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OPTIMIZATION PROBLEMS IN THERMAL SOLAR PLANTS
Session 2.18.
1632
OPTIMIZATION PROBLEMS IN THERMAL SOLAR PLANTS A. ADELL, Laboratoire de Chimie-Physique, Université des Sciences et Techniques du Languedoc, 34060, Montpellier, France ABSTRACT The aim of this work is to answer such elementary questions as "what is an op timization problem and how to solve it?" The case of thermal solar plants has received a special attention because of the obvious importance of optimization for their industrial development. The principal concepts of the literature ha ve been presented while introducing some new ones such as "quality factors". KEYWORDS Optimization problems, solar plants, Thermoeconomics, quality factors, systematic decrease of exergy. Plant optimization means a complex set of mathematical and technical operations in order to satisfy some predetermined criteria such as maximizing the return on the financial investment, minimizing pollution and so on. OPTIMIZATION CRITERIA The optimization criteria concerning the apparatuses of transformation of en ergy are generally classified into three groups (Maczeck, 1980): thermodynamic socio-economic and technical. The technical optimization criteria proceed from the conditions imposed by ter chnical considerations like the minimum bulkiness, the best means of detecti ng the escapes of working fluid, the minimum of corrosion... . The economic optimization criterion is the most important for the owner of the machine who wishes to maximize the benefit of his investment. But when the de cision is to be taken by, for example, a public body, other socio-economic criteria must be taken into account, e.g. minimizing foreign currency cost, maximizing the number of created jobs,minimizing the energy and materials in puts ... . The thermodynamic optimization criterion has to be considered when the deci sion is to be made by a heat engineer whose only concern is the thermo-energetic behaviour of the system. GLOBAL OPTIMIZATION In practice several criteria intervene simultaneously. This is treated in the so-called global optimization (Le Goff, 1979). In global optimization some criteria can be contradictory such as minimizing the selling price and maximizing the reliability. It is the politic decider's responsability in important so cio-economic projects, to asses the respective weight that must be given to each criterion for establishing priorities. In order to achieve global opti mization, three types of consideration must be taken into account:
Session 2.18.
1633
1- complying with the technical specifications of the project. The tech nical specifications fix the objectives to be reached and indicate the value or the constraints that must be verified by certain parameters; they also contain, more or less implicitly, some optimization criteria with eventually some indications on their respective weight. 2- choosing a particular kind of apparatus the working principle of whi ch allows the satisfaction of the objectives of the technical specifications, and considering some hypotheses concerning the expected behaviour of the sys tem. These hypotheses may be technical in nature which allows the modelling of the operational behaviour of the real machine in a theoretical or empirical way, e.g. the nature of irreversibilities, the life-time of the machine parts ... . These hypotheses may also have a socio-economic nature, e.g. national or world-wide evolution of material or energy costs; then they correspond to the choice of an economic model. All these hypotheses constitute "bets" whi ch all the knowledge of "the man of the art" will tend to minimize the risk.of. 3- taking into account additional constraints imposed by the environment of the system; the word "environment" been taken in a wide sense: can the de signer realize the parts of the machine at the required dimensions himself or must he buy standard industrial elements? what are the chemical nature and the degree of purity of the available constituents? what are the ambient meteoro logical conditions?... . Thus the global optimization can be done and discussed within scenarios which involve various choices: technological choices concerning the type of machine, technico-economic choices linked to the bets considered and political or tech nocratic choices of the respective weight that should be given to the criteria. OPERATIONAL, CONSTRUCTIONAL AND DESIGN PARAMETERS The practice of optimization can be done by two sorts of operations: a mathe matical treatment of an a priori problem, allowing the solution of the optimum value for certain parameters of the system, or technical operations which mu st be done on the machine during its working,among which automatic control constitutes the principal element. The parameters that should be taken into account in order to achieve the op timization are either those which define the conditions of the working and of the environment from a physical as well as a socio-economic viewpoint, or th ose that characterize the machine constituents (elements or working fluids): chemical nature, geometry, cost... . The behaviour of these parameters may be continuous, discrete or simply qua litative; they may also be time-dependent in non stationary systems such as solar energy apparatuses. The knowledge of the qualitative parameters — chemi cal nature and degree of purity of fluids, chemical nature and quality of the surface or type of manufacture of pipes... — is equivalent to the implicit knowledge of a great number of coefficients among which some play an important part in the optimization according to the hypotheses considered. These coeffi cients generally have a discrete value,as in the case of viscosity and heat transfer coefficients and for the cost of the type of thermal insulator, for instance. The usual method to proceed to the optimization of these qualitative parameters consists of an a priori discussion of the result expected for dif ferent initial choices. The inverse method is of course theoretically possi ble and may lead to interesting information delimiting the possible choices, but generally does not directly result in a concrete solution. So the choices of the qualitative parameters are most often included in the planned scenario. The set of parameters have been classified into three categories in the li terature: operational, constructional, and design parameters (Maczeck, 1980). DECISION VARIABLES AND OBJECTIVE FUNCTIONS
1634
Session 2.18.
