Exergy Method: Technical And Ecological Applications [Paperback ed.] 1853127531, 9781853127533

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Exergy Method Technical and Ecological Applications

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Exergy Method Technical and Ecological Applications By

Jan Szargut

Exergy Method Technical and Ecological Applications

Jan Szargut Silesian University of Technology, Gliwice, Poland

Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-85312-753-3 ISSN: 1369-7331 Library of Congress Catalog Card Number: 2004113017 No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2005. Printed in Great Britain by Athenaeum Press Ltd, Gateshead. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Contents Foreword Preface Introduction Nomenclature

xi xiii xv xvii

CHAPTER 1 Exergy balance and exergy losses

1 1.1. Definition of exergy and exergy losses ................................................ 1 1.2. Exergy balance and exergy efficiency .................................................. 4 1.3. Exergy and anergy ............................................................................... 9 1.4. Typical irreversible phenomena ........................................................... 9 1.5. Anthropogenic and natural exergy losses ......................................... 16 Exercises ........................................................................................................ 18

CHAPTER 2 Calculation of exergy

19 2.1. Types and components of exergy ...................................................... 19 2.2. Physical exergy .................................................................................. 20 2.3. Chemical exergy ................................................................................. 21 2.4. Enthalpy of devaluation ..................................................................... 34 2.5. Chemical exergy of organic fuels ....................................................... 35 2.6. Chemical exergy of solutions ............................................................. 37 2.7. Exergy of thermal radiation ................................................................ 38 2.8. Nuclear exergy ................................................................................... 41 2.9. Exergy losses in thermal plants .......................................................... 42 Exercises ........................................................................................................ 54

CHAPTER 3 Cumulative exergy consumption and partial exergy losses 3.1. 3.2.

57 Definition of the cumulative exergy consumption (CExC) .......................................................................... 57 The problem of CExC of the human work ........................................... 58

3.3. 3.4. 3.5. 3.6. 3.7.

A set of input-output equations ........................................................ 59 Cumulative exergy efficiency ............................................................. 60 Cumulative and partial exergy losses ................................................. 63 Net coefficients of consumption ........................................................ 64 Sequence method for the evaluation of partial exergy losses ........................................................................... 67 Exercises ........................................................................................................ 72

CHAPTER 4 Practical rules for improving thermodynamic imperfection

75 Exercises ........................................................................................................ 90

CHAPTER 5 Depletion of non-renewable natural resources; thermo-ecological cost

91 Definition of the thermo-ecological costs .......................................... 91 Evaluation of the thermo-ecological cost of deleterious waste products ................................................................ 91 5.3. Balance equations .............................................................................. 92 5.4. Influence of the interregional exchange ............................................. 93 5.5. Calculation of the thermo-ecological cost .......................................... 95 5.6. Sustainability index .......................................................................... 102 Exercises ...................................................................................................... 103 5.1. 5.2.

CHAPTER 6 Economic applications of exergy

105 Exergo-economics ............................................................................ 105 Optimization of the thermo-ecological cost ...................................... 108 Optimization of the exergetic cost .................................................... 114 Correction of the economic optimization ......................................... 115 Influence of thermodynamic imperfection on the investment cost .......................................................................... 115 6.6 Evaluation of the natural mineral capital and freshwater resources of the Earth .................................................... 116 Exercise ........................................................................................................ 119 6.1. 6.2. 6.3. 6.4. 6.5.

CHAPTER 7 Application of exergy for determining the pro-ecological tax 7.1. 7.2. 7.3.

121 Necessity of a new tax ..................................................................... 121 Structure of the pro-ecological tax ................................................... 122 Pro-ecological tax resulting from the use of machines and installations ............................................................... 125

7.4. Burdening of imports and exports .................................................... 126 7.5. Discussion ....................................................................................... 127 Exercises ...................................................................................................... 128

Appendix Solutions of exercises References Index

129 143 153 161

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Foreword It is a great pleasure, and an honour for me, to have the opportunity to write a foreword to this latest monographic work by Professor Szargut. When I first met him in person (in the 1980s) I had already got to know him through his work actually it is through one of his papers, (Brennstoff-Wärme-Kraft, v.19, n.6, 1967), that I became interested in the concept of exergy and in its possible applications to the thermodynamic evaluation of industrial processes. Throughout my academic years, I can honestly say that his work was a constant stimulus and a consistent source of inspiration. There is no field of exergy analysis that Jan Szargut has not contributed to - from the determination of the exergy value of materials, to the definition of a rational performance parameter (the exergetic efficiency), to the coupling of exergetic and economic considerations in determining the “optimal” design or operating point of a process or of a technological production chain, he has published an extensive number of seminal theoretical works and innumerable applications. He has worked (and often published) with most of the historical innovators of the exergy field including, amongst others, Viktor Brodyanskyi, Wolfgang Fratscher, Richard Gaggioli and Dominick Sama. In my view, Professor Szargut’s most enduring contributions are his fundamental work on the definition of a “standard reference environment” (Chapter 2 and Appendices) and his great intuition about the Cumulative Exergy Content (CEC, Chapter 3). The two are linked of course to each other, and all present research in both fields is still in essence based on his work. The CEC has proven to be a very fruitful tool, both for theoretical and practical reasons - it constituted the first rational basis for a completely coherent evaluation method of every energy- and material conversion process. One of the great gifts of Jan Szargut is his scientific perseverance, by which I mean his ability to further expand and deepen his own research. And this is very clear in this book. The CEC was conceived as an “industrial” method, and since his first introduction (in the 1980s), substantial advances, especially in the field of Thermo-Economics, led to the agreement that the neglected environmental externalities do play an important role in the definition of “optimal” designs. Here we see that Szargut elaborated on his original concept, and, with a remarkable intellectual effort presents it in such simple terms so as to make it seem obvious, he further extends the CEC to account for the “depletion of the non-renewable natural resources”. This is by no means a trivial task, but he

explains it here (Chapter 5) in an easy yet rigorous style, basing it all on fundamental principles (Basic equations, Section 5.3). From his early interests in the possible implications of the exergy analysis to the problems of social economy, Professor Szargut developed an original version of the otherwise well-known Exergo-Economics (Chapter 6), a theory that links the irreversible losses (exergy destruction) to the capital costs associated with their avoidance. Though the two of us do not agree here in our conclusions and in some non-secondary details of the methodology, still I must admit that he presents a very strong point for his side of the story. And he goes - again!- one step further, when he develops (Chapter 7) a rational, complete and well-founded proposal for a new system of taxation that takes into account not the monetary income, but the “ecological cumulative burden” that a commodity (goods or a service) places on the environment. All of us who work in the field of Energy System Analysis owe something to Jan Szargut, and I wish that the younger readers of this book, who cannot fully appreciate this simple fact because they cannot put it in a proper historical perspective, will take not only my word for this strong statement, but also that of a large number of colleagues whose esteem for him has remained unchanged through the years. As witnesses, if needed, I call on more than 200 papers he published, and the large number of his former students who are all now leaders in their respective fields. A final word of warning - this book is not an easy read! If it seems so, read it over, and you will appreciate, as I did, the somewhat hidden depth and wide implications of virtually all of its fundamental statements. Read it over. And over. And you will discover, as I did, that most of what you do now, Jan Szargut did - or thought of - before. And he managed to do it, somehow, better. With friendship, respect and gratefulness for what he has taught me until now, and with the certainty to learn more from him in the future, Enrico Sciubba University of Roma 1, “La Sapienza”

Preface The exergy method makes it possible to detect and quantify the possibilities of improving thermal and chemical processes and systems. The introduction of the concept “thermo-ecological cost” (cumulative consumption of non-renewable natural exergy resources) generated large application possibilities of exergy in ecology. The economy of non-renewable natural exergy is one of the most important problems considered in the present book. The book contains a short presentation of the basic principles of exergy analysis and discusses new achievements attained in the last 15 years. The most important new problems, discussed so far only in scientific journals, are: calculation of the chemical exergy of all the stable chemical elements, discussion of the global natural and anthropogenic exergy losses, practical guidelines for improvement of the thermodynamic imperfection of thermal processes and systems, development of the determination methods of partial exergy losses in thermal systems, discussion on the thermo-ecological cost, a general method for the optimization of the operational and design parameters aiming at the minimization of the depletion of non-renewable natural resources, sustainability index of the natural environment, evaluation of the natural mineral capital of the Earth, application of exergy for the determination of a pro-ecological tax, substituting the existing personal taxes. The book assumes a knowledge of basic thermodynamics, and therefore, is appropriate for graduate students and also for engineers practicing in the field of energy and ecological management. The book may be also helpful in scientific research. The book has been prepared thanks to the support of the Institute of Thermal Technology of the Silesian University of Technology (Gliwice, Poland), directed by Prof A. ZiĊbik. I would like to express my gratitude to Prof Enrico Sciubba (University “La Sapienza”, Rome, Italy) for his valuable suggestions for improving the text of the book, and to Dr W. Stanek (Institute of Thermal Technology, Gliwice) whose scientific results enriched the content of the book. Jan Szargut 2005

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Introduction All kinds of human activities are supported by a consumption of natural resources, both non-renewable and renewable ones. The exhaustion of nonrenewable natural resources is very dangerous for the future of humankind and therefore a measure should be introduced to evaluate these resources, and methods should be devised to estimate their exhaustion. The ability of natural resources to run thermal, chemical and biological processes, results from the deviation of their state and composition from the thermodynamic equilibrium, with the prevailing components of the natural environment. Hence, the ability to perform maximum work in the conditions of natural environment can be accepted as a measure for the evaluation of natural resources. The quantity defined in such a way has been termed exergy. This term was introduced by Z. Rant [42], whose works began a very intensive development in this field of technical thermodynamics. Exergy analysis has been developed as a result of the application of the second law of thermodynamics in investigations of the thermodynamic imperfection of thermal processes. Forerunners of exergy analysis were G. Gouy, A. Stodola, J.W. Gibbs and F. Bošnjakoviü. Gouy [23] and Stodola [55] formulated independently the law determining the loss of the ability to perform work due to thermodynamic irreversibility. Gibbs [22] introduced the concept of maximum work of chemical reactions. Keenan [30] introduced the concept of availability. Bošnjakoviü analyzed the deleterious impact of irreversibilities and propagated the idea of “fighting” them [7]. During the decade 1960–1970, exergy analysis was developed mainly in Europe, but after 1980 the interest in this analysis increased considerably in America. The bibliography of exergy analysis contains thousands of papers, among them many monographic books [104]. Among the authors of monographic books the following ones may be mentioned: J. Szargut and R. Petela [60]; I. Nerescu and V. Radcenko [39]; W.M. Brodyanskyi [9]; W. Fratzscher W.M. Brodyanskyi and K. Michalek [19]; T.J. Kotas [31]; J. Szargut, D.R. Morris and F. Steward [69]; W.M. Brodyanskyi, M.V. Sorin and P. Le Goff [10].

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Nomenclature aij,auj Aij A,a B, b b* Bm Bs c c dij Di,Dr E fij fuj f g G, g Gj fG H, h HL,HH mi mij

coefficient of gross consumption of the major product of ith process or of uth by-product, related to the complex of useful products of the considered jth process, containing a unit of its major product coefficient of the net consumption of the major product of ith process per unit of the major product of the considered jth process square matrix of the coefficients Aij, aij exergy of the stream of matter crossing the immovable system boundary (total and specific), J; J/kg, J/mol cumulative consumption of primary exergy per unit of the considered product, J/kg, J/mol exergy of stream of matter crossing the movable system boundary, J exergy of the system, exergy of the matter contained within the system boundary, J specific heat capacity, J/(kg·K), J/(mol·K) velocity related to the Earth’s surface, m/s auxiliary consumption coefficient, taking into account the import of semi-finished products per unit of the jth major product monetary value of the ith exported and rth imported product, unitary diagonal matrix coefficient of by-production of the substituted major product of ith process per unit of the major product of the considered jth process coefficient of production of the uth by-product per unit of the major product of the considered jth process square matrix of the coefficients fij gravitational acceleration, m/s2 Gibbs free energy (total and specific) J; J/kg, J/mol annual production of the jth product standard free energy of formation at normal temperature, J/kg, J/mol enthalpy (total and specific) J; J/kg, J/mol lower and higher heating value J/kg, molarity of the ith component of solution, mol/kg H2O amount of the ith machines or installations applied in the jth production process,

M (MHL) n N p pkj Pk Pr Q r r R Re s ssu S, s SNkj T V, v wk W x X yi zi j Nj b* i   B E j m

molecular mass, kg/kmol, g/mol lower heating value of gaseous fuels, J/mol substance amount, mol electric power pressure, Pa amount of the kth aggressive component of waste products rejected to the environment per unit of the jth major product annual production of the kth aggressive component of waste products rejected to the environment in the considered region Prandtl number amount of heat, J velocity related to the system boundary, m/s annual rate of discount gas constant, J/(kg·K), J/(mol·K) Reynolds number velocity of the system boundary related to the Earth’s surface, m/s replacement ratio in units of the sth substituted major product per unit of the uth by-product entropy (total and specific) J/K; J/(kg·K), J/(mol·K) net cumulative consumption of the kth semi-finished product per unit of the jth product temperature on thermodynamic scale, K volume (total and specific), m3; m3/kg, m3/mol monetary coefficient of ecological damages per unit of kth aggressive waste product, $/kg, $/mol mechanical work, J quality of saturated vapor height, m molar fraction of the ith species in a solution mass fraction of ith species in a solution immediate gross consumption of primary exergy per unit of the jth product, together with its by-products, J/kg, J/mol immediate net consumption of primary exergy per unit of the jth product, J/kg, J/mol cumulative exergy loss burdening all the links of fabrication of the considered product activity coefficient, (molarity scale) symbol of loss (B = exergy loss, p = pressure loss) symbol of increase (B = exergy increase, T = temperature increase) exergy (exergetic) efficiency energy efficiency specific thermo-ecological cost of the jth product, J/kg, J/mol thermo-ecological cost of the exported products, per monetary unit, J/$

k i 

cumulative exergy consumption of non-renewable resources due to the emission of unit of kth waste product, J/kg, J/mol life time of the ith machine or installation operating with rated capacity, years pro-ecological tax

Subscripts a B ch d D E el f F i,j,k,p i m m n N nu ph q s u u w 0

output related to exergy chemical input driving related to energy related to electricity friction fuel order number of the process and its major product internal mechanical moving standard, conventional net value (per unit of the major product), nuclear physical related to the heat source related to the system useful order number of the by-product waste ambient, related to the reference level

Superscripts  . – *

standard state at normal temperature dot above the symbol denotes a quantity related to the time unit dash above the symbol denotes a local (differential) value cumulative quantity

Abbreviations CExC CExE CExL DCP

cumulative consumption of primary exergy cumulative exergy efficiency cumulative exergy loss domestic consumption product, $/year

HP-plant heat-and-power plant LNG liquid natural gas NLUE natural losses of utilizable exergy PExL partial exergy loss VAT value added tax

Chapter 1 Exergy balance and exergy losses 1.1. Definition of exergy and exergy losses In fig. 1.1 a hydraulic and a thermal power plant are compared. The hydraulic plant exploits the difference between the levels of water in the upper and lower reservoir. Similarly, a thermal plant exploits the temperature difference between the hot heat source and a cold heat sink. There is, though, a great difference between the two plants: the hydraulic one can (neglecting viscous and mechanical friction) convert into work the total potential energy of water taken from the upper reservoir. However, as Carnot discovered (1824), a thermal power plant (even operating without any losses) can convert into work only a portion of the heat taken from the hot source. The law of Carnot has the form: W =Q

T1 T 2 T1

Figure 1.1: Comparison of a hydraulic and thermal power plant.

(1.1)

2 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

where T1 and T2 are the absolute temperature values of the hot source and the cold sink. The heat from the hot source can be best utilized if a natural (“free” and practically unlimited) cold sink can be used. The natural environment represents such a heat sink or source. Hence, the “quality” of heat is not constant; it depends on both the absolute temperatures of the heat source and of the natural environment. This quality can be expressed by means of the maximum ability to perform work between the two given heat reservoirs: W max = Q

T  T0 T

(1.2)

where T0 = absolute ambient temperature (T–T0)/T = dimensionless Carnot-factor characterizing the quality of heat taken from the source with a constant temperature. The amount of the performed work can exceed the value resulting from eqn (1.2), only if an artificial heat sink is used, which must, be created and maintained by means of other valuable kinds of energy. Equation (1.2) applies only to ideal reversible processes. According to the second law of thermodynamics, all real processes are irreversible. In real processes, the amount of performed work is always smaller than that resulting from eqn (1.2). Hence, eqn (1.2) characterizes the maximum attainable amount of performed work. Particular kinds of energy differ also in their ability to be transformed into other kinds of energy. For example, internal energy can be only partially transformed into mechanical energy (kinetic or potential) or into mechanical work. It is worth stressing that the ability of some streams of matter to drive thermal processes (for example, the stream of compressed air) cannot be characterized in terms of energy (the energy of compressed air at ambient temperature equals the energy of atmospheric air). The ability to perform mechanical work has been accepted as a measure of the quality of various kinds of energy, characterizing their ability to be transformed into other kinds of energy. This ability depends not only on the composition and state parameters of the considered matter (determining its energy), but also on the composition and state parameters of the matter commonly appearing in the environment in which the considered transformation process takes place. The mentioned environmental parameters should determine the reference level for the calculation of the discussed quality index. This quality index for energy has been termed by Z. Rant [42] exergy. It expresses the maximum work output attainable in the natural environment, or a minimum work input necessary to realize the reverse process. A second version proposed by Riekert [47] is very convenient and can be formulated as follows: Exergy expresses the amount of mechanical work necessary to produce a material in its specified state from components common in the natural environment, in a reversible way, heat being exchanged only with the environment.

