EURO Summer Course onSustainable Assessment of Clean Air Technologies

Lectures 1-3

Fundamentals of Thermoeconomics

Antonio Valero*, Luis Serra, Javier UcheCIRCE, Center of Research for Energy Resources and Consumption

Centro Politécnico Superior, Universidad de ZaragozaZaragoza, Spain

Aim

The aim of this lecture is to introduce Thermoeconomics as an analytical andpowerful tool for the cost accounting, diagnosis, improvement, optimization anddesign of energy systems using the combination of Second Law ofThermodynamics and Economics.

OBJECTIVES

The lectures will comprise the following objectives:

§ A vision about the importance of Thermoeconomics as a tool for improvingenergy systems and prevent damage on envioronment.

§ Introduce the basic concepts of exergy, cost, exergetic and monetary costs,fuel, product, unit exergetic consumption and efficiency.

§ Describe the process of cost formation and distinguish between the physicaland the thermoeconomic plant models. The productive structure and itsmathematical representation.

§ Analyze the basic concepts to diagnose and optimize energy systems usingthermoeconomics.

* Director of CIRCE and Chair on Thermal Systems

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CONTENTS

1. Introduction to Thermoeconomics

2. Basic Concepts.

2.1. The Concept of Cost.

2.2. Fuel, Product and Unit Exergetic Consumption.

2.3. Physical and Thermoeconomic Plant Models.

3. Calculating Thermoeconomic Costs

3.1. Economic Resources and Thermoeconomic Costs

4. Thermoeconomic Applications to Complex Energy Systems .

4.1. Operation Thermoeconomic Diagnosis

4.1.1. Technical Exergy Saving

4.1.2. Impact on Resources Consumption

4.1.3. Malfunction and Dysfunction Analysis

4.1.4. Intrinsic and Induced Malfunctions

4.2. Thermoeconomic Optimization

KEYWORDS

Thermoeconomics, diagnosis, optimization, costs, exergy, exergetic cost,irreversibility, malfunction, fuel, product.

GLOSSARY

Thermoeconomics: Science that combines Thermodynamics and Economics inorder to avoid the natural resources consumption in processes.

Malfunction: Effect of an inefficiency in a/several process units of a system.

Exergy: Amount of available energy in a physical process.

Cost: Amount of resources to obtain a product.

1. Introduction to Thermoeconomics

As the human population grows, our finite world is becoming smaller and naturalresources are more and more scarce. We must conserve them in order to

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survive and Thermoeconomics plays a key role in this endeavor. We should findout how energy and resources degrade, which systems work better, how toimprove designs to reduce consumption and prevent residues from damaging theenvironment. Thermoeconomics and its application to engineering energysystems can help to answer these questions.

The production process of a complex energy system (e.g. a complex powerplant) can be analyzed in terms of its economic profitability and efficiency withrespect to resource consumption.

An economic analysis can calculate the cost of fuel, investment, operation andmaintenance for the whole plant but provides no means to evaluate the singleprocesses taking place in the subsystems nor how to distribute the costs amongthem.

On the other hand, a thermodynamic analysis calculates the efficiencies of thesubsystems and locates and quantifies the irreversibilities but cannot evaluatetheir significance in terms of the overall production process.

Thermoeconomic analysis combines economic and thermodynamic analysis byapplying the concept of cost (originally an economic property) to exergy (anenergetic property), (see e.g. Valero et al. (1986)). Most analysts agree thatexergy is the most adequate thermodynamic property to associate with costsince it contains information from the second law of thermodynamics andaccounts for energy quality (Tsatsaronis (1987, 1998), Gaggioli and El-Sayed(1987), Moran (1990)). Exergetic efficiency compares a real process to areversible one, (i.e. an ideal process of the same type). An exergy analysislocates and quantifies irreversibilities in a process. Exergy basedthermoeconomic methods are also referred to as “exergoeconomics”(Tsatsaronis and Winhold (1985)).

The physical magnitude connecting physics (thermodynamics) and economics isentropy generation or, more specifically, irreversibility. This represents the“useful” or available energy lost or destroyed (exergy destruction) in all physicalprocesses. All real processes in a plant are non-reversible and, as aconsequence, some exergy is destroyed and some natural resources areconsumed and lost forever, which creates cost. All natural resources have aneconomic cost: the more irreversible a process, the more natural resources areconsumed (higher energetic cost) and the higher the required investment (higherthermoeconomic cost). If we can measure this thermodynamic cost byidentifying, locating and quantifying the causes of inefficiencies in realprocesses, we can provide an objective economic basis using the cost concept.

Thus, thermoeconomics assesses the cost of consumed resources, money andsystem irreversibilities in terms of the overall production process. Consumed

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resource cost involves resources destroyed by inefficiencies and helps to pointout how resources may be used more effectively to save energy. Money costsexpress the economic effect of inefficiencies and are used to improve the costeffectiveness of production processes.

Assessing the cost of the various streams and processes in a plant helps tounderstand the process of cost formation, from the input resource(s) to the finalproduct(s). This process can solve problems in complex energy systems thatcannot normally be solved using conventional energy analysis based on the FirstLaw of Thermodynamics (mass and energy balances only), for instance:

1. Rational price assessment of plant products based on physical criteria.

2. Optimization of specific process unit variables to minimize final product costsand save resource energy, i.e. global and local optimization.

3. Detection of inefficiencies and calculation of their economic effects inoperating plants, i.e. plant operation thermoeconomic diagnosis.

4. Evaluation of various design alternatives or operation decisions andprofitability maximization.

5. Energy audits.

Specific examples of these applications will be given. Many reports also providespecific information about thermoeconomic applications (Lozano and Valero(1993), Tsatsaronis (1994), Lozano, Valero and Serra (1996), Valero et al.(1994), Bejan et al. (1997), Valero and Lozano (1997), Valero, Correas andSerra (1999), Lozano et al. (1994), Frangopoulos (1987), Von Spakovsky andEvans (1993), El-Sayed and Tribus (1983), El-Sayed (1988)).

Thermoeconomic methods can generally be subdivided into two categories(Tsatsaronis (1987)), those based on cost accounting (e.g. Exergetic CostTheory, Lozano et al. (1993), Average-Cost-Approach, Bejan et al. (1997),Last-In-First-Out Approach, Lazzareto and Tsatsaronis (1997)) and those basedon optimization techniques (e.g. Thermoeconomic Functional Analysis,Frangopoulos (1987), Engineering Functional Analysis, von Spakovsky andEvans (1993), Intelligent Functional Approach, Frangopoulos (1990)). Costaccounting methods help to determine actual product cost and provide a rationalbasis for pricing, while optimization methods are used to find the optimum designor operating conditions.

