrease of the exergy destruction in the HRSG could be cost effective for the entire system, even if this would increase the investment costs associated with this component. The exergy destruction in the HRSG can be reduced by decreasing the values of 7"6 and TT. A decrease in the value of 7"7 also results in a decrease in the exergy loss from the total system. In terms of the decision variables, temperatures 7"6 and T7 may be reduced by increasing T3 and/or decreasing T4 at fixed values of the remaining decision variables.

The relatively high value o f f in the air preheater suggests a reduction in the investment costs of this component. This can be achieved by decreasing T3. It should be noted, however, that changes suggested by the evaluation of this component should only be considered if they do not contradict changes suggested by components with a larger value of Co + Z.

Summarizing the foregoing conclusions, the following changes in the design variables are expected to improve the cost effectiveness of the system.

Increase the value of Z~ as suggested by the evaluation of the combustion chamber and HRSG.

Decrease the pressure ratio p2/pl (and thus p4/ps) and the isentropic efficiencies ~/~c and t/s,, as suggested by evaluation of the air compressor and gas turbine.

Maintain T4 fixed, since we get contradictory indications from the evaluations of the combustion chamber on one side and the gas turbine and HRSG on the other side.

Second iteration Contemplating the effects of changes made in accord with the above list in the values of the

design variables used in the first iteration, the following new values are selected for the second iteration:

T3 = 870 K, T4 = 1520 K (unchanged), p2/p~ = 9, t/so = 85%, r/st = 85%.

The new values of the thermoeconomic variables for each component are summarized in Table 3. In the new design, the combustion chamber, the gas turbine and the air compressor also have the highest values of the sum d?t, + 2 and are, therefore, still the most important components from the thermoeconomic viewpoint. The high cost rate associated with the combustion chamber can be reduced by increasing the values of 7"3 and T4. In the evaluation of the cogeneration system we

1540 TSATSARONIS and MORAN: EXERGY-AIDED COST MINIMIZATION

Table 3. Component data for the second iteration case: (7"3 = 870 K; 7"4 = 1520 K; p:/p~ = 9; q~ = 0.85; r/st = 0.85)*. The values given in parentheses refer to the third iteration case considered here (T3 = 910 K; T4 = 1480 K; p2/p~ = 7;

r/~ = 0.83)**

E Eo yo cv co C~D 2 CD + 2 r f Component [%] [MW] [%] [S/G J] [S/G J] [S/h] [S/h] IS/h} [%1 [%1

Combustion 80.3 25.93 29.77 10.50 13.26 980 72 1052 26.3 6.8 chamber (81.3) (27.47) (29.92) (9.42) (11.71) (931) (55) (986) (24.35) (5.5) Gas turbine 94.9 3.18 3.66 13.26 16.97 152 647 799 28.0 81.0

(94.3) (3.69) (4.01) (11.71) (13.75) (155) (296) (451) (17.45) (65.6) Air 92.1 2.34 2.69 16.97 23.96 143 546 689 41.2 79.2 compressor (90.5) (2.99) (3.25) (13.75) (18.38) (148) (324) (472) (33.61) (68.7) HRSG 66.6 6.40 7.35 13.26 25.60 305 261 566 93.1 46.1

(67.6) (6.10) (6.65) (11.71) (23.51) (257) (284) (541) (100.74) (52.5) Air 84.7 3.15 3.62 13.26 18.94 150 206 356 42.9 57.8 preheater (85.6) (4.97) (4.90) (11.71) (16.53) (190) (275) (464) (41.20) (59.2) Overall 49.1 41.01 47.09 4.57 21.80 675 1922 2597 377.0 74.0 plant (46.6) (44.79) (48.79) (4.57) (19.06) (736) (1424) (2160) (317.17) (65.9) *For the overall plant in the new design case we have C'p.,o, = $3355/h and ~'L.,o, = ~'7 = $157/h. **For the overall plant in the last design case we have ~P.~o, = $2934/h and C'L.,o, = C'7 = $167/h.

should, however, consider that the value of this sum for the combustion chamber will always be the highest among the (~D + Z values for the components of the cogeneration system.

The gas turbine now has the highest f value. The reduction in this value from 82.7% in the base design to 81.0% in the new design is relatively small compared with a target value of below 75%. This observation suggests: (1) a significant decrease in the values of r/s, and/or p2/p~, that is a decrease greater than the decrease in these variables in the previous step: from 86 to 85% and from I0 to 9, respectively; and (2) a reduction in the value of T4. Note that the decrease in the T4 value contradicts the corresponding suggestion from the combustion chamber.

The high values of the exergoeconomic factor f and the relative cost difference r for the air compressor suggest a decrease in the values of the decision variables p2/p~ and r/~. The anticipated increase in the exergetic efficiency of the HRSG (see first iteration) was not realized because of the interdependence of the components: the reduction in the values of p:/p~, q~ and qs, for the compressor and the turbine leads to an increase in the temperature differences (and, therefore, a decrease in the exergetic efficiency) of the HRSG. Thus, the HRSG thermoeconomic evaluation suggests that the 7"3 value increases and the 7"4 value decreases.

The relatively high value of f in the air preheater suggests a reduction in the T3 value. As noted in the first iteration, however, changes suggested by the evaluation of this component should only be considered if they do not contradict changes suggested by components with a higher value of the sum ~"D "31- J~'.