Optimization problems arise when we have to answer some essential questions concerning a project; these questions relate to the determination of extrema: which machine is cheaper? how to improve an existing machine? what is the best technological channel? Among these questions the principal is: what value sho uld be given to the parameters of the system in order to realize the best plant? To solve mathematically an optimization problem we must first determine which are the parameters of the system that have to be known to answer the asked question. These parameters constitute the decision variables of the questions. In order to formulate mathematically the questions of the optimization problem a quantity called the objective function (o.f.) is associated with each crite rion contained in the questions. These o.f. must satisfy an extremum condition. The o.f. are explicitly expressed by means of physical or economic laws that apply to the model of the scenario, as a function of certain parameters which are the decision variables of the criterion. Beside these o.f. associated with the optimization criteria, there are other o.f. with no extremum conditions: they are the parameters constrained by the technical specifications or by the environment. These quantities must in their turn constitute or depend on new variables which, together with the decision variables of the criteria, form the primary decision variables also called
essential variables. These essential variables are necessary to answer fundamental points such as: how to fabricate the machine and what are the conditi-, ons for its operation?
In order to achieve the global optimization that is to satisfy simultaneously all the questions while respecting the objectives and the constraints of the project, a set of coupled mathematical relations must be solved. These rela tions contain a great number of interdependent variables. This interdependen ce entails a great complexity for optimization problems, furthermore, as has been seen, some variables can have a discrete value which varies with the scenario and so have to be optimized case by case; this increases the length of calculations, complicates the discussions but affords an opportunity for future innovations. PARTIAL OPTIMIZATION Partial optimization is characterized by a limited number of mathematical re lations with which is associated a limited number of variables. Partial opti mization may be envisaged either to treat separately a few questions the an swer of which offers an intrinsic interest, or to bring about a simplification of the problem of global optimization. In the latter case, partial optimiza tion becomes a method of uncoupling the mathematical relations of the problem. Among the simplifying methods, the well-known method using what is called "zone decomposition" (El Saved, 1970) allows the reduction of the problem of global optimization of a complex machine to the problem of partial optimiza tion in each element or functional zone of the machine. This method becomes of great effectiveness when the concept of available energy — also called exergy <— is used. However this method leads to an approximate general solut ion because it neglects the interdependence existing between certain variables of each zone. One of its main advantages is to legitimize the use of finer thermodynamic models in each zone, contrary to the global optimization which more often can only be done using coarse or over-simplified models, e.g. " quasi-perfect machine", "basic cycle"... . This is what Tribus, initiator of the method, has justified by saying: "It is much more important to be able to survey the set of possible systems approximately than to examine the wrong system exactly". Thermodynamic optimization is another well-known example of partial optimiza tion involving the use of only one criterion and a limited number of decision variables.
Session 2.18.