Unfortunately, there are almost as many nomenclatures as theories. This causesconfusion, complicates method comparison and impedes the development ofthermoeconomics in general (Tsatsaronis (1994)). The Structural Theory ofThermoeconomics (Valero et al. (1992,1993)) provides a general mathematicalformulation using a linear model which encompasses all thermoeconomic

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methodologies. The most systematic and widespread methodologies (see above)use exergy to linearly apportion costs when two or more coproducts appear, andtheir results can be reproduced using the Structural Theory (Erlach (1998),Erlach et al. (1999)). For this reason, all concepts and procedures explained hereare based on the general and common mathematical formalism of the StructuralTheory.

This introductory section on the fundamentals of thermoeconomics is divided intothree parts. First the basic concepts needed to perform and understand thethermoeconomic analysis of complex energy systems are presented. Specialattention has been paid to explaining the thermoeconomic cost concept. Oncethe average and marginal costs are defined, in the second part their meaning,relationship and calculation procedures are fully explained with examples.Finally, the third part describes some applications of thermoeconomic analysis asapplied to operation diagnosis and optimization of complex energy systems.

2. Basic Concepts

All thermoeconomic methodologies use costs based on the Second Law ofThermodynamics when solving engineering problems. In this section, the costconcept is explained together with all the new basic concepts, including fuel,product and thermoeconomic models needed to perform a thermoeconomicanalysis of a plant.

2.1. The concept of cost

The cost of a flow in a plant represents the external resources that have to besupplied to the overall system to produce this flow. Thermoeconomic analysisdistinguishes between exergetic costs and monetary costs.

The exergetic cost (Valero et al (1986)) of a mass and/or energy flow is theunits of exergy used to produce it, e.g. the exergetic cost of the net power is theexergy provided by the natural gas to generate the electrical power delivered tothe net by the co-generation plant (see Figure 1). These costs are a measure ofthe thermodynamic efficiency of the production process generating these flows.The unit exergetic cost of a mass and/or energy flow represents the amount ofresources required to obtain one unit of exergy. Thus, if the unit exergetic costof the electricity is three, three units of plant exergy resources are consumed toobtain one exergy unit of electrical power.

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HRSG

Compressor

Combus tor

Turbine

7AirGasesNatural GasWorkWater/Steam

1

2 3

40

5 6

8

1

2

4

3

Figure 1. Physical structure of the co-generation plant.

The monetary cost takes into account the economic cost of the consumed fuel(i.e. its market price) as well as the cost of the installation and the operation ofthe plant and defines the amount of money consumed to generate a mass and/orenergy flow. These costs are a measure of the economic efficiency of aprocess. Similarly, the unit monetary cost (also called unit exergoeconomiccost or unit thermoeconomic cost) of a mass and/or energy flow is the amountof monetary units required to obtain one unit of exergy.

We can further distinguish between average costs, which are ratios andexpress the average amount of resources per unit of product, and marginalcosts, which are a derivation and indicate the additional resources required togenerate one more unit of the product under specified conditions.Mathematically they are defined as:

unit average cost:

i

o*

BB

k = (1)

unit marginal cost:

conditionsi

0*

BB̂

k

∂∂

= (2)

The average costs are only known after production, when we know how manyresources were used and the production obtained. The average cost is notpredictive. Knowing the average unit cost of a product does not provide the cost

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of a production process P + ∆P. Thermoeconomic cost accounting theoriescalculate average costs and use them as a basis for a rational price assessment,under physical criteria, of the internal flows and the products of the plant.

Marginal costs can be used to calculate additional fuel consumption when theoperating conditions are modified. Thermoeconomic optimization methods(Frangopoulos (1990, 1997)), Von Spakovsky and Evans (1993)) are based onmarginal costs when solving optimization problems.

2.2. Fuel, product and unit exergetic consumption.

A productive purpose, a certain good or service to be produced, can be definedfor every plant. In order to generate this product, some resources have to besupplied to the plant and are consumed in the process. For example, in the co-generation plant, natural gas is supplied to the plant to generate electric powerand process steam.

A productive purpose expressing a process unit function in an overall productionprocess can be defined for each process unit. The productive purpose of aprocess unit measured in terms of exergy is called product. To create thisproduct, another exergy flow(s) is consumed. The flow of exergy which isconsumed in the process unit during the generation of its product is calledfuel(s).

Real process exergy is destroyed in any process. That is, part of the fuel exergyis destroyed during product generation. Using the definitions of fuel and product,the exergy balance for a process unit can be formulated as:

F P I= + (3)

Therefore, the fuel required to generate a certain amount of a product dependson the amount of irreversibility (exergy destroyed). The fuel exergy required togenerate one exergy unit of product is defined as unit exergetic consumption k :

PF

k = (4)

It is a measure of the thermodynamic efficiency of the process and equals onefor reversible processes and is greater than one for all real processes. The moreirreversible a process, the higher the value of the unit exergetic consumption.Combining equation (4) with the exergy balance on a fuel/product basis(Equation 3), the unit exergetic consumption k can also be formulated as:

kIP

= +1 (5)

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The reciprocal of the unit exergy consumption is defined as the exergeticefficiency η. It is equal to one for reversible processes and is less than one forall real processes.

η =PF

= −1IF

(6)

Fuel and product definitions for some typical process units in a dual-purposepower and desalination plant are shown in Table 1. The fuel-product definitionfor the process units of the co-generation plant (Figure 1) are shown in Table 2.

Process unit Fuel Product

Boiler Natural gas Exergy difference betweenthe generated steam flowand the entering waterflow

Pump Work to drivepump/compressor

Exergy supplied to theworking fluid

Turbine withoutextraction

Exergy removed fromworking fluid during theexpansion

Generated work

Turbine withextraction

Exergy removed fromworking fluid during theexpansion

Generated work

Generator Mechanical work Electric Work

Heat exchanger Exergy removed from thehot flow

Exergy supplied to the coldflow

Table 1. Fuel and product definitions for typical power plant units.

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No. Subsystem Fuel Product Technicalproductioncoefficients

1 Combustor F1 = B1 P1 = B3–B2 kcb = F1/P1

2 Compressor F2 = B5 = Wcp P2 = B2–B0 kcp = F2/P2

3 Turbine F3 = B3–B4 P3 = B5 + B6 = Wcp+ Wnet

kgt = F3/P3

4 HRSG F4 = B4 P4 = B7 = BHEAT kHRSG = F5/P5

5 Junction P1 = B3–B2

P2 = B2–B0

Pj1 = B3 r1 = P1/Pj1

r2 = P2/Pj1

6 Branching 1 Pj1 = B3 F3 = B3–B4

F4 = B4

7 Branching 2 P3 = B5 + B6 =

Wcp + Wnet

F2 = B5 = Wcp

B6 = Wnet

Table 2. Fuels and Products of the process units of the co-generation plant.

