Chemical Engineering Science 58 (2003) 4291–4305www.elsevier.com/locate/ces

Globally optimal power cycle synthesis via the In%nite-DimEnsionAlState-space (IDEAS) approach featuring minimum area

with %xed utility�

Lealon L. Martin, Vasilios I. Manousiouthakis∗

Department of Chemical Engineering, UCLA, Los Angeles, CA, 90095, USA

Received 16 July 2001; received in revised form 31 May 2002; accepted 15 July 2002

Abstract

This paper demonstrates the use of the In%nite DimEnsionAl State space (IDEAS) approach in synthesizing optimal power cyclesfeaturing minimum heat exchange area. IDEAS is used to synthesize power cycle networks which include splitters, mixers, pumps,turbines, and heat exchangers and feature a single or multiple working 8uid(s). The overall synthesis goal is to minimize heat exchangearea requirements, while delivering a %xed percentage of the maximum net power obtainable from a given set of hot and cold utilities. Theglobal optimality of the obtained power cycle network con%guration is guaranteed, since IDEAS gives rise to convex (linear) programs.The power of the proposed approach is demonstrated on a case study involving the generation of electricity by a bottoming cycle with apure ammonia working 8uid. Real thermodynamic data for pure ammonia and rigorous equipment models are employed in carrying outthe proposed optimization.? 2002 Elsevier Ltd. All rights reserved.

Keywords: Power cycle synthesis; Minimum area; IDEAS; Optimization

1. Introduction

There can be several feasible con%gurations for a powercycle operating within the same source and sink tempera-ture limits. Of these, only a few con%gurations will deliverthe best performance, in terms of thermodynamic e<ciency,power output, utility pro%ts, capital costs, total annualizedpro%t, or area. A method for the synthesis of power cyclesmust be able to choose cycle con%guration and operatingparameters that deliver the best cycle performance in one ofthe aforementioned terms. Currently, most power cycles aresynthesized by modifying sections of existing designs to al-leviate shortcomings that can be determined by pinch or sec-ond law analysis. Conventional 8owsheet optimization andanalysis methodologies, such as pinch analysis or secondlaw costing strategies (El-Sayed & Tribus, 1985a, b) thathave been employed in the aforementioned studies, providereasonable, thermodynamically e<cient starting points for

� First presented at the 1999 AIChE Annual Meeting Paper 48c.∗ Corresponding author. Department of Chemical Engineering, Univer-

sity of California, 5531 Boelter Hall 405, Hilgard Avenue, Los Angeles,CA 90024-1592, USA. Tel.: +1-310-825-9385; fax: +1-310-825-2394.

E-mail address: vasilios@ucla.edu (V. I. Manousiouthakis).

cycle improvement. The obtained designs may, however, besub-optimal in the economic sense because either the 8ow-sheet structure or most of its variables are %xed. Therefore,it is likely that optimal con%gurations may be overlooked.In this work, a systematic formulation for the design of op-timal power cycles is proposed.

2. The optimal power cycle synthesis problem

We de%ne optimal power cycle synthesis as follows:

Given a set of hot utility streams with their 8ow rates, spe-ci%c heats, inlet temperatures, outlet temperature bounds,and associated utility costs; a set of cold utility streamswith their speci%c heats, inlet and outlet temperatures; aworking 8uid with known thermodynamic properties andassociated utility costs; and models for heating, pump-ing and expansion operations; synthesize a thermo-dynamically feasible power cycle—consisting of heatexchangers, turbines, and pumps—that minimizes someobjective (such as heat exchange area (MA)), while sat-isfying some constraints (such as delivering a knownpercentage of the net power maximum).

0009-2509/03/$ - see front matter ? 2002 Elsevier Ltd. All rights reserved.PII: S0009 -2509(02)00526 -2

4292 L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305

The proposed optimization scheme can quantify rigor-ously the tradeoI between heat exchange area and netoperating (utility) pro%ts from power cycle operations.IDEAS provides a general design framework that con-siders all feasible cycle designs. Furthermore, IDEASgives rise to convex (linear) program formulations, thusguaranteeing the global nature of the obtained optima.The proposed IDEAS methodology will be demon-strated subsequently in a power cycle synthesis casestudy.To ensure the applicability of the proposed method we

will follow a two-step procedure to the optimal power cyclesynthesis problem. First, we will identify the maximum de-liverable power without explicitly incorporating area in theproblem formulation. Once the maximum operating pro%tis identi%ed, then a heat exchange area minimization prob-lem will be formulated, subject to a constraint requiringthe power generated to be a percentage of the maximumdeliverable power. Repeated solutions of this problem forvarious percentage values give rise to a rigorous tradeoIbetween heat exchange area and operating e<ciency. It isimportant to point out that this optimization problem isconvex (linear), thus ensuring the global nature of theobtained optimum.

3. Thermodynamic property data

In the proposed research, we aim at covering the wholerange of operations (both sub-critical and supercritical) fora pure substance working 8uid in our eIort to identify glob-ally optimal cycles. It is thus imperative that the employedthermodynamics be beyond question.For exact thermodynamics of pure substances and select

binary mixtures, the NRTL and Chen models for the equa-tion of state (EOS) provide accurate data. Recent studiesby Kutney et al. (1997) have shown good empirical modelsfor supercritical process modeling using EOS derived fromhard sphere volume translated Van der Waals equations ofstate. Performance analysis in recent years for the Kalina cy-cle have used the modi%ed Benedict–Webb–Rubin (BWR)equations of state (Uehara et al., 1994). This model hasshown capability to predict properties of ammonia + waterbinary mixtures of up to 20 MPa and 650 K. Most of theearlier predicted models are applicable only in a restrictedrange or allow calculation of only a limited number ofthermodynamic properties. Moreover, none of these modelshave been %tted to measurements in the single and two-phaseregions.Given the potential industrial application, a number of

scienti%c institutions, for example NIST and DECHEMA,have undertaken eIorts to provide comprehensive ther-modynamic databases/models for pure and multicom-ponent systems. Along these lines, a workshop was re-cently organized on ammonia-water mixtures (Friend &Haynes, 1996). A large number of experimental and the-

oretical models were compiled and cross-checked. Theseresults were formally published in a recent publication(Tillner-Roth & Friend, 1998a, b). In addition, thermo-dynamic data for pure ammonia systems in the sub- andsupercritical phase regimes are available in the NISTThermo-physical Properties of Fluids Systems Database athttp://webbook.nist.gov/chemistry/8uid.Tillner-Roth and Friend (1998a, b) have proposed a

fundamental equation of state for the Helmholtz free en-ergy of the ammonia–water system. They incorporateaccurate wide-ranging equations of state for the purecomponents (ammonia and water) and describe the en-tire thermodynamics of the mixture from the vapor tothe liquid, including the VLE by a single mathemat-ical equation. A number of thermodynamic properties(e.g., internal energies, enthalpy, entropy, speci%c vol-umes, heat capacities, etc.) can be evaluated from thismodel and can be used to describe the state space opera-tors. In our work, we have employed the NIST ammoniadatabase.