EXERGY BALANCE AND EXERGY LOSSES 3

In comparison to energy (which is a function of the state of the considered matter only) exergy is a function of the state of the considered matter and of the common components of the environment. Exergy does not satisfy a law of conservation. Every irreversible process causes an irrecoverable loss of exergy GB, which can be determined by means of the Gouy-Stodola law [23, 55]: įB = T0 6'S

(1.3)

where T0= environmental temperature 6'S = sum of the entropy increase values of all the kinds of matter taking part in the process. As Sama [48] has stated, some degree of exergy loss has to be accepted to reduce the capital investment. For example, a smaller heat transfer area in a heat exchanger can be attained for a higher temperature difference between the heating and the heating fluid in every point of the heat transfer surface. But any temperature difference during the heat transfer generates an exergy loss. Hence, accepting exergy loss should always have some economic justification. If such a justification does not exist, it indicates that the exergy loss results only from an error in the art of engineering. On the other hand, every exergy loss causes a decrease of the useful effect of the process or an increase in the consumption of its driving means (if the required useful effect is given). Hence, the presence of an exergy loss always indicates the possibility of a thermodynamic improvement of the process, but the profitability of such an improvement should be checked by means of an economic analysis (because this is usually connected with some increase of capital investment). Exergy losses can be divided into internal and external ones. Internal exergy losses are connected with the internal parts of the system; they appear inside the control boundary of the system and can be easily calculated by means of eqn (1.3). External exergy losses result from the discharge of waste products of the process into the environment. The exergy of waste products is destroyed in the environment. Hence, the external exergy loss equals the exergy of the considered waste product. The calculation of external exergy losses by means of eqn (1.3) is difficult; it is easier to determine them by means of the exergy of waste products. External exergy losses indicate that the primary causes of exergy losses often act far from the place where the exergy loss appears. It may be difficult to decide which is the primary cause of an exergy loss. For example, the external exergy loss when combustion gases flow out of the boiler can be interpreted as a result of irreversible phenomena inside the boiler or as the result of the absence of their utilization behind the boiler. Internal exergy losses can be divided into technical and structural ones [9]. Technical exergy losses result from the technical imperfection of parts of the installation, and can be decreased by improving the performance of the relevant components. Structural exergy losses cannot be substantially

4 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

diminished without changing the structure or principle of the process. For example, if the heat capacities of the fluid streams in a heat exchanger differ, the temperature gap between these streams differs in each single cross-section of the heat exchanger and cannot be eliminated even if an infinitely large heat transfer area is used. The main task of exergy analysis is to detect and to quantify the causes of thermodynamic imperfections. These causes are characterized by connected exergy losses.

1.2. Exergy balance and exergy efficiency Exergy is exempt from the law of conservation, so that the exergy balance must be closed by the internal exergy loss [56]: B d = ' B s + 6 ' B q + B a u + B a w +W + G Bi

(1.4)

where Bd= exergy delivered to the system with the streams of matter Bau, Baw = exergy carried out from the system by useful products and waste products 'Bs = increase of exergy of the system 'Bq = increase of exergy of the head source being in contact with the system W = work performed by the system GBi = internal exergy loss, due to irreversibilities inside the system. Exergy carried away with waste product constitutes the external exergy loss. The exergy increase of the external heat source can be determined by means of the Carnot factor: 'Bq =  Q

T  T0 T

(1.5)

where Q = heat extracted from the external heat source T= temperature of the heat source (measured at the system boundary). For steady-state processes, in which the flow velocities, chemical composition and thermal parameters are constant at all points of the system, the exergy of the system remains constant and the exergy balance can be expressed as a time rate of the components: B d = B au + B aw + 6'B q + W + įBi

(1.6)

where the dotted terms denote quantities per unit time. The exergy balance can be presented in the form of a flow diagram, fig. 1.2. This diagram was introduced by Szargut [56] in 1956 and by Grassmann

EXERGY BALANCE AND EXERGY LOSSES 5

Figure 1.2: Flow diagram of the exergy balance.

[24] in 1959. The width of the arrows is proportional to the exergy values. The width of the band of the internal exergy loss increases inside the system boundary from zero to the value resulting from eqn (1.3). The exergy balance can also be presented together with the energy balance in a common flow diagram, fig. 1.3 [3]. The arrows with thick lines represent the energy flows. The hatched arrows represent the exergy flows, and the crosshatched areas—the exergy losses. Figure 1.4 presents combined flow diagrams of simplified typical heat machines. No exergy flow is associated with the heat rejected to the environment. The work delivered by the prime mover is smaller than the driving heat. The heat delivered to the heated room by the heat pump is greater than

Figure 1.3: Common flow diagram of energy and exergy balances of a steadystate flow process.

6 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Figure 1.4: Common flow diagrams of energy and exergy balances for simple heat machines.

the driving work, but the delivered exergy is very small. In a refrigerator the flow of the useful heat extracted from the refrigerated room and of the exergy delivered to this room have opposite directions. The ratio of the useful exergy effect to the consumption of the driving exergy constitutes the exergy efficiency, also called rational efficiency or effectiveness:

ȘB =

Bau  Bdr + 'Bsu + Ȉ'Bqu +Wu BD  Ȉ'BqD  WD

(1.7)

where BD, Bdr= exergy of the delivered driving materials (for example, fuels) and of the non-driving raw materials

EXERGY BALANCE AND EXERGY LOSSES 7

'Bsu= useful exergy increase of the system –'Bsu, 'Bqu= exergy decrease of the source of driving heat and increase of the serviced source of heat WD,Wu= driving work and useful performed work. The exergy efficiency is always smaller than unity. In some processes its value is negative. For example the cooling tower of the steam power plant does not produce any positive exergy effect and its exergy efficiency calculated by means of eqn (1.7) would be negative. In such cases the degree of thermodynamic perfection Kp can be calculated to characterize the irreversibility of the process. It can be defined as the ratio of the useful exergy output to the exergy input. When calculating Kp the transit exergy should be omitted [9]. It comprises the exergy components not taking part in the thermodynamic changes (for example, the chemical exergy of the solvent when analyzing chemical processes in solutions). Example 1.1. The exergy efficiency of a thermal prime mover can be determined as the ratio of the performed mechanical work to the exergy decrease of the source of driving heat. Hence, it expresses the ratio of the energy efficiency of the considered prime mover to the energy efficiency of a Carnot machine: ȘB =

Ș W T = E QD T  T0 ȘE max

(1.8)

where QD= driving heat, T = the temperature of the source of driving heat. Similarly, the exergy efficiency of a compressor heat pump absorbing the bottom heat from the environment can be determined as the ratio of the exergy increase of a heated room to the driving mechanical work. Hence, it expresses the ratio of the coefficient of performance COP of a real heat pump to COPmax of the Carnot heat pump: ȘB =

Qu Tu  T0 COP = WD Tu COPmax

(1.9)

where WD = driving work, Qu = useful heat, Tu = the mean temperature of the carrier of useful heat, COP = Qu/WD. Equation (1.9) should not be applied if the bottom heat is not taken from the environment. The exergy drop of the bottom heat source should be taken into account, according to the definition, eqn (1.7):

ȘB =

Qu Tu  T0 Tu  T0 COP = Tb  T0 Tu T0 Tu COP + ( 1  COP ) W D + Qb Tb Tb

(1.10)

8 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

where Qb = heat extracted from the bottom source, Tb = temperature of the bottom source. Example 1.2. In a complex (multi-purpose) process, the ratio of the energy efficiency of a real process to that of a reversible process does not usually express the exergy efficiency because the proportion of the useful effects in the real process differs from that in the reversible process. For example, the internal exergy efficiency of a heat-and-power plant producing useful mechanical work Wu and useful heat Qu is expressed as follows: Tu  T0 Tu T  T0 QD T

Wu + Qu ȘB i =

(1.11)

where the symbols: see eqns (1.8) and (1.9). The influence of the exergy losses appearing in particular parts of the process on its useful effect Bu may be examined by means of the coefficients of sensibility:

Vk = 

wBu wK = B w(įBk ) wH k

(1.12)

where GBk,Hk = total and relative exergy loss in the kth part of the process. The application of sensibility factors can be illustrated by the analysis of a chain process comprising three links. The first link is fed with the driving exergy BN, the third one delivers the useful effect, which can be expressed as: Bu = BN ȘB1 ȘB 2 Ș B 3

(1.13)

Exergy losses in the individual links are įB1 = BN (1  ȘB1 ), įB2 = BN Ș B 1 (1  ȘB 2 ), įB3 = BN Ș B 1 ȘB 2 (1  ȘB 3 )

(1.14)

At constant exergy efficiency values of each link, we can express the coefficients of sensibility as follows:

V 1 = ȘB 2 ȘB 3 , V 2 = ȘB 2 , V 3 = 1.

(1.15)

Hence, the influence of particular exergy losses increases in the course of the processes. It is smallest at the beginning of the chain of processes, and largest at the end.

EXERGY BALANCE AND EXERGY LOSSES 9

1.3. Exergy and anergy Exergy is sometimes defined as a transformable part of energy (Rant [43], Baehr [3]). This definition is unsatisfactory because of quite different features of energy and exergy. For example, energy may be calculated from an optional reference level. Only one requirement should be fulfilled: the assumed conventional reference level should be the same for all portions of matter taking part in the considered process. The reference level of exergy is determined by the parameters of the natural environment. A degree of some convention appears also in this case, but it is small and exactly quantifiable (for example, the assumption of the conventional normal ambient temperature, required in some cases). A negative value of energy does not have any essential meaning. For example, the chemical enthalpy calculated by means of the enthalpy of formation is usually negative. On the other hand, a negative value of physical exergy means that the analyzed state cannot be attained without a consumption of driving exergy. As a consequence of the cited definition of exergy, Rant [43] and Baehr [3] introduced a concept of anergy, defined as a non-transformable part of energy. Hence, energy is the sum of exergy and anergy. The reference level of anergy is optional, similarly as that of energy. Hence, anergy can be negative. It would be very difficult to explain the physical meaning of a negative value of anergy. The use of the concept of anergy is very inconvenient when the temperature of the considered portion of matter is lower than the ambient one. In this case, the extraction of some amount of energy from this matter (for example, by refrigeration) causes an increase of its exergy and decrease in anergy. Hence, the interpretation of exergy as a part of energy is not logical. Therefore, the discussed definition of exergy and the concept of anergy are not used in the present book.

1.4. Typical irreversible phenomena 1.4.1. Friction The main causes of exergy losses are friction, heat transfer with a finite temperature gradient, throttling, diffusion, combustion and other chemical reactions. For example, the exergy loss due to the mechanical or hydraulic friction can be expressed as follows: įB = Q f

T0 T

(1.16)

where Qf = heat of friction T = temperature of the body absorbing the heat of friction. Figure 1.5 presents an irreversible adiabatic expansion. The area under the curve represents the heat of the hydraulic friction absorbed by the working fluid. The

10 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Figure 1.5: Heat of friction and exergy loss in an adiabatic flow machine. sum of the entropy increments is expressed by the difference s2–s1. Hence, the exergy loss is represented by the area of the rectangle below the line of the ambient temperature. It is larger than the heat of friction when the temperature of the body absorbing the heat of friction is lower than the ambient one. This effect appears in refrigerators. The hydraulic friction during the flow of the chilling fluid not only decreases its refrigeration ability but also increases the demand for compression work. 1.4.2. Irreversible heat transfer The exergy loss due to the irreversible heat transfer is given by the formula:

d(į B ) = d Q

T1  T2 T0 T1 T2

(1.17)

where

dQ= elementary amount of transferred heat T1,T2= temperature of the bodies exchanging heat (the warmer and the colder one). The lower the temperature in the process, the greater is the deleterious influence of irreversibility. On the other hand, the higher the temperature in the process, the larger are the admissible temperature differences in the heat transfer processes. Therefore, in metallurgical ovens the temperature difference between the heating gases and the heated body may amount to some hundred degrees, but in condensers of steam turbines or refrigerators it is not larger than few degrees.

Example 1.3. The thermal power of a heat pump is 10kW. Its driving electric power is 2.85kW. The mean temperature of the heating fluid is 323K, and of the heated room 293K. The ambient temperature is 273K.

EXERGY BALANCE AND EXERGY LOSSES 11

According to eqn (1.9) the minimum driving power N d min of the heat pump in the considered conditions would be: N d min =

10 u 20 = 0.68kW . 293

Hence, a considerable part of the real driving electric power (2.17kW) is lost due to the irreversible course of the processes connected with the operation of the heat pump. One of the important sources of exergy losses is the irreversible heat transfer between the heating fluid and the heated room. The exergy loss due to this irreversible heat transfer is įB =

10 u 30 u 273 = 0.86kW 323 u 293

Hence, the considered exergy loss (which can be interpreted as a loss of driving electric power) amounts to 30% of the driving power. The temperature distribution in heat exchangers is usually analyzed by means of the diagram enthalpy stream—temperature (fig. 1.6a). It is convenient to introduce the exergy temperature defined as follows: T ș =1  0 (1.18) T In the diagram enthalpy stream-temperature in the exergy scale (fig. 1.6b) the area between the curves of the heating and the heated medium represents the exergy loss due to the irreversible heat transfer, according to the formula:

Figure 1.6: Temperature distribution and exergy loss in a heat exchanger.

12 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

§ 1 1 ·  = d Q ( șh  șc ) d(į B ) = d Q T0 ¨ © Tc Th ¸¹

(1.19)

where Th, Tc, Th, Tc are the exergetic and the absolute temperature of the hot and cold fluid. 1.4.3. Throttling Throttling is a particular case of irreversible adiabatic expansion (the performed work is equal to zero). The elementary exergy loss can be expressed as follows: d(įB ) = 

T0  V dp T

(1.20)

where V is the volumetric flow rate of the fluid. The application of eqn (1.20) to an ideal gas indicates that the exergy loss due to its throttling is independent of its temperature: p1 p2

įB = n RT0 ln

(1.21)

where n denotes the molar flow rate of the gas. 1.4.4. Diffusion Diffusion appears in many technical and natural processes. In the simplest case of diffusion the initial temperature and pressure of two solutions is the same, but the chemical composition is different. The adiabatic mixing of these solutions is irreversible. It is impossible to separate the final solution so as to restore the initial state without some consumption of exergy. The exergy loss due to diffusion expresses the minimum work required to restore the initial state. In the case of two ideal solutions (denoted as 1 and 2) with the same initial temperature and pressure the exergy loss due to diffusion can be expressed as follows: ª įB =  RT0 «(n1 + n2 ) ¦ yi lnyi n1 i ¬

where

¦y

i1

lnyi1  n2

i

¦y

i2

i

º lnyi 2 » ¼

(1.22)

yi = molar fraction of the ith component n1, n2 = number of moles of the mixed solutions.

Example 1.4. The oxygen-enriched blast delivered to a blast furnace (containing 30% O2 per mole) is produced by mixing technical oxygen (con-

EXERGY BALANCE AND EXERGY LOSSES 13

taining 95% O2 per mole) with atmospheric air (21% O2 per mole). The consumption of technical oxygen per mole of the enriched blast results from the balance of oxygen: y B = 0.95 nx + 0.21(1  nx )

where nx = consumption of technical oxygen, in moles per mole blast, yB = molar fraction of oxygen in the enriched blast. From the oxygen balance it results that nx = 0.122. From eqn (1.22) results the exergy loss due to the irreversible mixing of technical oxygen with atmospheric air. Assuming T0 = 290K we obtain: 290u8.3143[0.3 ln 0.3+0.7 ln 0.7 – 0.122(0.95 ln 0.95+0.05 ln 0.05) –0.878(0.21 ln 0.21+0.79 ln 0.79)] = 326.4kl/kmol The calculated value expresses the loss of driving exergy that would appear if the production of technical oxygen were reversible. In real processes the loss of driving exergy would be considerably greater. The technology of the immediate production of enriched blast is known, but as yet it is too expensive. 1.4.5. Combustion Combustion is a complex process composed of several irreversible steps: mixing of the fuel with the oxidizer (diffusion of oxygen), chemical reaction, heat transfer from reacting molecules to other molecules, mixing of combustion products with the remaining part of combustion gases (diffusion). Usually, combustion runs simultaneously with heat transfer to the walls of the combustion chamber. The total exergy loss in an adiabatic and isobaric combustion process represents a considerable part of the fuel exergy. Figure 1.7 presents an exergy analysis of the adiabatic and isobaric combustion of elementary coal, the air ratio being stoichiometric [60]. The area below the isobar 3–4 of combustion gases represents the heating value of the fuel, and that between the isobar 3–4 and the environmental isotherm—the physical exergy of combustion gases. Point 1 represents the state of the combustion reactants (fuel + air), and point 2 of the combustion products in equilibrium with the environmental air. The area under 1–2 represents the chemical exergy of fuel, and under 1–5 the total exergy loss in the adiabatic combustion process. The total exergy loss amounts to 124 840 kJ/kmol C (38.4% of the chemical exergy of fuel). The total exergy loss increases with the oxidizer excess. Point 4c represents the state of combustion gases after the thermal dissociation of CO2. After the temperature of combustion gases has been lowered, their state returns to the isobar 3–4. The additional exergy loss due to dissociation is small.

14 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Lior [35] analyzed the components of the exergy loss due to the particular steps of the combustion process. He came to the conclusion that the exergy loss due to the chemical reaction of oxidation is relatively small, and amounts only to about 5–6% of the chemical exergy of fuel, constituting about 15–18% of the total exergy loss. Mixing the combustion reactants causes 8–10% of the total exergy loss. The main cause of exergy loss is the internal heat exchange (72– 77% of the overall exergy loss). The exergy loss in an adiabatic combustion process can thus be reduced by preheating the combustion reactants. As shown in fig. 1.8 the enthalpy increase of combustion gases equals that of the reactants, while the entropy increase of combustion gases is smaller than that of combustion reactants. Therefore, the total exergy loss becomes smaller. 1.4.6. Absorption and emission of thermal radiation The absorption of thermal radiation on a solid surface is always connected with the emission of thermal radiation from the surface. The simultaneous absorption and emission of thermal radiation are irreversible if the temperature values of the emitting and absorbing surfaces are different [40]. The energy flux Ơb associated with the thermal radiation of a black surface is expressed as follows: 4 E b = A ı T = Aeb

(1.23)

where A= area of the emitting surface, V= 5.667 u 10–8 W/(m2 K4) the Stefan–Boltzmann constant, eb = density of the energy flux of thermal radiation.

Figure 1.7: Exergy loss in an adiabatic and isobaric combustion process of elementary coal.

Figure 1.8: Influence of the preheating of combustion reactants on the exergy loss in adiabatic combustion of elementary coal.