2.3. Physical and thermoeconomic plant models

A plant is analyzed using a physical model with a set of equations to describe thephysical behavior of the process units. It calculates parameters such astemperatures, pressures, efficiencies, power generated etc. to describe thephysical state of the plant. Depending on the analysis, a decision has to be takenon the detail required i.e., which flows and process units are to be considered.The process units for the analysis do not necessarily correspond to physicalunits. Various parts of the installation can be combined into one process unit andphysical units can be further disaggregated. It is important to choose anappropriate aggregation level that properly defines the behavior of each processunit and its purpose in the overall production process. The physical structure(see Figure 1) depicts the process units, mass and connecting energy flowsconsidered in the physical model.

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F1 =B1

P1 = B3 – B2

P2 = B2– B0

Wnet

P4 = B7 = Bheat

F2 = B5

F3 = B3 – B4

F4 = B4

Pj1 = B3

b1

b2

Comb

Comp

HRSG

Turb

j1

P3

Figure 2. Productive Structure of the co-generation plant.

The minimum physical data required in a thermoeconomic analysis aretemperatures, pressures, mass flow rates and compositions of all mass flowstogether with the heat and power rates of the energy flows considered. Usuallyall this information is fully or partially obtained from the physical model of theplant. But it is not strictly indispensable if all the required data are measuredplant data, collected directly from the plant data acquisition system.

Nevertheless, when pricing all mass and energy flows in the thermoeconomicanalysis, it is absolutely necessary to define a thermoeconomic model of theplant which considers the productive purpose of the process units, i.e. thedefinitions of fuels and products and the distribution of the resources throughoutthe plant. The productive model can be graphically depicted by the productivestructure diagram (Figure 2).

In this scheme, the flows (lines connecting the equipment) are the fuel and theproduct of each subsystem. Each “real“ piece of equipment in the plant has anoutlet flow (product) and an inlet flow (fuel). The capital cost of the units is alsoconsidered as an external plant resource and is represented as inlet flowscoming directly from the environment (not considered in Figure 2). Since the fuelof a process unit can be the product of another and the product of a process unitcan be the fuel of several subsystems, two types of fictitious devices areintroduced: junctions (rhombs) and branching points or branches (circles). In ajunction, the products of two or more process units are joined to form the fuel ofanother process unit. In a branching point, an exergy flow (fuel or product in theproductive structure –see Figure 2-) is distributed between two or more processunits. Sometimes the productive structure can be simplified (with the sameresults) by merging the junctions and branches in a new fictitious process unitcalled junction-branching point. For the sake of simplicity, the explanation of thefundamentals of thermoeconomics will be made using the productive structuredepicted in Figure 2.

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The productive structure is a graphical representation of resource distributionthroughout the plant. Thus, its flows are fictitious and are not necessarilyphysical flows. While each plant has only one physical structure to describe thephysical relations between the process units, various productive structures canbe defined depending on the fuel and product definitions as well as decisions onhow the plant resources are distributed among the process units.

Thus, the thermoeconomic model (mathematical representation of the productivestructure) is a set of mathematical functions called characteristic equations,which express each inlet flow as a mathematical function of the outlet flows forall the productive structure process units and a set of internal parameters xl:

Bi = gi(xl, Bj) i = 1,…,m–s (7)

where the index i refers to the input flows of the process unit l, the index jrefers to the output flows of the process unit l, and m is the number of flowsconsidered in the productive structure. Every flow is an input flow of a processunit and an output flow of another process unit or the environment. For the flowsinteracting with the environment, we define:

Bm-s+1 = ωi i = 1,…,s (8)

where s is the number of system outputs, and ωi is the total system product, i.e.an external variable which determines the total product. The characteristicequations for the system in Figure 2, are shown in Table 3:

Nº Processunit

Entry Outlet Equation

1 Combustor F1 P1 F1 = gF1 (x1, P1) = k

cb P1

2 Compressor F2 = Wcp P2 F2 = g F2 (x2, P2) = k

cp P2

3 Turbine F3 P3 = Wgt F3 = g F3 (x3, P3) = k

gt P3

4 H.R.S.G. F4 P4=BHEAT

=ω4 F4 = g F4 (x4, P4) = k

HRSG P4 = k

HRSG ω4 = k

HRSG B

HEAT

5 Junction 1 P1 , P2 Pj1 P1 = g P1 (x5, Pj1) = r1 Pj1 = r1 (F3+F4)

P2 = g P2 (x5, Pj1) = r2 Pj1 = r2 (F3+F4)

6 Branching 1 Pj1 F3, F4 Pj1 = g Pj1 (x6, F3, F4) = (F3+F4)

7 Branching 2 P3 F2, Wnet P3 = g P3 (x7, F2, ω

3) = F2 + ω3 = W

cp + W

net

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Table 3. Characteristic equations of the co-generation plant.

The inlet and outlet flows of the productive structure units are extensivemagnitudes, which are the product of a quantity (usually mass flow rate) and aquality (specific magnitude). The magnitudes applied by most thermoeconomicmethodologies are exergy (Tsatsaronis (1987)), negentropy (Frangopoulos(1983)) and money. Other magnitudes, like enthalpy or entropy, can also beused.

The internal variables appearing in the thermoeconomic model depend on thebehavior of the subsystem and they are presumably independent of mass flowrates. This implies that relations like efficiencies or pressure and temperatureratios -which are mainly independent of the quantity of the exiting flows- can beused as internal parameters.

Note that the main objective of the productive structure, and hence of thethermoeconomic model, consists on sorting the thermodynamic magnitudesrelated to the physical mass and energy flow-streams connecting the plantsubsystems, in a different way that the equations modeling the physical plantbehavior do, in order to explicitly determine for each subsystem its energyconversion efficiency.

It is important to take in mind that, as it was already explained, thermoeconomicsconnects thermodynamics, which is a phenomenological (black box analysis)science, with economics. That is, by sorting the thermodynamic properties of thephysical mass and energy flow-streams of a plant, which in turn provide theenergy conversion efficiency of each subsystem, thermoeconomics analyzes thedegradation process of energy quality through an installation, i.e.thermoeconomics evaluates the process of cost formation.

