1 3.19.2.0 Thermoeconomic Analysis Antonio Valero and César Torres Center of Research for Energy Resources and Consumption, Centro Politécnico Superior, Universidad de Zaragoza, Spain. Keywords: Thermoeconomics, Exergy, Cost Accounting, Optimization of Energy Systems, Energy Auditing, Diagnosis of Energy Systems. Contents 1. Introduction .................................................................................................................. 1 2. An historical overview ................................................................................................. 4 3. What is the Exergy Cost? ............................................................................................. 6 4. Cost Accounting ........................................................................................................... 9 4.1. The process of cost formation ............................................................................. 14 4.2. The principle of non equivalence of the irreversibilities ..................................... 16 5. Thermoeconomic Diagnosis ....................................................................................... 16 6. Thermoeconomic Optimization .................................................................................. 19 7. Final reflections and conclusions ............................................................................... 24 Glosary ........................................................................................................................... 27 Nomenclature.................................................................................................................. 29 Bibliography ................................................................................................................... 30 Summary This paper introduces the basis of thermoeconomics analysis. It provides an in-depth summary of the state-of-art and the progress that has been made in this field. The concept of exergy costing and the cost formation process are presented. A brief introduction of the applications of thermoeconomic analysis, that will be developed in the articles of this chapter, is made. It includes cost accounting, design, optimization and diagnosis of energy systems. 1. Introduction The increasing demand of natural resources by current energy conversion technologies and the concern for the impact on the environment due to emission, waste disposal and signs of global warming has brought the creation of new disciplines that help to understand how to improve the design and operation of energy systems and prevent residues from damaging the environment.
Transcript
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3.19.2.0
Thermoeconomic Analysis
Antonio Valero and César Torres
Center of Research for Energy Resources and Consumption, Centro Politécnico Superior, Universidad de Zaragoza, Spain.
Keywords: Thermoeconomics, Exergy, Cost Accounting, Optimization of Energy Systems, Energy Auditing, Diagnosis of Energy Systems.
Summary This paper introduces the basis of thermoeconomics analysis. It provides an in-depth summary of the state-of-art and the progress that has been made in this field.
The concept of exergy costing and the cost formation process are presented. A brief introduction of the applications of thermoeconomic analysis, that will be developed in the articles of this chapter, is made. It includes cost accounting, design, optimization and diagnosis of energy systems.
1. Introduction The increasing demand of natural resources by current energy conversion technologies and the concern for the impact on the environment due to emission, waste disposal and signs of global warming has brought the creation of new disciplines that help to understand how to improve the design and operation of energy systems and prevent residues from damaging the environment.
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Thermoeconomics is, in its widest possible sense, the science of natural resources saving that connects physics and economics by means of the Second Law of Thermodynamics.
A thermal power plant or a chemical plant are examples of energy systems formed from a set of subsystems or processes. These systems interact with their environment, consuming some external resources, which are then transformed into products. The final purpose of this transformation is to increase the economic utility.
The production process of a complex energy system can be analysed in terms of its economic profitability and efficiency with respect to resource consumption. An economic analysis can calculate the cost of fuel, investment, operation and maintenance for the total plant or even individual components but provide no means on how to allocate costs among them and its products. On the other hand, thermodynamic analysis let us calculate the efficiency of the individual process of the plant and locates and quantifies the irreversibilities but it cannot evaluate their significance in terms of the overall production process.
Thermoeconomic analysis combines economic and thermodynamic analysis by applying the concept of cost, originally an economic property, to exergy (see Exergy, Energy System Analysis and Optimization). Most analysts agree that exergy is an adequate thermodynamic property to which allocate cost because it accounts for energy quality. The exergy of a thermodynamic flow is the minimum amount of technical work needed for its production, from the reference environment. Once the reference environment is defined, exergy is the thermodynamic function of state which makes possible to formulate the equivalence between different energy and/or matter flow streams of a plant. Two flows are thermodynamically equivalent, that is, it is theoretically possible to get one from the other without additional consumption of energy resources if, and only if, they have the same exergy. Exergetic efficiency compare a real process to an ideal process, i.e. reversible, of the same type. An exergy analysis locates and quantifies irreversibilities in a process.
The physical magnitude connecting thermodynamics and economics is entropy generation or more specifically irreversibility. It represents the “useful” energy lost or destroyed, in all physical processes, and it has been used for pinpointing the true inefficiencies of industrial processes. Since all common processes in an actual plant are not reversible, exergy is destroyed and some natural resources are consumed and lost forever, which involves a cost in economic terms. The more irreversible a process is, the more natural resources are consumed.
The exergy balance accounts for the degradation of the exergy. The incoming exergy will always be greater than the leaving one:
This expression only keeps in mind the irreversibilities of the process. The purpose of this process is set by means of the definition of its efficiency. This is to say, that there is an implicit classification of the flows crossing the boundary of the system: the flows that are the production objective, the resources required to carry out the production and those that are residual. This information is not implicit in the second law and is the most important conceptual leap separating and at the same time uniting physics with economics.
is of utmost importance because it places “purpose” in the heart of thermodynamics.
The concept of efficiency defined as
Efficiency = Product / Resource
is older than thermodynamics and measures the quality of a process. The desireto produce a certain product is external to the system, and must be defined beforehand. Once this has been done, the design of the system and its functional structure will fit the aim of using available resources (capital, raw material, man power,…). Every definition of efficiency demands a comparison of the product obtained with the resources needed to obtain it. Its inverse value is:
Unit Consumption = Resources / Product
This expression is also a definition of the unit average cost when resources refer to the overall plant instead of individual processes. This concept is the key of thermoeconomics. A logical chain of concepts can be established, see Figure 1, which allows connecting physics with economics.
Purpose Second LawEfficiency
ExergyCost
Economic Cost
Figure 1. Logical chain of thermoeconomic concepts
Thermoeconomics deals with man-made energy systems. Its efficiency is a purposive concept and so they are the thermoeconomics analyses.
Thus, thermoeconomics assesses the cost of consumed resources, money and system irreversibilities in terms of the overall production process. They help to point out how resources may be used more effectively in order to save them. Money costs express the economic effect of inefficiencies and are used to improve the cost effectiveness of production processes. Assessing the cost of the flow streams and processes in a plant helps to understand the process of cost formation, from the input resources to the final products.
These analyses can solve problems related to complex energy systems that could not be solved by using conventional energy analyses. Among other applications thermoeconomics are used for:
• Rational prices assessment of plant products based on physical criteria.
• Optimisation of specific process unit variables to minimize the final product cost, i.e. global and local optimisation.
• Detection of inefficiencies and calculation of their economic effects in operating plants, i.e. plant operation thermoeconomic diagnosis.
• Evaluation of various design alternatives or operation decisions and profitability maximization.
• Energy audits.
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These applications of thermoeconomic analysis are briefly analysed in this article, and will be developed in detail in the articles of thisTopic.
2. An historical overview For introduction to this section, the contributions of Gaggioli and El Sayed (see bibliography) have been followed. The history of Thermoeconomics is relevant to understand how the current state of art has been reached (see bibliography). Also, its history helps us to know why it is very closely related with second law analysis and the development of the concept of exergy. It is generally agreed that the basic concept of exergy was first formed, independently, by Gibbs in U.SA, and Maxwell in England in the last quarter of the 19th century.
The first proposal in the literature to use second law analysis for costing purposes was a paper of Keenan in 1932. While he does not do exergy costing therein, he refers to it explicitly as the means for appropriately apportioning cost associated with the cogeneration of electric power and steam for distribution. Engineers thought that “obviously” the fuel cost should be allocated to the steam and the power in proportion to their energy content. The result, however, was the cogenerated electricity costed in this manner, was far less expensive than electricity produced in conventional power plants. Keenan pointed out that the value of the steam and the electricity lied in the “availability” not in their energy. Although the proposal was based on Second Law analysis, it is strangethat in practice the use of second law concepts was circumvented.