EXERGY BALANCE AND EXERGY LOSSES 15

When considering the thermal radiation of a closed surface, the density of the energy flux of thermal radiation is inversely proportional to the second power of the distance from the emitting surface. The entropy flux associated with the stream of black surface radiation is 4 4 E b Sb = A ı T 3 = 3 3 T

(1.24)

In order to calculate the energy flux of the radiation of a gray surface, eqn (1.23) should be multiplied by the surface emissivity H. Eqn (1.24) can be also multiplied by the emissivity of a gray surface, but only when the emissivity does not depend on the wave length. Example 1.5. The energy flux of solar radiation over the atmosphere is eSE = 1.395 kW/m2. The density of the energy flux decreases with the second power of the distance from the Sun. Hence, near the Earth, above its atmosphere the density of the energy flux of solar radiation is: 2

§ R· § 695 500 · eSE = eS ¨ ¸ = ı TS4 ¨ © L¹ © 149 500 000 ¸¹

2

where R = radius of the Sun, L = distance from the Sun surface to the Earth. After introducing the value of eSE the equivalent temperature of the Sun’s surface can be calculated: TS = 5807 K, if its emissivity is H = 1. Entropy generation connected with a simultaneous emission of thermal radiation of a black surface and absorption of a stream of radiation of another black body contains a negative component expressing the entropy of absorbed radiation and two positive components expressing the entropy of the emitted radiation and the entropy increase of the heat sink represented by the absorption surface due to the absorption of heat [40]. Considering two parallel black plates we obtain, per unit of the absorption surface: ª4 T 4  Ta4 º ¦ ' Sa = ı « Ta3  Te3 + e » Ta ¬3 ¼





ª1 § T 4· º = ı « Ta3 + Te3 ¨ e  ¸ » © Ta 3 ¹ ¼ ¬3

(1.25)

where Te and Ta are the absolute temperature values of the emitting and absorbing surface.

16 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

In order to calculate the exergy loss due to the simultaneous absorption and emission of thermal radiation, eqn (1.25) should be multiplied by the ambient temperature. Example 1.6. The temperature of the emitting surface can be higher or lower than that of the absorbing surface. Let us consider a system of two parallel black plates with the temperature values T1 = 1500K, T2 = 500K. The ambient temperature is T0 = 300K. When considering the first plate it should be assumed that T1 = Ta, T2 = Te. From eqns (1.24) and (1.3) it results į b 1 = 17 kW/m2. When considering the second plate, we should introduce T1 = Te, T2 = Ta. In this case we obtain į b 2 = 96.3 kW/m2. In both cases the exergy loss is positive. The sum of the calculated exergy losses expresses the total exergy loss appearing in the considered system, that is in accordance with eqn (1.17).

1.5. Anthropogenic and natural exergy losses Besides exergy losses derived from human activity, huge losses within the Biosphere result from the irreversibility of natural phenomena (the absorption of solar radiation, the emission of thermal radiation, the irreversible heat transfer from inside the Earth to its surface and the braking of the planetary motion). The irreversible absorption of solar radiation and emission of infrared radiation constitute the prevailing part of these losses. It is worth stressing that natural exergy losses are the main cause of the formation of the non-equilibrium natural environment [87]. According to the laws of non-equilibrium thermodynamics, irreversible phenomena can be accompanied by coupled processes partially decreasing the effects of irreversibility. These coupled processes can lead to the formation of stationary dissipative structures, for example biological living structures. Hence, the huge natural exergy losses connected with solar radiation, are the very cause of the existence of life on Earth. Exergy losses occurring within the zone of human existence, located near the Earth’s surface, represent the most accessible natural resources of renewable exergy. These losses may be distinguished by the term “natural losses of utilizable exergy NLUE” [88]. Every utilization of these resources denotes a shift of exergy losses from the sphere of natural to the sphere of anthropogenic exergy losses. For example in hydro-power plants about 1 TW (1012 W) of natural exergy losses is shifted into the zone of anthropogenic ones. The energy power of solar radiation reaching the upper layer of the Earth’s atmosphere is about 174,000 TW [52]. From this radiation about 30% is reflected by upper layers of the atmosphere (without any exergy loss), about 20% is absorbed in the atmosphere and re-emitted in the form of infrared radiation, about 24% of this energy warms the surface of continents and oceans (about 0.5% of this quantity is transformed into the exergy of wind, waves and seacurrents [27, 36]. About 23% of the total solar radiation is used for the evaporation of water (about 0.4% of this quantity is transformed into the potential exergy of water droplets in clouds, then about 0.012% into potential

EXERGY BALANCE AND EXERGY LOSSES 17

exergy of water in rivers). About 0.02% of the total stream of solar radiation (energy about 40 TW, exergy about 37 TW) is absorbed by plants and partially transformed into their chemical energy (energy about 2.5 TW, exergy about 2.9 TW) [21]. About 1 TW of the chemical energy of plants is used by humankind for its needs. From the energy power of solar radiation reaching the Earth’s surface (87 000 TW after Smil [52]) about 10% are reflected, and only a very small fraction is accumulated in the form of natural resources (for example, in the form of peat). Almost the total amount of the absorbed solar radiation is, after numerous transformations, emitted again in the form of infrared radiation. As a result of the greenhouse effect, a considerable flow of radiation circulates between the Earth’s surface and the atmosphere. Therefore, the total energy flow of the radiation absorbed by the Earth’s surface (equal to the emitted energy flow of infrared radiation, amounting to 181,000 TW) is larger than the immediately absorbed energy flow of solar radiation [26]. Assuming a mean emissivity of the Earth’s surface equal to 0.9 we can calculate its mean equivalent temperature (289K). The energy flow of the secondary radiation of the atmosphere amounts to 102,700 TW [26]. Hence, the energy stream of the radiation reaching the Earth’s surface amounts to 114,100 TW. From this quantity results the equivalent temperature of the upper layers of the atmosphere (251K). In order to calculate the natural losses of the utilizable exergy connected with the absorption of solar radiation and the emission of infrared radiation, the entropy balance of the Earth’s surface should be formulated, under steady state conditions: E E · 4 § E ¦ 'S = S0  SS  Sa  Sr = ¨ 0  S  a ¸  Sr 3 © T0 TS Ta ¹

(1.26)

where: S0 , SS , Sa , Sr = entropy flow of own infrared emission, absorbed solar radiation, absorbed secondary radiation of the atmosphere, and relict radiation of the cosmic space E 0 , E S ,E a = energy flow of the mentioned kinds of radiation. The relict radiation (remaining after the Big Bang) has a spectral composition corresponding to the temperature 2.73K. The energy flow of the Earth’s surface radiation transmitted immediately to the cosmic space amounts to 7000 TW [26]. The corresponding flow of relict radiation reaching the Earth’s surface amounts to 5.6u10–5 TW. Its entropy flow is very small and may be neglected. It has been assumed that the accumulation of energy near the Earth’s surface is very small. From eqn (1.26) we obtain the sum of natural exergy losses burdening the absorption and emission on the Earth’s surface: ¦ 'S = 272 TW / K, į B = 78.500 TW

18 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

The calculated sum of NLUE is greater than the exergy of the solar radiation absorbed by the Earth’s surface. The remaining part of the con-sidered losses results from the irreversible radiation exchange between the Earth’s surface and the cosmic space. Natural exergy losses may be compared with the effects of human activity. The energy power of chemical and nuclear fuels extracted from natural resources is about 12 TW. (About 6 TW of the chemical energy of fuels and 1 TW of nuclear energy are used for the production of electricity [52].) This power is almost totally destroyed in irreversible production and consumption processes. Only some small part remains accumulated in buildings, machines and installations, but their exergy is continuously destroyed by corrosion and weathering. Additional losses of exergy appear in anthropogenic processes of the consumption of plants. The estimated sum of anthropogenic exergy losses per time unit is about 13 TW. Hence, they are about 6,000 times smaller than the NLUE. This proportion reflects the scale of the available natural exergy resources.

Exercises 1.1. Calculate the exergy efficiency of the prime mover if its energy efficiency amounts to KE = 0.388. The temperature values of the feeding heat source and the environment are Tf = 850K, T0 = 283K. 1.2. The prospectus of an industrial compressor heat pump gives the ratio of the useful heat to the consumption of driving electricity 3.9 if the temperature of the environment (the source of bottom heat) is T0 = 258K and of the heated room Tu = 350K. Check the credibility of the information given. 1.3. The irreversibilities in a considered Carnot prime mover result only from the temperature differences between the heat sources and the working fluid. The temperature values of the feeding heat source and of the environment are Tf = 900K, T0 = 300K. The temperature differences are 'T = 20K. The power of the prime mover is 10 MW. Calculate the rates of exergy losses. 1.4. A stream of pure CO2 with ambient temperature and pressure mixes with atmospheric air. The atmospheric parameters are T0 = 283K, p0 = 0.1MPa. The partial pressure of CO2 in the atmosphere amounts to 33 Pa. Calculate the exergy loss due to the irreversible mixing per 1kg of CO2 . Assume that CO2 has the properties of an ideal gas. 1.5. A piece of ice (1kg) with the initial temperature T1 = 263K melts, and the obtained liquid water reaches the ambient temperature T0 = 293K. The specific heat capacities of ice and liquid water are ci = 2.14 kJ/(kg K), cw = 4.18 kJ/(kg K); the enthalpy of melting is L = 333.4 kJ/kg. Calculate the exergy loss due to the irreversibility of the process.

Chapter 2 Calculation of exergy 2.1. Types and components of exergy Exergy of a stream of matter crossing an immovable system boundary (flow exergy) is a principal quantity appearing in exergy balances. This quantity can be divided into its components, the most important being: kinetic exergy, potential exergy, physical exergy bph, chemical exergy bch and nuclear exergy bnu [56]. Kinetic exergy should be calculated by means of the velocity c related to the Earth’s surface. Potential exergy should be determined by means of the height X above the lowest level prevailing near the considered device. Per 1kg of the considered substance, we obtain: b

c2  gX  bph  bch  b nu 2

(2.1)

where g is the gravitational acceleration. The exergy Bs of the substance contained within the system boundary (internal exergy) differs from flow exergy (of the substance stream crossing the system boundary), like the enthalpy from the internal energy [56]: Bs

B  V ( p  p0 )

(2.2)

where V = volume of the considered part of the system p, p0 = pressure of the considered substance and of the atmospheric air. Exergy of the substance stream crossing a moving system boundary (with a constant linear velocity s) is Bm

s B  V ( p  p0 ) r

(2.3)

where r is the velocity of the substance stream related to the system boundary. In eqn (2.3) it has been assumed that the direction of the velocity s is opposite to that of the velocity r.

20 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

2.2. Physical exergy Physical enthalpy, entropy and exergy are calculated from the state determined by the ambient temperature T0 and pressure p0 without any change of the chemical composition of the considered substance. The specific physical exergy bph is expressed as: bph

hph  T0 s ph

(2.4)

where hph,sph denote the specific physical enthalpy and entropy. Physical exergy can be divided into a part depending on pressure and another depending on temperature. The temperature-dependent part corresponds to the variations of enthalpy andentropy at p0. T

bphT

('hp0  T0 's p0 ) |

(2.5a)

T0

In the case of ideal gases eqn (2.5a) takes the form bphT

¦yh i

phi

i

 T0 ¦ yi 's pi

(2.5b)

i

Under the additional assumption of a constant thermal capacity the temperature-dependent part of the physical exergy becomes: bphT

§ T· c p ¨ T  T0  T0 ln ¸ T0 ¹ ©

(2.5c)

where cp denotes the specific thermal capacity at constant pressure. The temperature-dependent part of the physical exergy is always positive and equals zero at environmental temperature. Hence, below the environmental temperature it increases when the temperature of the considered substance decreases. The pressure-dependent part increases with pressure and is negative when the pressure is lower than the ambient one. As the temperature-dependent part is computed at constant ambient pressure and temperature increasing from T0 to T, so the pressure-dependent part corresponds to the isothermal variations of enthalpy and entropy at T = const: p

bphp

('h  T0 's )T

const

|

(2.6a)

p0

In the case of ideal gases the formula becomes: bphp

RT0 ln

p p0

(2.6b)

CALCULATION OF EXERGY 21

Figure 2.1: Reduced physical exergy of a perfect gas.

The pressure-dependent part of the physical exergy of a solid or liquid substance can usually be calculated under the assumption that its specific volume v does not depend on the pressure: bphp

v( p  p0 )

(2.6c)

Figure 2.1 presents the reduced dimensionless physical exergy of a perfect gas.

2.3. Chemical exergy Chemical exergy expresses the exergy content of the substance at environmental temperature and pressure [57]. Its value results from the difference of composition of this substance in relation to the commonly appearing components of the environment. Some authors maintain that chemical exergy should be calculated from the level of an equilibrium environment [105, 106]. Such an assumption would facilitate the calculation of the chemical exergy, because the state of every component of an equilibrium environment could be used as a reference level for the calculation of the chemical exergy. It is, however, worth stressing that the state of the natural environment is far from thermodynamic equilibrium, and the attainment of such a dead state is impossible because of the continuous transmission of the solar radiation representing not only a flow of energy but also a flow of entropy. Some part of the transmitted energy is accumulated in the form of the chemical energy of plants, the potential energy of liquid water, etc., but the prevailing part is dissipated. The absorption of solar radiation connected with the

22 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

emission of infrared radiation of the Earth’s surface (characterized by a much greater entropy) is irreversible, causes huge exergy losses (about 113u106 GW [88]) and generates components that are not in equilibrium with the prevailing components of nature. Ahrendts [1] calculated the equilibrium composition of the environment at a given ambient temperature and pressure. He treated the natural environment as a closed system with a given temperature and pressure. He assumed various values of the thickness of the external layer of the lithosphere (1m, 10m, 100m, 1000m), considered 17 chemical elements and 692 of their compounds, and took into account the total mass of each element. Calculations of the equilibrium composition of the environment yielded values very different from those measured in the field: the differences are extremely large for the atmosphere (see table 2.1). In an equilibrium state, the oxygen would be mainly combined in nitrates solved in seawater, and the concentration of free oxygen in the atmosphere would amount to some ppm only. In order to calculate the chemical exergy in a non-equilibrium environment, Szargut [57] introduced the concept of reference species. In chemical reactions only the number of atoms (the amount of moles) of each individual element does not change. Therefore, for every element taking part in a chemical reaction a separate reference species containing the considered element should be accepted. To prescribe the lowest (but appearing in nature) reference level of chemical exergy, the most common components of the natural environment should be chosen as reference species [59, 70]. It should be stressed that the reference species of particular elements are independent of each other. It is impossible to formulate a chemical reaction comprising only reference species. Therefore, the problem of chemical equilibrium between reference substances does not exist. After selecting the reference species, a reference reaction can be formulated for every chemical compound. In this reaction, except the considered compound (or element), only reference substances appear: influent refeence species supplementing the considered compound and effluent reference species resulting from the reaction. Table 2.1: Equilibrium composition of atmospheric air in a system with an external layer of the lithosphere 100m thick T0=298 K, p0=77 kPa. Component N2 H2O Ar CO2 O2

Molar fraction 0.9451 0.0397 0.0117 0.0033 2.6 ppm

CALCULATION OF EXERGY 23

Example 2.1. Formulation of the reference reaction. The following reference species have been chosen: CO2 for carbon, O2 for oxygen and CaCO3 for calcium. The reference reaction for the element Ca has the following form: Ca + ½ O2 + CO2 Ÿ CaCO3 The reference species O2 and CO2 appear as the influent and CaCO3 as the effluent reference species. The ambient temperature and pressure are not constant. Usually changes of these parameters influence, though slightly, the value of chemical exergy. The chemical composition of the natural environment is also not the same in various places. Calculations of the chemical exergy are usually difficult. In order to prepare the tables of chemical exergy, the concept of standard chemical exergy has been introduced [59]. It is calculated at normal temperature and pressure (25qC, 1 Atm) and conventional standard concentration of the reference species in the environment. This standard concentration results from the mean composition of atmospheric air, sea water and the external layer of the Earth’s crust. The standard chemical exergy of a chemical compound or element can be calculated by means of the exergy balance of a reversible standard reference reaction. The reactants taking part in a standard chemical reaction appear separately in standard state. The normal standard state denotes additionally a normal temperature. bchq

q q ' r G q  ¦ Bchk  ¦ Bchj k

where

' r G q = normal standard free energy of the reference reaction

¦ B ,¦ B q chk

k

(2.7)

j

q chj

= sum of the normal standard chemical exergies of the effluent and

j

influent reference species. Example 2.2. Calculation of the standard chemical exergy of a chemical element. The normal standard free energy of the reference reaction cited in Example 2.1 is 'rGq= –738.6 kJ/mol CaCO3. The normal standard values of the chemical exergy of the reference species are 16.3 kJ/mol CaCO3, 19.87 kJ/mol CO2 and 3.97 kJ/mol O2. According to eqn (2.7) the standard chemical exergy of the element Ca is bchq

738.6  16.3  19.87  1.5 u 3.97 729.1 kJ/mol

The formulation of the reference reaction and the application of eqn (2.7) are required only for chemical elements. After calculating the normal standard chemical exergy of the chemical elements, the normal chemical exergy of every chemical

24 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

compound can be calculated by means of the exergy balance of the reversible normal standard reaction of formation of the considered compound bchq

q ' f G q  ¦ nel bchel

(2.8)

el

q = normal standard chemical exergy of the compound and element where bchq , bchel nel = number of moles of the elements per unit of the compound 'fGq = Gibbs normal standard free energy of formation of the compound.

Example 2.3. Calculation of the standard chemical exergy of a chemical compound. Gibbs normal standard free energy of formation of the compound SO3 is 366.5 kJ/mol. The normal standard exergies of the elements are 609.6 kJ/mol S and 3.97 kJ/mol O2. According to eqn (2.8) the normal standard bchq

366.5  609.6  1.5 u 3.97

249.1 kJ/mol.