4. The In�nite DimEnsionAl State space (IDEAS)approach

In%nite DimEnsionAl State space (IDEAS) has been putforward as a tool for the optimal synthesis of chemicalprocess networks. A bibliographic survey of IDEAS appli-cations to optimal process network synthesis via IDEAS canbe found in the literature (Justanieah & Manousiouthakis,2000; Martin & Manousiouthakis, 2000a, b, c). Within theIDEAS framework, the synthesis of the overall networkis decomposed into the synthesis of an operator network(OP), where process technologies (unit operations) and/ortheir aggregate eIects are represented, and a distributionnetwork (DN), where mixing, splitting, recycling, and by-passing is allowed within stream families (sub-DNs). TheDN has four sets of junctions, one set for external feedstreams, a second set for %nal processed streams, a set forOP inlet streams, and %nally a set for OP outlet streams.Mass and energy conservation laws are enforced at alljunctions, while unit operations—with the appropriate pro-cess and equipment models—are included in the OP. Byusing the IDEAS approach, we can eIectively describe allprocess networks. Furthermore, the resulting mathematicalformulations exhibit convex (linear) feasible regions. Con-sequently, as long as objective functions are chosen to beconvex, all local solutions of these problems are globallyoptimal. Moreover, IDEAS represents the only systematicdesign methodology which provides a complete and gener-alized process network. These general properties are treatedin detail in recent work by Wilson and Manousiouthakis(2000). Therefore, we can identify optimal network con-%gurations with minimum utility, linear total annualizedcost objectives, while guaranteeing global optimality of thedesign.

L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305 4293

The key development behind IDEAS is the use of aprocess operator whose domain and range lie in in%nite(rather than %nite) dimensional spaces. This process repre-sentation allows both consideration of all possible processnetworks for an a priori given set of technologies, and theformulation of convex (linear) process network synthesisproblems that guarantee global optimality of the obtainedsolutions.Traditionally, process operations have been considered to

take inlet stream information (such as 8ows, component con-centrations, enthalpy, and so on) and transform it to similaroutlet stream information. The resulting process operatorswere nonlinear, giving rise to non-convex optimal networksynthesis formulations.IDEAS represents a radical departure from this approach.

It considers that the process operator takes extensive (quan-tity) inlet stream information (that is, 8ow), available at anypossible intensive (quality) inlet stream (that is, componentconcentration, enthalpy, and so on) and unit operation de-sign parameter (that is, residence time, number of transferunits, and so on) conditions, and transforms it to extensiveoutlet stream information (that is, 8ow) available at thecorresponding intensive outlet stream conditions providedby the unit operation model. When viewed in this manner,it is easy for the reader to verify that the resulting IDEASprocess OP is linear for any chemical process. This is the di-rect result of the following property of chemical processes:when their inlet 8ow rates are increased proportionally(without altering the other intensive inlet conditions), theiroutlet 8ow rates are also increased by the same proportion,while their intensive outlet conditions remain unaltered, aslong as appropriately de%ned design parameters are keptconstant. Heat exchangers, mass exchangers, reactors, distil-lation columns, and all other chemical processes satisfy thisproperty.Having described the conceptual foundation of the IDEAS

approach, we are now ready to present its mathematicalformulation.

4.1. IDEAS variable sets

A graphical representation of an IDEAS process networkis shown in Fig. 1. The DN contains N “families”or setsof streams which are only allowed to mix within their ownsub-DN. The ith family has �u(i) external inlet streamswhich are quanti%ed by the sequence pair, (ui; �i). Let ui(j)and �i(j) be the jth terms of this sequence pair for fam-ily i. Then, ui(j) represents the scalar extensive quantity(8ow) variable at condition j, while �i(j) represents thevector intensive quality (pressure, temperature, enthalpy,component concentration, and so on) information at condi-tion j. Considering the total mass 8ow rate entering eachfamily to be non-negative and %nite, and given that thenumber of inlet streams for each family, �u(i) is %nite, itthen holds that ui ∈’+0 ⊂ ‘+1 . The number of qualities

(DN)

(DN)

(DN)

1

2

N

i = 1,...,N

DN

OP (OP)

(OP)

(OP)M

2

1

(x

(w

(u ,α )i i

(y ,β )

,γ )

z

i i

i i

,τ )

i

Fig. 1. In%nite DimEnsionAl state-space (IDEAS) representation forchemical process networks featuring multiple families (N ) and multipletechnologies (M).

that are needed to describe a stream may vary from familyto family and is given by nq(i), yielding �i ∈ ‘nq(i)∞ . With-out loss of generality and for mathematical simplicity, itis assumed that �i(j)∈Qnq(i) ⊂ Rnq(i). The external outletstreams are described similarly by the sequence pair (yi; �i),where yi ∈ ‘+1 ; �i ∈ ‘nq(i)∞ , and �i(j)∈Qnq(i) ⊂ Rnq(i). Sum-marizing the notation

u= {ui}Ni=1; ui = {ui(1); ui(2); : : : ; ui(�u(i))};

y = {yi}Ni=1; yi = {yi(1); yi(2); yi(3); : : : ; };

�i(j) = [�1i (j); �2i (j); : : : ; �

nq(i)i (j)]T;

�i(j) = [�1i (j); �2i (j); : : : ; �

nq(i)i (j)]T:

In this work, we consider an OP consisting of M tech-nologies, the mth of which possesses nd(m) design param-eters. For simplicity, each unit operation is restricted toa single inlet stream from each family. Multiple OP inletstreams from the same stream family can be easily ac-commodated with simple modi%cations of the subsequentdiscussion.Capturing the action of the OP requires a sequence pair

that belongs to a fundamentally diIerent space than thoseof the DN external and internal outlets. Information pertain-ing to the OP inlet is given by the sequence pair (w; �). Theterms of the %rst sequence w={w(k)}∞k=1 are N -dimensionalvectors whose entries w(k)=[wi(k); i=1; : : : ; N ] representthe quantity (8ow) variables from each of the N familiesentering the OP at condition k. Considering the total mass8ow rate entering the OP from all families, over all condi-tions, to be %nite, it then holds that wi ∈ ‘+1 and w∈ ‘+N

1 .The terms of the second sequence, �={�(k)}∞k=1 are �(k)=[�1(k); : : : ; �N (k); R(k); D(k)], where �i(k); i = 1; : : : ; N is

4294 L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305

an nq(i)-dimensional vector representing the qualityinformation entering the OP from the ith family atcondition k; R(k) is an (N − 1) × N matrix represent-ing the linear interrelations among the 8ow variableswi(k); i = 1; : : : ; N entering the OP at condition k; D(k)is a vector, whose dimension is equal to the numberof design parameters for the technology m employedat condition k. Notation for these quantities is givenbelow

�i(k) = [�1i (k); �2i (k); : : : ; �

nq(i)i (k)]T;

D(k) = [D1(k); D2(k); : : : ; Dnd(m)(k)];

R(k) =

R1;1(k) R1;2(k) : : : R1;N (k)

R2;1(k) R2;2(k) : : : R2;N (k)

: : : : : : : : : : : :

RN−1;1(k) RN−1;2(k) : : : RN−1;N (k)

:

Since the entries of �(k) are considered to be bounded, thefollowing holds; �i ∈ ‘nq(i)∞ ; i= 1; : : : ; N; R∈ ‘(N−1)×N