1635
WORKING PARAMETERS, QUALITY FACTORS, DEGREE OF FREEDOM Two types of situation occur during the working of machines of transformation of energy:— thermodynamic equilibrium states (or steady flows), for which the passage from one state to another defines the transformation between these -- mass and energy transfers which characterize an evolution between these two equilibrium states. These two situations concern the whole system, that is the machine itself (in ternal situation) as well as its thermodynamic environment (external situation). THE working parameters are used to describe these two situations. These paramaters are composed of the variables of equilibrium states and of the transfer variables. They are interconnected by two sorts of relations: .fundamental laws that directly link the parameters together; .phenomenological laws that may have a theoretical or empirical basis and that involve various coefficients depending on the experimental conditions, according to the hypotheses considered in the scenario. In the case of a machine for the transformation of energy, the exergetic effi ciency is the principal optimizable o.f. because contrary to the energetic ef ficiency, it has an intrinsic physical meaning independently of the tempera ture of the heat exchanges. In simple cycle flow machines, efficiency is affected by two types of decrease: on the one hand the systematic decrease of exergy and on the other hand the occasional decrease of exergy; decrease meaning either a loss of exergy with a corresponding decrease of energy, or a dissipation of exergy with conservation of energy and creation of an equal amount of anergy. These decreases may be internal or external. The decreases of exargy are associated with what we have called the quality factors'. These quality factors depend on two kinds of parameters of intensive or extensive nature. EXQJtg&tic, VJLkkJLpOütLonk'. if one considers that the irreversible phenomena responsible for dissipations are in their linear domain, then the quality fac tors connected to the exergetic dissipations depend on the intensive phenome nological coefficients that link the cause of the phenomena (generalised affi nities) to their effects (generalised fluxes), e.g. thermal conductivity, dif fusion, viscosity... coefficients. The quality factors also depend on extensi ve coefficients such as the coefficients of dimension or shape, e.g. heat ex changers surfaces, diameter and surface state of pipes, shape of ajutages... . EXQAgOJXCL ZoòòZA'. they can also be connected with quality factors depending on two kinds of intensive or extensive coefficients; for instance in the case of the use of heat conveying fluid, these coefficients would be the specific heat and the mass of this fluid. A constituent of a machine is said to have perfect quality if during the wor king of the machine no dissipation or loss of exergy due to this constituent intervenes. This situation can be reached by assigning certain conditions to only one of the two components of the quality factors. The systematic decreases of exergy derive from the choices made in the scena rio; effectively these choices exercise constraints on some quality factors e.g. choice of a technology (real machine with no perfect Carnot cycle...), choice of certain qualitative parameters (nature of working fluid, type of condenser, particular ajutage...). It must be noted that it is theexistence of these systematic decreases of exergy that gives justification to the thermo dynamic optimization. The occasional decreases of exergy come from the existence of the non cons trained quality factors. The maximizing of the exergetic effiency as a function of the non constrained quality factors is a trivial operation that consists in taking the value of the quality factor corresponding to an infinite quality. So the non constrai ned quality factors do not constitute real decision variables for the
Session 2.18.
1636
thermodynamic optimization. The primary decision variables which allow the complete satisfaction of all the energetic objectives, are the thermodynamic décision variables. The know ledge of these variables allows us to answer the two essential questions: how to fabricate the machine and how to make it work in order to satisfy all the energetic objectives that can be demanded from this type of machine?
The thermodynamic degree of freedom is the number of independent
thermodynamic
decision variables. More precisely, it is the minimum number of the primary decision variables that must be known to make it possible to fabricate a ma chine and to make it work so that this machine should satisfy all the energe tic objectives and this within a scenario corresponding to a technological choice with no occasional degradations of exergy and without any environmen tal constraints. Thus the thermodynamic optimization consists in finding the particular value of each of the thermodynamic decision variables correspon ding to the greatest exergetic efficiency of the machine. ECONOMIC OPTIMIZATION OF THERMAL SOLAR PLANTS Economic optimization -— the term thermo-economic would be more appropriate— consists in optimizing the actual cost of the project and the actual total financial value of the production, simultaneously. If the decision variables of the economic optimization intervene in both criteria — as is the case for most of them — then the unique criterion to be considered will be the maximi zing of the benefit. This is accomplished by seeking the maximum value for the following o.f.: actual value of the production / actual cost of the project. Thermodynamic optimization — the term thermo-energetic would be more ap propriated — of a machine for the transformation of energy, is done by taking the maximizing of energetic production as the optimization criterion. As alre ady discussed, this thermodynamic optimization is a partial optimization that can present an intrinsic interest, but that can also be done with the aim of simplifying the calculus of the global economic optimization of the project. It has also been seen that the thermodynamic optimization consists in deter mining the energetically optimum value of the thermodynamic decision varia bles within a particular scenario. In order to carry out the thermodynamic optimization, one must take for the non constrained quality factors a value that corresponds to an infinite quali ty — that is only systematic decreases of exergy have to be considered. Then, to achieve the global optimization of the system, the value of the quality factors responsible for the occasional decreases of exergy have to be sought. This is accomplished by a technico-economic optimization; but, and that is the interest of the method, the value found by the thermodynamic optimization can be taken as the value of the thermodynamic decision variables. The calculations may next be continued by an iteration repeating the first calculus of the thermodynamic optimization and using the new value found for the quality fac
tors. That amounts in effect
to a transfer
from occasional
to systematic
de-
creases of exergy. The justification of this method is based on the- fact that generally the variation of the thermodynamic decision variables has little influence on the costs and much on the efficiency, while the variation of the quality factors has little influence on the efficiency and much on the costs. In the case of thermal solar plants, the thernt>dynamic optimization presents an increased interest: according to the "gratiutousness" of the solar energy, the financial cost of a solar plant only covers the investment and maintenance costs of each of its elements among which the solar collector takes a very important part. To a first approximation, it can be said that the total cost of a project is proportional to the collecting area S. Besides, the selling price of the energetic production of the machine^which is proportional to this production, is then proportional to the product W.S, where hi is the mean value of the global solar efficiency during the irradiation. The o.f. linked to the
Session 2.18.