For elements commonly occuring in atmospheric air, the main components of air can be accepted as reference species (O2, N2, CO2, H2O, D2O, Ar, He, Ne, Kr, Xe). The specific chemical exergy of gaseous reference species results from its molar fraction in the air: bchrs

 RrsT0 ln zrs 0

(2.9)

where Rrs,zrs0 designate the gas constant of the reference species and its molar fraction in the environment (in the air when considering gaseous reference species). For gaseous substances containing only gaseous reference species the formulation of a reference reaction and the use of the tables of the standard chemical exergy is not necessary. The calculation of the chemical exergy may take into account only the change of the concentration of the components of the considered substance: Bch

nT0 R ¦ yi ln i

yi y0 i

(2.10)

where n = amount of moles of the substance yi, y0i = mol fraction of the ith component in the considered substance and in the atmospheric air R = universal gas constant. When calculating the chemical exergy of liquid or gaseous water, it should be taken into account that at ambient temperature the exergy of saturated vapour does not depend on its quality. When considering processes of humid air, the reference state for the chemical exergy of liquid or gaseous water may be determined by means of the

CALCULATION OF EXERGY 25

local ambient parameters. For locations remote from the sea the concentration of water vapour in the ambient air may determine the chemical exergy of water. In this case, according to eqn (2.10), the chemical exergy of saturated liquid water or vapour can be calculated by means of the relative humidity of atmospheric air: bchc 0

bchcc0

RwT0 ln

1

(2.11)

M0

where bchc 0 , bchcc0 = chemical exergy of saturated liquid and saturated vapour at ambient temperature Rw = gas constant of water M0 = relative humidity of atmospheric air. The chemical exergy of liquid water or water vapour should additionally include the work of the expansion of liquid water from the ambient pressure to the pressure of saturated vapour at ambient temperature: bchw

v c ( p0  ps 0 )  RwT0 ln

1

(2.12a)

M0

It is, however, not convenient to calculate separately the physical and chemical exergy of liquid water or water vapour. The sum of these quantities can be expressed as follows:

bw

h  h0c  T0 ( s  s0c )  RwT0 ln

1

M0

(2.13)

where h0c , s0cc = specific enthalpy and entropy of saturated liquid water at the ambient temperature. The values of saturated vapour at ambient temperature may be also introduced into eqn (2.13). In countries where fresh water is scarce, its production by desalination of seawater may be necessary. In such a case the concentration of pure water in the seawater may determine the reference level of the chemical exergy of water. The following formula expressing the chemical exergy of pure water (equal to the minimum desalination exergy) results from the assumption that pure water contained in seawater may be treated as a component of an ideal solution. Taking into account the molar fraction z of salt in seawater, we obtain bchw

v c ( p0  ps 0 )  RT0 ln

1 1 z

(2.12b)

The first component of eqn (2.12b) may be neglected. For other elements, solid reference species present in the external layer of the Earth’s crust (products of weathering) and ionic (or molecular) reference species

26 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

present in the sea water have been proposed [38, 59, 64, 66]. Similarly to eqn (2.9), the chemical exergy of solid or ionic reference species should depend on their concentration in the environment. It would not be acceptable to assume a zero value for the chemical exergy of solid reference species because it would denote, that they are commonly accessible in the environment and could be gained without any consumption of work. The external layer of the Earth's crust is a very complicated mixture of solid solutions. It is not possible to calculate exactly the concentration exergy of solid reference species. Szargut [59] proposed to evaluate the concentration exergy of solid reference species by means of the formula (2.9) valid for ideal solutions. The conventional mean molar fraction of solid reference species in the external layer of the Earth’s crust results from geochemical data [41] and from the evaluated mean molecular mass M0 of the upper layer of the Earth’s crust [59]: zrs 0

l n0i ci M 0 li

(2.14)

where noi = mean molar concentration of ith element in the continental part of the Earth’s crust li = number of the atoms of ith element in the molecule of the reference species ci = fraction of the element appearing in the form of reference species. The value M0=135.5 kg/kmol results from the composition of a mean sample of the lithosphere, calculated by Szargut [66] according to the geochemical data. The values of ci have been assumed conventionally according to geochemical information about the appearance of the assumed reference species. It is not necessary to determine exactly the value zni because it appears under the logarithm in eqn (2.9). Multiplication of zni by 100 changes the value resulting from eqn (2.9) by only 10 kJ/mol. The values for the elements Cr, Ir, Pd, Sn and Ti have been corrected according to the data of Taylor and McLennan [91, 92] and calculation results of Ranz-Villarino et al [44]. The exergy values of Al and Mn are corrected according to Rivero and Garfias [109]. Example 2.4. Calculation of the standard chemical exergy of a solid reference species. The mean molar concentration of the element Ca in the external layer of the Earth’s crust is 1.03u10–3. The fraction of the element Ca appearing in the form of the reference substance CaCO3 has been assumed as c=0.01. Hence, according to eqn (2.14) the conventional mean molar fraction of the reference species in the lithosphere is zrs 0

1.03 u 103 u 0.01 u 135.5 1.4 u 103

CALCULATION OF EXERGY 27

According to eqn (2.9) the standard chemical exergy of the reference species is bchq

8.3143 u 10 3 u 298 u ln0.0014 =16.3 kJ/mol

The assumption of solid reference species from among the compounds most stable in normal conditions, is not always reasonable. Sometimes the formation of most stable compounds is kinetically blocked and therefore they do not commonly appear in the environment. For example the nitrates NaNO3, KNO3, Ca(NO3)2 are most stable but do not commonly appear in the environment [1]. Therefore, the negative value of the chemical exergy of the mentioned nitrates should be accepted. The compounds of very rare elements should not be accepted as reference species either because of a small likelihood of their formation. However, the chemical exergy of commonly appearing components of the lithosphere should not have a negative chemical exergy. Components dissolved in sea water can be convenient as reference species because their mean concentration in sea water is sufficiently well known for many elements. However, only for dissolved molecules and for monocharged and bicharged ions the theory of thermodynamic functions is sufficiently exact. Therefore, only such kind of reference species is to be accepted. Sometimes the use of reference species dissolved in sea water results in negative values of the standard chemical exergy of solid compounds which are relatively common on the Earth's surface. Such unsatisfactory results have been obtained in the case of elements of the second column in the periodic system and moreover for some compounds of F and Mn. In such cases the assumption of solid reference species was unavoidable [71]. The formula expressing the standard chemical exergy of elements with reference species dissolved in seawater has been proposed by Morris [37, 38, 69]: bchq

ª º 1 q j « ' f Grq  zbchq H2  ¦ vk bchk  RTn [2.303 z (pH)  ln mn Ȗ]» 2 k ¬ ¼

(2.15)

where

j = number of reference ions or molecules derived from one molecule of the element under consideration ' f Grq = normal standard free energy of formation of the reference species

z = number of elementary positive charges of the reference ion Qk = number of molecules of additional elements present in the molecule of the reference species bchq H2 , bchq k = standard chemical exergy of hydrogen gas and of the kth additional element, mn = conventional standard molarity of the reference species in seawater, J = activity coefficient (molarity scale) of the reference species in seawater, pH = 8.1; exponent of the concentration of hydrogen ions in seawater.

28 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

The activity coefficient of single ions can be calculated by means of the DebyeHückel equation:  log Ȗ i

Azi2 I 1  ai B I

(2.16)

where A = 0.51 kg1/2 mol–1/2 for water at 25qC B = 3.287u109 kg1/2 m–1 mol–1/2 for water at 25qC ai = effective diameter of the ion I = ionic strength of the electrolyte. The ionic strength of the electrolyte results from the following equation: I

1 ¦ mi zi2 2 i

(2.17)

where mi = molarity of the ion, mol/kg H2O zi = number of elementary electric charges on the ion. The ion Cl– prevails among the negative ions in seawater, and therefore, the data for chlorides can be assumed as activity coefficients of the positive ions Na+, K+. The activity coefficients of the negative ions Cl– and SO--4 can be estimated in reference to the predominant positive ion Na+. The following ionic and molecular reference species dissolved in sea water have been accepted:  AgCl , Bi(OH)3 aq, BiO , Br  , CdCl2 aq, Cl , Cs , Cu   , HAsO 4 , HPO 4 ,      HgCl4 , JO3 , K  , Li  , MoO , PbCl2 aq, Rb ,SO 4 , Na , Ni 4 ,SeO 4 , WO 4 , Zn .

Example 2.5. Calculation of the standard chemical exergy with an ionic reference species. The reference ion of sulfur is SO 24  . Its normal standard free energy of formation is –744.6 kJ/mol, and j = 1, z = –2. An additional element in the reference ion is oxygen, nk = 2. The values of the standard chemical exergy are 236.09 kJ/mol H2 and 3.97 kJ/mol O2. The conventional standard molarity of the reference ion in seawater is m = 0.0117, and its activity coefficient is J = 0.11. According to eqn (2.15) the standard chemical exergy of sulfur is 744.6  236.09  2 u 3.97  8.3143 u 103 u 298 u [2.303 u 2 u 8.1  ln(0.0117 u 0.11)] 609.6 kJ/mol

The calculated values of the standard exergy of elements are cited in table 2.2. Table 2.3 contains unsatisfactory negative values obtained by means of the ionic reference species for some compounds of the elements of the second column in the periodic system. The value of the normal chemical exergy of Au calculated by means of ionic reference species is surprisingly small. On the other hand, the value for graphite obtained by means of the ionic reference species differs only slightly from that resulting from the gaseous reference species

Order no

Chemical element Chem Concentration symbol in Earth’s crust nch kmol/kg

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2 Ag Al Ar As Au B Ba Be Bi Br C Ca Cd Ce Cl Co Cr Cs Cu D

3 3.7u10–10 3.05u10–3 – 2.4u10–8 2u10–11 9.3u10–7 3.1u10–6 3.1u10–7 8.1u10–10 3.1u10–8 1.7u10–5 1.03u10–3 1.8u10–9 4.3u10–7 3.7u10–6 3.5u10–7 1.5u10–6 2.2u10–5 8.7u10–7 –

Table 2.2: Standard chemical exergies of the elements. Reference species Chemical state Standard c, J U, m, x formula chem. exergy

bchq 4 AgCl2– AlSiO5 Ar HasO42– Au B(OH)3 BaSO4 Be2SiO4 BiO+ Br– CO2 CaCO3 calcite CdCl2 CeO2 Cl– CoFe2O4 K2Cr2O7 Cs+ Cu2+ D2O

5 aq s g aq s aq s s aq aq g s aq s aq s s aq aq g

6 J=0.6 c=0.01 – J=0.1 c=0.5 J=1.0 c=0.01 c=0.01 J=0.6 J=0.68 – c=0.01 J=1.0 c=0.02 J=0.68 c=0.002 c=0.0l J=0.6 J=0.2 –

7 m=2.7u10–9 x=2.1u10–3 p=0.906 m=2.1u10–8 x=1.4u10–9 m=3.4u10–4 x=4.2u10–6 x=2.1u10–7 m=1u10–10 m=8.7u10–4 p=0.0335 x=1.4u10–5 m=6.9u10–11 x=1.2u10–6 m=0.5657 x=9.5u10–8 x=1u10–6 m=2.3u10–9 m=7.3u10–10 p=3.42u10–4

kJ 8 – 15.3 11.69 – 50.5 – 30.7 38.1 – – 19.87 16.3 – 33.8 – 40.1 34.3 – – 31.2

Free exergy of formation

Chemical element State Standard chem. exergy

' f Gq kJ/mol 9 –215.5 –2 441.0 0 –714.7 0 –968.8 –1 361.9 –2 033.3 –146.4 –104.0 –394.36 –1 129.0 –359.4 –1 024.8 –131.26 –1 032.6 –1 882.3 –282.2 +65.5 –234.55

bchq 10 s s g s s s s s s Br2l s,gr s s s,J Cl2g s,III s s s D2g

kJ/mol 11 70.2 795.7 11.69 494.6 50.5 628.5 775.1 604.4 274.5 101.2 410.26 729.1 293.8 1054.6 123.6 312.0 584.7 404.4 134.2 263.8

Order no

Chemical element Chem Concentration symbol in Earth’s crust nch kmol/kg

1 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

2 Dy Er Eu F Fe Ga Gd Ge H He Hf Hg Ho I In Ir K Kr La

3 1.8u10–8 1.7u10–8 7.9u10–9 3.3u10–5 1.0u10–3 2.2u10–7 3.4u10–8 1.4u10–8 – – 1.7u10–8 4.0u10–10 7.2u10–9 3.9u10–9 8.7u10–10 1.04u10–13 5.4u10–4 – 2.2u10–7

Table 2.2: Standard chemical exergies of the elements (continued) Reference species Chemical state Standard c, J U, m, x formula chem. exergy

bchq 4 Dy(OH)3 Er(OH)3 Eu(OH)3 CaF2.3Ca3(PO4)2 Fe2O3 Ga2O3 Gd(OH)3 GeO2 H2O He HfO2 HgCl42– Ho(OH)3 IO3– In2O3 IrO2 K+ Kr La(OH)3

5 s s s s s s s s g g s aq s aq s s aq g s

6 c=0.02 c=0.02 c=0.02 c=0.01 c=0.1 c=0.02 c=0.02 c=0.05 – – c=0.05 J=0.1 c=0.02 J=0.6 c=0.05 c=0.005 J=0.64 – c=0.02

7 x=4.9u10–8 x=4.6u10–8 x=2.lu10–8 x=2.2u10–5 x=6.8u10–3 x=3.0u10–7 x=9.2u10–9 x=9.5u10–8 p=2.2 p=4.85u10–4 x=1.2u10–7 m=3.4u10–10 x=1.0u10–8 m=5.2u10–7 x=2.9 u10–9 x=3.6u10–12 m=1.04u10–2 p=9.7u10 x=6.0u10–7

kJ 8 41.7 41.9 43.8 26.6 12.4 37.2 40.2 40.1 9.49 30.37 39.5 – 43.9 – 48.7 75.1 – 34.36 35.5

Free exergy of formation

Chemical element State Standard chem. exergy

' f Gq kJ/mol 9 –1 294.3 –1 291.0 –1 320.1 –12 985.3 –742.2 –998.6 –1 288.9 –521.5 –228.59 0 –1 027.4 –446.9 –1 294.8 –128.0 –830.9 –185.6 –282.44 0 –1 319.2

bchq 10 s,D s s F2g s,D s s,D s H2g g s,D 1 s,D I2s s s s g s,D

kJ/mol 11 975.9 972.8 1003.8 504.9 374.3 514.9 969.0 557.6 236.09 30.37 1062.9 115.9 978.6 174.7 436.8 256.7 366.6 34.36 994.6

Order no

Chemical element Chem Concentration symbol in Earth’s crust nch kmol/kg

1 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

2 Li Lu Mg Mn Mo N Na Nb Nd Ne N1 C Os P Pb Pd Pr Pt Pu

3 2.9u10–6 2.9u10–9 9.6u10–4 1.7u10–5 1.6u10–8 – 1.03u10–3 2.2u10–7 1.9u10–7 – 1.3u10–6 – 5.0u10–13 3.4u10–5 7.7u10–8 4.7u10–12 5.8u10–8 2.6u10–11 6.2u10–20

Table 2.2: Standard chemical exergies of the elements (continued) Reference species Chemical state Standard c, J U, m, x formula chem. exergy

bchq 4 Li+ Lu(OH)3 Mg3SiO10(OH)2 MnO2 MoO42– N2 Na+ Nb2O3 Nd(OH)3 Ne Ni2+ O2 OsO4 HPO42– PbCl2 PdO Pr((OH)3 PtO2 PuO2

5 aq s s s aq g aq s s g aq g s aq aq s s s s

6 J=0.68 c=0.02 c=0.06 c=0.01 J=0.1 – J=0.68 c=0.0l c=0.02 – J=0.2 – c=0.005 J=0.1 J=1.0 c=0.005 c=0.02 c=0.005 c=0.01

7 m=2.5u10–5 x=7.9u10–9 x=2.6u10–3 x=2.3u10–5 m=1.1u10–7 p=75.78 m=0.474 x=1.5u10–7 x=5.1u10–7 p=1.77u10–3 m=1.2u10–7 p=20.39 x=3.4u10–13 m=4.9u10–7 m=4.2u10–11 x=3.2u10–12 x=1.6u10–7 x=1.8u10–11 x=8.4u10–20

kJ 8 – 46.2 14.8 26.5 – 0.72 – 39.0 35.9 27.19 – 3.97 71.62 – – 65.6 38.8 61.3 108.9

Free exergy of formation

Chemical element State Standard chem. exergy

' f Gq kJ/mol 9 –294.0 –1 259.6 –5 543.0 –465.2 –836.4 0 –262.05 –1 766.4 –1 294.3 0 –45.6 0 –305.1 –1 089.3 –297.2 –82.5 –1 285.1 –83.7 –995.1

bchq 10 s s s s s N2g s s s,D g s O2g s s,w s s s,D s s

kJ/mol 11 393.0 945.7 626.1 487.7 730.3 0.72 336.6 899.7 970.1 27.19 232.7 3.97 368.1 861.4 232.8 146.1 963.8 141.0 1100.0

Order no

Chemical element Chem Concentration symbol in Earth’s crust nch kmol/kg

1 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77

2 Ra Rb Re Rh Ru S Sb Sc Se Si Sm Sn Sr Ta Tb Te Th Ti Tl

3 4.4u10–15 1.05u10–6 5.4u10–12 9.7u10–12 1.0u10–12 8.1u10–6 1.6u10–9 1.1u10–7 6.3u10–10 1.0u10–2 4.0u10–8 1.7u10–8 4.3u10–6 1.1u10–8 6.3u10–9 1.4u10–11 4.0u10–8 5.9u10–5 2.2u10–9

Table 2.2: Standard chemical exergies of the elements (continued) Reference species Chemical state Standard c, J U, m, x formula chem. exergy

bchq 4 RaSO4 Rb+ Re2O7 Rh2O3 RuO2 SO42– Sb2O5 Sc2O3 SeO42– SiO2 D quartz Sm(OH)3 SnO2 SrCO3 II Ta2O5 Tb(OH)3 TeO2 ThO2 TiO2 T12O4

5 s aq s s s aq s s aq s s s s s s s s s s

6 c=0.05 J=0.6 c=0.01 c=0.005 c=0.005 J=0.11 c=0.001 c=0.05 J=0.1 c=0.3 c=0.02 c=0.2 c=0.05 c=0.0l c=0.02 c=0.005 c=0.05 c=0.02 c=0.01

7 x=3.0u10–14 m=1.4u10–6 x=3.6u10–12 x=3.3u10–12 x=6.08u10–13 m=1.17u10–2 x=1.1u10–10 x=3.7u10–7 m=1.2u10–9 x=0.407 x=l.lu10–7 x=4.6u10–7 x=2.9u10–5 x=7.5u10–9 x=1.7u10–8 x=9.5u10–12 x=2.7u10–7 x=1.6u10–4 x=1.5u10–9

kJ 8 77.2 – 65.3 65.5 69.5 – 56.8 36.7 – 2.2 39.7 36.2 25.9 46.4 44.9 62.9 37.5 21.7 60.4

Free exergy of formation

Chemical element State Standard chem. exergy

' f Gq kJ/mol 9 –1 364.2 –282.4 –1 067.6 –299.8 –253.1 –744.6 –829.3 –1 819.7 –441.4 –856.7 –1 314.0 –519.6 –1 140.1 –1 911.6 –1 314.2 –270.3 –1 169.1 –889.5 –374.3

bchq 10 s s s s s s,rh s,III s,D s s s,D s,w s s s,D s s,D s,II s,D

kJ/mol 11 823.9 388.6 559.5 179.7 318.6 609.6 438.1 925.2 346.5 854.9 993.6 551.9 749.8 974.0 998.4 329.2 1202.6 907.2 194.9

Order no

s1 78 79 80 81 82 83 84 85 86

Chemical element Chem Concentration symbol in Earth’s crust nch kmol/kg 2 Tm U V W Xe Y Yb Zn Zr

Table 2.2: Standard chemical exergies of the elements (continued) Reference species Chemical state Standard c, J U, m, x formula chem. exergy

bchq

3

4

2.8u10–9 1.1u10–8 2.7u10–6 8.2u10–9 – 3.7u10–7 1.7u10–8 1.1u10–6 1.8u10–6

Tm(OH)3 UO3.H2O V2O5 WO42– Xe Y(OH)3 Yb(OH)3 Zn2+ ZrSiO4

5 s s s s g s s aq s

6 c=0.02 c=0.01 c=0.02 J=0.1 – c=0.02 c=0.02 J=0.2 c=0.1

7 x=7.6u10–9 x=1.5u10–8 x=3.7u10–6 m=5.6u10–10 p=8.7u10–6 X=1.0u10–6 x=4.6u10–8 m=1.7u10–8 x=2.4u10–5

kJ 8 46.3 44.7 31.0 – 40.33 34.2 41.9 – 26.4

Free exergy of formation

Chemical element State Standard chem. exergy

' f Gq kJ/mol 9 –1 265.5 –1 395.9 –1 419.6 –920.5 0 –1 291.4 –1 262.5 –147.3 –1 919.5

bchq 10 s s,III s s g s,D s,D s s,D

kJ/mol 11 951.7 1196.6 720.4 827.5 40.33 965.5 944.3 339.2 1083.4

34 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

2.4. Enthalpy of devaluation In order to facilitate a comparison of the energy and exergy balance, the concept of enthalpy of devaluation has been introduced by Szargut [57, 59]. It expresses the chemical energy of a chemical compound calculated from the level of conventional reference substances commonly appearing in the environment. For many chemical elements, the same reference substances have been assumed as for the chemical exergy. The enthalpy of devaluation is a part of the enthalpy of a material stream. The values of the normal enthalpy of devaluation of pure chemical compounds are reported in table I at the end of this book (together with the values of the normal chemical exergy). In contrast to the chemical exergy, the enthalpy of devaluation of reference substances is always zero. Hence, the assumed reference species for the enthalpy of devaluation can be easily found in table I. For example, CaCO3 is the reference species of Ca; CaSO4˜2H2O (gypsum) is the reference species of S; CaCO3˜MgCO3 represents the reference species of Mg; and Ca3PO4 the reference species of P. Similarly NaCl is the reference species of Cl, and Na2SO4 is the reference species of Na. Table 2.3. Standard chemical exergies of solid substances, resulting from the data of reference species dissolved in seawater.