∞ , andD∈ S ⊂ ‘nd∞ where nd =maxm nd(m).OP-outlet stream information for family i is given by

the sequence pair (xi; �i). Unlike the OP-inlet sequencepair, (xi; �i) belongs to a space similar to the space ofthe DN-inlets and outlets, namely xi ∈ ‘+1 ; �i ∈ ‘nq(i)∞ , and�i(j)∈Qnq(i) ⊂ Rnq(i).

x = {xi}Ni=1; xi = {xi(1); xi(2); xi(3); : : : ; };

�i(j) = [�1i (j); �2i (j); : : : ; �

nq(i)i (j)]T:

The %nal set of variables needed to mathematically formu-late IDEAS are the cross-8ow streams inside the DN. TheDN has four sets of conditions, a set of external DN-inletconditions, a set of external DN-outlet conditions, a set ofOP-inlet conditions, and a set of OP-outlet conditions. Thereare four sets of cross communication variables, each rep-resented by a sequence triplet quantifying stream 8ow rate,destination, and source as follows: DN-inlet to DN-outlet(zyui; �i; �i), DN-inlet to OP-inlet (zwui; �; �i), OP-outlet toDN-outlet (zyxi; �i; �i), and OP-outlet to OP-inlet (zwxi; �; �i).Since the total mass 8ow from the external inlets and thetotal mass 8ow to the OP are both %nite and the DN mustsatisfy a mass balance at any condition, zwui ∈ ‘+1 (2). Simi-larly, since the total mass 8ow leaving through the externaloutlets and the mass 8ow leaving OP to the DN are both%nite and mass balances are required to be satis%ed through-out the DN, it holds that zyxi ∈ ‘+1 (2); zyui ∈ ‘+1 (2), andzwxi ∈ ‘+1 (2).

4.2. The IDEAS convex feasible region

IDEAS formulations are constrained based on theproperties of the DN, properties of the OP, network spec-i%cations, and design criteria. There are two classes ofconstraints present in the DN; conservation of mass andconservation of quality related entities (for example, en-ergy). Mass conservation equations are enforced at all activeconditions in every family sub-DN and are given by Eqs.(1)–(4). Quality constraints are necessary only where mix-ing operations are possible, namely, at active DN externaloutlet and active OP outlet conditions. These are given byEqs. (5)–(6)

ui(j) =∞∑k=1

zwui(k; j) +∞∑o=1

zyui(o; j); j = 1 : : : �u(i); (1)

xi(n) =∞∑k=1

zwxi(k; n) +∞∑o=1

zyxi(o; n); n= 1 : : :∞; (2)

yi(o) =�u(i)∑j=1

zyui(o; j) +∞∑n=1

zyxi(o; n); o= 1 : : :∞; (3)

wi(k) =�u(i)∑j=1

zwui(k; j) +∞∑n=1

zwxi(k; n); k = 1 : : :∞; (4)

�i(o)yi(o) =�u(i)∑j=1

�i(j)zyui(o; j)

+∞∑n=1

�i(n)zyxi(o; n); o= 1 : : :∞; (5)

�i(k)wi(k) =�u(i)∑j=1

�i(j)zwui(k; j)

+∞∑n=1

�i(n)zwxi(k; n); k = 1 : : :∞; (6)

where i=1; : : : ; N . Technology model equations for the OPblock(s) are given by Eqs. (7) and (8).

xi = Fiw ∀i = 1; : : : ; N; (7)

R(k)w(k) = 0 ∀k = 1; : : : ;∞; (8)

where Fi : ‘+N1 → ‘+1 is a linear operator that transforms

the OP inlet 8ows into the ith family’s OP outlet 8ows.Speci%cally, the element Fl

i (j; k) represents the (linear)

L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305 4295

contribution of wl(k) on xi(j).

xi(1)

xi(2)

...

xi(j)

...

=

F1i (1; 1) : : : FNi (1; 1) F1i (1; 2) : : : FN

i (1; 2) : : : : : :

F1i (2; 1) : : : FNi (2; 1) F1i (2; 2) : : : FN

i (2; 2) : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : :

F1i (j; 1) : : : FNi (j; 1) F1i (j; 2) : : : FN

i (j; 2) : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : :

w1(1)

...

wN (1)

w1(2)

...

wN (2)

...

...

...

:

The elements of each matrix, Fi, are determined through so-lution of the modeling equations for all the unit operationsinvolved. Eq. (8) constrains the 8ows at each OP-inlet con-dition k; (wi(k); i= 1; : : : ; N ), to speci%ed ratios (elementsof R(k)) based on the quality information of unit operationinlet conditions. Eqs. (9) and (10) are space de%nitions forthe various 8ow vectors and 8ow sequences in the formula-tion. They serve as 8ow positivity constraints

ui ∈’+0 ; yi; xi; wi ∈ ‘+1 ; w∈ ‘+N1 ; i = 1; : : : ; N; (9)

zwui ∈ ‘+1 (2); zwxi ∈ ‘+1 (2); zyui ∈ ‘+1 (2);

zyxi ∈ ‘+1 (2); i = 1 : : : N: (10)

4.3. Formulation of the IDEAS optimal process synthesisproblem

Having presented the governing equations for the IDEASconceptual framework, we are now poised to set up thegeneral IDEAS synthesis formulation. To do so, we %rstprovide a description of the objective function. For theapplication of IDEAS presented in this paper, the lin-ear objective � is used, as shown in Eq. (11). The term i(j)ui(j) is the linear contribution of the external inletstreams to the objective function, where i ∈ ‘∞. Simi-larly, i(j)yi(j); "i(j)xi(j), and !i(j)wi(j) are the lin-ear contributions of the external outlet, OP outlet, andOP inlet to the objective function, respectively—where i; "i; !i ∈ ‘∞. The %nal term is the linear contribution ofthe internal DN cross-8ow streams to the objective function,$yui(j; k)zyui(j; k); $yxi(j; k)zyxi(j; k); $wui(j; k)zwui(j; k),and$wxi(j; k)zwxi(j; k)where$yui; $wui; $yxi,and$wxi ∈ ‘∞(2).The objective function can now be written in complete

form

�=N∑i=1

∞∑j=1

[ i(j)ui(j) + i(j)yi(j) + !i(j)wi(j)

+ "i(j)xi(j) +∞∑k=1

$yui(j; k)zyui(j; k)

+∞∑k=1

$yxi(j; k)zyxi(j; k) +∞∑k=1

$wui(j; k)zwui(j; k)

+∞∑k=1

$wxi(j; k)zwxi(j; k)

]: (11)

Combining the feasible region generated by Eqs. (1)–(8),the objective function de%ned by Eq. (11), and the variablesets described by Eqs. (9) and (10), the general chemicalprocess network synthesis formulation, within the IDEASconceptual framework, can be written as an in%nite dimen-sional linear program (P2):

(P2)inf

u;w; x; y; z�

subject to (1–10):(12)

For an extensive discussion on the important characteris-tics and attractive features of (P1), the reader is referred tothe literature (Wilson & Manousiouthakis, 2000). Next wedemonstrate how IDEAS can be used for the synthesis ofcost-optimal power cycles.