Depending on the analysis scope each subsystem can be identified with aseparate piece of equipment, a part of a device, several process units or eventhe whole plant. Sometimes the objective consists on analyzing a plant in a deepdetail. In this case it is advisable, if possible, to identify each subsystem with aseparate physical process (heat transfer, pressure increase or decrease andchemical mixture or reaction) in order to locate and quantify, separately ifpossible, each thermal, mechanical and chemical irreversibility occurring in theplant. If the objective consists on analyzing a macro-system composed ofseveral plants, probably in this case the more convenient approach is considereach separate plant as a subsystem.

Thus, thermoeconomics always performs a systemic analysis, no matter howcomplex the system is, basically oriented to locate and quantify the energyconversion efficiency. It is out of the scope of thermoeconomics to model the

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behavior of the process units, which is made by the mathematical equations ofthe physical model.

Even though (it is out of the scope of thermoeconomics simulate the behavior ofthe subsystems), it is very important build the thermoeconomic model withphysical meaning. This is the reason, as already explained, of defining differentthermoeconomic models for the same plant. Depending on the aggregation leveland on the nature of the thermoeconomic equations the model will contentphysical information about the actual system behavior with different accuracydegrees. The obtained results from a very rough thermoeconomic model, withoutany physical sensitivity related with the actual behavior of the plant, probably willbe useless.

The more extended thermoeconomic methodologies use exergy linear equationsin their thermoeconomic models, because they present practical (the model issimpler and for this reason easier to understand when applied to very complexenergy systems) and conceptual advantages, as it will be explained. Moreover,in many real plants it is possible to find an aggregation level where the systemand subsystems linearly behave with enough accuracy, under an engineeringpoint of view (Valero et al. (1999).

Thus, if the characteristic equations are first grade homogeneous functions withrespect to the subset B, of independent variables (as linear equations do), that is:

λBi = gi(λB1,…λBj, xl) λ∈ℜ (9)

Euler´s Theorem states that the homogeneous function of first order verify:

ss

22

11

ll

il

l

il

l

ii B

Bg

...BBg

BBg

B

∂∂

++

∂∂

+

∂∂

= l1,…,ls in Sl (10)

or using the marginal consumption notation,

∑∈

=lSj

jiji BB κj

iij B

g∂∂

=κ i=1,...,m l=1,...,n. (11)

This property means that the input of a process unit varies at the same rate asits outputs. Note that this property does not imply that the function must belinear.

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For instance, a Cobb-Douglas function z = a xα y(1-α), is also a homogeneousfirst order function.

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κij are the technical production coefficients and represent the portion of the i-thprocess unit production:

j

iij B

g∂∂

=κ (12a)

The sum of κij coefficients of a unit is the unit exergy consumption of that unit:

j

j

j

n

0iin

0iijj P

F

P

F===

∑∑ =

=κκ (12b)

It can be identified three types of linear characteristic equations:

1. Those connecting each fuel of a process unit to its corresponding product:

Fi = κij Pj as for instance F1 = gF1 (x1, P1) = kcb P1 (13a)

There is one such equation for each process unit's fuel. These types ofequations are generated in the pieces of equipment and they inform about:

(i) the productive function of each process unit, i.e. its production (product)

(ii) what the process unit needs (fuel) to develop its productive purpose, and

(iii) the thermodynamic efficiency of the process in the process unit.

2. Structural equations model how the resources consumed by the plant aredistributed through the plant process units. They show how the process units areconnected from a productive point of view. Structural equations arecharacteristic equations to describe the productive model of junctions andbranches, e.g.:

P1 = gP1 (x5, Pj1) = r1 Pj1 = r1 (F3+F4) (13b)

3.When the capital cost of the equipment is also considered in the analysis, athird type of characteristic equation is required; costing equations. Theseequations are very often not linear, but in the case of these equations this is aminor problem, because they can be linearized for different operation intervals.They relate the investment cost of the process unit with thermodynamicvariables and its product. They express the amount of resources needed to build,install, maintain (etc.) a process unit.

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For example, a costing equation proposed by El–Sayed (1996) for a MSFdesalination unit is on the form:

1.0t

5.0t

75.0n PTTQ1002.0Z −−− ⋅⋅⋅⋅⋅= ∆∆∆ (13c)

The diagram of the productive structure is also called a Fuel/Product diagram(Torres et al. (1999)) because in most cases the lines connecting the pieces ofequipment represent the fuels and products of the different units. Thus, thecharacteristic equations (see Table 3) using the Fuel – Product notation can alsobe written as:

∑+==

n

1jij0ii BBP i = 0,1,…,n (14)

This equation shows how the production of a process unit is used as fuel byanother unit or as a part of the total plant production. In the above expression,Bij is the production portion of the i-th process unit that fuels the j-th processunit, and Bi0 represents the production portion of the process unit i leading to thefinal plant product (the subscript 0 refers to the environment, which isconsidered another process unit interacting with the plant).

Equation (14) can be expressed in terms of the unit exergetic consumptions as:

∑+==

n

1jjij0ii PBP κ i = 0,1,…,n (15)

In matrix notation it can also be expressed as:

PKPPP += s (16)

where Ps is a (n×1) vector whose elements contain the contribution to the finalproduction of the system Pi0 obtained in each process unit, and ⟨KP⟩ is a (n×n)matrix, whose elements are the unit exergy consumption κij. This expressionhelps to relate the production of each process unit as a function of the finalproduction and the unit consumption of each process unit:

sPPP = where ( ) 1D

−−≡ KPUP (17)

In the same way, we can express the irreversibility of each process unit as:

sPII = where ( ) PUKI DD −≡ (18)

while the total resources of the system may be obtained as:

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set

TF PPκ= (19)

where ( )n001et ,...,κκκ ≡ , is a (n×1) vector whose elements contain the unit

consumption of the system-input resources.

3. Calculating Thermoeconomic Costs

Once the thermoeconomic model has been defined and the characteristicequations corresponding to the productive structure of the system are known,the costs of all flows in the productive structure can be easily calculated.

There are two different types of thermoeconomic costs: average costs andmarginal costs (equations 1 and 2). It is important to note that (as discussedbelow) the average and marginal costs coincide when the characteristicequations of the thermoeconomic model are first grade homogeneous functions(Serra (1994), Reini (1994), Uche (2000)).

This result is very important since both costs can be calculated using the sameprocedure. Marginal costs are a derivative (see equation 2) and can becalculated by applying the chain rule of the mathematical derivation. Similarly,average costs can also be obtained from the rules of the mathematical derivationapplied to the thermoeconomic model when the characteristic equations are firstgrade homogeneous functions.