The use of the second law and specially entropy and exergy in thermal and mechanical engineering has been resisted for a long time, because of the complicated means whereby they have been developed and explained. On the other hand, the first law analysis and the energy magnitude have been used because they are conceptually comfortable. However, while it appeared “obvious” that the attribution of the fuel cost to co-generated steam and electricity should be in proportion to their energies, to the cost accountant this resulted in gross inconsistencies. A method known as “Lost kilowatts” was adopted as rational for overcoming the inconsistencies of energy costing. The basis of the method is that the co generated electricity cost is obtained as if it had been produced in a condensing turbine and the remaining costs are assigned to the steam.
The idea of coupling exergy and cost streams was first introduced by M. Benedict in 1948 in a seminar at M.I.T. He determined the total cost attributable to the irreversibilities of an air separation plant and used this cost for “optimal design”. Unfortunately, the contents of the seminar were not published until the 80’s. The early work by Keenan and Benedict was extended by his students, specially E.P. Gyftopoulos, who worked on the concept of availability.
The interest to formulate the interaction between cost and efficiency was first highlighted by Tribus and Evans at UCLA, in the early 60’s. They were studying desalination processes, and making exergy analysis, which led them to the idea of exergy costing and its applications to engineering economics, for which they coined the word “Thermoeconomics”. The essence of the Evans-Tribus procedure was to trace the flow of money, fuel cost and operation and amortized capital cost through a plant, associating the utility of each stream with its exergy. Y. El-Sayed, professor of Mechanical Engineering in Egypt, corresponded with Evans and Tribus in connection with their work in desalination and he came to work with them in the U.S.A. They published in 1970 a key paper, multiple times referenced, called “Thermoeconomics
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and the Design of Heat Systems”, where the mathematical foundation for thermal system optimization is given.
Also in the sixties M. Obert and R. Gaggioli were working in the optimal design of power plant steam piping, They proposed costing the steam exergy at a value to that of power produced, penelizing irreversibilities for electricity which therefore, will not be produced. R. Gaggioli directed, in the University of Wisconsin, the Ph.D. Theses of G. Reistad (1970) and W. Wepfer (1979) on “Second Law Costing” methods, that include the definition of rules to provide a rational distribution of the cost.
Meanwhile in Europe, since the fifties, a largenumber of works in second law analysis has been developed, specially in East Europe. In 1952, Rant introduced the name of exergy, just as we know it currently, defined as external useful work in opposition to the energy (internal work). Other outstanding authors are: Beyer, Baehr, Brodiansky, Szargut, Knoche among others. Some of the works, that also included thermoeconomic analysis, were compiled by T.J. Kotas in 1985 in the book “The Exergy Method of Thermal Plant Analysis”, that is one of the basic references in exergy analysis and thermoeconomics.
The comprehensive effort to apply thermoeconomics to the analysis, optimization and design of thermal systems did not start until the eighties. It startedin 1985, when R. Gaggioli, who led the Systems Analysis Technical Committee of the Advanced Energy Systems Division (AESD) of the American Society of Mechanical Engineers (ASME), gathered and strived to broaden the participation of non U.S. scientists and research groups that are working in advanced second law analysis and thermoeconomics. Under this idea a series of AESD annual international meetings was created, that focused on modern aspects of thermal sciences with particular emphasis on engineering thermodynamics including exergy analysis and thermoeconomics. The first one took place in Rome in 1997, and was chaired by Enrico Sciubba and Michael Moran.
It is at that time thatthermoeconomics actually took off. The interest and works regarding to thermoeconomic analysis highly increased: G. Tsatsaronis (1985), introduces the key concept of Fuel and Product. C. Frangopoulos (1983) and M. Von Spakovsky (1986), whose Ph.D theses directed by R. Evans applied and formalized the autonomous method of Evans and El-Sayed. In 1986 A. Valero and co-workers published another key paper “A General Theory of Energy Saving” where the Theory of Exergy Cost was introduced.
The “International Mechanical Engineering Congress & Exposition” (IMECE) formerly “Winter Annual Meeting” of ASME and its non-US counterpart conferences on Efficiency, Cost, Optimization and Simulation of Energy Systems” (ECOS) are the key references to follow the current state-of-art since its beginnings. ECOS and closely related conferences have been held in Italy, China, Greece, Spain, Poland, Turkey, Germany, France, Japan, Mexico and Scandinavian countries. In these countries, among others, there are active research groups in advanced energy systems including exergy analysis and thermoeconomics.
Thermoeconomic methods are generally subdivided in two categories, those based on cost accounting, e.g. Exergy Cost Theory, Average Cost Approach, Last-In-First-Out Approach (see bibliography) and those based on optimization techniques e.g. Thermoeconomic Functional Analysis, Engineering Functional Analysis. Cost accounting methods help to determine the actual cost of products and provide a rational basis for pricing, while optimisation methods are used to find optimum design or operation conditions.
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In the 90’s an important work starts in order to achieve a greater standardization and formalism. Many articles published (see bibliography) compare, analyze and unify the different methodologies (see Structural Theory of Thermoeconomics).
One of the most interesting initiatives is the project CGAM (1993), led by: C. Frangopoulos, G. Tsatsaronis, A.Valero and M. Von Spakovsky (see bibliography) whose objective was to show how the methodologies of each group of research could be applied by solving a predefined and simple problem of optimization of gas turbine cycle. In the final analysis, the aim was the unification of the different methodologies.
In the same direction, in the year 2001 another project called TADEUS (in honor to Proffesor Tadeus Kotas) was initiated. Its aim is to apply procedures from different research groups in thermoeconomic analysis to the diagnosis of the energy system malfunction and inefficiencies. The objective of this new effort is to establish the common concepts and nomenclature, compare the results and highlight the main characteristics of each approach.
3. What is the Exergy Cost? To introduce thermoeconomics let us start from the simplest physical concept of cost asking ourselves: What is the exergy content of a beautifully designed bohemian glass? Or that of a stone sculpture? Or that of gold? The answer is zero in practical terms. Many things we value, thermodynamics does not. The source of value may be or may not be related to its exergy content, even for the case of fuels. The only thing that physics can do is to assess the physical cost of objects, i.e. the amount of energy units required to produce a given product, namely embodied energy.
The concept of embodied energy comes from the 70s, when it was a great concern with the first global energy crisis. The problem with energy is the lack of techniques to allocate values of embodied energy when two products are produced simultaneously. A more precise concept then came: the “exergy cost” proposed by Valero or the “cumulative exergy consumption” proposed by Szargut (see bibliography) which are in fact the same concepts as embodied energy but using exergy.
Cost could be defined as the amount of resources needed to obtain a functional product. From one hand, resources take a general meaning. From the other hand, cost is associated to the purpose of production. It is associated neither to price nor to the resources that could be saved if the production process were less efficient or more conventional one.
Cost is an emergent property. It cannot be measured as a physical magnitude of a flow stream as temperature or pressure, it depends of the system structure and appears as an outcome of the system analysis. Therefore, it needs precise rules for calculating it from physical data. Cost is a property that cannot be found in the product itself.
The exergy cost of a mass or energy stream is the amount of exergy required to produce it. For example in the case of the cogeneration plant depicted in Figure 3, the exergy cost of the net power is the exergy provided by coal or natural gas to generate the electrical power delivered to the network by the cogeneration plant.
The unit exergy cost of a mass or energy stream represents the amount of exergy required to obtain a unit of exergy of the product stream. If iE represents the exergy of the i-th product stream and *
iE its exergy cost, the unit average exergy cost is written as:
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*
* ii
i
EkE
= (1)
The monetary or exergoeconomic cost, takes into account the economic cost of the consumed fuel cF (i.e. its market price $/MJ) as well as the cost of the installation and the operation of the plant, Z ($/s), and defines the amount of money to generate a mass and/or energy flow. The economic cost balance could be written as: P F FC c E Z= + (2)
These costs measure the economic efficiency of a process. Similarly, the unit monetary cost (also called exergoeconomic cost) of mass and/or energy flow is the amount of monetary units per unit of exergy required to obtain the referred flow:
F FP
P
c E ZcE
+= (3)
One can further distinguish between average cost, which is a ratio and express the average amount of resources per unit of product, and marginal cost which indicates the additional resources 0E required to obtain one additional unit of a product stream iE under specified conditions, defined as:
* 0i
i cond
EkE
⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠ (4)
Figure 2 shows the differences between average and marginal cost for a generic costing function. Other types of cost should be also considered: the opportunity cost refers to the highest rate of return one could obtain per extra unit available for some resource. This concept can be used for allocating cost to by-products. The abatement cost is defined as the amount of resources required to eliminate an undesired output, waste or residue.