5. Pure ammonia power cycle design case study

In the case study, we consider the synthesis and design of abottoming cycle. We seek to synthesize a power cycle design

4296 L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305

Table 1Stream data for the power cycle synthesis case study

Utility data

Number of working 8uid families 4Number of hot utility families 1Number of cold utility families 1

Working 8uid (ammonia,i = 1; : : : ; 4)

Number of inlets, �u(i) 0Low pressure (i = 1) 10:3 bar (subcritical)Intermediate pressures (i = 2; 3) 50:0 bar; 105:0 bar (subcritical)High pressure (i = 4) 200:0 bar (supercritical)Regenerator heat transfer 0:04 kW m−2 K−1

coe<cientUtility costs, cu(i) None

Hot utility (hot gas, i = 5)Number of inlets, �u(5) 1Supply 8owrate, u5(1) 496 kg s−1

Supply temperature 795 KTarget temperature Variable (291–795 K)Speci%c heat capacity 1:1001 kJ kg−1K−1Boiler heat transfer coe<cient 0:04 kW m−2 K−1Utility costs, cu(5) None

Cold utility (cooling water, i = 6)Number of inlets, �u(6) 1Supply 8owrate, u6(1) VariableSupply temperature 291 KTarget temperature Fixed at 297 KSpeci%c heat capacity 4:184 kJ kg−1 K−1Condenser heat transfer 2:969 kW m−2 K−1

coe<cientCooler heat transfer coe<cient 0:597 kW m−2 K−1Utility costs, cu(6) 502:20 $ kg−1

that utilizes pure ammonia to recover heat from exhaust gasfrom a combustion turbine. The potential economic bene%tof implementing such an operation has been underscoredin the works of Kalina, Leibowitz, Markus, and Pelletier(1991) and Tetelbaum (1993).In employing the IDEAS approach, we consider a DN

consisting of six stream families (N = 6), namely a hot ex-haust gas utility family, a cooling water utility family, andfour isobaric ammonia families. To simplify computations,mixing is restricted to streams with similar quality proper-ties. Stream data and speci%cations are supplied in Table1. We also consider an OP consisting of three technologyblocks (M =3), namely a heat exchanger network, a turbinenetwork, and a pump network. Technology cost data aresupplied in Table 2. Next we discuss modeling, sizing, andcosting of all the equipment types needed for a power cycle.We then couple these cost and equipment models with avail-able thermodynamic data (discussed earlier) and the designspeci%cations given in Table 1 and apply the general IDEASproblem formulation in (P2) to synthesize a globally opti-mal power cycle. Finally, we analyze the resulting design(s)and make comparisons with “perfect” power cycle designs(Table 3).

Table 2Technology cost data for the power cycle case study

Technology data

Number of technologies: 3

Technology 1 (heat exchange)Cost of operation, cw;1(1) NoneProduct sales price, cw;2(1) NoneEquipment cost, cw;3(1) 507:0 $ m−2

Technology 2 (vapor expansion)Cost of operation, cw;1(2) NoneProduct sales price, cw;2(2) 390 $ kW−1

Equipment cost, cw;3(2) 651:80 $ kW−1

Technology 3 (liquid compression)Cost of operation, cw;1(3) 390 $ kW−1

Product sales price, cw;2(3) NoneEquipment cost, cw;3(3) 1827:02 $ kW−1

Table 3Comparison data between “perfect” and ammonia power cycles

& = 0:79 “Perfect” IDEAS “Perfect”Lnet = 89; 898 kW cycle cycle cycle

(condensing) (GPP-575) (cooling)

Cold utility 8owrate, (kg s−1) 6043 5665 6057.5Hot gas outlet temp. (K) 352 370 352Thermodynamic e<ciency 0.37 0.33 0.37TT 1h (K) 79 70 70TT 2h (K) 55 58.0 48.5TT 1c (K) 6 9 12TT 2c (K) 4 4 8Total heat (m2=1000) 101 119 129Transfer AreaAnnualized (10 year) CC, ($=106) 11.0 12.8 12.4

i

w (k)x ( j )

(k)δ

w (k)

( j ) Cold Sideγ (k)δ

D (k)

1

Hot Sidei

3

1

i 2

( j )γi 11

( j )xi 11

2i2

i2i 22

Fig. 2. Heat exchanger equipment model.

5.1. Process unit modeling

Our proposed system requires three technologies(M =3)—heat exchangers, pumps and turbines—integratedto achieve power generation. Heat exchanger, turbine, andpump equipment are shown in Figs. 2–4.Our cycle employs only single component 8uids (pure

ammonia, hot gas, cooling water). Therefore, we need only

L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305 4297

Fig. 3. Turbine equipment model.

Fig. 4. Pump equipment model.

specify pressure and speci%c enthalpy data for each ammoniafamily, and only speci%c enthalpy for the utility families;thus, nq(1)=nq(2)=nq(3)=nq(4)=2 and nq(5)=nq(6)=1.These vectors can be written in explicit form

�i(j) =

[P�i(j)

H �i(j)

]; �i(j) =

[P�i(j)

H �i(j)

];

�i(k) =

[P�i(k)

H �i(k)

]; �i(j) =

[P�i(j)

H �i(j)

];

where i = 1; : : : ; 4 and

�i(j) = [ H �i(j) ]; �i(j) = [ H �i(j) ];

�i(k) = [ H �i(k) ]; �i(j) = [ H �i(j) ];

where i = 5; 6. According to the Gibbs–Duhem equation,other quality information (8uid temperature, speci%c en-tropy and so on), can be de%ned in terms of the vectors

given above; for example, T�i(j)=T (H �i(j); P�i(j)) andS�i(j)=S(H �i(j); P�i(j)).We are now in the position to completely de%ne unit

operation models for the technology blocks. Let us recall theOP inlet condition vectorD(k), which contains the necessarydesign parameters for the technology employed at the kthjunction. The %rst element of this vector, D1(k), designatesthe technology (either 1, 2, or 3) employed at condition k.D2(k) designates the primary family of the unit operationat junction k. The term primary family is used to denotethe stream family used as a basis for equipment capacityand sizing parameters (speci%c power, speci%c area, etc.)and equipment and utility cost coe<cients. For convention,hot stream families are primary for heat exchangers, highpressure families are primary for turbines, and low pressurefamilies are primary for pumps.