According to the previous premises, the cost of the plant resources can bedefined as:

i

e

1i

*i,oo BkB ∑

== (20)

where e, is the number of system inputs, and k*o,i is the unit cost of the –i–

external resource.

Each flow, as a process unit input, is a function (defined by the characteristicequation) of a set of internal variables, x, external variables ω and the outputflows of the process unit. The cost of the plant resources is then a function ofeach flow, the set of internal variables of each process unit and the final productof the plant B0 = B0(Bi, x, ω ), according the relations (7) and (8).

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When calculating the variation of the resources consumed in the plantconcerning a flow, the chain rule can be applied:

*i,0

i

0 kBB

=∂∂ i = 1,… ,e (21a)

∑≠= ∂

∂

∂∂

=∂∂ m

ij1j i

j

j

0

i

0B

g

BB

BB i = e+1,… ,m (21b)

The expression i

0

BB

∂∂

represents the marginal costs which evaluate the additional

consumption of the resources, when an additional unit of the flow –i– isproduced, under the conditions that the internal variables, x, do not varythroughout this process.

We can denote these marginal costs as k*i, and

j

iij B

g∂∂

≡κ the marginal

consumption of flow –i– to produce the flow –j–, then we can rewrite theprevious expressions, as:

*i,0

*i kk = i = 1,… , e (22a)

∑≠=

=m

ij1j

*jji

*i kk κ i = e+1,… , m (22b)

Note that the unit exergetic cost of each fuel entering the plant is unity becausethere is no energy quality degradation nor exergy destruction at the verybeginning of the productive process. Hence, the amount of exergy consumed toobtain each plant’s fuel is its own exergy content and therefore its unit exergeticcost equals one.

It can easily be proved that the cost of each flow P*ij of the productivestructure using the Fuel/Product notation is:

ij*

i,P*ij BkP = (23)

And the exergetic cost of the product of each process unit is the same as thecost of the resources needed to obtain it, hence:

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∑===

n

0jji

*j,P

*i

*i BkFP i = 1,…,n (24)

This cost equation can also be expressed in terms of the unit exergeticconsumptions:

∑+==

n

1j

*j,Pjii0

*i,P kk κκ i =1,…,n (25)

which can be used to obtain the unit exergetic cost of the flows appearing in theproductive structure diagram as a function of the unit exergetic consumption ofeach process unit.

Then, if the characteristic equations and the marginal consumptions for eachprocess unit are known, the marginal cost k

* for each flow can be obtained by

solving the system of linear equations (25).

Example 1

For the example of a co-generation plant (Figure 2), equations (21a), (21b) canbe written as:

1

1*F F

Bk

1 ∂∂

=

*P

2

3

3

1

2

1*F 32

kFP

PB

FB

k =∂∂

∂∂

=∂∂

=

*P

3

1j

1j

1

3

1*F 1j3

kF

P

PB

FB

k =∂

∂

∂∂

=∂∂

=

*P

4

1j

1j

1

4

1*F 1j4

kF

P

PB

FBk =

∂

∂

∂∂=

∂∂=

cb*F

1

1

1

1

1

1*P kk

PF

FB

PBk

11=

∂∂

∂∂=

∂∂=

cp*F

2

2

2

1

2

1*P kk

PF

FB

PB

k22

=∂∂

∂∂

=∂∂

=

gt*F

3

3

3

1

3

1*P kk

PF

FB

PB

k33

=∂∂

∂∂

=∂∂

=

20

HRSG*F

4

4

4

1

4

1*P kk

PF

FB

PBk

44=

∂∂

∂∂=

∂∂=

1*P2

*P

1j

1

1

1

1j

2

2

1

1j

1*P rkrk

PP

PB

PP

PB

PB

k121j

+=∂∂

∂∂

+∂∂

∂∂

=∂∂

=

*P

net

3

3

1

net

1*W 3net

kWP

PB

PBk =

∂∂

∂∂=

∂∂=

The thermoeconomic model (characteristic equations) of an energy systemcontains the mathematical dependence between the resources consumed andplant flows (products and internal flows). It is therefore possible to define a setof linear equations to calculate the costs of every flow of the plant's productivestructure. Note that these equations show the process of cost formation on theproductive structure.

The proposed procedure to calculate the marginal cost of all the flows of a plantis general and valid for any thermoeconomic formulation that uses equationsconnecting inlet and outlet flows of each process unit.

Just as –k*– was defined as a marginal cost when production is modified, wecan also obtain the marginal cost when the internal variables x are modified.Similarly, applying the chain rule, we get:

∑= ∂

∂=

∂∂ m

1i i

j*j

i

0x

gk

xB

(26)

This equation expresses the effect on additional resource consumption when aninternal parameter xi is modified and is the basis for the thermoeconomicdiagnosis (explained in detail below). To determine the physical model of thesystem, a set of equations must be defined which relate the internal and externalvariables to the thermodynamic laws: mass, energy and entropy balances.

The most developed thermoeconomic optimization methodologies (Frangopoulos(1987, 1990), Von Spakovsky et al. (1993)), use the Lagrange multipliersoptimization method to calculate the marginal costs defined in the previoussection. It can easily be proved (Serra (1994), Reini (1994)) that the Lagrangemultipliers are the marginal costs defined in equation (2), i.e:

i

0i B

B∂∂

=λ i = 1, ...,m (27)

This multiplier represents the variation of the objective function B0 concerningthe state variable Bi.

21

3.1. Economic resources and thermoeconomic costs

Thermoeconomic cost calculation considering the process unit capital cost Z, issimilar to the above method but should be explained in more detail. The capitalcost of each process unit Z can be considered an external flow of plantresources from the environment to the process unit (see Figure 3). This willrepresent the monetary units per second needed to compensate the depreciation,maintenance cost and so on, of the process unit.

B j

BhBi

x l

ZlEconomicResources

B 0

Z l ( Bl, B

j, Bh)=

Figure 3. Economic resources scheme.

According to marginal cost analysis, Z represents an environmental resource andcan be handled in the same mathematical way as energy resources. The amountof resources consumed when manufacturing a device are, in fact, resourcesconsumed to obtain the plant products. Some authors (Brodyansky et al. (1993),Le Goff (1979)) have developed methodologies to evaluate the total amount ofresources consumed when building a process unit. Then the marginal unit cost∂Z/∂B, can be considered a marginal consumption κzj.