00 P (kW)
CP (
c/kW
h)
Total Cost
Average Cost
MarginalCost
Figure 2. Marginal and Average Cost
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Then, what is the exergy cost of electricity? A first answer would be 3 units of exergy, because we suppose that it is produced in a conventional thermal power plant burning fuel-oil or coal, with an efficiency of 33%. If we consider electricity at the consumer’s point, we must add the losses in transport. This adds a 10% to the unit cost giving now a cost of 3.3 units. However coal or fuel-oil must be mined, treated and transported to the plant. Additionally, each time we produce electricity we produce wastes that need to be disposed. Disposal of wastes requires more exergy: Flue Gas Desulfuration, Special NOx burners and CO2 capture and storage. Different studies indicate that electricity of thermal origin at the consumer’s point can range between 4.5 and 5.5, which means that we need 4.5-5.5 units of exergy of coal in mine to produce one unit of electrical exergy. On the other hand, electricity is a “raw fuel” used in most industrial processes. We can question ourselves what is the physical cost measured in exergy units of all manufactured products as well as the exergy needed for their use, maintenance, repair and disposal. This comprehensive analysis is named Exergy Life Cycle Assessment (see bibliography) and provides average exergy costs because it is focused in obtaining round numbers which could be used as ecological indices of sustainability.
However, the problem comes up when two or more products, by-products and residues are produced simultaneously. How to allocate costs? We need rules mathematically supported rather than considering concepts of uselfulness. We need to look inside the system in order to understand the process of cost formation, by identifying the internal relationships of all the structure components.
Indeed, the main problem of allocating costs has been to find a function that adequately characterises every one of the internal flows in a system and distributes cost proportionally. This function needs to be universal, sensitive and additive. That is, it needs to have an objective value for every possible material manifestations, it needs to be able to vary when these manifestations do so and each internal flow property needs to be represented additively. There is a wide international consensus that the best function, at least for energy systems, is exergy, which can contain in its own analytical structure the flow history.
If we face the problem of calculating approximate average costs, we probably can stop our analysis by disaggregating our system at not very detailed level. Also, and commonly, cumulative exergy consumption analysis does not go into process details but focuses on the overall exergy consumption.
Not only physical components can be disaggregated, exergy itself can be disaggregated into its mechanical, thermal, chemical or physical components that have their own history of formation. Each sub-process has consumed its resources to produce a particular increase of pressure, temperature or composition. The flow history and its physical cost can always be reconstructed i.e. the amount of given resources to produce the flow. By a systematic account of these consumed resources, we can associate a physical cost to each identified flow, representing the sum of the resources needed to produce it under the given circumstances.
The problem with allocating costs to flow bifurcations can be solved classically. If the bifurcation does not affect the quality of the bifurcated flow, the costs are proportional to the quantity of each exiting flow. If the bifurcation affects quality, we must perform a detailed analysis of the change of exergy components and their proportion to allocate a cost to the bifurcated flow. Each exergy component is a reflection of a qualitative flow property which, in turn, has passed on to the flow throughout the history of the productive process with a specific resource consumption cost.
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Therefore three conditions are needed to allocate costs. First the definition of the boundaries of the system. Second, a structure of the system in which all the components or processes are described in terms of black boxes interacting to each other through energy flows ( or more generalized: energy, economic or information flows). And third the definition of the purpose of production for each and every component.
In thermoeconomics the words: history, degradation, exergy, quality, cost, resource, consumption, purpose and causality are related between them. In the cost formation process, it is essential to analytically search for the locations and physical mechanisms that make up a specific productive flow. The resources are used up to provide physico-chemical qualities to the intermediate products until a finished product is obtained. The main problem to be solved using exergy is how to measure and homogenize the accounting of these qualities.
The product cost obtained in this way is an average value, that it is does not distinguish between the first and the last units produced. Neither does it provide reasons to explain why the sequence of sub-processes is as it is. There are no a priori reasons to add qualities to a productive flow in a determined order and the analysis of the average costs does not provide it per se. It only reflects the facts as they are produced and takes them into account. It is only an additional knowledge and an engineering good practice of the analyst that can justify why, for example, first we compress, then we heat and then we change composition and successively so on.
At a given condition of the plant, that is, an instantaneous photograph of it, the resources and the products are counted analytically and numerical indices are given to each identified flow, which we call average costs. The problem now is to find a use for these values. Evidently they are useful but cannot be overestimated since they do not reflect the cost evolution of the next productive unit nor do they give us hints about why the process is like it is nor if it can be improved. Nonetheless, they are useful for benchmarking, that is to compare situations, either between two productive processes that are not too different or two states of the same process.
4. Cost Accounting For introducing this section, the contributions of Tsatasaronis, Kotas and Gaggioli were mainly followed (see bibliography).
Cost accounting consists of procedures for determining, or better for estimating, the total cost of production per unit of output for each product from a thermal system (e.g., for the electricity, steam, hot water, chilled water,…). All of the capital and operating costs, which are incurred to operate a thermal system, must be allocated to the final products. Thus, for each product there are the direct costs, those which are clearly attributable to the product such as resources and materials devoted specifically to it, and there are all the other indirect costs. A main challenge to the cost accountant is to assign each indirect cost in an equitable manner.
The purpose of cost accounting could be stated in broad terms as:
• Determining the actual cost of products.
• Providing a rational basis for pricing products and/or evaluating their profitability.
• Providing means for controlling costs.
• Forming a basis for operating decisions and their evaluation.
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For a cogeneration plant, see Figure 3, it is clearly important to determine what it costs to produce each unit of electricity and steam. These costs are significant, in as much as the price of each product will be affected, though generally not determined, by the cost of producing each. And clearly the profitability of individual products requires knowledge of the cost of each one. The determination or estimation of product costs are relevant not only for plants in operation but also for plants at the design stage.
There is not only interesting to find the costs of final products, it is also valuable to have them for internal products. Then the build up of costs for each final product could be traced through the energy system. If each cost of the internal streams of the system is assessed, it could be used, by comparing them with standard or reference cost values, to control and to avoid excessive resources consumption. This is the purpose of thermoeconomic diagnosis.
The determination of all costs of the streams is also useful to make trade-off analysis of the economics of the subsystem components. In an existing plant, such analysis can be used, for example, for maintenance and retrofit decisions, as well as for developing and implementing operation and control strategies. Likewise, it can be used for the discovery of improvements on system concept and optimising the design of a particular component and or the system as a whole.
The cost balance for the total system or for each component of the system could be formulated as: i i j j
i IN j OUT
c E Z c E∈ ∈
+ =∑ ∑ (5)
where iE represents the exergy of the inputs flows of the system/component and jE the exergy of the output flows. ic represent the unit costs of the flows, that are known for the inputs, and jc must be determined for the outputs.
In the case of a single product (output) of the plant, the unit cost of the product can be determined by Eq. (5). In the case of a multi-product plant, the balance equation is not sufficient. Additional criteria are required to determine the relationship between the unit costs of the different products. This is where exergy can be used as a basis for cost allocation of the products.
We will illustrate this with the example of a simple cogeneration plant depicted in Figure 3.