5.1.1. Technology 1: heat exchanger (HEX) modelThe HEX technology block consists of four process unit

types (boilers, condensers, coolers, and regenerators) whichare characterized by three possible stream interactions: heatexchange between a hot utility stream and a working 8uidstream (boiler), heat exchange between a cold utility streamand a working 8uid stream (condensers and coolers), andheat exchange between working 8uid streams (regenera-tors). The process model for all of these unit types is asfollows: consider the OP-inlet junction k, which enters theHEX technology block (m= 1) and has inlets from familyi1 and family i2; let also the outlet conditions be j1 in familyi1 and j2 in family i2 (see Fig. 2). The resulting modelingequations are as follows:

D1(k) = 1; (13)

D2(k) = i1; (14)

D3(k) = LH(k)=H �i1 (k)− H �i1 (j1); (15)

where nd(D1(k))=nd(1)=3; and D3(k) (the unit’s capacityparameter) is the speci%c heat load of the HEX unit, LH(k),with respect to the primary family. Further, we can evaluateand specify elements of R(k) as follows:

Ri1 ;i1 (k) =− H �i1 (k)− H �i1 (j1)

H �i2 (j2)− H �i2 (k); i1¡i2; (16)

Ri1 ;i2 (k) = 1; i1¡i2; (17)

Ri2 ;i1 (k) =− H �i1 (k)− H �i1 (j1)

H �i2 (j2)− H �i2 (k); i2¡i1; (18)

Ri2 ;i2 (k) = 1; i2¡i1; (19)

Ri; i(k) = 1; i ¡max(i1; i2); i = min(i1; i2); (20)

Ri; i+1(k) = 1; i¿max(i1; i2): (21)

4298 L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305

All other elements of R(k) are zero. The corresponding el-ements of the F operator are

Fi1i1 (j1; k) = 1; (22)

Fi2i2 (j2; k) = 1: (23)

We can then establish that: the log mean temperature diIer-ence, .(k), and the speci%c surface area, A(k) are

.(k) =(T�i1 (k)− T�i2 (j2))− (T�i1 (j1)− T�i2 (k))ln (T�i1 (k)− T�i2 (j2))=(T�i1 (j1)− T�i2 (k))

; (24)

A(k) =D3(k)

.(k)U (k); (25)

where U (k) is the overall heat transfer coe<cient(kW m−2 K−1). The speci%c heat transfer area for the unitA(k) is calculated while accounting for 8uid heat capacitychanges in the unit. We can also establish two other dimen-sionless parameters—NTU(k), the number of transfer unitsand 11(k), the heat exchanger eIectiveness (e<ciency).They are de%ned (Incoprera & DeWitt, 1996) as

NTU(k) =Tdf(k).(k)

(26)

11(k) =

1−exp[−NTU(k)(1−C(k))]

1−C exp[−NTU(k)(1−C(k))]; C(k)¡1;

NTU(k)1 + NTU(k)

; C(k) = 1;

(27)

where

Tdf(k) = max(T�i1 (k)− T�i1 (j1); T �i2 (j2)− T�i2 (k)); (28)

Cr(k) =min([

T�i1 (k)− T�i1 (j1)T�i2 (j2)− T�i2 (k)

];

×[T�i2 (j2)− T�i2 (k)T�i1 (k)− T�i1 (j1)

])(29)

are the maximum heat driving force, Tdf(k), and the heatcapacity ratio, Cr(k).

5.1.2. Technology 2: turbine expander (TUR) modelNow consider that the OP inlet junction k enters the TUR

technology block (m = 2) and has one inlet stream fromhigh pressure family i1 and one outlet stream to low pressurefamily i2; let also the unit outlet stream condition be j. Theresulting model is as follows:

D1(k) = 2; (30)

D2(k) = i1; (31)

D3(k) = LT (k)=H �i1 (k)− H �i2 (j); (32)

where nd(D1(k)) = nd(2) = 3 and D3(k) (the unit’s capac-ity parameter) is the speci%c power output (load) for theTUR unit, LT (k), with respect to D2(k). In addition, we canevaluate and specify elements of R(k) and Fi1 as follows:

Ri; i(k) = 1; i ¡ i1; (33)

Ri; i+1(k) = 1; i¿ i1: (34)

All other elements of R(k) are zero, and the correspondingelements of the F operator are

Fi1i2 (j; k) = 1: (35)

We now de%ne the turbine e<ciency, 12(k). Let Wj denotethe outlet condition at which isentropic expansion betweenfamilies i1 and i2 is achieved, i.e. S�i1 (k) = S�i2 ( Wj)

12(k) =H �i1 (k)− H �i2 (j)

H �i1 (k)− H �i2 ( Wj): (36)

5.1.3. Technology 3: pump compressor (PMP) modelFinally, consider that the OP inlet junction k enters the

PMP technology block (m=3) and has one inlet stream fromlow pressure family i1 and one outlet stream to high pressurefamily i2; and consider the unit outlet junction condition tobe j. The resulting modeling equations are as follows:

D1(k) = 3; (37)

D2(k) = i1; (38)

D3(k) = LP(k)=H �i1 (k)− H �i2 (j); (39)

where nd(D1(k))=nd(3)=3 andD3(k) (capital cost capacityparameter) is the speci%c power input (load) required for thePMP unit, LP(k), with respect to 8ow from D2(k). Elementsof R(k), and Fi1 are as follows:

Ri; i(k) = 1; i ¡ i1; (40)

Ri; i+1(k) = 1; i¿ i1: (41)

All other elements of R(k) are zero, and the correspondingelements of the F operator are

Fi1i2 (j; k) = 1: (42)

We now de%ne the pump e<ciency, 13(k). The Wj denotesthe outlet condition at which isentropic expansion betweenfamilies i1 and i2 is achieved, i.e. S�i1 (k) = S�i2 ( Wj).

13(k) =H �i1 (k)− H �i2 ( Wj)

H �i1 (k)− H �i2 (j): (43)

5.2. Case study cost objectives

The convex (linear) objective function of (P2) is ableto capture a variety of practically relevant objectives. Forexample, a utility cost objective is represented by setting i="i=!i=$yui=$wui=$yxi=$wxi={0} in Eq. (11) and also

L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305 4299

i= {0}, except for the %rst �u(i) terms which represent thecost coe<cients for each of the external DN-inlet streams.The objective function given by Eq. (11) can also handle netutility pro%t, annualized capital cost, and total annualizedpro%t functions using appropriately selected !i’s, as long asequipment costs are directly proportional to their capacityparameter. Accordingly, we write the IDEAS minimum area(MA) objective for the power cycle synthesis problem, a%nite estimation of Eq. (11), as follows:

(P3) infu;y;x;w;z

N∑i=1

∞∑k=1

!i(k)wi(k) (44)

subject to Eqs. (1)–(10) and

N∑i=1

�u(i)∑j=1

ui(j)H �i(j)−∞∑o=1

yi(o)H �i(o)

= &Lmaxnet ;

where & is the fraction of the maximum deliverable netpower (Lmaxnet ) required in the %nal power cycle design. Thecomputing of Lmaxnet is discussed in the appendix. Values of!i(k) are given below

!i(k) =

{A(k) if D1(k) = 1;

0 if D1(k) = 2; 3:

5.3. Results and discussion

Using data from Tables 1 and 2, we were able to suc-cessfully employ the IDEAS methodology to identify theglobally minimum heat exchange area (MA) of an ammo-nia power cycle featuring a given power output. To accom-plish this goal, %rst the aforementioned data were employedto identify the maximum deliverable power of the systemas 113; 794 kW. The supporting calculations are shown inthe appendix. Rigorous tradeoI curves between convergedminimum annualized capital cost and the percentage of themaximum deliverable power, &=1%; : : : ; 90%; and betweenconverged minimum total heat exchange area and the per-centage of the maximum deliverable power are given inFigs. 5 and 6. From Figs. 5 and 6, in8ection in the curve canbe observed near & = 79%. Beyond this percentage, mini-mum annualized capital costs begin to explode; thus makingthis range clearly unattractive. So, we now consider capitalcost optimal designs that deliver 89; 898 kW of power, or&= 79%.Due to the large scale nature of IDEAS formulations,

candidate design conditions were considered in an iterative,sequentially increasing manner. As the number of candidateconditions was increased, successive numerical approxima-tions of (P3) converged to a global minimum CC solution.A convergence plot of CC vs the number of candidate con-ditions is shown in Fig. 7. For illustrative purposes, the opti-mal power cycle design using 149 conditions (GPP-149) is

0

50000

100000

150000

200000

250000

300000

350000

400000

0 10 20 30 40 50 60 70 80 90

Hea

t Exc

hang

e A

rea

(m2 )

Percentage of Maximimum Deliverable Net Power

Fig. 5. Heat exchange area minima versus Lorentz percentage (&).