1637
economie criterion is then proportional to hi: o.f. proportional to hl.S/S = hi Thus, except in the vicinity of the optimum economic solution,each variation of a decision variable that increases the exergetic efficiency also increases the economic efficiency, so that the search for the optimum conditions of the exergetic efficiency tends to the global economic optimization of the project. THERMOECONOMIC ANALYSIS The thermoeconomic analysis of a machine for the transformation of energy con sists in finding out where and how loss and degradation of energy occur,how they depend on the choices of the parameters of the system and how they influ ence one another. The economic analysis consists in determining where and how much material and energy come into orout the system and how much it costs. Considering the interdependence between the energetic and economic fluxes le ads to the so-called thermoeconomic analysis (also called Thermoeconomics)which has been introduced by Tribus, Evans and El Sayed (1962 ,1970), and has become of frequent use in the study of industrial processes. The utilization of exergy for studying and optimizing the transformation of energy processes is now a well-known and effective tool(Auracher, 1984). This concept has been used in solar technologies in particular for collectors (Schölten, 1984), solar thermal engines (Vokaerd, 1981) and for solar refrige ration systems like liquid absorption machines (Anan, 1984) or solid adsorp tion machines . (Adell, 1983). Tsatsaronis (1984) has proposed the term Exergoeconomics as better than Thermoeconomics to characterize the combination of economic and exergetic analyses. CONCLUSION Some concepts of the literature have been presented in order to answer such elementary questions as: what is an optimization problem? how can we solve them? According to the diversity of the problems which contain an optimization condition, it is often difficult to give definitions general enough to be ap plicable to all situations. The principal domain of application that we have constantly kept in mind is the optimization of the machines for the transfor mation of energy and essentially those that use renewable sources of energy. We have emphasized the interest of thermodynamic optimization as a partial optimization which permits the simplification of the global economic optimi zation in the case of thermal solar plants. REFERENCES Adell, A. (1983) Optimisation du fonctionnement des systèmes de réfrigération solaires à adsorption solide. International Congress on Refrigeration,Paris Anan, D.K. (1984)Second law analysis of solar powered absorption cooling sys' terns. A.S.M.E. J. of^ Solar_Energy Engineering, 106, 291-298. Auracher, H. (1984")T Fun3¥m*erita*r~a*spects oT~ exergy~~applications to the analy sis and optimization of energy processes.Heat Recovery Systems, 4, 323-27. El Sayed, Y.M. and R.B. Evans (1970); Thermoeconomics and the design of heat systems. J.jDfJSngin. forJPower, 92, 27-35. Le Goff, F. (f979j*En^gTl^ique industrielle, 1 et 2 ,Techn. et D o c , Paris Maczeck, K. (1980) Application of various criteria for optimizing refrigerating plants. In Saving of Energy,International Institute of Refrigeration, Paris Schölten, W.B. (1984) 'Ä comparizon of exergy delivery capablities of solar col lectors. A.S.M.E. J. of Solar Energy Engineering, 106, 490-493. Tribus, M. and R.B. Evans Tî962)' The~Thermoeconomics of sea water conversion. University of California, Los Angeles Report N°62-63. Tsatsaronis, G. (1984) Combination of exergetic and economic analysis in energy conversion processes.Eur. Cong, on En. Economics and man. Ind.,Algarve,Port. Vokaerd, D.and Jf Bougard (1981) Solar refrigeration Engines, I.S.E.S.Brighton
1632
OPTIMIZATION PROBLEMS IN THERMAL SOLAR PLANTS A. ADELL, Laboratoire de Chimie-Physique, Université des Sciences et Techniques du Languedoc, 34060, Montpellier, France ABSTRACT The aim of this work is to answer such elementary questions as "what is an op timization problem and how to solve it?" The case of thermal solar plants has received a special attention because of the obvious importance of optimization for their industrial development. The principal concepts of the literature ha ve been presented while introducing some new ones such as "quality factors". KEYWORDS Optimization problems, solar plants, Thermoeconomics, quality factors, systematic decrease of exergy. Plant optimization means a complex set of mathematical and technical operations in order to satisfy some predetermined criteria such as maximizing the return on the financial investment, minimizing pollution and so on. OPTIMIZATION CRITERIA The optimization criteria concerning the apparatuses of transformation of en ergy are generally classified into three groups (Maczeck, 1980): thermodynamic socio-economic and technical. The technical optimization criteria proceed from the conditions imposed by ter chnical considerations like the minimum bulkiness, the best means of detecti ng the escapes of working fluid, the minimum of corrosion... . The economic