First we will show that exergy and not energy is appropriate to take as the basis for costing co-generated products. Energy costing rules consider that a unit of heat is equivalent to a unit of work, then the equations to determine the cost of heat and work could be written as:
a a aF W Qa aW Q
C C CC C W Q⎧ = +⎪⎨ =⎪⎩
(6)
As aforementioned, Keenan was the first author that proposed to use exergy or available energy to allocate costs. The exergy costing equations are:
b b bF W Q
b bW Q Q
C C CC C W E⎧ = +⎪⎨ =⎪⎩
(7)
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If we allocate costs in proportion to their energy content the unit cost of electricity is 1.21 units of fuel, that is 2.5 times less expensive than electricity produced in a conventional Rankine cycle. Furthermore this value is not sensible to the variation on the quality of the steam produced or to any malfunction that would occur in the turbine. According to this criteria it is the same to produce 1 kW of thermal energy than 1 kW of mechanical energy.
Turbine
Generator
Boiler
1
4
5 6
Pump
1
2 3
4
Z1 = 10 c/s
η = 85%
ηis = 90%
Z2 = 0.6 c/sZ1 = 0.3 c/s
η = 97 %
T2 = 460 ºC
P2 = 50 bar
T3 = 252 ºC
P3 = 10 bar
Figure 3. Physical Diagram of a simple cogenaration plant
Figure 4 shows a curve comparing the cost of steam using energy and exergy costing criteria to allocate costs in the cogeneration plant, for different steam pressures. The irrationality of energy costing is evident from the behaviour of the energy costing curve at low turbine exhaust pressures. At a pressure of 1 bar, for example, the unit cost on a energy basis for steam is close to 3 c/kg, even though such steam has very limited usefulness. Unlike exergy costing, energy costing does not make distinctions about the usefulness of energy transfer. We need cost accounting criteria that will be sensible to the quality and degradation that occurs in the system.
Exergy cost accounting provides a wide and clear vision of the use and degradation of energy and in consequence of natural resources. Exergy provides a thermodynamic value of any energy stream with respect to reference conditions, and can be rigorously obtained from the laws of thermodynamics, which allows precise measurements. Exergy analysis provides two important messages: one is that it allows to quantify and locate thermodynamic losses. The second is, it allows us to concentrate on the relevant part of the energy, namely “useful” energy. Exergy may regarded as a measure of the capacity of a given form of energy to produce work. It is also exergy which is lost, or consumed, in order to make a particular process, and it is therefore reasonable to assess the price of energy products on the basis of its exergy content.
Figure 4. Energy vs. Exergy Cost of Cogenerated Steam
The basic concepts underlying the explicit determination of auxiliary equations based on exergy criteria were presented, for example (see bibliography) by Gaggioli and Wepfer in 1980. These methods or rules are called the equality, the extraction and the by-product methods. They required a judgment regarding the purpose of each unit of the plant and distribute the costs proportionally to exergy contents of the flows.
We illustrate these methods using the turbine of the cogeneration plant. The balance cost Eq. (5) for this component is: 2 2 2 3 3 5 5c E Z c E c E+ = +
Now there is a problem since, assuming the cost of the inlet steam 2c is known, the cost balance equations have two unknowns, namely 3c and 5c . Thus an assumption must be made regarding the allocation of the cost to the two outputs: shaft power 5E and exhausted steam 3E .
In the equality method, the generation of the two products is considered to have the same priority, so the cost of the high pressure steam and the capital cost of the turbine are charged to the generation of both outputs (products) proportionally to their exergies, leading to the relationship:
2 2 23 5
3 5
c E Zc cE E
+= =+
In the extraction method, it is considered that the purpose of the turbine is to generate shaft power and thus the whole cost of the turbine and the irreversibility occurring in it are charged against it. This results in assuming the unit cost of exergy of steam entering the turbine to be the same as that of that of the steam leaving it, thus:
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3 2
2 3 2 25
5
( )c c
c E E ZcE
=− +=
In the by-product method the cost of one of the outputs is assumed to be known. It could be assumed that the generation of process steam is essential even if no electric power is generated. Therefore, the exergy of the process steam is costed as if it were produced alone in a hypothetical low pressure boiler at the required pressure and temperature. Therefore, the cost of electricity is determined by the cost balance. If the cost of electricity calculated by this method proves to be competitive with the purchase of electricity, then a decision in favor of co-generation should be made. An equivalent procedure could be made in case of estimating the electricity cost.
The “Exergy Cost Theory” developed a general procedure (see The Process of Cost Formation) for the determination of the auxiliary equations. One key step in the application of this procedure is to make a rational judgment on how the efficiency should be defined for each component of the plant. That is, for each unit to decide which inlet streams should viewed as fuel streams and which as feedstocks. And for the outlets which are products, and which are unspent fuel. To make these definitions is equivalent to deciding what the productive purpose of each component is, and choosing between the aforementioned extraction and equality methods. The other key step is to decide which streams should be viewed as residue streams, either wastes or by-products, for which the unit cost can be assigned rather than evaluated. This is equivalent to the by-product method or making the assignment to zero. These procedures are also patterned in part from the ideas of Fuel and Product developed by Tsatsaronis and Winhold (see bibliography). Furthermore the method is formulated in a mathematical manner which is significantly useful for computer implementation.
Table 1. Exergy and Exergoeconomic costs for the cogeneration plant flows
Nr E(MW) E* (MW) k* (kW/kW) C(c/s) c(c/MJ)
1 100 100 1 45 0,45
2 33,96 100 2,94 55 1,62
3 23,36 68,79 2,94 37,84 1,62
4 0 0 1 0
5 9,97 31,21 3,13 17,76 1,78
6 9,67 31,21 3,23 18,06 1,87
The system of equations, which determine the cost of heat and work for the cogeneration plant as well as for internal flows, are:
1
4
2 2 1 1 4 4 1
3 3 5 5 2 2 2
6 6 5 5 3
2 3 0
fuel
water
c cc c
c E c E c E Zc E c E c E Z
c E c E Zc c
==
− − =+ − =
− =− =
Table 1, shows the exergy and exergoeconomic cost of all flows considered in the simple cogeneration plant, in the case of the turbine the extraction rule is used. In this
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case the cost of electricity is higher than the cost of steam, because in the first case we include the irreversibilities of the turbine and the generator.
Figure 5 depicted a cost-irreversibility diagram obtained form the results on Table 1, which shows how the cost is formed. The cost of steam produced on the boiler is distributed in proportion to its exergy content. A 70% is allocated to generated steam, the rest is allocated to cost of the electricity generated. The cost of the steam includes its exergy content plus a 70% of the irreversibility generated in the boiler, meanwhile the cost of electricity includes its exergy content plus a 30% of the irreversibilities on the boiler plus the irreversibilities on turbine and generator. This is the reason why the unit cost of electricity is 15,4% higher than the cost of steam. The 9,9% is due to irreversibilities and 5,5% to investment cost of the turbo-generator.
F
1
2 3
CQ = 68,79 MW
100 MWEQ = 23,36 MW
CW = 31,21 MW
W = 9,67 MW
Exergy
AcumulatedIrreversibility
Figure 5. Irreversibility-Cost Diagram of Cogeneration Plant
4.1. The process of cost formation
Cost accounting methodologies are based on cost allocation rules, which attribute to the useful product of each system component the sacrifice of resources required to obtain it, and distribute its costs proportionally to the exergy. They are mainly numerical techniques that calculate values in an accurate way, but they lack a mathematical structure. Therefore, it is not easy with this blind information to identify the process of cost formation. In order to do that, general relationships are required that relate the overall efficiency of the plant and the cost of the products with the efficiency and irreversibilities of each individual component that form the system.
In order to analyze how the costs in a productive process are formed, we consider a sequential process, as it is shown in Figure 6. Let us name the exergy of the resources (fuel) of each component as F and its product as P. Correspondingly F* and P* will name the exergy cost (MW) of the fuel and product streams.