Ann

ualiz

ed C

apita

l Cos

t ($/

yr)

Lorentz Percentage

Fig. 6. Annualized capital cost versus Lorentz percentage (&).

12.5

13

13.5

14

14.5

15

15.5

100 150 200 250 300 350 400 450 500 550 600

Tot

al A

nnua

lized

Cos

t ($

in m

illio

ns)

Number of Candidate Conditions

Fig. 7. IDEAS CC convergence plot for the power cycle case study.

provided in tabular form in Tables 4–8.This design possessesexcellent heat recovery characteristics as shown in Fig. 8.It is able to largely overcome the ine<ciencies resultingfrom the evaporation of ammonia at constant temperature.

4300 L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305

Table 4Heat exchanger data (1–10) for the optimal GPP-149 cycle

w(k) P�(k)∗ T�(k)∗ T�i(j)∗

L(k) A(k) 11k Flow vector (bar) (K) (K) (kW) (m2)

1[4:3 95:7 0:0

0:0 0:0 0:0]

[10:3

50:0

] [354:00

296:0

] [314:0

297:0

]451.48 58.73 0.69

2[1:9 42:8 0:0

0:0 0:0 0:0]

[10:3

50:0

] [354:00

297:0

] [314:0

298:0

]202.60 27.27 0.70

3[11:7 243:7 0:0

0:0 0:0 0:0]

[10:3

50:0

] [394:00

299:0

] [354:0

300:0

]1157.59 69.29 0.42

4[4:6 95:7 0:0

0:0 0:0 0:0]

[10:3

50:0

] [394:00

300:0

] [354:0

301:0

]454.35 27.58 0.43

5[4:6 95:7 0:0

0:0 0:0 0:0]

[10:3

50:0

] [394:00

301:0

] [354:0

302:0

]456.26 28.10 0.43

6[4:6 95:7 0:0

0:0 0:0 0:0]

[10:3

50:0

] [394:00

302:0

] [354:0

303:0

]456.26 28.52 0.43

7[0:0 0:0 0:0

35:7 0:0 144:8]

[200:0

1:0

] [314:00

291:0

] [292:0

297:0

]3634.37 1071.19 0.96

8[0:0 0:0 0:0

25:3 78:2 0:0]

[200:0

1:0

] [714:00

795:0

] [754:0

755:0

]3439.55 2097.29 0.49

9[20:4 23:1 0:0

0:0 0:0 0:0]

[10:3

50:0

] [394:00

296:0

] [314:0

333:0

]4187.06 535.79 0.82

10[0:0 29:2 0:0

0:0 78:2 0:0]

[50:0

1:0

] [554:00

755:0

] [714:0

595:0

]13758.21 8624.12 0.80

The design features 6 turbines, 2 pumps, and 25 heat ex-changers and demonstrates heat integration among the iso-baric ammonia families (10:0 bar with 50:0 bar and 10:0 barwith 200:0 bar) in the HEX operator. The converged mini-mum heat exchange area included 575 candidate conditions(GPP-575) and was 119 × 103 m2. The heating curves forthe converged solution (GPP-575) are given in Fig. 9.To appreciate the eIectiveness of the obtained MA glob-

ally optimal ammonia cycle, we will compare it to the opti-mal MA of a perfect cycle that employs an imaginary 8uidwith the following characteristics:

(1) Constant vapor and liquid heat capacity, whose ratiocan be chosen at will.

(2) Vapor and liquid isobars of the 8uid can be arbitrar-ily close to each other at a point of the H–S diagramthat can be chosen at will (so that no pumping work isrequired to switch isobars).

(3) The heat transfer coe<cients between this 8uid and theavailable hot and cold utilities are the same as thosebetween the ammonia and these same utilities.

Under these premises, a cycle of this imaginary 8uid canbe placed within the available hot and cold utilities so as tooptimize area, while at the same time ensuring a given levelof power production. The associated optimization problemis detailed and solved in the Appendix for two perfect cycles.In the %rst perfect (condensing) 8uid cycle, we consider 8uidcondensation at the condenser exit, such that heat exchangetakes place between the cooling water and turbine exhaust(vapor phase). In the second perfect 8uid cycle (cooling),8uid condensation takes place at the condenser entrance.Here the heat exchanger unit acts eIectively as a cooler.Solution data for the converged optimal design (GPP-575)and the two perfect cycles can be found in Table 3. Fromthe table, we see that there is little diIerence between the

L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305 4301

Table 5Heat exchanger data (11–20) for the optimal GPP-149 cycle

w(k) P�(k)∗ T�(k)∗ T�i(j)∗

L(k) A(k) 11k Flow vector (bar) (K) (K) (kW) (m2)

11[0:0 0:0 0:0

13:3 81:4 0:0]

[200:0

1:0

] [314:00

555:0

] [514:0

355:0

]17908.23 14123.09 0.83

12[99:4 0:0 0:0

0:0 0:0 4856:5]

[10:3

1:0

] [314:00

291:0

] [295:0

297:0

]121917.61 8995.26 0.83

13[72:9 0:0 0:0

30:7 0:0 0:0]

[10:3

200:0

] [554:00

292:0

] [314:0

514:0

]44317.13 16695.51 0.92

14[0:0 0:0 0:0

43:0 246:4 0:0]

[200:0

1:0

] [314:00

595:0

] [554:0

355:0

]65042.85 42350.20 0.85

15[19:3 0:0 0:0

0:0 0:0 1024:3]

[10:3

1:0

] [354:00

291:0

] [295:0

297:0

]25713.08 1639.84 0.94

16[0:0 0:0 0:0

26:7 143:8 0:0]

[200:0

1:0

] [314:00

635:0

] [594:0

355:0

]44289.93 24767.33 0.87

17[0:0 0:0 0:0

5:1 24:5 0:0]

[200:0

1:0

] [292:00

675:0

] [634:0

315:0

]9690.59 7526.81 0.94

18[0:0 3:7 0:0

0:0 6:8 0:0]

[50:0

1:0

] [634:00

795:0

] [714:0

675:0

]897.87 384.50 0.75

19[0:0 0:0 3:7

0:0 7:1 0:0]

[105:0

1:0

] [634:00

795:0

] [714:0

675:0

]940.77 401.57 0.75

20[0:0 0:0 0:0

5:1 10:5 0:0]

[200:0

1:0

] [634:00

795:0

] [714:0

675:0

]1391.56 590.78 0.75

IDEAS power cycle operating conditions (with a pure am-monia working 8uid) and the better of the two perfect work-ing 8uid cycles. This comparison further substantiates thepower of the IDEAS approach to identify MA optimal powercycles with real (in this case, ammonia) working 8uids.