Summarizing the foregoing suggestions from the thermoeconomic evaluation of each component, the following changes in the decision variables are expected to improve the cost effectiveness of the cogeneration system:

Increase the value of 7"3 as suggested by the evaluations of the combustion chamber and HRSG. Decrease the pressure ratio p2/pJ and the isentropic efficiencies r/~ and qst as suggested by the

evaluations of the air compressor and gas turbine. Decrease the temperature 7"4 as suggested from the evaluations of the gas turbine and the HRSG.

Third iteration To illustrate the effect of the suggested changes in the decision variables on the overall costs,

we use the following new set of values for the design variables:

T~=910K; T0=1480K; p2/p~=7; ~/= --- 0.83; and r/~,=0.83.

The results from the thermoeconomic analysis for the last set of values are summarized in Table 3 by the values given in parentheses. A comparison of the corresponding values shown in Table 3 demonstrates the improvement in the cost effectiveness of the last design case. As a result of these changes, the value of the objective function ~p.,o, is reduced from $3355/h to $2934/h. The t~7 value

TSATSARONIS and MORAN: EXERGY-AIDED COST MINIMIZATION 1541

has increased from $157/h in the new case, to $167/h in the last case. This increase is, however, outweighed by the decreases in the values of CD.k + Zk.

Additional iterations

Some additional iterations, conducted in a similar way, are necessary to further decrease the value of the objective function and/or establish a nearly-optimal design case. The cost-optimal values of the decision variables calculated using Box's complex method and the corresponding computer program from Ref. [3] are:

T~=910.2K; T4=I463.0K; p2/pl=5.77; r/so=0.811; and qst=0.845.

With these values we obtain for the objective function Cv,tot = $2870/h. For the cost-optimal case, the exergetic efficiency of the overall system is 45.0%, the cost rate associated with the exergy loss is $205/h, and the pinch temperature difference in the heat-recovery steam generator is 49.7 K.

THE BENEFITS OF THE THERMOECONOMIC OPTIMIZATION

Complex thermal systems cannot always be optimized using mathematical optimization techniques. The reasons include incomplete models, system complexity and structural changes:

Some of the input data and functions required for the thermodynamic and, particularly, the economic model might not be available or might not be in the required form. For example, it is not always possible to express the purchased-equipment costs as a function of the appropriate thermodynamic decision variables.

Even if all the required information is available, the complexity of the system might not allow a satisfactory mathematical model to be formulated and solved in a reasonable time.

The analytical and numerical optimization techniques are applied to a specified structure of the thermal system. However, a significant decrease in the product costs may be achieved through changes in the structure of the system. It is not always practical to develop a mathematical model for every promising design configuration of a system. More importantly, analytical and numerical optimization techniques cannot suggest structural changes that have the potential of improving the cost effectiveness.

The usual approach to the optimization of such complex systems is to iteratively optimize subsystems and/or ignore the influence of some structural changes and decision variables. An alternative to this approach is the iterative thermoeconomic optimization technique presented here. This technique improves the engineer's understanding of the interactions among the system variables, and generally reveals opportunities for design improvements that might not be detected by other methods. It enhances the knowledge, experience and intuition of design engineers, but does not substitute for engineering creativity.

REFERENCES

1. Bejan, A., Tsatsaronis, G. and Moran, M., Thermal Design and Optimization. Wiley, New York, 1996. 2. EPRI Technical Assessment Guide (TAG), Electric Power Research Institute, TR-100281. Vol. 3, Revision 6.

1991. 3. Kuester, J, K. and Mize, J. H., Optimization Techniques with Fortran. McGraw-Hill, New York, 1973.

APPENDIX in defining the exergetic efficiency and other key variables it is necessary to identify both a product and a fuel for the thermodynamic system being analyzed. The product represents the desired result produced by the system. Accordingly, the definition of the product must be consistent with the purpose of purchasing and using the system. The fuel represents the resources expended to generate the product and is not necessarily restricted to being an actual fuel such as natural gas, oil or coal. Both the product and the fuel are expressed in terms of exergy. An exergy loss is handled similarly. Using these concepts an exergy rate balance for component k of an overall system at steady state consisting of a number of components may be expressed as

/~Fk = Ep.k + ED,k + EL,{ (A1)

where F: fuel, P: product, D: destruction, L: loss. Exergy losses are associated with discarded matter and/or heat transfer to the surroundings.

1542 TSATSARONIS and MORAN: EXERGY-AIDED COST MINIMIZATION

For component k, the exergetic efficiency is then

~p., _ (t?o., + EL.,) E, = ~ = 1 (A2) EF.k

The exergy destruction ratio and the exergy loss ratio are, respectively,

/~D.k yD~ = L'r .... (A3)

yL., = EF .... (A4)

where ~PF.to, is the exergy rate of the fuel provided to the overall system. The cost rate of exergy destruction is

(;r,.,=cv.,Eo.,. (EP., fixed) (A5)

where Cv.k: cost per unit of fuel exergy. The relative cost difference for component k is

CP,k - - CF,k r k - - - -

CF,*

=cF.,(Eo.k + &, ) + (2~ ~ + 2 '~) cv.*Er,,* (A6)

= 1 - e , + 2'/ , I + 2", M Ek CF,k EP,,

where cv., : cost per unit of product exergy, 2c~: cost rate associated with capital investment, Z~, M : cost rate associated with operating and maintenance expenses.

The exergoeconomic factor of component k is

2, f i = Z, + cv., (ED., + Er.,) (A7)

where 2, = 2 cx + 2 M.