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TF 1F2F
rFnF
1P2P
rP
rI
nP TP
1κ 2κ rκ rκ
Figure 6.- Sequential processes system diagram
The unit exergy costs of the product of a generic unit –i– can be expressed as follows: * * * * *
, , ,P i i i i i F i i i F i ik P P F P k F P k k= = = = (8)
and the unit exergoeconomic cost ($/MJ) as: , , , ,( )P i P i i F i i i F i i i ic C P C Z P c k Z P= = + = + (9)
The increment of the unit cost in a process could be expressed as: * * *
, , ,P i F i F i i ik k k I P− = (10)
and ( ), , ,P i F i F i i i ic c c I Z P− = + (11)
These equations reveal that the unit cost of the product is always greater than the unit cost of the fuel. Any difference is calculated in terms of the irreversibility and capital costs.
As in thermoeconomics all the magnitudes are in proportion to exergy, all the reasoning can be made using exergy costs (kW) instead of the more difficult to obtain exergoeconomic costs measured in monetary units ($/s). Then from now on, we develop the concepts using the monetary stripped exergy cost.
In the case of the sequential system, where the product of a process is the fuel of the next one, the exergy cost of the product of the process –i–, could be written as:
*
1
i
i i rr
P P I=
= +∑ (12)
It shows that the exergy cost contains both the exergy of the product, and all the irreversibilities generated in the processes required to obtain it. Note that the left hand side of Eq. (12) has an economic meaning, meanwhile the right hand side of this equation is purely physics. This connects thermodynamics with economics.
On the other hand, Eq. (13) shows, that the unit exergy cost of the product of a component equals the product of the unit exergy consumption of the components taking part in the production:
*,
1
i
P i rr
k k=
= ∏ (13)
These expressions could be generalized for any plant, it does not matter how complex is it. (see Symbolic Exergoeconomics Analysis of Energy Systems).
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For example, if we consider a full example of the cogeneration plant, including feedwater pump recirculation, its cost formulae are as follows:
* *1 2 3 1, ,
2 3 4 2 3 4
(1 ) (1 )1 1P W P Q
r k k k r kk krk k k rk k k
− −= =− −
where –r– is the ratio of mechanical exergy to the total exergy of the high pressure steam. The denominator of these formulae represent the contribution of recirculation. If this value is close to zero the system could be considered as a sequential process.
Once these formulae are obtained, it is possible to know easily the effect of the variation of the efficiency of a component on the overall system. This question is analyzed in the next section.
4.2. The principle of non equivalence of the irreversibilities
The exergy balance of the overall system is given by T T TF P I= + . If the quantity of final product TP , is kept fixed, any variation in the design or in operation of the plant will cause an increase or decrease in the consumption of the resources equal to the variation of the total irreversibility, T TF I∆ = ∆ .
We could suppose that this change is due only to the variation in the exergy consumption of the unit –i–, given by ik∆ , then, the variation of resources entering the system could be written as: *
1 1 1 ,T i i i n T F i i iF k k k k k P k k P− +∆ = ∆ = ∆ (14)
Under the conditions stated, the unit exergy cost of fuel *,F ik and the product iP of the
unit –i– will remain constant, and the increase of the resources consumption will be given by i i i iF I P k∆ = ∆ = ∆ . A first conclusion from this analysis state that there is no equivalence among the irreversibilities of the process of a system. The more advanced the process is, the larger is the impact on resources consumption for the same increase of the local irreversibilities. Therefore, the unit exergy cost is a measure of the impact of local malfunctions in terms of overall resources consumption. A second outcome is that when modifying the efficiency of a process, a variation of the production demand occurs and the irreversibilities of all units that precede it will change. This effect is called structural disfunction.
The application of these ideas to performance tests and energy audits in actual installations will allow to identify and quantify the causes of the increase of the consumption resources in a system. That is the basis of thermoeconomic diagnosis.
5. Thermoeconomic Diagnosis The general objective of energy audits and diagnosis is to discover the degradations in energy systems. There are two main techniques to energy system diagnosis of deteriorations:
• Thermo-mechanical monitoring conditions are usually adopted in power plants in order to predict failures.
• Thermodynamic monitoring methodologies are mainly suitable to analyze anomalies causing reduction of system efficiency.
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Thermoeconomic diagnosis has its foundations on second law analysis and its aim is the detection of efficiency deviation, its economic impact, the identification and the location of its main causes. Thermoeconomic diagnosis procedures are based on the comparison between two working states: the actual operating state and a reference operating state, corresponding to the system without any deterioration or operation anomaly, and the fact shown in previous section, that irreversibilities have a different fuel impact depending on the location of the irreversibility in the process chain.
The difference between thermoeconomic analysis and conventional second law analysis for energy audit management stems from the following ideas:
Technical exergy saving: Not all exergy saving which are thermodynamically possible can be achieved in practice. In fact, the real saving potential of each unit is limited by technical and/or economic constraints. Thus, for a subsystem or installation, the technical exergy saving can be expressed as 0I I I∆ ≡ − , where 0I is the design base irreversibility and ∆I is the irreversibility increase when plant operating under design conditions supplies the same total product. Then it will be possible to express the additional consumption of overall resources as follows: 0
T T T jj
F F F I∆ = − = ∆∑ (15)
Malfunction and disfunction: The malfunction of a process is defined as the variation of its irreversibilities due to a degradation of its efficiency. The disfunction of a process is defined as the variation of its irreversibility due to the changes in its production demand.
The exergy balance for a component –i– can be written as ( 1)i i i i iI F P k P= − = − . If the operating conditions of the plant vary, the efficiency and the amount of product change. Then, the technical exergy saving could be broken down into as: 0 ( 1)i i i i i i ijI P k k P MF DF∆ = ∆ + − ∆ = + (16)
where iMF represents the malfunction of the i-th process, and ijDF the disfunction generated by the j-th process on the i-th. (see Thermoeconomic Diagnosis of Energy Systems)
Malfunction Cost is the effect of a local malfunction on the overall consumption of resources feeding a plant, given by: * * 0
,i F i i iMF k k P= ∆ (17)
and Fuel Impact is the effect of all local malfunctions on the overall consumption of resources feeding a plant.
Equation (17) is valid for the case of a sequential process, while the general equation of fuel impact is presented and discussed in Thermoeconomic Diagnosis of Energy Systems. For the general case the technical exergy saving coincides with the sum of the malfunction cost of all components: *
T j jj j
F I MF∆ = ∆ =∑ ∑ (18)
Figure 7 shows that a variation of 1% in the isentropic efficiency of the cogeneration turbine, causes a fuel impact of 0.24 MW, and compares the values of the technical exergy saving and malfunction costs for each component It explains that the malfunction is a better parameter than the irreversibility variation, in order to locate the
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causes of fuel impact. In case of technical saving analysis the causes of the fuel impact are located in the boiler, meanwhile in the case of the malfunction cost analysis, they are located in the turbine and in the additional steam produced to maintain the electricity production.
0 50 100 150 200 250
Boiler
Turbine
Generator
SteamProduction
Fuel Impact (kW)
Irreversivility
Malfunction
Malfunction Cost
Figure 7. Fuel Impact and Technical Saving
The efficiency variation of a component may have different causes, either external to the plant (variation of environmental conditions, plant production demand and fuel quality) or internal, which are the presence of anomalies due to the component degradation (also called intrinsic malfunctions), efficiency variation induced by modification of the component operation conditions, and control system intervention.
The development of thermoeconomic diagnosis begun with the work of the research group led by A. Valero (see bibliography). The mathematical formulation of the fuel impact formula, was developed in several works: Lozano et al. (1994), Reini et al. (1995) and Torres et al. (1999). The major effort was concentrated in developing procedures to localize anomalies and quantify effects both in the component where the malfunction is originated and those indirectly affected by malfunctions, and the definitions and calculations procedures for concepts such as: intrinsic malfunction, induced malfunction, disfunction and malfunction cost (see Thermoeconomic Diagnosis of Energy Systems)
The TADEUS problem, an acronym of Thermoeconomic Approach to the Diagnosis in Energy Utility Systems, is a project aimed at integrating various experiences accumulated by several groups of researchers working in thermoeconomic diagnosis. As a result a set of articles were published in 2004, showing different approaches, each one having particular characteristics that are, nonetheless, complementary to each other.