6. Conclusions

In this work, we demonstrated the application of IDEASto the synthesis of area-optimal power cycles. In applyingthe IDEAS approach to power cycle design, we have de-veloped a systematic methodology which guarantees theglobal optimality of the obtained designs. The methodologywas demonstrated in a case study involving a pure am-monia working 8uid, bottoming cycle. The methodologyemployed real thermodynamic data and yielded a convergedsolution for the in%nite problem using a %nite numberof junctions. Comparisons with other power cycles em-

ploying perfect (though imaginary) working 8uids, ascer-tained the e<ciency and optimality of the obtained IDEASdesigns.

Notation

N number of stream familiesM number of technologiesi; i1; i2 family indicatorsm technology indicatorj; j1; condition indicatorsj2; k; o; nu; y; w; x extensive quantity (8ow) variableszyui; zyxi; extensive quantity (cross-8ow) variableszwui; zwxi

�; �; �; �; � intensive quality information (pressure, tem-perature, etc.)

D OP design parameters

4302 L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305

Table 6Heat exchanger data (21 and above) in the optimal GPP-149 cycle

w(k) P�(k)∗ T�(k)∗ T�i(j)∗

L(k) A(k) 11k Flow vector (bar) (K) (K) (kW) (m2)

21[0:0 39:9 0:0

0:0 81:0 0:0]

[50:0

1:0

] [594:00

795:0

] [714:0

635:0

]14255.62 6139.91 0.80

22[0:0 0:0 0:0

26:7 62:8 0:0]

[200:0

1:0

] [594:00

795:0

] [714:0

635:0

]11052.92 4657.08 0.80

23[0:0 0:0 0:0

43:0 110:3 0:0]

[200:0

1:0

] [554:00

795:0

] [714:0

595:0

]24278.68 10027.47 0.83

24[0:0 0:0 0:0

14:7 57:8 0:0]

[200:0

1:0

] [514:00

795:0

] [754:0

595:0

]12725.93 5037.15 0.85

25[0:0 0:0 0:0

29:3 81:4 0:0]

[200:0

1:0

] [514:00

795:0

] [714:0

555:0

]21489.88 8485.19 0.85

Table 7Turbine information for the optimal GPP-149 cycle

w(k) P�(k)∗ T�(k)∗ P�i(j)∗

T�i(j)∗

L(k) 12(k)k Flow vector (bar) (K) (bar) (K) (kW)

1[0:0 72:9 0:0

0:0 0:0 0:0][50.0] [714.00] [10.3] [554.0] 31145.22 0.90

2[0:0 0:0 45:9

0:0 0:0 0:0][105.0] [634.00] [10.3] [394.0] 25059.80 1.00

3[0:0 0:0 29:2

0:0 0:0 0:0][105.0] [634.00] [50.0] [554.0] 5453.66 0.93

4[0:0 0:0 3:7

0:0 0:0 0:0][105.0] [714.00] [50.0] [634.0] 776.58 0.89

5[0:0 0:0 0:0

39:9 0:0 0:0][200.0] [754.00] [50.0] [594.0] 16192.57 0.96

6[0:0 0:0 0:0

78:8 0:0 0:0][200.0] [714.00] [105.0] [634.0] 15531.37 1.00

Table 8Pump information for the optimal GPP-149 cycle

w(k) P�(k)∗ T�(k)∗ P�i(j)∗

T�i(j)∗

L(k) 13(k)k Flow vector (bar) (K) (bar) (K) (kW)

1[118:7 0:0 0:0

0:0 0:0 0:0][10.3] [295.00] [50.0] [296.0] −777:77 1.00

2[0:0 118:7 0:0

0:0 0:0 0:0][50.0] [309.00] [200.0] [314.0] −3483:91 0.86

R matrix of linear interrelations between w8ows

F matrix of linear interrelations between w andx 8ows

� IDEAS objective functionv; ; $; !; " contributions of extensive quantities to the

objective functionP pressure

L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305 4303

800

700

600

500

400

300

Tem

pera

ture

(K

elvi

n)

200000150000100000500000Heat Load (kiloJoules)

PLOT 1

Hot Gas Utility

Cold Water Utility

High Pressure Ammonia

Low Pressure Ammonia

Fig. 8. Composite T-H diagram for the CC optimum power cycle design at 149 conditions.

T temperatureH speci%c enthalpyS speci%c entropy. log mean temperature diIerenceA speci%c surface area (heat transfer)U overall heat transfer coe<cient1 unit e<ciency (eIectiveness)NTU number of transfer units (heat exchanger)Tdf maximum heat driving forceCr heat capacity ratio& percentage of maximum deliverable powerLT speci%c powerLmaxnet maximum deliverable power

Spaces of in:nite sequences

‘1 sequences of real numbers with %nite abso-lute sum:

∑∞i=1 |a(i)|¡∞

‘∞ sequences of real numbers with %nite maxi-mum absolute value: supi=1:::∞|a(i)|¡∞

‘n1 sequences of real n-dimensional vectors with

%nite absolute sum:∑∞

i=1

∑nj=1 |aj(i)|¡∞

‘n∞ sequences of real n-dimensional vec-

tors with %nite maximum absolute value:maxi=1:::∞maxj=1:::n |aj(i)|¡∞

7 sequences of real numbers with the %rst Nelements non-zero

‘1(m) m-tupled sequences of real numbers,f(k1; : : : ; km), with %nite absolute sum:∑∞

k1=1 · · ·∑∞

km=1 |f(k1; : : : ; km)|¡∞

Other sets

Rn n-dimensional vectors with all elements realQn n-dimensional vectors with all elements real,

rationalRn×K n× K matrices with all elements real

Acknowledgements

Financial support from the National Science FoundationAward Nos. GER-9554570, CTS-9528653, CTS-9876489,UCLA Project ’88, and the U.S. Department of Educationis gratefully acknowledged.

Appendix

In synthesizing power cycles, it is extremely important to%rst establish net power generation limits, for a power cycleoperating between a hot utility with known 8ow rate (wh),heat capacity (cp;h), and inlet temperature (T inh ) and a coldutility with known heat capacity (cp;c), and inlet (T inc ) andoutlet (T outc ) temperatures. This maximum net power, Lmaxnet ,can be evaluated through %rst and second law analysis.From %rst law principles, we can obtain an expression for

the maximum net power that can be produced from such asystem

Lmaxnet = whcp;h(T inh − T outh )− wccp;c(T outc − T inc ): (A.1)

4304 L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305

800

700

600

500

400

300

Tem

pera

ture

(K

elvi

n)

200000150000100000500000Heat Load (kJ)

PLOT 2

Hot Gas Utility

Cold Water Utility

High Pressure Ammonia

Low Pressure Ammonia

Fig. 9. Composite T-H diagram for the CC optimum power cycle design at 575 conditions.