For the process unit depicted in Figure 3 the characteristic equations are:

Bi = f(Bj, κij) (28a)

Z j = Z(Bj, κzj) (28b)

And the cost of the product is:

zjij*j

j

j*i

j

i*j k

B

Zk

BB

k κκ +=∂

∂+

∂∂

= (29)

If Zj is proportional to the production of the unit, or in other words itscharacteristic function is first order homogeneous, the marginal cost is equal tothe average cost. But, unfortunately Zj is a non-linear function of the productionin most cases.

22

4. Thermoeconomic applications to complex energy sytems

Having defined the tools needed for a thermoeconomic analysis of a complexsystem, some applications to thermoeconomic diagnosis and optimization can bepresented. The methodology is presented together with a simple application.

4.1 Operation thermoeconomic diagnosis

Diagnosis is the art of discovering and understanding signs of malfunction andquantifying their effects. In the case of Thermoeconomics, the effect of amalfunction is quantified in terms of additional resources consumed to obtain thesame production, both in quality and in quantity.

The main problem in energy system diagnosis can be summarized in thefollowing question: Where, how and which part of the consumed resources canbe saved by keeping the quantity and quality of the final products constant? Toanswer these questions, we need:

(a) Procedures that accurately determine the state of the plant.

(b) A theory to provide the concepts and tools to understand and explain thecauses of this state.

The methodology presented here applies Structural Theory to provide the toolsto investigate the causes of the irreversibilities and the cost formation process.

In order to clarify the explanation of the proposed method we use a simpleexample (a more complex one can be found in Lerch et al. (1999)), the co-generation plant depicted in figure 1, whose design and operational exergy flowvalues are shown in Table 4. The plant has a co-generation gas turbine cycleand uses the turbine outlet gases as thermal energy in a heat recovery steamgenerator that produces steam (flow #7) together with the electric energyproduced in the turbo-generator (flow #6).

Flow (kW) 1 2 3 4 5 6 7 8

Design 11 781 2704 9614 3831 2977 2500 2355 388

Operation 11 914 2758 9753 3887 3056 2500 2355 424

Table 4. Design and operation exergy flow values of the co-generation plant(Figure 1)

23

4.1.1. Technical exergy saving

Once the exergy flows have been supplied by an appropriate performance testor a model simulator, the irreversibilities in each productive unit can be obtainedfrom the exergy balance. But not all exergy losses can be saved in practice. Infact, the potential exergy saving is limited by technical and/or economicconstraints. It also depends on the decision level that limits the actions to beundertaken. In contrast to conventional thermodynamic analysis,Thermoeconomics assumes a reference situation of the plant operating underdesign conditions. From this perspective, in the plant of Figure 1, we see thatonly 133 kW, of the 7.06 MW of total irreversibilities can be saved with respectto design conditions.

Therefore, the additional fuel consumption can be expressed as the differencebetween the resource consumption of the operating plant and the resourceconsumption for a reference or design condition with the same productionobjectives:

0TTT FFF −=∆ (30)

and it can be broken up into the sum of the irreversibilities of each process unit:

( ) ∑ ∆=∑ −=∆=∆==

n

1jj

n

1j

0jjTT IIIIF (31)

However, even though the methods based on Second Law Analysis (Kotas(1985)) and Technical Exergy Saving are useful to quantify the additional fuelconsumption, they fail when trying to identify the real causes of the additionalresources consumption.

4.1.2 Impact on resources consumption

The Fuel / Product diagram of the co-generation plant is shown in Figure 2. Thisdiagram can be simplified by merging junction 1 and branching point 1 in a newfictitious process unit called junction – branching point (see Figure 4). This newproductive structure is slightly different than Figure 2, and is more compact.

The characteristic equations of this new productive structure are obtained as inthe previous section applying equation (16)

PKPPP += s

For the sake of simplicity we did not consider thermal and mechanical exergiesas separate entities. Two auxiliary variables also appear r1 = (B3-B2)/B3 and r2= B3/B2, which correspond to the part of the fuel of the turbine and the heat

24

recovery steam generator (HRSG) coming from the combustor and thecompressor respectively. Flow #8, produced in part in the combustor and in thecompressor, also leaves the system as a residue. Only a part of the enteringgases to the turbine: B3-B8 are used as a fuel of other process units of thesystem. Therefore, only a part of the combustor’s and compressor’s product isused as a fuel for other process units (useful product). Accordingly, Figure 4shows the chosen disaggregation scheme of the system and the Fuel/Productvalues for the design conditions are shown in Table 5. The F-P definition isshown in Table 6.

(1)

(3)-(2)

(2)(3)-(4)

(4)-(8)

(6)

(5)

(7)

1

3

2

4

(8)

Figure 4. Fuel/product diagram for the co-generatiron plant shown in Figure 1

F0 F1 F2 F3 F4 Total

P0 0 11 781 0 0 0 11 781

P1 0 0 0 4156 2474 6631

P2 0 0 0 1627 968 2595

P3 2500 0 2977 0 0 5477

P4 2355 0 0 0 0 2355

Total 4855 11 781 2977 5783 3443

Table 5. Fuel and energy flows (kW) in design conditions for the co-generationplant shown in Figure 1

Nº Process unit Fuel Product Residue

25

1 Combustor B1 B3-B2

2 Compressor B5 B2

3 Turbine B3-B4 B6

4 HRSG B4-B8 B7 B8

Table 6. Fuel and product definition corresponding to Figure 5.

In order to bring together the problem of the impact of resources consumptionwith thermoeconomic diagnosis we need to know the increase of the unit exergyconsumption of each process unit of the plant. A performance test or a simulatorprovides the real values of the unit consumptions which are then compared withthe design values.

The values of the unit exergetic consumption increase are found as:

∆κij = κij (x) − κij (x0)

Table 7 shows the ∆κij values for the plant in Figure 1.

eκ∆ 0.4006 0.0000 0.0000 0.0000

0.0000 0.0000 -0.1667 0.3857

0.0000 0.0000 0.1593 0.4636

0.0000 1.1147 0.0000 0.0000

KP∆

0.0000 0.0000 0.0000 0.0000

k∆ 0.4006 1.1147 -0.0074 0.8493

Table 7. Increase of unit exergetic consumption (100∆κij).