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6. Thermoeconomic Optimization For intruducing this section the contributions of Frangopoulos, El-Sayed and Von Spakovsky were mainly used (see bibliography).
Thermal system design required to answer questions such as: What processes or equipment items should be selected and how should they be arranged? What is the preferred size of a component? What is the best temperature, pressure, flow rate, and chemical composition of each stream in the system? To answer these questions, engineers need to formulate an appropriate optimization problem.
The first step in the definition of optimization problem is to define clearly the boundaries of the system to be optimized. All the subsystems that significantly affect the performance of the system under study should be included in the optimization problem. The selection of criteria on the basis of which the system design will be evaluated and optimized is the key element in formulating an optimization problem. Optimization criteria may be economic (total capital investment, total annual levelized costs, annual levelized net profit,…) technological (thermodynamic efficiency, production time, production rate, fuel consumption,…) and environmental (rates of emitted pollutants). An optimized design is characterized by a minimum or maximum value, as appropriate for each selected criterion.
Another essential element in formulating the optimization problem is the selection of the design variables that adequately characterize the possible design options. In selecting these variables, it is necessary to include all the important variables that affect the efficiency and the cost effectiveness of the system. Each component and the system as a whole is defined by a set of quantities. Some of them are fixed by external conditions (e.g. environmental pressure and temperature, fuel price) and are called parameters. The remaining are variables, i.e. their value may change during the optimization procedure. Those variables, the values of which do not depend on other variables or parameters, are called independent or design variables. The rest can be determined by the solution of a set of equality constraints and they are called dependent variables.
The mathematical model for an optimization problem consists of:
• An objective function to be minimized.
• A set of equality constraints.
• A set of inequality constraints.
Thermoeconomic optimization methods use as a primary criterion an optimization performance measure: minimize the total levelized cost of the system products that includes the cost of external fuel resources, capital investment and maintenance cost. Also multicriteria optimization and environmental constraints can be considered.
The objective function expresses the optimization criterion as a function of the design variables: 0 ( ) ( ) ( )e T i
iMin C x c F x Z x= +∑ (19)
where ec represents the unit cost/price of the external resources, TF the exergy of the external resources, and iZ the capital and maintenance cost of each component.
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In order to define the equipment cost functions iZ , we will assume the investment costs to increase with increasing capacity and decreasing exergy consumption (or increasing efficiency), and can be represented, by the following relation:
,0,0
,0
+βi
iii i i i
i i
kZ Z P
k k
α
γ⎛ ⎞= ⎜ ⎟⎜ ⎟−⎝ ⎠
(20)
where ik is the unit consumption of the component, iP the product (size) of the component and ,0iZ is the part of the cost independent of the product. Both production and efficiency are functions of the design variables.
Another approach of costing functions that handles itself in optimization is: ,i a i iZ c A= (21)
where ac is the unit manufacture cost, and A represents a characterizing dimension of the component, and it is expressed in terms of thermodynamic variables, following a typical Cobb-Douglas production function: j
i i jj
A xαυ= ∏ (22)
where iυ is a characteristic parameter of the characteristics of the device. The number of thermodynamic parameters depends on the process; these are essentially sizing parameters (mass rate, heat rate or power) and efficiency parameters (adiabatic or isentropic efficiencies, pressure losses, heat transfer temperature differences,…). An example of cost function (see Cost Functions of components for optimal system design) for a heat exchanger will be: 1 0.15 0.15
m t sA Q T P Pυ − − −= ∆ ∆ ∆
The exchanger surface A is expressed in terms of the heat load, the logarithmic mean temperature difference and the pressures losses on the shell side and the tube side.
01
InvestementCost
Unit Consumption k
Cos
t ($)
Fuel Cost
TotalCost
Figure 8.- Costing Functions
21
The balance between thermodynamic measures and capital expenditures is an economic feature, which applies to the thermal system as a whole and to each component individually. The costs of resources usually vary to the opposite direction of the cost of equipment with respect to the design variables. An improvement on the structure or the efficiency of the equipment implies a reduction of the resources consumption but an increase of the capital investment. Figure 8 shows a graph of a generic function for the cost of production.
The equality constraints are provided by appropriate thermodynamic and cost models as well as appropriate boundary conditions. These models must include the flow rate and energy balances for each component, relations associated with the engineering design, such as the local efficiencies of the components. The model adopted by thermoeconomic optimization (see Structural Theory of Thermoeconomics) relates the input (fuels) of each components with its outputs and design variables: ( , )i i jE g E x= (23)
The model can also contain inequality constraints that specify the allowable operating ranges, the maximum and minimum performance requirements, and bounds on the availability of resources. When optimum is reached with only equality constraints, we obtain the shadow costs, one for each independent variable. If an inequality constraint is active in the optimum, the cost becomes an opportunity cost for the constrained variable (see bibliography).
There are cost optimization procedures which make no use of the exergy concept. So cost-effectiveness of every change carried out on a plant component must be assessed in terms of the overall system parameters, e.g. its effect on the consumption of fuel resources. This makes optimization very complex and computer time consuming. With thermoeconomic optimization these difficulties may be overcome. For example, with proper themoeconomic analysis and under certain conditions, the decomposition is applicable, which facilitates the solution of the problem, because it allows the optimization problem of the whole system to be decomposed into a set of optimization problems of subsystems or components, which are of smaller dimension (i.e. they have fewer independent variables) and can be solved more easily.There was basically, at the beginning, two different thermoeconomic approaches: the structural method that use the local unit cost of the irreversibilities (see Design Optimization of Power and Cogeneration Systems), and the autonomous method introduced by R. Evans and Y. El-Sayed in 1970 that is the starting point of other state-of-the-art techniques. (see Optimization Methods for Energy Systems and Functional Analysis).
We will illustrate this last method using a sequential process (fig. 6). Also, the optimization will be restricted to one independently variable - ix - per component. With the output of the system PT constant, the objective function is the total cost, per unit of time and is given by:
0 1 1 11
( , )n
i i ii
C c E Z E x+=
= +∑ (24)
where 1c ($/MJ) is the unit exergoeconomic cost of the fuel input, 1E the exergy input rate, and iZ the cost of capital investment per unit of time of each component of the system.
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Assume that the optimum configuration of each component, and hence of the whole system, can be obtained by choosing the values of - ix -, that satisfy the condition of minimum for the objective function (24), given by:
0 0 1, ,i
C i nx
∂ = =∂
… (25)
With iE as independent variables, n+1 constraining equations relating these variables are required:
1
1
( , ) 1, ,i i i i
n s
E g E x i n
E constω+
+
⎫= = ⎪⎬
= = ⎪⎭
… (26)
As a result a Lagrangian function L becomes the objective function instead of (24):
( )1 1 1 1 1 11 1
( , , ) ( , ) λ ( , ) λ ( )n n
i i i i i i i i n s ni i
L c E Z E x g E x E Eλ ω+ + + += =
= + + − + −∑ ∑E x (27)
If we introduce the function: 1 1 1( , ,λ ) ( , ) λ ( , )i i i i i i i i i i iE x Z E x g E x+ + +Γ = + (28)
the Langrangian function (27) could be rewritten as:
( ) ( )1 1 1 1 1 1 11
( , , ) λ ( , ,λ ) λ λ ωn
i i i i i i n si
L c E E x Eλ + + + +=
= − + Γ − +∑E x (29)
Minimum 0C occurs when the partial derivatives of L with respect of x and E variables vanish. From the condition 0iL E∂ ∂ = , we get:
1 1
11 1
λ
λ λi ii i
i i
cZ g
E E++ +
=⎧⎪ ∂ ∂⎨ = +⎪ ∂ ∂⎩
(30)
These equations determine the values of the Lagrange multipliers λi or shadow costs, that take the role of local marginal costs of exergy inputs of each component.
From the condition 0iL x∂ ∂ = , the minimum is obtained as:
iλ 0i i i
i i i
Z gx x x
∂Γ ∂ ∂= + =∂ ∂ ∂
(31)
From the previous equations, it is shown that the system may be decomposed into autonomous subsystems, whose local functions are defined in Eq. (28) and they are optimized for the local variables ix .