Chemical formula

Reference species of the elements

1

2

Au C

AuCl2–, Cl– HCO3–, O2 g

CaCO3 (calcite)

Ca2+, CO2 g, O2 g

CaCO3.MgCO3

2+

mol–1

Normal chemical exergy bchq kJ mol–1

3

4 0 0

2+

Ca , Mg , CO2 g, O2 g 2+

Ca2(PO4)2

CaCO3 s, Mg , CO2 g, O2 g Ca2+, F–, HPO42–, O2 g CaCO3 s, F–, HPO42–, O2 g Ca2+, HPO42–, O2 g

CoFe2O4

Co2+, Fe2O3 s, O2 g

CaF2.3Ca3(PO4)2

Free energy of formation ' f G q kJ

2+

Co3O4

Co , O2 g

Mg3Si4O10(OH)2

Mg2+, SiO2 s, O2 g, H2O g

15.4 404.5

–1 129.0

–0.4

–2 163.6

–7.8

–2 163.6 –12 985.3 –12 985.3 –3 885.3

8.9 –199.0 –32.0 –9.4

–1 032.6

–27.5

–772.8

–19.1

–5 543.0

–30.5

–465.2

–39.2

(talc) MnO2

Mn2+, O2 g

CALCULATION OF EXERGY 35

The enthalpy of devaluation of the compounds of C,H,O and N equals their lower heating value. The enthalpy of devaluation of the compounds of C,H,O,N and S is slightly higher than the heating value because the enthalpy of devaluation of S (725.42 MJ/kmol) is higher than its heating value (296.83 MJ/kmol).

2.5. Chemical exergy of organic fuels The chemical exergy of gaseous and liquid organic fuels can be determined by means of the tables of the standard chemical exergy or, according to Fan and Shieh [17], by means of the group contribution method (if the chemical formulae of their components are known). This is, however, not possible in the case of the majority of solid and liquid organic fuels consisting of complex solutions and mixtures of many (usually unknown) compounds. An exact calculation of the chemical exergy is not possible in this case. An approximate calculation method proposed by Styrylska and Szargut [58] is based on an analogy with the chemical exergy of pure organic substances. After calculating the chemical exergy of several organic substances, approximate formulae have been derived expressing the ratio E of their chemical exergy to the lower heating value as a function of the atomic ratio of the elements C, H, O, N, S. The most important formulae are: — For solid C,H,O,N compounds ȕ 1.0347  0.0140

ȕ

H O N  0.0968  0.0493 ; C C C

O  0.5 C

(2.18a)

1.044  0.016(H/C)  0.3493(O/C)[1 0.0531(H/C)]  0.0493(N/C) O ;  2 (2.19a) 1 0.4124(O/C) C

— For liquid C,H,O,S compounds ȕ 1.047  0.0154

H O S§ H·  0.0562  0.5904 ¨1  0.175 ¸ ; C C C© C¹

O 1 C

(2.20a)

For technical fuels it is more convenient to introduce the mass fractions into eqns (2.18a), (2.19a) and (2.20a). — For bituminous coal, lignite, coke and peat, it results from eqn (2.18a):

ȕ 1.0437  0.1896

zH 2 zC

 0.2499

zO2 zC

 0.0428

z N2 zC

(2.18b)

36 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

— For wood, from eqn (2.19a): 1.0412  0.2160 ȕ

zH 2 zC

zO 2 ª zH 1  0.7884 2 « zC ¬ zC zO2 1  0.3035 zC

 0.2499

zN2 º »  0.0450 z C ¼

(2.19b)

— For liquid technical fuels, from eqn (2.20a): ȕ 1.041  0.1728

zH 2 zC

 0.0432

zO2 zC

 0.2169

zH · zS § 1  2.0628 2 ¸ zC ¨© zC ¹

(2.20b)

The values resulting from eqns (2.18a)–(2.20a) can be applied only to the combustible portion of the technical fuel. The lower heating value of the combustible substance of a moist technical fuel results from the formula: H Lc

H L  Lzw

(2.21)

where HL = lower heating value of moist technical fuel L = enthalpy of water vaporization zw = mass fraction of water in moist fuel. Equations (2.18a) and (2.19a) have been determined without taking into account the sulfur content, because of the lack of sufficient thermo-chemical data. The chemical exergy of solid fuels containing sulfur can be approximately calculated under the additional assumption that sulfur appears as a free element: bch

( H L  Lzw )ȕ  (bchS  H S ) zS  bcha za  bchw z w

(2.22)

where zS, za = mass fraction of sulfur and ash bch S, bch a, bch w = standard chemical exergy of sulfur, ash and liquid water. For liquid fuels, the calculation proceeds as follows: bch

( H L  Lzw )ȕ  bchw z w

(2.23)

In cases in which an approximate value suffices, the values from table 2.4 can be used. The standard chemical exergy of pure organic substances can be calculated by means of the group contribution method. Per 1 mol of the considered substance: bchq

¦gb

i gi

i

(2.24)

CALCULATION OF EXERGY 37

Table 2.4. Ratio of the standard chemical exergy of organic fuels to the lower and higher heating value. Fuel

bch /HL

bch /HH

1.09 1.17 1.06 1.15 1.07 1.04 1.00 0.98

1.03 1.04 1.04 1.05 0.99 0.94 0.85 0.97

Hard coal Lignite Coke Wood Liquid HC-fuels Natural gas (high CH4) Coke-oven gas Blast-furnace gas

where gi = number of ith groups in the molecule under consideration bgi = contribution of the ith group to the standard chemical exergy of the compound. The values of bg i are reported in table II at the end of the book. Example 2.6. Calculation of the standard chemical exergy by means of the group contribution method. The molecule of liquid methyl-alcohol contains one CH3 group (with bg = 651.46 kJ/mol) and one OH group with bg = –33.97 kJ/mol). The standard chemical exergy of liquid methyl-alcohol is bchq

752.03  33.97

718.06 kJ/mol

According to [69] the exact value is 718.0 kJ/mol.

2.6. Chemical exergy of solutions The chemical exergy of solutions can be calculated by means of the partial specific exergies of the solutes: Bch

¦n b i

chi

(2.25a)

i

where ni = number of moles of the ith component bchi = partial molar chemical exergy of the component. The partial molar chemical exergy can be expressed in terms of the activity ai of the component: bchi

bchi  RT0 ln ai

(2.26)

38 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

In the case of ideal solutions, the activity equals the molar fraction, and the specific chemical exergy per 1 mole of solution may be expressed as follows: bch

¦ yb

i chi

i

 RT0 ¦ yi ln yi

(2.25b)

i

The last term expresses the exergy loss due to mixing. Table III at the end of the book contains the values of the standard chemical exergy of species dissolved in an ideal aqueous solution (molarity 1 mol/kg H2O).

2.7. Exergy of thermal radiation The exergy of the Earth’s infrared radiation, calculated with the assumption that the Earth's surface is black, amounts to zero, and represents the reference level for the calculation of the exergy of thermal radiation. The exergy flux of the black surface emission per unit of the absorbing surface, results from eqns (1.23) and (1.24), and from the scheme of the reversible prime mover presented in fig. 2.2. The absorbing surface of the prime mover, parallel to the black emitting surface has the area of 1m2. The prime mover absorbs the energy flux ơb and emits the energy flux ơb0 the “waste” radiation of a black surface with the environmental temperature T0. The energy balance of the prime mover is closed by the stream of zero-exergy heat q0 delivered from the environment. The work performed by the prime mover expresses the exergy flux of the black surface radiation: bb

eb  q0  eb 0

(2.27)

From the condition of reversibility of the considered prime mover and from eqn (1.24) it results: sb 

q T0

sb 0

or

4 eb q0  3 T T0

4 eb 0 3 T0

Figure 2.2: Reversible prime mover fed by a black radiation.

(2.28)

CALCULATION OF EXERGY 39

From eqns (2.27) and (2.28) it results: bb eb

1

4 T0 1 § T0 ·  3 T 3 ¨© T ¸¹

4

(2.29)

The same result was obtained by Petela [40] with a different method. eqn (2.29) does not depend on the emissivity of the environment. Equation (2.29) may be also applied to gray surfaces if their emissivity does not depend on the wavelength. When considering the exergy of the total radiation flux, not only the emission but also the reflected radiation should be taken into account. In exact calculations, the distribution of the radiation brightness over the wavelength should be known. Usually, the actual radiation flux may be replaced by the emission of a black surface. The equivalent temperature of this black surface can be calculated from the condition of the equality of the energy fluxes. The spectral distribution of solar radiation on the Earth’s surface depends on the composition of the atmosphere and the path length through the atmosphere determined by the position of the Sun. Example 2.7. Exergy of the solar radiation. Assuming the mean temperature of the Earth’s surface as 300K we obtain from eqn (2.29) the ratio of exergy to energy of solar radiation above the atmosphere: 0.9311. A more exact value, taking into account the spectral distribution of solar radiation above the atmosphere, is 0.9327 [40]. At the Earth’s surface the discussed ratio depends on the ambient terrestrial temperature, on the position of the Sun and on the composition of the atmosphere. At T0=289 K we obtain from eqn (2.29) the ratio 0.9336. Hence, the exergy flow of solar radiation immediately absorbed by the Earth’s surface amounts to 73,100 TW. Example 2.8. Exergy balance of the Earth’s surface and atmosphere. Figure 2.3 presents the exergy flows appearing above the Earth’s surface and connected with the absorption of solar radiation and the emission of infrared radiation [88]. The mean exergy flow of solar radiation reaching the external layers of the atmosphere amounts to 162,400 TW. The reflection of solar radiation from the upper layers of the atmosphere (48,700 TW) does not evoke any exergy losses. Some part of the solar exergy is being destructed in the atmosphere (32,400 TW) due to the absorption of solar radiation and re-emission of infrared radiation. This exergy loss has not been classified to the NLUE because the considered phenomenon occurs outside the zone of human existence. The exergy flow of solar radiation heating the continents and oceans amounts to 43,200 TW. A prevailing part of this radiation is transformed into infrared radiation of the Earth’s surface, partially transmitted to the cosmic space and partially absorbed by the atmosphere and re-emitted to the Earth’s surface, causing the greenhouse effect. Only a small part of the radiation heating the Earth’s surface (about 370 TW [27]) is transformed into mechanical exergy of

40 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Figure 2.3: Distribution of exergy flows above the Earth’s surface.

the wind, waves and sea-currents. Some part causes a weathering of rocks, producing components necessary for the vegetation of plants [52]. The exergy flow used for evaporation is about 38,100 TW. A small part of this exergy is transformed into the potential exergy of water droplets or ice particles contained in clouds. The mass of evaporated water (assuming a mean temperature of 25qC) amounts to 15u109 kg/s. Assuming the mean height of clouds as 2000m, the potential exergy power of the clouds is 300 TW. A prevailing part of this exergy is destroyed by rain- and snowfalls. Only a very small part (5 TW) is transformed into the potential exergy of rivers and lakes [27]. The rivers and lakes are also a renewable source of fresh water, necessary for human life (see point 6.6.2). The radiation absorbed by the vegetation of plants (only the active part) has an energy about 40 TW [27], (exergy about 37 TW). Only a small part of it is transformed into the chemical exergy of plants (about 2.5 TW of energy and 2.9 TW of exergy). About 1 TW of the chemical exergy of plants is consumed by humans. The left part of fig. 2.3 represents exergy losses resulting from the radiation exchange between the Earth and cosmic space. This space emits the long-wave

CALCULATION OF EXERGY 41

relict radiation. Its spectral composition represents a very low temperature 2.73K, and reaches the Earth continuously from all directions. It is worth remembering that at a temperature lower than the ambient one the flow directions of energy and exergy are opposite to each other. Therefore, the infrared emission of the Earth (whose exergy equals zero) causes an inflow of exergy, which may be interpreted as exergy flow of the relict radiation of the space. The positive exergy of this cold radiation results from the possibility to build a prime mover using the ambient thermal radiation as a hot source and the cold radiation as a cold sink of heat. The exergy flow of the cold relict radiation is not small. Its energy flow is very small, and amounts to 3.15u10–6 W/m2. However, the ratio of exergy to energy is very high. From eqn (2.29) at T0 = 289 K it results that: br er

41.9 u 106 ,

br

147 W/m 2

The total flow of exergy of relict radiation reaching the Earth’s surface results from the area of the Earth’s surface. Introducing the main radius of the Earth we obtain: B r

4S u 63674402 u 147

74900 TW

The exergy flow of the relict radiation is 1.02 times larger than the exergy flow of solar radiation absorbed by the Earth’s surface. The exergy of the relict radiation does not produce any lasting energy effect on the Earth’s surface. Only the temperature of the upper layers of the atmosphere drops. The secondary radiation of these layers also has a positive exergy (about 5200 TW) generated thanks to the destruction of the exergy of relict radiation. Only a small part of the exergy flow of the mentioned kinds of cold radiation is absorbed by the Earth's surface. A prevailing part is destroyed outside the zone of human existence. From the balance of natural losses of utilizable exergy it results that only 7% of the considered losses (5400 TW) are due to the destruction of exergy of the relict radiation and cold radiation of the upper layers of the atmosphere. The exergy flows of solar radiation have also been analyzed by Wall and Gong [106], but the exergy inflow resulting from cold space radiation has not been taken into account.

2.8. Nuclear exergy The energy of nuclear fuels is of very high quality because it intrinsically corresponds to a very high temperature. Hence, it can be assumed that the exergy of nuclear raw materials equals the available energy. It can be determined from the concentration of fissionable and fertile components in the raw material and from the energy of fission. The exergy efficiency of thermal nuclear power plants should take into account only the exergy of fission of fissionable elements. The mean exergy of

42 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

fission per nucleus of 235U, 233U and 239Pu amounts to about 200MeV. Therefore, the specific exergy (per kg) of the fissionable element can be expressed as follows: b=19.3  1012

1 M

’qV’Ž

(2.30)

where M is the molecular mass of the fissionable isotope. Natural uranium contains 0.72% of fissionable 235U. The nuclear exergy of natural uranium resulting from eqn (2.30) amounts to 584 000 MJ/kg (whereas its chemical exergy is only 5 MJ/kg).

2.9. Exergy losses in thermal plants 2.9.1. Compressor An isothermal compression of a perfect gas does not change its enthalpy nor its internal energy. The amount of driving energy equals the amount of heat rejected to the environment. Hence, the energy efficiency of an isothermal compressor is equal to zero. The energy analysis cannot characterize the thermodynamic imperfection of a compressor cooled during the compression of the fluid. Only an exergy analysis can correctly represent the useful effect of the compression. The compression of a gas or vapour can be realized to increase the pressure according to a specified design value, to overcome the flow resistance or to increase the temperature (for example in a gas turbine cycle). In the first two cases—only the isothermal increase of exergy represents the useful effect of the process. When analyzing the compression of a perfect gas, the following formula expresses the exergy efficiency of the compressor:

ηBT 

& 0 ln p2 p1 nRT | W& D |

(2.31)

where n& = molar flow rate of the compressed fluid p1, p2 = initial and final pressure of the fluid | W& D | = driving mechanical power. Thus, the exergy component of the compressed fluid resulting from its increased temperature, represents the external exergy loss of the system. In the third case, the exergy efficiency should take into account also the temperature increase after the compression. For a perfect gas:

ηB 

n&[c p (T2 − T0 − T0 ln T2 T0 )  RT0 ln p2 p1 ] | W& D |

(2.32)

Example 2.9. Exergy balance of a two-stage piston compressor of air with an interstage and cylinder cooling.