If further, the system is assumed isolated, the total entropychange of the system becomes equal to the sum of the utilitystream entropy changes

TStot = TSutil = TSh +TSc = 0; (A.2)

where TSc=wccp;c ln T outc =T inc and TSh=whcp;h ln T outh =T inhfor hot/cold utility streams with constant speci%c heat.For the generated power to be maximized, it must hold

that

T outh = T inc : (A.3)

The resulting T–H diagram for the cycle is shown in Fig.10. The maximum net power can be written as

Lmaxnet = whcp;h[(T inh − T inc )− f(T outc − T inc )]; (A.4)

where f = ln T inh =T incln T outc =T inc

= wccp; c

whcp; h.

Let us now consider a second cycle, shown in Fig. 11,which delivers a percentage, &, of Lmaxnet . Here WT outh ¿T inc .The perfect working 8uid has 8owrate wf; vapor and liq-uid heat capacities cp;hf and cp;cf, respectively; undergoesisobaric heating from a to b; isentropic expansion from bto c; isobaric cooling from c to a; and perfect (isentropic,isothermal, isenthalpic) compression at a. The net power forthe cycle is

Lnet =wfcp;hf[(Tb − Ta)− f′(Tc − Ta)]

= &Lmaxnet = Qh − Qc;

where

Qh = whcp;h(T inh − WT outh ) = wfcp;hf(Tb − Ta); (A.5)

Tcout

Thin

Thout

=

1

Tcin

Tem

pera

ture

Enthalpy (Heat Duty)

hw

cw p,cc

cp,h

1

Fig. 10. T-H diagram for an isolated reversible cycle operating betweenone hot utility and one cold utility.

Qc = Wwccp;c(T outc − WT inc ) = wfcp;cf(Tc − Ta); (A.6)

f′ =ln Tb=Ta

ln Tc=Ta=

wfcp;cf

wfcp;hf: (A.7)

The minimum capital cost of the cycle can then be identi%edbased on turbine and heat exchanger equipment only. Sincewe know the turbine load, &Lmaxnet , (and thus the turbine cap-ital cost)—we need only minimize the area of the two heatexchangers. The resulting minimum area formulation is asfollows:

(P4) minWT outh ;Ta;Tb;Tc

Qh

UhTT 1h−TT 2hln TT 1h =TT 2h

+Qc

UcTT 2c −TT 1cln TT 2c =TT 1c

(A.8)

L. L. Martin, V. I. Manousiouthakis / Chemical Engineering Science 58 (2003) 4291–4305 4305

Thout

Tcout

Tcin

Thin

Tc∆ 2

Tc∆

Th∆ 1

Th∆ 2

cp,hwh

1

cp,cw

{c

1

cp,hfwf

1

cp,cfwf

1

Enthalpy (Heat Duty)

Tem

pera

ture

c

}

b

}

1{a

Fig. 11. T-H diagram for an isolated reversible cycle operating betweenone hot utility and one cold utility.

subject to

TT 1h = T inh − Tb; (A.9)

TT 2h = WT outh − Ta; (A.10)

TT 1c = Ta − T inc ; (A.11)

TT 2c = Tc − T outc ; (A.12)

Qh = &Lmaxnet

[Tb − Ta

(Tb − Ta)− (Tc − Ta)f′

]; (A.13)

Qh = whcp;h(T inh − WT outh ); (A.14)

Qc = &Lmaxnet

[Tc − Ta

(Tb − Ta)1=f − (Tc − Ta)

]; (A.15)

f′ =ln Tb=Ta

ln Tc=Ta; (A.16)

TT 1h ;TT 2h ;TT 1c ;TT 2c ; Qh; Qc¿ 0; (A.17)

f′¿ 1: (A.18)

This is a non-convex optimization problem with three de-grees of freedom (Ta; Tb; Tc) whose global solution can befound through explicit search.

References

El-Sayed, Y., & Tribus, M. (1985a). A theoretical comparison of theRankine and Kalina cycles. In Analysis of Energy Systems: Designand Operation: ASME Journal, 97–102 AES-Vol I.

El-Sayed, Y., & Tribus, M. (1985b). Thermodynamic properties of water–ammonia mixtures: Theoretical implementation for use in power cyclesanalysis. In Analysis of Energy Systems: Design and Operation:ASME Journal, 89–95 AES-Vol I.

Friend, D. G., & Haynes, W. M. (1996). Report of the workshopon thermophysical properties of ammonia/water mixtures. NationalInstitute of Standards and Technology Interagency Report.

Incoprera, F., & DeWitt, D. (1996). Introduction to heat transfer. NewYork: Wiley.

Justanieah, A., & Manousiouthakis, V. (2000). Globally optimal multi-component membrane networks synthesis. In Energy and theenvironment topical conference proceedings, American Institute ofChemical Engineers, process Design for Pollution Prevention, Session44.

Kalina, A. I., Leibowitz, H. M., Markus, D. W., & Pelletier, R. I.(1991). Further technical aspects and economics of a utility-sizekalina bottoming cycle. ASME Papers 91-GT-365.

Kutney, K. H. ., et al. (1997). Hard sphere volume translated van derwaals equations of state for supercritical process modeling. FluidPhase Equilibria, 128(1–2), 149–171.

Martin, L., & Manousiouthakis, V. (2000a). Globally optimalammonia/water power cycle network synthesis via in%nite dimensionalstate-space (ideas) optimization. In Energy and the environmenttopical conference proceedings, American Institute of ChemicalEngineers, process Integration of Material and Energy, Session 47.

Martin, L., & Manousiouthakis, V. (2000b). Minimum utility costsfor non-isobaric azeotropic distillation networks. In Energy andthe environment topical conference proceedings, American Instituteof Chemical Engineers, design of Thermodynamically E<cientDistillation Networks, Session 48.

Martin, L., & Manousiouthakis, V. (2000c). Pure ammonia powercycle synthesis featuring a mtac objective via ideas. In Energy andthe environment topical conference proceedings, American Instituteof Chemical Engineers, management of Energy and EnvironmentalResources, Session 50.

Tetelbaum, S. (1993). High e<ciency supercritical bottoming cycle.Latest Advances in Power Generating Facilities: Design, Operationand Maintenance and Environmental Improvements, 22, 57–62(aSME PWR).

Tillner-Roth, R., & Friend, D. G. (1998a). Helmholtz free energyformulation of the thermodynamic properties of the mixture(water + ammonia). Journal of Physical and Chemical ReferenceData, 27(1), 63–96.

Tillner-Roth, R., & Friend, D. G. (1998b). Survey and assesment ofavailable measurements on thermodynamic properties of the mixture(water-ammonia). Journal of Physical and Chemical ReferenceData, 27(1), 45–62.

Uehara, H., Ikegami, Y., Fukugawa, H., & Uto, M. (1994). Performanceanalysis of OTEC system using Kalina cycle (Thermodynamiccharacteristics of cycle). Transactions of the Japan Society ofMechanical Engineers, 60(578), 3519–3525.

Wilson, S., & Manousiouthakis, V. (2000). IDEAS approach to processnetwork synthesis: Application to multicomponent MEN. A.I.Ch.E.Journal, 46(12), 2408–2416.