A special case appears when capital cost and the exergy input increase linearly with the exergy output of the components: , 1( )i Z i i iZ f x E += and 1( )i i i ig k x E += , then Eq. (30), is rewritten as:
1 1
1 ,
λλ ( ) λ ( )i Z i i i i i
cf x k x+
=⎧⎨ = +⎩
(32)
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where ik represents the unit exergetic consumption, and therefore λi represents the unit average exergoeconomic cost. The general methodology is fully explained in the articles Functional Analysis and Structural Theory of Thermoeconomics.
There are several approaches to Thermoeconomic optimization, that were presented in a set of articles in 1993, as a result of the project CGAM.
The Exergoeconomic Optimization Approach, proposed by Tsatsaronis uses an iterative design improvement procedure that does not aim at calculating the global optimum of a predetermined objective function, as the conventional optimization methods do, but tries to find a “good” solution for the overall system design. The basic idea, lies in a commonly accepted concept from the cost view point: at constant capacity for a well designed component, group of components, or total system, a higher investment cost should correspond to a more efficient component and vice versa.
The Functional Analysis proposed for C. Frangopoulos and the Engineering Functional Analysis proposed by M. von Spakovsky used the method of the Lagrange multipliers and decomposition procedures.
A. Valero and coworkers, present a similar approach, but propose to use the unit average exergy costs instead of the Lagrange multipliers.
Y. El- Sayed proposed also, in order to avoid problems with the isolation of the decision variables (see Design Optimization of Power and Cogeneration Systems) to divide the decision variables {x} into local variables {xL} and global variables {xG}, in general the number of global variables is much smaller than the local variables, iterate to find the local optimum of each component respect its local variables and the global optimum respect to the global variables.
The decomposition strategy is based on the Principle of Thermoeconomic Isolation (TI) introduced by R. Evans in 1980: A component of a thermal system is thermo-economically isolated form the rest of the system if its production iP and the unit cost of the resources λi are known quantities and independent from the rest of the component variables.
It is an ideal condition which cannot be fully achieved for most of real systems, but the more the TI conditions are fulfilled the fewer iterations are required to achieve the optimal solution for the whole system. Therefore the thermoeconomic model of the system is subdivided or decomposed into subgroups. Each subgroup is optimized in turn, according to a sequential process, iterating around the system until the system’s internal economy converges, within prescribed tolerances, to a single set of values. Decomposition may only approach the global optimum since the degree of thermoeconomic isolation of the independent variables, the choice of the subgroups and their functions, and the nature of the dependent variables greatly affects how close the approach will be. Nonetheless, the advantages of this strategy facilitates the optimal design of individual units in highly interdependent complex systems, and let the designers to concentrate their efforts on designing the variables of single components, while resting assured that these efforts improve the overall system.
These ideas could be used to design a strategy for local and global optimization problems, that is described in Figure 9.
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Define productive structure
Define Thermodynamic Model
Select Decision Variables
Divide decision variables into local and global variables{XL,1 , ...,XL,N,XG}
Define Cost Functions
Chose initial values for all design variables
Until convergence criteria is satisfied
Solve termodynamic model
For each component of the system
Compute the unit cost of products
Find the optimun of the local function Γ irespect the local set of variables {XL,i}
Find the optimun of global cost function C0respect the set of global variables {XG}
Figure 9.- Thermoeconomic Optimization Strategy
7. Final reflections and conclusions Everyday life is immersed in economics. Our society tends to substitute human and natural values by economic ones, i.e. prices. But capital is not the only resource. To say the contrary is an arrogant way of reducing nature and society to nothing. We put prices almost to everything even though we don’t know its value. So, what is the alternative? We can approach to know the value of things and living entities, if we try to reproduce or replace them. When doing this, we realize how difficult it is. How many resources –of all kind of types– are needed to get them? How much knowledge is necessary to understand the mechanisms by which resources can be converted into entities? These questions give a sense of ethics of conservation because we should not destroy what we don’t know how to construct it.
This is why it is so important to know costs measured as a general sacrifice of resources, independently of the numeraire we use to quantify them. However, as it is well known, the value of cost has its drawbacks. It depends on the limits of the system, it also depends on the value –exergy in our case– and on the purpose (efficiency) of each and every subsystem that interrelates the productive structure of the system. Relativism is a main feature of resources accounting.
Besides of that, we have also another problem: are the costs of things stable with production? In other words, when we say that the cost of electricity is three, does this value change when we demand more electricity? Everybody knows that in small production intervals the average cost of the product remains almost constant. So, what
25
does it mean small production intervals? This depends on the purpose of our study. In a macro level, average costs can be used without much error. On the contrary if we are interested in optimizing a power plant, we realize that the cost of electricity is very sensitive to the efficiency of the plant components.
The purpose of the study advises the accounting methodology i.e. –the accuracy of results as well as the relaxation of the rules to assess costs–. Thermoeconomics is another viewpoint of matter analysis in Physics. When we try to understand the heat and mass transfer mechanisms in a heat exchanger, we use the techniques provided by the heat and mass transfer sciences. These sciences use and interpret pressure and temperature as overall values of properties that spatially and temporally change. The understanding of a system interconnection and efficiency is better attained using thermodynamics rather than the sciences of heat and mass transfer. Thermodynamics helps to understand how energy is used and degraded in the different parts of the system. But thermodynamics requires an input-output abstraction of the system. Some information can be obtained from its analyses, some other cannot. There is no contradiction between the different types of analysis. They focus on different aspects of reality. And the “micro” level science –heat and mass transfer- is capable to explain the magnitudes used at the next level science, for instance the interpretation of pressure, temperature, energy and so on by the transfer sciences. There is not a disruption in knowledge but continuity.
When we focus on exergy accounting, we are at the next (aggregated) level of knowledge, now connected to thermodynamics. Thermoeconomics uses thermodynamics but differs in the type of problems it tries to solve as well as the use of new concepts –like cost– borrowed from economics.
Exergy could be considered as the bridge between thermodynamics and cost accounting methodologies. This is, because exergy connects with intensive properties like pressure, temperature, energy and so on and on the other side it can be rigorously defined and its cumulative consumption calculated. When we investigate all the properties of exergy costs we are giving a conceptual bridge between classical Physics and Economics. On the other hand, and as explained before, we are giving a basis for a general resources accounting methodology that can use any other numeraire or even relax the rules of accounting themselves. That could be the case of emergy whose main interest is to know how many units of solar energy are needed to produce any product.
Second law costing is deeply related to causality and not so related, as it might be expected, to internal price allocation. In our point of view, when we are to assess the impact that a malfunction has on the consumption of resources, you only need to provide information about the structure of the plant and the efficiency of the components. Exergy cost accounting is not so interested in arguments about how many rewards you get by selling the products or by saving money because of joint production instead of a separate one. Cost accounting propositions like "heat as a by-product" or "work as a by-product" and so on, are needed when we are analyzing the price formation process or better said, the price propagation process. But, when we are interested in the cost formation process throughout the structure and we understand cost as a sacrifice of resources, there is no doubt what are we finding.
There is a claim of the lack of words and for the dangerous use of the word "cost". In fact, cost has different meanings for different people and practitioners. Cost is most of the times related to money and not so much to physical resources, but in its broader meaning cost is measured in resources, in time, in lives, etc. Besides of that, "cost" is in
26
many cases mistaken as "price" synonym. We mostly agree on its conceptual difference but we forget it when we apply rules for cost accounting that include external considerations about the finality of the production, or the market's utility of the products we obtain.
So, what is the purpose of cost accounting? The answer must be deeply related to the physical behavior of the system. Cost should be related to physical measurements like mass flow rates, pressures, temperatures and compositions, then, related to actual irreversibilities occurring in the system and then to causes. This is the only way that we can provide physical roots to the accounting techniques. From this point we may use –or not– the values of exergy costs as a basis for price setting, both external and internal prices. Exergy cost accounting provides a set of indicators about how much impact on resources has a perturbation on a given structure. Modern management techniques that use internal prices to buy and sell intermediate products within the company can base its price-settings on exergy costs but should not be taken directly instead of prices. Internal economy techniques are one thing and exergy cost accounting another.