CALCULATION OF EXERGY 43

The following parameters result from thermal measurements: ambient air: p0 = 0.1 MPa, T0 = 291 K, relative humidity M0 = 78%, air before the first stage: p1 = 99 kPa, T1 = 291 K, V1 = 0.92 m3/s, air after the first stage: p2 = 0.27 MPa, T2 = 368 K, air before the second stage: p3 = 0.265 MPa, T3 = 300 K, M3 = 100%, air after the second stage: p4 = 0.82 MPa, T4 = 393 K, cooling water in the first stage: G I = 2.15 kg/s, TI = 292 K, TII = 296 K, cooling water in the interstage cooler: G = 1.06 kg/s, TIII = 292 K, TIV = 308 K, c

cooling water in the second stage: G II = 2.06 kg/s, TV = 292 K, TVI = 297 K, driving electric power | N el | = 330 kW, heat losses of the compressor cylinders immediately to the environment amount to 8% of the heat transferred to the cooling water, efficiency of the electric motor KM= 0.92. The assumption has been made that after the interstage cooler the compressed air does not contain any droplets of liquid water. The condensate of air humidity flowing out of the interstage cooler has a temperature equal to that of the cooled air. Exergy loss in the electric motor:

G B M

(1  KM ) N el

26.4 kW

26.4 kW.

Mechanical exergy loss in the compressor:

G B m

KM N el  Wi

where the internal driving power of the compressor | Wi | results from the energy balance of both cylinders: | Wi | H 2  H 1  H 4  H 3  1.08('H wI  'H wII )

where 'H wI , 'H wII = enthalpy increase of the cooling water stream in the first and second stage. The flow rate of humid compressed air is 0.03753 kmol/s. The exergy efficiency of the compressor set should be calculated according to eqn (2.31). The internal exergy losses result from the exergy balances of particular links of the considered system. The external exergy loss comprises the physical exergy of warm cooling water, the exergy of condensate formed in the interstage cooler, and a component of exergy of the compressed air resulting from its elevated temperature (only the increase of pressure is the aim of compression in the con-sidered case). The relative exergy losses are related to the driving electric power:

44 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

exergy efficiency of the system exergy loss in the electrical motor mechanical exergy loss in the compressor internal exergy loss in the first stage internal exergy loss in the interstage cooler internal exergy loss in the second stage external exergy losses

57.90% 8.00% 9.55% 6.76% 2.76% 8.97% 6.06%.

2.9.2. Steam or water boiler The energy efficiency of a steam or water boiler expresses the ratio of the enthalpy increase of the heated fluid to the chemical energy of fuel. Its value is high, and amounts to 70–95%. However, the quality of chemical energy is much higher than that of the physical energy of the produced hot fluid. Therefore, the exergy efficiency of a boiler is much lower than the energy efficiency. This difference results from two main causes: (1) the combustion process is irreversible (see section 1.3.5.), (2) the irreversible heat transfer between the combustion gases and the heated fluid causes large exergy losses. The exergy efficiency of the steam or water boiler can be expressed as follows:

KB

G [h2  h1  T0 ( s2  s2 )] B F

G (h2  h1 )(1  T0 Tm ) D E ch

KE D

§ T0 · ¨©1  T ¸¹ m

(2.33)

The flow diagram of the considered exergy balance is presented in fig. 2.4.

Figure 2.4: Exergy balance of a two-stage air compressor.

CALCULATION OF EXERGY 45

where

D = ratio of the chemical exergy of fuel to its chemical energy KE = energy efficiency of the boiler Tm = mean thermodynamic temperature of the heated fluid during heat absorption Tm

h2  h1 s2  s1

(2.34)

In Europe, the lower heating value of fuel is used to determine the energy efficiency of a boiler. In this case, the ratio D in eqn (2.33) should be related to the lower heating value. Exergy losses in a combustion process can be lowered by increasing the combustion temperature. However, simultaneously the exergy loss increases due to the irreversible heat transfer between the combustion gases and the heated fluid. Therefore, the combustion air is preheated only to increase the energy efficiency of the boiler by lowering the outlet temperature of the combustion gases. An improvement of the exergy efficiency of the boiler can be attained by increasing the mean thermodynamic temperature of the heated fluid. This method is applied in steam power plants (increasing the temperature of live steam, secondary superheating). In water boilers for district heating systems, an increase of the mean temperature of the heated water would not be expedient, because it would increase the exergy losses in all subsequent heat transfer processes downstream. The exergy efficiency of steam boilers usually amounts to 35–50%, and that of water boilers to 15–25%. Example 2.10. Exergy balance of a fluidized-bed steam boiler. A fluidized-bed steam boiler fed with hard coal is schematically presented in fig. 2.5. The combustion air is preheated by means of bleed steam taken from the steam turbine. The combustion air preheaters and their fans are located outside the system boundary. Therefore, limestone is delivered to the boiler to absorb the sulfur oxides formed in the combustion room. The measurement data determining the energy and exergy balance of the boiler are as follows: ambient air: p0 = 0.0994 MPa, T0 = 298 K, relative humidity M0 = 60%, primary combustion air: nap = 2.137 kmol/s, pap = 0.1178 MPa, Tap = 312 K, secondary combustion air: nas = 0.831 kmol/s, pas = 0.1085 MPa, Tap = 302 K, steam: G = 72.91 kg/s, pst = 13.72 MPa, Tst = 811.5 K, st

feed water: Tfw = 490.3 K, pfw = 14.89 MPa, fuel: lower heating value Hl = 19 106 kJ/kg, chemical composition: c = 0.5019, h = 0.035, s = 0.0139, o = 0.0950, n = 0.0072, w = 0.229, a (ash) = 0.118, ash from the boiler: total amount per fuel unit at = 0.1933 kg/kg flying ash: fraction in the total ash gfa = 0.88, content of combustibles cfa = 0.0087, content of CaO gCaO fa = 0.2209,

46 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Figure 2.5: Scheme of a fluidized-bed steam boiler fed with hard coal.

bottom ash: fraction in the total ash gba = 0.12, content of combustibles cba = 0.0087, content of CaO gCaO ba = 0.4520, temperature Tba = 1107.8 K, limestone: flow rate G ls = 0.795 kg/s, combustion gases: temperature Tcg = 400.5 K, oxygen molar content (related to dry gas) [O2] = 0.0539 immediate heat loss: H = 0.0061 (related to the chemical energy of fuel), fan of fluidization air: driving power E el = 260 kW, electro-mechanical efficiency Kme = 0.92. The energy and exergy balance of the considered boiler is presented in fig. 2.6. The energy efficiency of the boiler is very high (93.6%), however, the exergy efficiency is considerably smaller; it amounts to 43.3%. The internal irreversible phenomena do not influence the energy balance of the boiler, they influence considerably, however, the exergy balance. The relative internal exergy loss taking into account the irreversibility of combustion and heat transfer between the combustion gases and water, and the immediate heat loss to the environment, amounts to 52.7%. The external energy- and exergy loss with combustion gases is relatively small. The external exergy loss of combustion gases also takes into account the chemical exergy resulting from the difference of the chemical composition of the combustion gases and ambient air, and depends mainly on the heightened concentration of CO2. 2.9.3. Thermal power plant fed by a chemical fuel The exergy efficiency of a thermal power plant fed by a chemical fuel is slightly smaller than its energy efficiency because the chemical exergy of the fuel is slightly higher than its heating value:

CALCULATION OF EXERGY 47

Figure 2.6: Energy (a) and exergy (b) balance of the fluidized-bed steam boiler.

ȘB

N BchF

N ĮE chF

KE Į

(2.35)

However, the distribution of losses in the exergy balance is different from that in the energy balance. The main energy loss in a steam power plant appears in the condenser where the expanded vapour transfers the low quality waste heat to the cooling water. In a gas-turbine power plant, the main energy loss results from the ejection of the expanded hot gas into the environment. In both cases, the main exergy loss is connected with the combustion process.

48 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

In a steam power plant the boiler is responsible for the low efficiency of the plant. In a gas-turbine power plant the high exergy loss in the combustion chamber results mainly from lowering the temperature of the working fluid before the gas turbine (the supply of excess air, the injection of steam or liquid water). The gas-turbine power plant provides thus better possibilities to improve the efficiency. The irreversible heat transfer between the combustion gases and the working fluid is eliminated. The maximum temperature of the working fluid before the turbine is less limited. However, the irreversibility of the rejection of waste heat to the environment is greater in the gas turbine plant because it is not possible to realize this process isothermally. The exergy losses in the boiler are much greater than the energy ones. The heat stream rejected to the environment is very large, but its quality is very low. The exergy balance indicates the causes of the large amount of waste heat. 2.9.4. Compression refrigerator and heat pump The main difference between the energy and exergy balance of a refrigerator lies in the different direction of the energy and exergy flow in the refrigerated cell. The extraction of heat from this cell means an increase of its exergy. Example 2.11. Energy and exergy balance of a simplified steam power plant Figure 2.7 presents a simplified scheme of a steam power plant. In fig. 2.8 the energy and exergy balances of this plant have been compared..

Figure 2.7: Simplified scheme of a steam power plant.

CALCULATION OF EXERGY 49

Figure 2.8: Energy (a) and exergy (b) balances of a steam power plant with a simplified scheme.

Usually, the compression refrigerator together with the electrical motors driving the main machine and the auxiliary equipment (fans and pumps) is included in the balance. The exergy efficiency of this complex unit may be expressed as a ratio of real COP to COPmax of the Carnot-refrigerator:

KB

Qr

T0  Tr 1 Tr EelD

COP COPmax

(2.36)

50 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

where Qr = heat extracted from the refrigerated cell. Tr = absolute temperature of the refrigerated cell. EelD = total amount of driving electricity. The largest exergy losses in the refrigerator result from irreversible heat transfer. Hence, intermediate heat carriers should be avoided. One of the advantageous working fluids in compression refrigerators is NH3. However, ammonia is explosive and poisonous. Therefore, it cannot be applied without an intermediate cold carrier between the machine and the refrigerated cell. Throttling of the working fluid after condensation also causes an exergy loss. Attempts to introduce a non-throttling cycle were not entirely successful [79]. Example 2.12. Energy and exergy balance of an ammonia compression refrigerator. Figure 2.9 presents a scheme of an ammonia compression refrigerator. Table 2.5 contains the parameters of working fluid and brine. Figure 2.10 presents the energy and exergy balances of the considered installation. The assumed ambient temperature is 20qC, and the temperature in the refrigerated room is –1qC. The mechanical efficiency of the compressor is Km = 0.83, the efficiency of the electrical motor KM = 0.9. The exergy efficiency of the considered complex is 23.6%. Its value could be increased by the application of a non-hazardous working fluid and elimination of the intermediate heat carrier (brine). The scheme of a compression heat pump is similar to that of a compression refrigerator. However, usually the ‘bottom’ heat is not extracted immediately from the ambient atmosphere. The designer looks after a bottom heat source warmer than the ambient air, to improve the COP. Thus, a formula analogical to (1.10) should be applied. That formula is based upon the assumption Qu = WD + Qb. It is not satisfied

Figure 2.9: Scheme of an ammonia compression refrigerator.

CALCULATION OF EXERGY 51

Table 2.5: Parameters of ammonia and brine in a refrigerator cycle Location 1 2 3 4 b1 b2

Flow rate kg/s 0.082 0.082 0.082 0.082 16.3 16.3

Pressure MPa

Temperature qC

0.27 1.1 1.1 0.27 – –

–10 119 25 –12 –5 –7

when we consider the complex of a heat pump together with the electrical motors driving the main machine and the auxiliary equipment. In this case the condition Qu < Qb +EelD appears because the heat losses of electrical motors might not be transferred to the carrier of useful heat. The exergy efficiency of the complex of

Figure 2.10: Energy and exergy balance of an ammonia compression refrigerator. C — compressor, Cn — condenser, T — throttling valve, E — evaporator, Cc — cooling chamber, MElL — mechanical and electrical losses, InL — internal exergy losses.

52 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

a compression heat pump may be based upon the assumption that only the mechanical losses in the machines are absorbed by the carrier of useful heat: ȘB

COP T T 1  (COP  Q M ) b 0 Tb

Tu  T0 Tu

(2.37)

where COP = Qu/Eel D. The exergy efficiency of a heat pump decreases when the ambient temperature grows. The exergy efficiency is usually considerably smaller than that of the compression refrigerator. It results mainly from greater exergy losses due to the irreversible heat transfer. Additional exergy losses are introduced by the intermediate heat carrier between the condenser and the heated room. A large temperature difference between the heating fluid and the heated room lowers also the exergy efficiency. Heat pumps are used for heating dwelling-houses. Usually the temperature of the heating fluid is 40–50qC. A considerable improvement of the exergy efficiency might be obtained by the introduction of a low-exergy internal installation (floor or wall heating, low-temperature radiators) [83]. Example 2.13. Figure 2.11 presents the exergy efficiency of a compression heat pump extracting the bottom heat from the ground with a constant temperature of 8qC and delivering the useful heat to the room with a temperature of 20qC. Constant values COP = 3.5 and KM = 0.9 have been assumed. At a constant value of COP the exergy efficiency depends on the ambient temperature. 2.9.5. Sulfuric acid plant Figure 2.12 presents the scheme of a plant producing sulfuric acid from liquid sulfur. Figure 2.13 informs about the internal and external exergy losses appearing in the particular parts of the plant (Rasheva and Atanasova [45]).

Figure 2.11: Exergy efficiency of a compression heat pump.

CALCULATION OF EXERGY 53

Figure 2.12: Scheme of a chemical plant producing sulfuric acid from liquid sulfur, after Rasheva and Atanasowa [45].

The external exergy losses result from the discharge of waste products to the environment. The largest external losses appear in the oven for burning sulfur and in the acid cooler. The exergy losses in the oven can be decreased by a better preheating of the combustion air (the actual temperature of the incoming air is only 55qC), and by increasing the pressure in the oven in order to apply the gas turbine. Internal exergy losses in the converter (where SO2 is oxidized to SO3) are difficult to avoid. Large internal exergy losses in evaporators result from large temperature differences between the hot gases and evaporating water. The pressure of water might be increased to produce steam with higher parameters. High exergy losses in the absorbers might be reduced by introducing a hot absorption system. Hot absorption water can be produced in the coolers of acid. The exergy efficiency of the investigated plant amounted to 55%. According to the results presented in fig. 2.13, following main improvment possibilities of the investigated process may be indicated [45]: — the internal exergy losses in the oven (1) may be decreased by increasing the temperature of the incoming combustion air; this would result, however, in increasing the temperature of the outlet gases; in order not to increase the internal exergy losses in the evaporator (10) it would be necessary to increase simul-taneously the pressure of the saturated steam produced in the evaporator (10); — the external exergy loss in the oven (1) may decreased by combining the process with a gas turbine utilizing the high temperature of the outlet gases; this would require, however, an increase of the pressure in the oven; — the internal exergy losses in the second evaporator (11) may be reduced by increasing the parameters of the produced steam.

54 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Figure 2.13: Exergy losses in a chemical plant producing sulfuric acid, after Rasheva and Atanasova [45]. 1 - oven for burning sulfur, 2 - converter, 3 - oleum absorber, 4 intermediate monohydrate absorber, 5 - final monohydrate absorber, 6 - oleum cooler, 7 - acid cooler, 8 - superheater, 9 - economizers, 10 and 11 - evaporators, 12 - heat exchangers.

— the internal exergy losses in the absorber (5) may be reduced by the use of the hot absorption system, used for heating the supplied water.

Exercises 2.1. Calculate the temperature-dependent and pressure-dependent component of the specific physical exergy of combustion gases containing 71.8% N2, 8.7% CO2, 17.5% H2O and 2% O2 (mol fraction). The parameters of combustion gases are p = 0.42 MPa, T = 1100 K. The specific physical enthalpy and entropy values are cited in table 2.6. Compare the exact value of the temperature-dependent component with the approximate value calculated by means of the mean specific heat capacity. 2.2. Calculate the specific chemical exergy of combustion gases containing 71.8% N2, 8.7% CO2, 17.5% H2O and 2% O2 (mol percent), assuming standard environmental parameters. Calculate the ratio of chemical exergy to the temperature-dependent component of the physical exergy determined in exercise 2.1. 2.3. Calculate the specific physical exergy of steam with the parameters p = 14 MPa, T = 560qC. The ambient parameters are p0 = 98 kPa, T0 = 293 K. 2.4. Calculate the specific physical exergy of liquid ammonia with the temperature of –15qC. Assume standard ambient parameters. 2.5. Calculate the mol fraction of SO2 in atmospheric air assuming standard

CALCULATION OF EXERGY 55

Table 2.6: Parameters of the components of combustion gases Component

N2 CO2 H2O O2

Amount of moles

Specific physical enthalpy J/mol

Specific physical entropy J/(mol K)

Specific heat capacity J/(mol K)

0.718 0.087 0.175 0.020

24 671 38 776 30 118 26 215

39.592 60.587 47.906 41.736

30.768 48.358 37.560 32.693

environmental parameters and the thermodynamic equilibrium between SO2 and the reference species. The standard normal exergy of SO2 is cited in table I. 2.6. Calculate the standard chemical exergy of Ca2SiO4 if the standard free energy of formation is known, 'f G0 = 2191.6 kJ/mol. The standard values of the chemical exergy of the elements are cited in table I, at the end of this book. 2.7. Calculate the standard chemical exergy of the element B from the data concerning the reference species cited in table 2.2. 2.8. Calculate the standard free energy increase for the reaction CaO + SO2 + 0.5 O2 o CaSO4 using the values of standard chemical exergy cited in table I. 2.9. Calculate the maximum work of the reaction CaCO3 + N2 + 2.5 O2 o Ca(NO3)2 + CO2 assuming the normal environmental temperature, standard partial pressures of the gaseous substances, pure CaCO3 and Ca(NO3)2. Notice that by means of this reaction work could be produced by means of the common components of the environment (CaCO3, O2, N2, CO2), if the irreversibility of the cited reaction could be sufficiently reduced. 2.10. Calculate the specific chemical exergy of bituminous coal with the following composition (mass percent): zC = 55%, zH2 = 4%, zO2 = 8%, z N2 = 1%, zS = 2%, zw = 10%, za = 20%. The lower heating value is 24 880 kJ/kg. Assume that ash contains only SiO2. Use standard ambient parameters. 2.11. Calculate the standard chemical exergy of liquid freon CHFCl2 by means of the contribution method of chemical groups (see table II, Appendix). 2.12. Calculate the standard chemical exergy of steel with the following composition (mass percent): zFe = 98%, zC = 0.6%, zMn = 0.7%, zS = 0.3%, zP = 0.1%, zSi = 0.3%. Assume that carbon appears in the form of Fe3C and Mn3C. 2.13. Calculate the exergy efficiency of a coal-fueled boiler heating the network water from 50qC to 110qC. The energy efficiency of the boiler amounts to 0.86. The

56 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

ambient temperature is –10qC. Take the ratio of the chemical exergy to the lower heating value of coal from table 2.4. 2.14. Calculate the exergy efficiency of a coal-fueled boiler producing steam with the parameters p = 14 MPa, T = 560qC. The feeding water has a temperature of 240qC. The energy efficiency of the boiler amounts to 0.93. The ambient temperature is 10qC. Take the ratio of the chemical exergy to the lower heating value of coal from table 2.4. 2.15. An absorption heat pump operates as an auxiliary part of a geothermal heating system. Calculate its exergy efficiency. The driving heat is delivered by means of hot water with initial and final temperature Tw1 = 160qC, Tw2 = 80qC. The energy efficiency of the considered heat pump is characterized by COP = 1.6. The bottom heat is extracted from the preliminarily utilized geothermal water cooled down in the heat pump from 42 to 27qC. The useful heat is delivered to the network water to heat it fromss 40 to 78qC. The ambient temperature is –5qC.