optimization criterion is the most important for the owner of the machine who wishes to maximize the benefit of his investment. But when the de cision is to be taken by, for example, a public body, other socio-economic criteria must be taken into account, e.g. minimizing foreign currency cost, maximizing the number of created jobs,minimizing the energy and materials in puts ... . The thermodynamic optimization criterion has to be considered when the deci sion is to be made by a heat engineer whose only concern is the thermo-energetic behaviour of the system. GLOBAL OPTIMIZATION In practice several criteria intervene simultaneously. This is treated in the so-called global optimization (Le Goff, 1979). In global optimization some criteria can be contradictory such as minimizing the selling price and maximizing the reliability. It is the politic decider's responsability in important so cio-economic projects, to asses the respective weight that must be given to each criterion for establishing priorities. In order to achieve global opti mization, three types of consideration must be taken into account:
Session 2.18.
1633
1- complying with the technical specifications of the project. The tech nical specifications fix the objectives to be reached and indicate the value or the constraints that must be verified by certain parameters; they also contain, more or less implicitly, some optimization criteria with eventually some indications on their respective weight. 2- choosing a particular kind of apparatus the working principle of whi ch allows the satisfaction of the objectives of the technical specifications, and considering some hypotheses concerning the expected behaviour of the sys tem. These hypotheses may be technical in nature which allows the modelling of the operational behaviour of the real machine in a theoretical or empirical way, e.g. the nature of irreversibilities, the life-time of the machine parts ... . These hypotheses may also have a socio-economic nature, e.g. national or world-wide evolution of material or energy costs; then they correspond to the choice of an economic model. All these hypotheses constitute "bets" whi ch all the knowledge of "the man of the art" will tend to minimize the risk.of. 3- taking into account additional constraints imposed by the environment of the system; the word "environment" been taken in a wide sense: can the de signer realize the parts of the machine at the required dimensions himself or must he buy standard industrial elements? what are the chemical nature and the degree of purity of the available constituents? what are the ambient meteoro logical conditions?... . Thus the global optimization can be done and discussed within scenarios which involve various choices: technological choices concerning the type of machine, technico-economic choices linked to the bets considered and political or tech nocratic choices of the respective weight that should be given to the criteria. OPERATIONAL, CONSTRUCTIONAL AND DESIGN PARAMETERS The practice of optimization can be done by two sorts of operations: a mathe matical treatment of an a priori problem, allowing the solution of the optimum value for certain parameters of the system, or technical operations which mu st be done on the machine during its working,among which automatic control constitutes the principal element. The parameters that should be taken into account in order to achieve the op timization are either those which define the conditions of the working and of the environment from a physical as well as a socio-economic viewpoint, or th ose that characterize the machine constituents (elements or working fluids): chemical nature, geometry, cost... . The behaviour of these parameters may be continuous, discrete or simply qua litative; they may also be time-dependent in non stationary systems such as solar energy apparatuses. The knowledge of the qualitative parameters — chemi cal nature and degree of purity of fluids, chemical nature and quality of the surface or type of manufacture of pipes... — is equivalent to the implicit knowledge of a great number of coefficients among which some play an important part in the optimization according to the hypotheses considered. These coeffi cients generally have a discrete value,as in the case of viscosity and heat transfer coefficients and for the cost of the type of thermal insulator, for instance. The usual method to proceed to the optimization of these qualitative parameters consists of an a priori discussion of the result expected for dif ferent initial choices. The inverse method is of course theoretically possi ble and may lead to interesting information delimiting the possible choices, but generally does not directly result in a concrete solution. So the choices of the qualitative parameters are most often included in the planned scenario. The set of parameters have been classified into three categories in the li terature: operational, constructional, and design parameters (Maczeck, 1980). DECISION VARIABLES AND OBJECTIVE FUNCTIONS
1634
Session 2.18.