Equation (19) is used to obtain the increment of the total resources of anoperating plant regarding the reference conditions:

PP ∆+∆=∆ et

et

TF κκ 0 (32)

The increase of the process unit production from equation (16) may beexpressed in terms of the unit exergy consumption as:

PKPPKPPP ∆+∆+∆=∆ 0s (33)

26

hence, applying equation (17), we obtain:

( )0s PKP??PP?P += (34)

If we want to analyze the fuel impact due to an increment of the exergy unitconsumption of the process units, equation (32) could be written as:

s*P

t0*P

t0e

tTF PPKPP ∆+∆+∆=∆ κκκ (35)

If no change in the total production of the plant is assumed, then:

( ) 0PKPk ∆+∆=∆ *P

te

tTF κ (36)

or in scalar format:

∑

∆∑=∆

= =

n

iiji

n

j

*jP,T PkF

1

0

0κ (37)

The ∆⟨KP⟩ matrix is the key to predict the impact on fuel of a physical variationof a parameter in the system.

Using the above equation, the additional resource consumption ∆FT (also calledFuel Impact; Reini (1994)) can be expressed as the sum of the contributions ofeach process unit.

The variation of the exergetic unit consumption of each process unit increases

its resources consumption and its irreversibilities in a quantity 0ijiPκ∆ , which we

call, malfunction. Consequently, this implies an additional consumption of

external resources given by 0iji

*jP, Pk κ∆ , which is also named the malfunction

cost. Therefore, the total fuel impact can be written as the sum of the fuelimpact or malfunction cost of each process unit, as shown in equation (37).

The proposed method provides the exact values of the additional resourceconsumption of each process unit malfunction for any operational state. Othermethods, such as the Theory of Perturbations (Lozano et al. (1996)), onlyprovide an approximate predictive value, based on marginal costs (Lagrangemultipliers) which is valid for an operating state close to the referenceconditions.

Figure 5 compares the fuel impact and the increase of irreversibilities or thetechnical exergy saving of each process unit and also compares (first column)the malfunction and the fuel impact for each process unit. Three malfunctions inthe plant are shown in the combustor, the compressor and the HRSG. Thelargest irreversibilities increase is in the combustor, but the largest fuel impact is

27

in the compressor. The question that arises is, What causes the irreversibilitiesincrease and the fuel impact, and how are they related?

0

20

40

60

80

Combustor Compressor Turbine HRSG

Fuel ImpactMalfunctionTechnical Saving

Figure 5. Fuel impact and technical saving.

4.1.3. Malfunction and dysfunction analysis

We have shown that there is no direct relationship between the increase of theirreversibilities and fuel impact. The more advanced the production process is,the greater the cost of the irreversibility malfunction and, as a consequence, thegreater its fuel impact.

Furthermore, the degradation of a process unit will force other process units toadapt their behavior in order to maintain their production conditions and modifytheir irreversibilities. Figure 6 shows how an increase of the unit consumption ofa process unit will not only increases the irreversibilities on it but also theirreversibilities of the previous process unit.

∆I2∆P1∆F2

∆I1∆F1

F1F2

2I

2P1P

1I

1 2

Figure 6. Malfunction and fuel impact.

The irreversibility increase of a generic system’s process unit is given by:

PUKPKI ∆−+∆=∆ )( DD0

D (38)

28

From the above expression, we can distinguish two types of irreversibilities:

Endogenous irreversibility or malfunction produced by an increase of the unitconsumption of the process unit itself:

∑ ∆=∆==

n

jjiiii i

PkPMF0

00 κ (39)

Exogenous irreversibility or dysfunction induced in the process unit by themalfunction of other subsystems, which forces it to consume more localresources to obtain the additional production required by the other process units:

iii PkDF ∆−= )1( (40)

The malfunction only affects the behavior of the process units; the dysfunctionis a result of how the process units adapt themselves to maintain the totalproduction. The dysfunction generated by a process unit is defined as:

DIi = ∆Fi – MFi (41)

Table 8 shows the malfunctions, dysfunctions, impact on fuel and increase ofirreversibility of the example analyzed here.

(kW) Combustor Compressor Turbine HRSG Total

DI 0.000 30.699 4.979 22.243 57.921

MF 26.562 28.925 -0.408 20.000 75.079

∆F 26.562 59.624 4.571 42.243 133.000

DF 46.664 6.849 4.408 0.000 57.921

MF 26.562 28.925 -0.408 20.000 75.079

∆I 73.226 35.774 4.000 20.000 133.000

Table 8. Malfunction and dysfunction table in kW.

4.1.4. Intrinsic and induced malfunctions

Using the above method we can identify and quantify malfunction effects. Forexample, we found three malfunctions in the gas turbine cycle (Figure 1): oneeach in the combustor, compressor and HRSG. But, What are the causes of themalfunctions? In fact, the actual operation values shown in Table 4 correspondto a 1% decrease in compressor isoentropic efficiency. This means that HRSGand combustor efficiencies can be changed by varying compressor efficiency.

29

How do we approach this problem? The relationship between operation andefficiency of the process units could be analyzed using a simulator. If all theplant process units were isolated, the efficiencies of those process units wouldbe independent variables (Lozano et al., 1996). So we will assume that there isan operating parameter xr affecting the efficiency of the i-th process unit of theplant and thus, in most cases, also indirectly affecting the efficiencies of theother plant process units.

Once the relationship between unit exergy consumption and the operatingparameters is known, the above methodology can be applied to distinguish theeffect of an operating parameter on the internal economy of a process unit, i.e.its malfunction and the cost of its malfunction.

Plant operating parameters could be classified according to their effect on theefficiency of the process units of the system:

Local variables: They mainly affect the behavior of the process unit related tothe variable, e.g, the isoentropic efficiency of a turbine. From a practical point ofview, a variable is considered local and therefore related to a subsystem. Thetotal fuel impact due to its perturbation is basically located in this process unit.

Global and/or zonal variables: This is the case when an operating parametercannot be associated with a specific process unit. We must identify them asoperating set points, environmental parameters and the production load or fuelquality.

In this lecture we will focus our analysis on local variables and how they affectadditional fuel consumption and the other plant process units. This analysis is, infact, the next step in the thermoeconomic diagnosis.

Unfortunately the problem of locating causality of losses in a structure is rathermore complex than locating malfunctions and dysfunctions.

When a plant unit deteriorates (when its behavior is degraded) its physicalvariables are modified, its efficiency is decreased and its unit exergyconsumption increases.

The unit exergy consumption increase of each process unit, due to the variationof an operating parameter xr, is:

)()( 00 xx ijrijrij x κκκ −∆+=∆ (42)

Therefore, it will be possible to approximate the malfunction of a process unit asthe sum of the contributions of each operating parameter:

30

∑ ∑ ∆≅=r

n

ji

rjii PMF

1

0κ (43)

According to the classification of operating parameters, the intrinsicmalfunction is that part of the process unit malfunction due to thedegradation/improvement of the process unit itself, which is, in turn, due tovariation of local operating parameters:

∑ ∑ ∆≡∈ =iLr

n

ji

rji

Li PMF

1

0κ (44)

A system malfunction or improvement does not only have consequencesupstream (by trying to see the variation in consumption of used resources) butalso downstream. Clearly the degradation or improvement of a system’s flowentry conditions will affect its efficiency to a greater or lesser extent. This willmodify the production and affect the next process unit.