Chapter 3 Cumulative exergy consumption and partial exergy losses 3.1. Definition of cumulative exergy consumption (CExC) Consumption of exergy connected with the fabrication of some considered product appears not only in the plant manufacturing the product, but also in all plants delivering semi-finished products and raw materials for the final production process. The total consumption of the exergy of natural resources connected with the fabrication of the condidered product and appearing in all the links of the network of production processes, has been called cumulative consumption of exergy CExC [67]. The analysis of the cumulative consumption of exergy can be considered as a development of the analysis of the cumulative consumption of primary energy. The definition of the cumulative consumption of energy or exergy is similar to that of the economic cost. Therefore, the first authors analyzing the cumulative consumption of energy introduced the term energy cost [11–13]. Monographic books presenting the problem of cumulative consumption of energy have been elaborated [6, 8]. An essential difference between the analysis of the cumulative consumption of energy and exergy should be stressed. The first one does not give any possibility of determining the influence of the imperfection of particular partial processes on the final result of the analysis, whereas the second enables us to determine partial exergy losses appearing in particular links of the considered system and so indicates the possibilities for the improvement of the chain of constituent processes. The CExC-values can be calculated separately for each particular kind of primary exergy or may comprise the sum of values of the non-renewable primary exergy or the sum of all the values of the primary exergy. Usually the second method is applied. When calculating the partial exergy losses appearing in the particular links of the energo-technological network, the calculations of CExC should comprise all the natural resources of exergy, the renewable and nonrenewable ones, because it is impossible to distinguish which exergy losses are

58 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

connected with the renewable and separately with the non-renewable exergy resources. CExC can be calculated by means of process analysis (sequence method) or by means of a set of balance equations. Process analysis (sequence method) begins in the final link of the fabrication of the considered product and runs through the processes of producing semi-finished products, the fabrication of machines and equipment until the extraction of raw materials from natural resources. The sequence method enables us to analyze only one product, but can be applied if the connections between the considered links of the technological network are weak (the consumption of the considered product in preceding links is small). The method of balance equations is more general. It requires more work because need for the formulation and solution of the set of balance equations. The balance equations based upon the statement that the CExC-value burdening the useful products of the process results from the sum of CExCvalues characterizing the delivered materials, semi-finished products and energy carriers.

3.2. The problem of CExC of the human work Some authors try to introduce the CExC of human work into the calculation of CExC of useful products [43]. However, the CExC of human work results from the total consumption of the primary exergy in the considered region. The sum of all the values of CExC burdening the human work equals the total consumption of the primary exergy. When calculating the CExC values burdening particular useful products, the CExC values burdening the used materials, semi-finished products and energy carriers are additionally introduced into the calculations. Hence, the consumption of primary exergy appears twice in the calculations. Szargut [76] elaborated a simple proof that it is not necessary to take into account the human labor when calculating the CExC-value because it would denote a double introduction of the same quantity. The mentioned statement results, for example, from the analysis of a simple system as presented in fig. 3.1. The system contains coal mines, industry and the consumption sector. Some part Ck of the produced coal is consumed immediately. The remaining part Ci is used in industrial processes. The consumption of industrial products in coal mines is Ic. In order to simplify the mathematical proof, the cumulative con-sumption of all the kinds of the primary exergy has been taken into account. If we introduce the cumulative consumption of exergy burdening the human work into the balance equations, we obtain for the analyzed system: bc*

bc  ( I c C )bi*  lc r , bi*

(Ci / I )bc*  li r ,

(3.1) (3.2)

where bc* , bi* = CExC-value burdening a unit of coal and of the industrial product bc = specific exergy of coal lc,li = specific consumption of human labor in coal mines and industry

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 59

Figure 3.1: Scheme of a simplified system. r = CExC-value burdening a unit of human labor Ci,Ic = consumption of coal in industry and of industrial products in coal mines C,I = total production of coal and industrial products. From eqns (3.1) and (3.2) it results: bc*

bc  > lc  ( I c / C )li @ r

(3.3)

1  ( I c / I )(Ci / C )

The total consumption of primary exergy can be expressed as: bi* I k  bc*Ck

bi* ( I  I c )  bc* (C  Ci )

Cbc  ( Lc  Li )r

(3.4)

where: Lc

lc C ,

Li

li I .

(3.5)

On the other hand, it is evident that the total consumption of primary exergy is Cbc. So the balance of CExC is closed, if the quantity r does not appear in eqns (3.1) and (3.2). Hence, the mentioned statement is correct.

3.3. A set of input-output equations The balance equation of CExC is based upon the statement that the CExC-value of the product of some considered link results from the CExC-values burdening the semi-finished products and by-product, and from the exergy of natural resources extracted from nature in the considered link. The set of balance equations takes the form [67]: b*j  ¦ f uj bu* u

¦a b ¦a * ij i

i

* uj u

u

b  Ej

(3.6)

60 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

where b*j , bi* , bu* = specific CExC-value burdening the major product of the jth and ith process, and the uth by-product aji,auj = coefficient of consumption of the ith semi-finished product and uth by-product per unit of the jth product fuj = coefficient of the by-production of the uth by-product per unit of the jth major product Ej = exergy of the natural resources extracted from nature in the jth process, per unit of the jth product. The terms representing the production and consumption of the by-products can be replaced by terms containing CExC of the ith major product substituted by the uth by-product, according to the principle of avoided expenditures (the by-product should be burdened with the consumption of semi-finished products and energy carriers avoided in the substituted process thanks to the utilization of the considered by-product) f uj

fij siu

, auj

aij siu

, ȡu

ȡi siu

(3.7)

where siu denotes the replacement ratio in units of the ith major product replaced by a unit of the uth by-product. After introducing (3.7), eqn (3.6) takes the form: b*j  ¦ ( fij  aij )bi*

ȕj

(3.8)

i

The consumption of machines and installations can be also taken into account in eqn (3.8), using their consumption coefficient: aij

mij

W iG j

(1  ui )

(3.9)

= number of ith machines or installations participating simultaneously in the fabrication of the jth product Gj = annual production of the jth product Wi = life time of the ith machine or installation ui = fraction of the value bi* which might be recovered by the use of secondary raw materials remaining after the wear of the considered machine or installation [15, 16].

where mi

3.4. Cumulative exergy efficiency When considering the production of materials or energy carriers, the cumulative exergy efficiency CExE (or the cumulative degree of perfection CDP) of the total

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 61

chain of processes leading from the natural resources to the considered product can be expressed by means of the ratio of useful exergy to the cumulative consumption of driving exergy: Ș*B

bu bD*

(3.10)

The cumulative consumption of exergy should in this case take into account all the kinds of primary exergy (non-renewable and the renewable ones). CExC should be calculated in a globally. Exemplary values of CExE are cited in table 3.1. CExE can be expressed in the form of the product of component efficiencies when the chain of production processes leading to the considered product does not contain strong feedbacks. Weak feedbacks can be taken into account by means of an iterative procedure. Example 3.1. The immediate exergy efficiency of some power plant fed with hard coal amounts to KBel = 0.36. The CExE value should additionally take into account the cumulative efficiency of fuel delivery (extraction and transportation) Ș*df , the efficiency of transformation and transmission of electricity Ktr and the cumulative exergy investment efficiency Ș*B inv (taking into account the consumption of materials and energy burdening the construction of the power plant. The consumption of electricity connected with the delivery of fuel and the construction of the plant is not high; thus the product of partial efficiency values can be applied: nB* el

ȘB el Ș*df Ștr nB* inv

(3.11)

The cumulative efficiency of investments can be expressed as follows: Ș*B inv

B *f IJt * B *f IJt  ¦ Binv i

(3.12)

i

where B *f = rated cumulative consumption of fuel exergy, per time unit Wt = total operation time of the plant with rated power B = cumulative consumption of the primary exergy burdening the con * inv i

struction of the particular parts of the plant. Usually the investment efficiency is neglected when calculating CexE. Assuming in the considered example Ș*df = 0.93, Ktr= 0.9, Ș*B inv = 0.96 we obtain Ș*B el = 0.29. CExE can be calculated also for the by-products; according to the principle

62 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table 3.1: Exemplary values of CExC and CexE. Material Exergy b

Units in Columns 2, 4

CExC

CExE %

Notes

1

2

3

4

5

6

Bituminous coal Lignite

1.09 HL 1.17 HL

MJ/kg MJ/kg

1.12 HL 1.20 HL

93.1 92.3

Metallurgical Coke Natural gas Gasoline Electricity

1.06 HL

MJ/kg

1.35 HL

74.2

1.04 HL 1.07 HL 1

MJ/kmol MJ/kg J/J

1.06 HL 1.32 HL 3.40

91.5 80.8 29.4

Name

Oxygen for Metallurgy Pig iron Converter steel Electric steel Steel pipes Hot-rolled steel

3.4

kJ/mol

237

1.4

8.75 8.04 8.04 7.04 7.04

MJ/kg MJ/kg MJ/kg MJ/kg MJ/kg

28.6 26.7 22.8 58.7 37.05

30.6 30.1 35.4 16.0 19.0

Cold-rolled steel

7.04

MJ/kg

47.6

14.8

Copper

2.11

MJ/kg

147.4

1.4

Zinc

5.19

MJ/kg

125.8

4.1

Aluminium

32.9

MJ/kg

250.2

13.2

Sulfur Ammonia

19.01 19.84

MJ/kg MJ/kg

30.2 48.2

62.9 41.1

Sulfuric acid Methanol Glass Cellulose

1.67 22.41 0.174 16.45

MJ/kg MJ/kg MJ/kg MJ/kg

11.1 73.1 33.4 60.0

15.0 30.7 0.5 27.4

high CH4 content from bituminous coal, energy efficiency 36%, at the place of consumption not compressed liquid liquid, 15% scrap liquid, 100% scrap 40% scrap in raw materials 40% scrap in raw materials electrolytic, from Cu2S electrolytic, from ZnS electrolytic, from Al2O3 bore-hole method semi-combustion of natural gas from sulfur panels from wood,waste products used as fuels

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 63

of avoided expenditures it should be done by means of CExE of the replaced product and the exergy replacement ratio: Ș*Bu

Ș*B s

(3.13)

S B su

where K*B s = CExE of the production of the replaced major product sB

su

= exergy replacement ratio, in units of exergy of the sth replaced major product per unit of exergy of the considered by-product.

Usually the replacement ratio is less than 100% and the CExE of the byproduct is greater than the CExE of the replaced major product. It denotes that the by-product should be burdened with a decreased consumption of driving exergy of the considered complex process, and a greater part of driving exergy should burden the major product, respectively. The cumulative driving exergy consumption B *D m burdening the major product of the complex process can be expressed as follows: * BDm

BD*  ¦ u

Bu Ș*Bu

(3.14)

where BD* is the cumulative driving exergy of the complex process. Hence, CExE characterizing the fabrication of the major product results from the relation: Ș*Bm

Bm BD*  ¦ u

Bu Ș*Bu

(3.15)

where Bm, Bu are the exergy of the major product and of the uth by-product. Example 3.2. Blast furnace gas (the by-product of the blast furnace process) can be used in heating ovens of steel plants. In this case it substitutes for the natural gas, but the exergy substitution ratio is smaller than 100% because blast furnace gas contains a greater fraction of non-combustible components, which decreases the efficiency of the heating oven. The exergy replacement ratio of natural gas by blast furnace gas amounts to about 85%. The CExE of the delivery of natural gas is about 87%. Hence, the CExE of the production of blast furnace gas amounts to 102%. The value exceeding 100% has only a mathematical meaning and should be used in calculations of the CExE burdening the production of liquid pig iron.

3.5. Cumulative and partial exergy losses The difference between the CExC-value and the exergy of the considered product can be partitioned into partial exergy losses PEL appearing in the particular links of the network of production processes [74]. Such a partition is possible if the

64 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

calculation of CExC comprises all the kinds of exergy extracted from nature, i.e. taken from non-renewable or renewable domestic or foreign sources: b*j

b j  ¦ įbkj

(3.16)

k

where b*j = specific cumulative consumption of exergy burdening the jth product

G bkj = partial exergy loss connected with the fabrication of the jth product and appearing in the kth link of the network of production processes. The analysis of the partial exergy losses indicates possibilities of decreasing the CExC-value. It can be attained by improving the links with great PEL-values, by changing the technology of production of some semi-finished products or by changing the kind of some semi-finished products. Partial exergy losses can be calculated only in a global scope or for products fabricated without a considerable consumption of imported materials and energy carriers. When analyzing the partial exergy losses, all the kinds of the primary exergy should be taken into consideration, including the renewable ones. The partial exergy losses cannot be divided into losses connected with non-renewable and renewable kinds of the primary exergy. Domestic and imported semi-finished products and energy carriers cannot be distinguished either. Hence, the analysis is based upon the determination of the cumulative consumption of exergy performed in a global scope. The analysis of partial exergy losses makes it possible to evaluate quantitatively the deleterious impact of all irreversible phenomena appearing in the energy- and technological network as a result of the fabrication of the considered product. Hence, we obtain complete information about the thermodynamically allowed possibilities of improving the network to decrease the consumption of primary exergy. This improvement can comprise the reduction of the irreversibility of most imperfect component processes, a change of the technology or a change of the applied semi-finished products and energy carriers. The analysis of partial exergy losses should not be excessively detailed because this would decrease the clearness of the results. The partial exergy losses appearing in the links preceding the final link can be determined without any partitioning into more detailed components and can even be cumulated with exergy losses appearing in previous processes.

3.6. Net coefficients of consumption The coefficients of consumption aij,Ej appearing in eqn (3.8) refer to a unit of the major jth product and include by-products. For example, in a heat-andpower plant the coefficient of coal consumption relates to a unit of useful heat together with the cogenerated amount of electricity. To calculate partial exergy losses the net coefficients of consumption of semi-finished products and natural

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 65

resources should be determined [74]: aij  ¦ f pj Aip

Aij

(3.17)

p

ȕ Nj

ȕ j  ¦ f pj ȕ N p

(3.18)

p

where Aij,Aip = coefficient of net consumption of the ith product per unit of the jth and pth major product ENj,ENp = coefficient of immediate net consumption of natural resources per unit of the jth and pth major product. Eqns (3.17), (3.18) can be presented in a matrix form: A

a ( E  f ) 1 ,

B T ( E  f ) 1

BNT

(3.19)

where E is a unitary diagonal matrix. The coefficients of net consumption can be used to calculate of the CExCvalues similarly as the coefficients of gross consumption: b*T

BNT ( E  A) 1

(3.20)

We can check the correctness of eqns (3.19) introducing them into eqn (3.20): b*T [ E  a ( E  f ) 1 ]

B T ( E  f ) 1

(3.21)

Multiplying eqn (3.21) by the matrix (E+f) we obtain the formula (3.8). The local net exergy loss GbNj related only to a unit of the major product of the jth process, results from the exergy balance of the jth process. It comprises the internal and external exergy losses:

¦ A b ȕ ij i

Nj

b j  įbNj

(3.22)

BNT  GBNT

(3.23)

i

or in a matrix form: bT ( E  A)

Partial net exergy losses result from the cumulative consumption of particular products per unit of the considered product: įbNkj

S Nkj įbNk

(3.24)

where SNkj denotes the cumulative net consumption of the kth product per unit of the jth product. The value of SNkj results from the matrix relation: SN

( E  A) 1

The sum of partial exergy losses fulfils eqn (3.16).

(3.25)

66 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Example 3.3. Calculating the partial exergy losses connected with the production of coke and liquid pig iron, a simplified system comprising a coal mine, an iron-ore mine, a source of natural gas, a power plant, a coking plant, an ore-sinter plant and a blast furnace plant has been investigated [73]. The coking plant is equipped with a dry-quenching installation and a steam turbine driving an electrical generator. The mentioned plants have only weak connections with other industrial plants. Four by-products appear in the considered system: coke-oven gas, tar and electricity produced in the coking plant and blast-furnace gas. It has been assumed that the coke-oven gas and tar substitute the natural gas in a proportion of 0.514 kmol n.g./kmol c-o.g. and 0.0499 kmol n.g./kg tar. The basic part of the blast-furnace gas substitutes natural gas in a proportion of 0.114 kmol n.g./kmol b-f.g. The peak part of the produced blast-furnace gas is burnt in the boiler house and substitutes a hard coal in the proportion 3.62kg coal/kmol b-f. g. The consumption of steel in coal mines has been substituted by the consumption of pig iron. The following values of specific exergy and of the coefficients of consumption and by-production have been assumed (the mentioned values of the substitution ratio have been taken into account): (1) Coal mine b1 = 24.7 MJ/kg, a11 = 0.0058 kg/kg, a31 = 0.000041 kmol/kg, a41 = 0.175 MJ/kg, a71 = 0.004 kg/kg, (2) Iron-ore mine b2 = 0.35 MJ/kg, a32 = 0.00023 kmol/kg, a42 = 0.0735 MJ/kg (3) Extraction of natural gas b3 = 802 MJ/kmol, a43 = 4.17 MJ/kmol. (4) Power plant b4 = 1 MJ/MJ, a14 = 0.156 kg/MJ. (5) Coke plant b5 = 28.95 MJ/kg, a15 = 1.60 kg/kg, a35 = 0.00458 kmol/kg, f35 = 0.0112 kmol/kg, a45 = 0.0958 MJ/kg, f45 = 0.391 MJ/kg. (6) Ore sinter plant b6 = 0.73 MJ/kg, a26 = 0.827 kg/kg, a36 = 0.0002247 kmol/kg, a46 = 0.1073 MJ/kg, a56 = 0.0612 kg/kg. (7) Blast furnace plant b7 = 8.75 MJ/kg liquid pig iron, a17 = 0.0366 kg/kg, f17 = 0.0214 kg/kg, a27 = 0.206 kg/kg a37 = 0.00441 kmol/kg, f37 = 0.00549 kmol/kg, a47 = 0.0568 MJ/kg, a57 = 0.571 kg/kg a67 = 1.594 kg/kg. Calculation results of partial exergy losses connected with the production of coke and pig iron are presented in table 3.2.