Optimization problems arise when we have to answer some essential questions concerning a project; these questions relate to the determination of extrema: which machine is cheaper? how to improve an existing machine? what is the best technological channel? Among these questions the principal is: what value sho uld be given to the parameters of the system in order to realize the best plant? To solve mathematically an optimization problem we must first determine which are the parameters of the system that have to be known to answer the asked question. These parameters constitute the decision variables of the questions. In order to formulate mathematically the questions of the optimization problem a quantity called the objective function (o.f.) is associated with each crite rion contained in the questions. These o.f. must satisfy an extremum condition. The o.f. are explicitly expressed by means of physical or economic laws that apply to the model of the scenario, as a function of certain parameters which are the decision variables of the criterion. Beside these o.f. associated with the optimization criteria, there are other o.f. with no extremum conditions: they are the parameters constrained by the technical specifications or by the environment. These quantities must in their turn constitute or depend on new variables which, together with the decision variables of the criteria, form the primary decision variables also called
essential variables. These essential variables are necessary to answer fundamental points such as: how to fabricate the machine and what are the conditi-, ons for its operation?
In order to achieve the global optimization that is to satisfy simultaneously all the questions while respecting the objectives and the constraints of the project, a set of coupled mathematical relations must be solved. These rela tions contain a great number of interdependent variables. This interdependen ce entails a great complexity for optimization problems, furthermore, as has been seen, some variables can have a discrete value which varies with the scenario and so have to be optimized case by case; this increases the length of calculations, complicates the discussions but affords an opportunity for future innovations. PARTIAL OPTIMIZATION Partial optimization is characterized by a limited number of mathematical re lations with which is associated a limited number of variables. Partial opti mization may be envisaged either to treat separately a few questions the an swer of which offers an intrinsic interest, or to bring about a simplification of the problem of global optimization. In the latter case, partial optimiza tion becomes a method of uncoupling the mathematical relations of the problem. Among the simplifying methods, the well-known method using what is called "zone decomposition" (El Saved, 1970) allows the reduction of the problem of global optimization of a complex machine to the problem of partial optimiza tion in each element or functional zone of the machine. This method becomes of great effectiveness when the concept of available energy — also called exergy <— is used. However this method leads to an approximate general solut ion because it neglects the interdependence existing between certain variables of each zone. One of its main advantages is to legitimize the use of finer thermodynamic models in each zone, contrary to the global optimization which more often can only be done using coarse or over-simplified models, e.g. " quasi-perfect machine", "basic cycle"... . This is what Tribus, initiator of the method, has justified by saying: "It is much more important to be able to survey the set of possible systems approximately than to examine the wrong system exactly". Thermodynamic optimization is another well-known example of partial optimiza tion involving the use of only one criterion and a limited number of decision variables.
Session 2.18.