Not only are there dysfunctions when there is an intrinsic malfunction. There arealso induced malfunctions, that can decisively affect the system's behavior. Forexample, using the throttle valve in a power plant can destroy a small additionalamount of exergy but the downstream effects on turbine efficiencies can bequite serious.

Thus, the difference between total process unit malfunction and intrinsicmalfunction is called induced malfunction. It is due to the degradation of otherplant process units which provoke a variation in the unit consumption of thatprocess unit:

Lii

Gi MFMFMF −= (45)

This phenomenon is not foreseen in classic linear thermoeconomic theory. Theaverage cost obtained from the most rigorous disaggregation analysis can neverpredict induced malfunctions and dysfunctions will only be predicted in caseswhere the hypothesis of linearity and continuity holds.

4.2. Thermoeconomic optimization

Many thermal systems are very complex due to the number of process unitsand/or its strong interdependence. This complexity makes difficult theoptimization of the system design and operation. The knowledge of the costs ofa system, which in the final instance give an economic meaning to the structuralinteractions between subsystems, allows us to formulate problems related totheir optimization and also to solve these problems, under certain conditions, in a

31

very symple way. This possibility allows us in turn to assume that with theappropiate calculation strategy the problem of the global optimization of the plantcould be reduced to a sequence of subsystem to subsystem optimization. Herewe describe strategies for optimizing complex systems as proposed by Lozano etal. (1996). They are based on sequential optimization from process unit toprocess unit using the Thermoeconomic Isolation Principle (Evans (1980)). Inthis section, we see how the cost of the resources consumed by the systemvaries when the unit of the cost of the resources consumed, the technicalproduction coefficients of the productive structure and/or external demand ofproducts vary. Once we know the relationship between the technical productioncoefficients and the design free variables the chain rule of derivation can beapplied to distinguish the effect of a design free variable on the internal economyof subsystems.

A process unit of a thermal system is thermoeconomically isolated from the restof the system if the product of the unit and the unit cost of its resources (internalproduct and/or external resources) are constant and known quantities. If a unitof a thermal system is thermoeconomically isolated, the unit may be optimized byitself (without considering the modifications of other variables of the rest of thesystem) and the optimun solution obtained for the unit coincides with theoptimum solution for the whole system.

Of course, T.I. (Thermoeconomic Isolation) is an ideal condition which cannotbe achieved in most of the real systems: Pj and k*

P,i change when designvariables of other process units change ,due to feedback. But the more constantPj and k*

P,i are, the closer to T.I. conditions and the fewer iteration loopsneeded to achieve the optimal solution for the whole system. Thus, the goal isnot to achieve T.I. but to approach it as much as possible in order to obtainmaximum advantages, which include:

(1) Improvements and optimal design of individual units in highly interdependentcomplex systems are greatly facilitated, as well as of whole systems.

(2) The designers can be specialized and their efforts concentrated ondesigning the variables of single units, while resting assured that theseefforts yield optimum design and/or improve the overall system

(3) The convergence of the solution is faster.

To optimize individual units, the objective function of the cost of product of theprocess unit –j– could be defined as:

k

j*

i,Pn

0iij

kPkkMin

∑=

(46)

32

where the unit cost of the input resources k*P,i and the production Pj are known

and constant.

In real world optimization problems, the design free variables do not necessarilycoincide with the technical production coefficients. In practice there will be afunction of the actual design free variables which can be named –x–

We say that a free variable x is a local variable of a subsystem –j– when theproduction coefficients κij of this subsystem only depend on x. When a designvariable is attached to several subsystems, the previous expression must beextended to all concerned subsystems.

To determine whether a design free variable is local or not and which processunits are involved, the cost resource impact of the design variables to eachprocess unit can be computed:

xPx

z

xkAC j,Pijn

0i

*i,P

xj,0 ∆

κ

∂

∂+

∂

∂= ∑

=(47)

and the ratio calculated:

ε jx =

∆C0 , jx

∆C0,ix

i =1

n

∑(48)

If this ratio is equal (or close) to 1, the design variable is local for process unit –j–, if it is equal (or close) to zero, the design variable is independent of thereferred j process unit. In other cases the design variable involves severalprocess units.

These ideas could be used to design a strategy for global optimization problems:

(0) Determine which variables are local and which are regional (involve severalprocess units)

(1) Determine a sequence for local optimization of each process unit

(2) Take an initial value of the design variables

(3) Calculate technical production coefficients and unit product cost

(4) Find optimum values for local variables

(5) Find optimum values for global variables

33

Iterate from (3) to convergence when design variables or unit product cost donot vary in the next iteration. In each iteration the unit cost of total product mustdecrease.

NOMENCLATURE

LATIN SYMBOLS

k*: Exergy unit cost.

B: Exergy flow (kW).

F: Fuel (kW).

P: Product (kW).

I: Irreversibility (kW).

k: Unit exergy consumption.

g: Characteristic function.

Z: Capital cost of a process unit ($).

T: Temperature (º C).

f: Function.

r: Exergy ratio.

MF: Malfunction generated in a component (kW).

DF: Dysfunction generated in a component (kW).

DI: Dysfunction generated by a component (kW).

C: Total economic cost ($/s).

X: Variable.

GREEK SYMBOLS

η: Exergetic efficiency.

ω: Total system product.

λ: Lagrange multiplier.

κ: Technical production coefficients.

α: Coefficient in the Cobb-Douglas function.

34

∆: Difference.

ε : Ratio (thermoeconomic optimization).

MATRICES AND VECTORS

x: Set of internal parameters

P: Product vector.

PS: Final product vector.

|P⟩: Product matrix operator.

⟨KP⟩: Matrix of unit exergy consumption.

I: Irreversibility vector.

|I⟩: Irreversibility matrix operator.

UD: Identity matrix.

KD: Diagonal matrix with the unit exergy consumption.

SUBSCRIPTS

o: outlet

i,j: index.

e: external (inlet).

t: total.

n: stage.

t: tubes.

SUPERSCRIPTS

-1: Inverse.

t: Transpose.

0: Design conditions.

L: Local.

G: Induced.

35

r: Operating parameter.

36

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