These are the ideas lying behind the exergy cost analysis. In the subsequent articles of this chapter, we will give a physical basis to the rules for calculating costs, we will give coherence to the connection of exergy cost with thermodynamics and we will understand under what conditions cost changes.
The connection between the subsystem level and the overall system takes now a crucial importance. Therefore we now concentrate on the relationship between efficiency change and impact on resources.
There exists a relationship between additional local irreversibility and additional consumption of resources (see Thermoeconomic Diagnosis of Energy Systems). The Eq. (17) is quite important because it justifies the practical reasons for internal cost accounting and thermoeconomic optimization based on decomposition strategies. In other words it answers the question of how many additional expenses you must pay because a malfunction appeared in a component. No more no less. The key word is propagation. And the idea of costing is to quantitatively assess the propagation effects on each and every subsystem in the plant.
From the point of view of thermoeconomic optimization, the exergy cost becomes the shadow exergy cost, or Lagrangean exergy multiplier, when the thermodynamic behavior of the plant is optimized. This connection is explained in the Structural Theory of Thermoeconomics.
Knowledge of the shadow costs helps designers react to possible changes to constraints. These constraints in the case of thermoeconomic analysis describe the exergy input-output structural relationships. The more advanced in the system is a process, the more impact on fuel consumption its local irreversibilities have (principle of non equivalence of the irreversibilities).
Therefore a crucial step in the optimization process, either in the design stage or in the operation stage, is the sensitivity analysis, and shadow costs are used to weight changes in the constraints. Relaxing the constraints leads to improvements in the optimum, or on the contrary, degradations of the optimun are assesed as the value of the shadow costs obtained.
A fascinating result is that the cost formation process, i.e. the formula that represents the relationship of any cost with the efficiencies of the subprocess and other related structural parameters have the same mathematical basis as that of the shadow costs.
27
Therefore the shadow costs obtained from the optimization problem provide the same results as those of the exergy cost obtained from the cost accounting methodologies (under given specifications as explained in the Structural Theory of Thermoeconomics).
This allows for the unification of thermoeconomic theories and for understanding why under what conditions they do not match. This is possibly the best theoretical contribution of thermoeconomic optimization.
Also, when we relate the exergy costs with the impact on resources feeding the plant, i.e. the Fuel Impact formula, we are bridging the thermoeconomic optimization and diagnosis. In other words if the loss of resources is thermodynamically and universally measured in terms of irreversibility, the relationship between the local and global irreversibility can be explained in terms of exergy cost and structural parameters (like efficiencies, bifurcations and production parameters).
Summarizing, these are the reasons, why the concept of exergy cost is of paramount importance on thermoeconomics:
• It relates cost, an economic concept, with irreversibility, a physical concept. But it is still measured in physical units and provides a rational basis for economic allocation rules.
• The formulae describing the cost formation process for a given system structure coincide with these obtained for the optimization problem. Thus conceptually connecting the optimization and the cost accounting methodologies. Also giving a way to understand why are the reasons for discrepancies. This is because it is easier to explain differences in formulae than in numbers.
• If the relationship between local irreversibility and the overall impact on resources is algebraically assessed through the exergy cost, we can understand and also have a solid and meaningful basis for thermoeconomic diagnosis.
Even now, when we dispose of powerful computers and numerical methods for optimization and simulation of real world process, that could overcome thermoeconomics. Thermoeconomics becomes a crystal clear and unique way to connect the physical universal measure of loss, i.e. irreversibility with the loss of resources at the overall system level and then to economics. This is perhaps the stronger reason for a continuing systematic research in this new science.
Glosary Abatement Cost: Amount of resources required to eliminate an undesired output or residue. Average (unit) cost: Average amount of resources consumed to obtain (a unit of) the product. Capital cost: Monetary value of a process unit. Characteristic equation: Equation describing exergy input(s) as a function of the exergy output(s) of a process unit. Cogeneration: It is the thermodynamically sequential production of two or more useful forms of energy from a single primary energy source. Constraints: Functional equalities or inequalities that have to be satisfied by the optimum solution in an optimization problem. Costing Functions: Functions which express the cost of a process unit in terms of thermodynamic and physical variables.
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Decomposition: Separation of an optimization problem (system) into subproblems (subsystems) in order to facilitate the solution.
Disfunction: Increment of the irreversibility of a process when production demand varies.
Embodied Energy: The amount of energy units required to produce a given product. Energy system: A system that transforms various energy forms to other energy forms, of which at least one is useful. Exergoeconomic cost: Amount of resources –apportioned in proportion to exergy-, expressed in terms of monetary units, to obtain a product. In a broad sense it is synonymous of thermoeconomic cost. Exergy (unit) cost: Amount of resources, expressed in terms of exergy, to obtain (a unit of) the product. Exergy: The maximum theoretical useful work that can be obtained if a thermodynamic system is brought into thermodynamic equilibrium with the environment, while the system interacts with the environment only. Functional analysis: The formal, documented determination of the functions of the system as a whole and of each unit individually.
Impact on fuel or fuel impact: Additional fuel consumption because of an inefficiency in a piece of equipment in a system. Independent variables: Those variables appearing in the objective function that do not depend on other variables or parameters; their optimum values, that is the values that minimize (or maximize) the objective function, are determined by the optimization procedure.
Induced Malfunction: Increment of the irreversibility of a process due to malfunction of other process of the system.
Intrinsic Malfunction: Increment of the irreversibility due to the degradation of the process itself. Irreversibility: Is a synonymous of exergy destruction, which represents the energy quality degradation occurring in whatever physical process. Irreversibility: That part of exergy that is destroyed in a process.
Malfunction Cost: Increment of external resources due to the malfunction of a component Malfunction: It represents the increase of irreversibility in a device suffering an inefficiency or efficiency degradation Marginal cost: Additional resources consumed for producing one more unit of the product under specified conditions. Objective function: A mathematical function, the minimization (or maximization) of which is the objective of optimization.
Opportunity Cost: The rate of degradation of the optimum per unit use of a nonoptimal variable in the design Optimization: The process of finding the conditions, i.e. the values of variables, that give the minimum (or maximum) of the objective function. Productive structure: Scheme of the function of the different process units and their interactions in a system. It represents how the resources consumed by the system are distributed among the different pieces of equipment and converted into the final plant product.
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Thermoeconomic Diagnosis: Part of the Thermoeconomic analysis which studies the causes of the malfunctions of the thermal processes and its impact in the consumption of natural resources. Thermoeconomic isolation principle: This determines whether a component behavior is independent or not with respect to the rest of the plant units. It can be defined as follows. A unit of a thermal system is thermoeconomically isolated from the rest of the system if the output product of the unit and the “prices” associated with the product and the inputs coming from other units of the system are constant and known quantities. Thermoeconomic model: Mathematical representation of the productive structure of a system.
Thermoeconomic Optimization: Part of the Thermoeconomic analysis which studies the optimum cost for design of thermal system.
Nomenclature Scalars Z Capital cost rate of a component ($/s)
*k Unit average exergy cost of C Exergoeconomic cost ($/s) c Unit exergoeconomic cost ($/MJ) E Exergy of a flow (kW) F Fuel exergy of a component (kW) I Irreversibility and exergy destruction of a component (kW) m Number of flows n Number of components P Product exergy of a component (kW) Q Heat W Work x Operational variable
Greek letters ∆ Increment κ Marginal exergy consumption η Exergy efficiency λ Lagrange multiplier; marginal cost Γ Local Cost Function
Subscripts 0 Related to environment e Inlet F, P Related to fuel or product i, j Indexes for component, flow number s Outlet T Total system Q, W Related to Head and Work
Superscripts * Exergy cost
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0 Reference values
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