1635
WORKING PARAMETERS, QUALITY FACTORS, DEGREE OF FREEDOM Two types of situation occur during the working of machines of transformation of energy:— thermodynamic equilibrium states (or steady flows), for which the passage from one state to another defines the transformation between these -- mass and energy transfers which characterize an evolution between these two equilibrium states. These two situations concern the whole system, that is the machine itself (in ternal situation) as well as its thermodynamic environment (external situation). THE working parameters are used to describe these two situations. These paramaters are composed of the variables of equilibrium states and of the transfer variables. They are interconnected by two sorts of relations: .fundamental laws that directly link the parameters together; .phenomenological laws that may have a theoretical or empirical basis and that involve various coefficients depending on the experimental conditions, according to the hypotheses considered in the scenario. In the case of a machine for the transformation of energy, the exergetic effi ciency is the principal optimizable o.f. because contrary to the energetic ef ficiency, it has an intrinsic physical meaning independently of the tempera ture of the heat exchanges. In simple cycle flow machines, efficiency is affected by two types of decrease: on the one hand the systematic decrease of exergy and on the other hand the occasional decrease of exergy; decrease meaning either a loss of exergy with a corresponding decrease of energy, or a dissipation of exergy with conservation of energy and creation of an equal amount of anergy. These decreases may be internal or external. The decreases of exargy are associated with what we have called the quality factors'. These quality factors depend on two kinds of parameters of intensive or extensive nature. EXQJtg&tic, VJLkkJLpOütLonk'. if one considers that the irreversible phenomena responsible for dissipations are in their linear domain, then the quality fac tors connected to the exergetic dissipations depend on the intensive phenome nological coefficients that link the cause of the phenomena (generalised affi nities) to their effects (generalised fluxes), e.g. thermal conductivity, dif fusion, viscosity... coefficients. The quality factors also depend on extensi ve coefficients such as the coefficients of dimension or shape, e.g. heat ex changers surfaces, diameter and surface state of pipes, shape of ajutages... . EXQAgOJXCL ZoòòZA'. they can also be connected with quality factors depending on two kinds of intensive or extensive coefficients; for instance in the case of the use of heat conveying fluid, these coefficients would be the specific heat and the mass of this fluid. A constituent of a machine is said to have perfect quality if during the wor king of the machine no dissipation or loss of exergy due to this constituent intervenes. This situation can be reached by assigning certain conditions to only one of the two components of the quality factors. The systematic decreases of exergy derive from the choices made in the scena rio; effectively these choices exercise constraints on some quality factors e.g. choice of a technology (real machine with no perfect Carnot cycle...), choice of certain qualitative parameters (nature of working fluid, type of condenser, particular ajutage...). It must be noted that it is theexistence of these systematic decreases of exergy that gives justification to the thermo dynamic optimization. The occasional decreases of exergy come from the existence of the non cons trained quality factors. The maximizing of the exergetic effiency as a function of the non constrained quality factors is a trivial operation that consists in taking the value of the quality factor corresponding to an infinite quality. So the non constrai ned quality factors do not constitute real decision variables for the
Session 2.18.
1636
thermodynamic optimization. The primary decision variables which allow the complete satisfaction of all the energetic objectives, are the thermodynamic décision variables. The know ledge of these variables allows us to answer the two essential questions: how to fabricate the machine and how to make it work in order to satisfy all the energetic objectives that can be demanded from this type of machine?
The thermodynamic degree of freedom is the number of independent
thermodynamic
decision variables. More precisely, it is the minimum number of the primary decision variables that must be known to make it possible to fabricate a ma chine and to make it work so that this machine should satisfy all the energe tic objectives and this within a scenario corresponding to a technological choice with no occasional degradations of exergy and without any environmen tal constraints. Thus the thermodynamic optimization consists in finding the particular value of each of the thermodynamic decision variables correspon ding to the greatest exergetic efficiency of the machine. ECONOMIC OPTIMIZATION OF THERMAL SOLAR PLANTS Economic optimization -— the term thermo-economic would be more appropriate— consists in optimizing the actual cost of the project and the actual total financial value of the production, simultaneously. If the decision variables of the economic optimization intervene in both criteria — as is the case for most of them — then the unique criterion to be considered will be the maximi zing of the benefit. This is accomplished by seeking the maximum value for the following o.f.: actual value of the production / actual cost of the project. Thermodynamic optimization — the term thermo-energetic would be more ap propriated — of a machine for the transformation of energy, is done by taking the maximizing of energetic production as the optimization criterion. As alre ady discussed, this thermodynamic optimization is a partial optimization that can present an intrinsic interest, but that can also be done with the aim of simplifying the calculus of the global economic optimization of the project. It has also been seen that the thermodynamic optimization consists in deter mining the energetically optimum value of the thermodynamic decision varia bles within a particular scenario. In order to carry out the thermodynamic optimization, one must take for the non constrained quality factors a value that corresponds to an infinite quali ty — that is only systematic decreases of exergy have to be considered. Then, to achieve the global optimization of the system, the value of the quality factors responsible for the occasional decreases of exergy have to be sought. This is accomplished by a technico-economic optimization; but, and that is the interest of the method, the value found by the thermodynamic optimization can be taken as the value of the thermodynamic decision variables. The calculations may next be continued by an iteration repeating the first calculus of the thermodynamic optimization and using the new value found for the quality fac
tors. That amounts in effect
to a transfer
from occasional
to systematic
de-
creases of exergy. The justification of this method is based on the- fact that generally the variation of the thermodynamic decision variables has little influence on the costs and much on the efficiency, while the variation of the quality factors has little influence on the efficiency and much on the costs. In the case of thermal solar plants, the thernt>dynamic optimization presents an increased interest: according to the "gratiutousness" of the solar energy, the financial cost of a solar plant only covers the investment and maintenance costs of each of its elements among which the solar collector takes a very important part. To a first approximation, it can be said that the total cost of a project is proportional to the collecting area S. Besides, the selling price of the energetic production of the machine^which is proportional to this production, is then proportional to the product W.S, where hi is the mean value of the global solar efficiency during the irradiation. The o.f. linked to the
Session 2.18.
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economie criterion is then proportional to hi: o.f. proportional to hl.S/S = hi Thus, except in the vicinity of the optimum economic solution,each variation of a decision variable that increases the exergetic efficiency also increases the economic efficiency, so that the search for the optimum conditions of the exergetic efficiency tends to the global economic optimization of the project. THERMOECONOMIC ANALYSIS The thermoeconomic analysis of a machine for the transformation of energy con sists in finding out where and how loss and degradation of energy occur,how they depend on the choices of the parameters of the system and how they influ ence one another. The economic analysis consists in determining where and how much material and energy come into orout the system and how much it costs. Considering the interdependence between the energetic and economic fluxes le ads to the so-called thermoeconomic analysis (also called Thermoeconomics)which has been introduced by Tribus, Evans and El Sayed (1962 ,1970), and has become of frequent use in the study of industrial processes. The utilization of exergy for studying and optimizing the transformation of energy processes is now a well-known and effective tool(Auracher, 1984). This concept has been used in solar technologies in particular for collectors (Schölten, 1984), solar thermal engines (Vokaerd, 1981) and for solar refrige ration systems like liquid absorption machines (Anan, 1984) or solid adsorp tion machines . (Adell, 1983). Tsatsaronis (1984) has proposed the term Exergoeconomics as better than Thermoeconomics to characterize the combination of economic and exergetic analyses. CONCLUSION Some concepts of the literature have been presented in order to answer such elementary questions as: what is an optimization problem? how can we solve them? According to the diversity of the problems which contain an optimization condition, it is often difficult to give definitions general enough to be ap plicable to all situations. The principal domain of application that we have constantly kept in mind is the optimization of the machines for the transfor mation of energy and essentially those that use renewable sources of energy. We have emphasized the interest of thermodynamic optimization as a partial optimization which permits the simplification of the global economic optimi zation in the case of thermal solar plants. REFERENCES Adell, A. (1983) Optimisation du fonctionnement des systèmes de réfrigération solaires à adsorption solide. International Congress on Refrigeration,Paris Anan, D.K. (1984)Second law analysis of solar powered absorption cooling sys' terns. A.S.M.E. J. of^ Solar_Energy Engineering, 106, 291-298. Auracher, H. (1984")T Fun3¥m*erita*r~a*spects oT~ exergy~~applications to the analy sis and optimization of energy processes.Heat Recovery Systems, 4, 323-27. El Sayed, Y.M. and R.B. Evans (1970); Thermoeconomics and the design of heat systems. J.jDfJSngin. forJPower, 92, 27-35. Le Goff, F. (f979j*En^gTl^ique industrielle, 1 et 2 ,Techn. et D o c , Paris Maczeck, K. (1980) Application of various criteria for optimizing refrigerating plants. In Saving of Energy,International Institute of Refrigeration, Paris Schölten, W.B. (1984) 'Ä comparizon of exergy delivery capablities of solar col lectors. A.S.M.E. J. of Solar Energy Engineering, 106, 490-493. Tribus, M. and R.B. Evans Tî962)' The~Thermoeconomics of sea water conversion. University of California, Los Angeles Report N°62-63. Tsatsaronis, G. (1984) Combination of exergetic and economic analysis in energy conversion processes.Eur. Cong, on En. Economics and man. Ind.,Algarve,Port. Vokaerd, D.and Jf Bougard (1981) Solar refrigeration Engines, I.S.E.S.Brighton