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 67

Table 3.2: Partial net exergy losses connected with the production of coke and liquid pig iron (in % of CExC).

Material

Coefficient of consumption

Process

net Aij

gross aij – fij Coke

coal mine iron ore mine

1.600 –

natural gas extraction

–0.00602

1.539 – 0.0011

unit

Partial net exergy loss %

kg/kg –

1.803 0.007

kmol/kg

0.002

coking plant

11.128

power plant

–

sinter plant

–0.2952

blast furnace plant

–

cumulative degree of

–

–

–

2.918

MJ/kg

0.048

–

–

0.160

–

–

83.934

0.0677

perfection Pig iron

coal mine iron ore mine natural gas extraction

0.0152 0.206

0.0366 0.206

–0.00108

–0.0038

kg/kg kg/kg

1.802 1.577

kmol/kg

0.077

coking plant

0.571

power plant

0.0568

0.571

kg/kg

10.251

sinter plant

1.594

0.0327

MJ/kg

6.684

1.594

kg/kg

10.351

–

–

34.315

–

–

34.943

blast furnace plant

cumulative degree of perfection

–

–

3.7. Sequence method for the evaluation of partial exergy losses The sequence method of the determination of PExL consists in the analysis of the subsequent links of the energy- and technological network, beginning with the final link and ending with the extraction of primary exergy from natural resources. It is convenient to present the results by means of a graphical scheme [25, 81]. The network can be divided into some levels [6]. On the first level the immediate consumption of semi-finished products and energy carriers in the final stage of the

68 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

production is analyzed, including the consumption of energy for transportation. On the second level the fabrication of semi-finished products and energy carriers for the first level should be considered, taking into account the immediate consumption of semi-finished products and energy carriers for this purpose, together with their transportation. On the third level the fabrication of machines and installations for the first level should be analyzed. The fourth level takes into consideration the fabrication of machines and installations for the production of machines and installations operating on the first level, together with the production of semi-finished products and energy carriers for the third level. The analysis of cumulative indices finishes usually on the second level. The application of the sequence method is easier if the values of CExC are known for the principal materials and energy carriers (for example, for hard coal and lignite, natural gas, coke-oven gas, pig iron and steel) [82]. The values cited by Szargut and ZiĊbik [81] change with the development of technology. The knowledge of the mentioned data permits the introduction into the sequence analysis the values of the CExL burdening the principal materials and energy carriers. A more detailed partitioning of these losses would require the use of the results of the sequence analysis of the production of principal materials and energy carriers. Example 3.4. Sequence analysis of the single-purpose production process. The sequence analysis of PExL in the production process with a singlepurpose final link, has been presented for the example of calcium carbide. The results of the sequence analysis of cumulative energy consumption, cited in [6] have been used. The following principal materials and energy carriers appear in this process: hard coal, coke, electricity and lime- stone. The graphical scheme of the sequence analysis of PExL is presented in Fig. 3.2. The summarized values are cited in Table 3.3. It contains the CExL burdening the production of electricity and coke. Therefore, the PExL-value determined for the extraction and transportation of hard coal relates only to the coal used for the production of electrodes. For the delivery of electricity and coke the CExE-values have been accepted (25.8% and 74.3%). Example 3.5. Sequence analysis of the cogeneration process. PExL in the cogeneration process were analyzed on the example of the production of network heat in a steam HP-plant [72]. The process substituted by the fabrication of a by-product (electricity in the considered case) was left out of consideration by means of the exergy balance formulated with the use of the net coefficients of the consumption burdening the production of heat only, eqn (3.17). The net consumption Pq of coal (per unit of time) burdening the production of heat results from the produced electric power and energy efficiency of the replaced power plant. If it can be assumed, that the efficiency of electricity transmission and delivery of coal is the same for the HP-plant and the substituted power plant, the following formula can be used:

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 69

Figure 3.2: Scheme of the sequence analysis of CExC and PExL for the production of calcium carbide [81].

Pq

P 

N el H L ȘEel

(3.26)

where P , Pq = total flow rate of coal consumed in the HP-plant and flow rate burdening the production of heat KEel = energy efficiency of the replaced power plant HL = lower heating value of fuel. It has been assumed that the electricity driving the network pumps is taken from the electrical network. The assumed heat losses in the main heating network are 15%, in the local network 2%. Per 1 GJth delivered to the consumers, 1.2 GJth should be produced in the HP-plant. The ratio of chemical exergy of coal to its lower calorific value is 1.09. The assumed water losses in the heating network per 1 h are as high as 0.7% of the volume of the pipes, the mean transportation distance is 4km, the pipe diameter 0.3m, the main thermal power delivered to the consumers is 35 MWth. The steel consumption for the maintenance of the pipes

70 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table 3.3: Partial exergy losses in the production of calcium carbide. Coefficient of cumulative consumption Process in natural units per 1 t of carbide Extraction and delivery of coal for the production of electrodes . . . . . . . . Production of coke . . . . Production and transmission of electricity. Extraction and delivery of lime stone. . . . . . . . Production of electrodes.. Burning of lime . . . . . . . Production of carbide . . . Exergy and cumulative exergy efficiency. . . . . . .

in units of exergy MJ/t carbide

Partial exergy loss

MJ/t carbide

% of CExC

56 kg 853 kg

1495 31 475

85 8121

3289.7 kWh

46 081

34 238.1 43.13

1845 kg 37 kg 985 kg 1000 kg

334 1645 6692 79 385

33.4 0.04 196.5 0.25 2987.6 3.76 10 813.5 13.62

–

–

22 910

0.11 10.23

28.86

results from an assumed durability of 10 years. The CExC value for the production of steel pipes is 60 MJ/kg. The driving power of the pumps is 160 kW in the main network and 80 kW in the local network. The graphical scheme in fig. 3.3 relates to the mean ambient temperature during the heating season in southern Poland. The calculation of exergy losses in the heat exchangers is based upon the following assumptions: — temperature of steam condensation 104qC, — temperature of hot and return water in the main network 78/42qC, — temperature of hot and return water in the local network 53/42qC, — temperature of the heated rooms 20qC. Similar calculations were performed for other values of the ambient temperature, assuming a constant ratio 0.27 of the counter-pressure power to the gross amount of heat. Table 3.4 contains the results of calculations. The PExL values burdening the production of electricity for driving the pumps have been cumulated. Similarly the CExL values connected with the delivery of coal for production of heat and burdening the production of steel are presented.

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 71

Figure 3.3: Scheme of the sequence analysis of CExC and PExL in the production of heat in the HP-plant [72].

72 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table 3.4: Partial exergy losses in the production and delivery of heat from a steam HP-plant, for the heating of rooms (in % of cumulative consumption of exergy). Ambient temperature, qC Process Extraction and transportation of coal for the production of heat . . . . . . . Production of heat in the HP-plant . . Production of electricity for driving the network pumps . . . Heat transfer in the main heat exchangers and transportation of heat in the main network. . . . . . . . . Heat transfer in local heat exchangers and transportation of heat in the local network . . . . . . . . Heat transfer in the internal installation . . . . . . . . . . . . . . Production of steel for the maintenance of pipes . . . . . . . . . . Cumulative exergy efficiency

–20

–11

–5

+2

+6

+12

5.1 56.6

5.2 51.0

5.4 56.7

5.6 63.1

5.8 66.0

6.2 68.1

2.8

3.4

3.7

4.2

4.5

6.4

9.3

13.2

13.0

12.4

12.7

13.0

4.3

4.7

3.8

3.1

2.4

1.6

11.1

11.9

9.2

6.0

4.3

2.1

0.7

0.8

0.9

1.0

1.1

1.2

10.1

9.8

7.3

4.6

3.2

1.4

Exercises 3.1. A compressor heat pump applied for the heating of rooms has the energy index COP = 3.5. The required temperature in the heated rooms is 22qC. The mean ambient temperature is 2qC. The cumulative exergy efficiency of the delivery of electricity is 0.29. Calculate the mean cumulative exergy efficiency of the considered heat pump. 3.2. Calculate the cumulative exergy efficiency of the absorption heat pump considered in exercise 2.15. Assume that the driving heat is produced in a boiler fired with natural gas. Its energy efficiency is 0.88. 3.3. The cumulative exergy consumption burdening the production of hotrolled steel tubes is 0.587 MJ/kg. The specific chemical exergy of steel is 7.04 MJ/kg. Calculate the cumulative exergy efficiency of the production of steel tubes. 3.4. The coal-fired steam HP plant equipped with a back-pressure steam turbine produces useful heat and electricity. The proportion of the produced electric power and thermal power is 0.3. The useful heat is absorbed by the network water heated from 55 to 110qC. The energy efficiency of the HP plant amounts to 0.9. The ambient temperature is –8qC. The cumulative exergy efficiency of the delivery of electricity from a replaced condensation steam power

CUMULATIVE EXERGY CONSUMPTION AND PARTIAL EXERGY LOSSES 73

plant is 0.29. Calculate the cumulative exergy efficiency of the production of useful heat. 3.5. Calculate the CExE for the production of methanol if the cumulative consumption of energy per ton of CH3OH was: 3308 MJ of electricity, 6082 MJ of heat, 554 MJ of the chemical energy of coke, and 45,348 MJ of the chemical energy of natural gas (expressed by the lower heating value). For fuels assume the values bch* H L from table 3.1. For electricity delivery Ș*B = 0.29 and for heat production Ș*B = 0.4. For the heat carrier the mean thermodynamic temperature Th = 425 K, the ambient temperature T0 = 283 K. 3.6. Calculate the partial exergy losses connected with the production of methanol. 3.7. Calculate the CExE for the production of ZnO-sinter if the consumption of energy and raw materials per ton ZnO was: 2076 MJ of electricity, 19,393 MJ of the chemical energy of bituminous coal, 48,649 MJ of the chemical energy of coke (chemical energy expressed by the lower heating value), 0.0077 t of hotrolled steel, 0.003 t of iron casting. For fuels assume the values bch* / H L from

table 3.1. For electricity delivery Ș*B = 0.29. For iron casting assume the value bch= 8200 MJ/t, Ș*B = 0.161.

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Chapter 4 Practical rules for improving thermodynamic imperfection The more rational utilization of energy is one of the cheapest methods of satisfying the energy demand of humankind. The aim of energy conservation is to satisfy all the demands for final energy by means of a smaller consumption of primary energy. The following main methods of energy conservation may be named: (1) reduction of the thermodynamic imperfection of energy processes, (2) improvement of the efficiency of energy-consuming installations and devices, (3) improvement of the operation of energy-consuming installations and devices, (4) improvement of the utilization of waste energy, (5) decrease of the consumption of materials, (6) decrease of the fraction of energy-consuming products in the overall industrial production, (7) improvement of the utilization of secondary materials obtained due to the wear of useful products. The solution of the first problem should be based on the principles of exergy analysis. Sama, Quian and Gaggioli proposed 13 common-sense 2nd Law guidelines ensuring the reduction of the thermodynamic imperfection of thermal processes [48]. These guidelines were supplemented by Szargut. The additional rules take into account the mutual influence of the particular links of energy systems. So, a set of 20 practical rules (or guidelines) have been formulated [75]: 1. Accept exergy losses only if they are indispensable for the reduction of investment expenditures. Exergy losses without any economic justification should be treated as the result of an engineer’s error. For example, exergy losses due to irreversible heat transfer cannot be eliminated from a heat exchanger because without temperature differences the heat transfer area would be infinitely large. On the other hand, the practical realization of a thermal process might deviate from the formulated rules if it were economically justified.

76 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

2.

Do not use excessively large or excessively small thermodynamic driving forces of the process operation. For example, the temperature differences between the heating and the heated fluid within a heat exchanger should not be too small (because the necessary heat transfer area would be too large) and not too large (because the efficiency of the entire process would be excessively reduced). The temperature difference between the heating and the heated stream should be approximately proportional to the absolute temperature.

3.

Minimize the mixing of streams with differences in temperature, pressure or chemical composition. The mixing of streams with different temperatures is equivalent to an irreversible heat transfer. The mixing of streams with different compositions appears, for example, in the production of air enriched with oxygen when technical oxygen (produced with a large consumption of driving energy) is mixed with atmospheric air. The immediate enrichment method would be thermodynamically more correct, but until today such methods are too expensive. In the chemical industry every recirculation of the exhausted material increases also the entropy generation and worsens the quality of the initial material. An important example of mixing streams with different compositions and/or temperatures appears in the recirculation process of the intermediate or outlet product. In many processes of chemical technology recirculation is used for a better utilization of the useful component contained in the outlet waste product. However, in this case the recirculation reduces the concentration of the active components in the inlet material, causing an increase of the necessary area or volume of the apparatus. In some processes the expected effect of recirculation is more important than the involved exergy loss. For example, the recirculation of combustion gases is applied in combustion processes to decrease the temperature and to reduce the formation of nitric oxides. However, as a negative effect, the mean temperature difference between the combustion gases and the heated matter becomes smaller, and that leads to an increase of the necessary heat transfer area. Another example of an unavoidable recirculation can appear in a suspension dryer. The necessary volumetric stream of the drying gas can be too small for the formation of a suspension of the dried solid particles. To attain a sufficient velocity of the drying gas, its recirculation can be applied.

Example 4.1. Figure 4.1 presents the scheme of a set of reactors in the process of esterification of acrylic acid by means of methyl alcohol. Every reactor operates with evaporation of the reactants. The active components (alcohol + acid + catalyzer) are separated from the vapour by condensation and turned back to the inlet of every reactor. Additionally, some part (up to 50%) of the exhausted fluid is recirculated from the outlet of the system to its inlet, to reduce the loss of the not

PRACTICAL RULES FOR IMPROVING THERMODYNAMIC IMPERFECTION 77

Figure 4.1: A set of reactors for the esterification of acrylic acid by means of methyl alcohol. Si , Wi — stream of the inlet- and outlet-fluid of the ith reactor, al+ac+ct — alcohol+acid+catalyzer, Ci — condenser and separator of the ith reactor, D — distributor, g — recirculation ratio.

utilized reactants. However, the concentration of the active reactants in the inlet fluid becomes smaller after recirculation, and the necessary volume of the reactors gets larger [108]. Example 4.2. The high temperature of the working fluid entering the gas turbine requires the cooling of the turbine blades. Usually an open system of cooling is applied. The compressed air taken from the outlet of the compressor (sometimes additionally cooled in an external cooler) flows in the channels inside the blades. The cooling air flowing out from the cooling channels (with an open outlet) mixes with the working fluid. The blade cooling decreases the efficiency of the turbine. The irreversible mixing of the cooling air with working fluid is the main cause of the efficiency decrease [89]. 4. Do not discard heat at high temperature to the ambient or the cooling water and do not heat the refrigerated stream with hot streams or with hot water. High-temperature waste heat has a positive exergy which should be utilized. The exergy of the refrigerated streams is also positive and may be utilized by the heat exchange with subambient process streams that need to be cooled. Example 4.3. Hot-rolled products in steel plants are usually cooled in open air. So their physical exergy is lost. In some plants the cooling process is performed within a chamber equipped with water-cooled walls where hot water or steam can be produced. Example 4.4. Liquid natural gas (LNG), after transporting by ship, should be evaporated before its pipeline transportation. The evaporation heat is usually taken from the environment. It is possible to utilize the physical exergy of LNG, for example for the production of electricity. In this case the atmospheric air

78 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

would be a higher heat source, and LNG would represent the lower heat source. The suitable cycle of the power plant might be realized by means of a substance having a critical temperature higher than the ambient one and its condensation pressure corresponding to the temperature of LNG—slightly higher than the ambient one. 5. All counter-current processes are generally thermodynamically more efficient than parallel ones. For example, at the same heat-transfer area the counter-flow heat exchanger ensures a higher temperature increase of the heated fluid than the parallelflow heat exchanger. At the same amount of transferred heat a parallel-flow heat exchanger requires a greater heat transfer area than the counter-flow heat exchanger. The application of the parallel current is acceptable only if the wall temperature in the heat exchangers has to be reduced or if rapid changes in stream temperatures are required. Example 4.5. Let us compare parallel-flow and counter-flow heat exchangers when the initial and final values of the temperature of the heat exchanging fluids are the same. Let us assume that the heating fluid has the initial and final temperature 500 and 150qC. The heated fluid has the initial and final temperature values 80 and 130qC (fig. 4.2). The logarithmic mean temperature difference amounts to 180.2K in the case of a counter-flow and 131.4K in the case of a parallel flow. Hence, the parallel-flow heat exchanger requires, in the considered example, a heat transfer area 37% larger than the counter-flow heat exchanger. 6.

When choosing streams for a heat exchange, try to match streams where the final temperature of one is close to the initial temperature of the other. This ensures that the maximum exergy is recovered in the heat exchange process or the heat transfer area can be reduced.

Figure 4.2: Temperature distribution in a parallel-flow and counter-flow heat exchanger.

PRACTICAL RULES FOR IMPROVING THERMODYNAMIC IMPERFECTION 79

Example 4.6. Let us consider a heat exchanger fed with hot combustion gases and heating two streams of fluid. The final temperature of the first stream is higher than the initial temperature of the second one (fig. 4.3a). To attain a smaller distance between the mentioned temperatures, a common part of the heat exchanger heating simultaneously both fluid streams can be realized (fig. 4.3b). This solution is in accordance with the pinch method [34]. Considering the temperature values cited in fig. 4.3, and assuming that the heat transfer coefficients are the same, the heat transfer area of the second version is smaller than that of the first one. The ratio of the values of the heat transfer area is 0.91. Example 4.7. Repowering of the conventional power plant fed with coal may be attained by means of the primary gas turbine fuelled with natural gas. Usually in the supplemented scheme the heat-recovery boiler of the gas turbine is used to preheat the feed water of the conventional power plant. The initial regenerative water preheaters fed with bleed steam are partially or totally switched off. However, the final temperature of the preheated water is considerably lower than the initial temperature of the outlet gases flowing from the gas turbine. Hence, in the hot part of the waste-heat boiler of the gas turbine the exergy losses are large. High temperature differences in the hot part of the waste heat boiler may be reduced by introducing into this boiler not only water preheating but also steam superheating [84], fig. 4.4. This solution improves the utilization of the expensive natural gas and the efficiency of the total plant. Figure 4.5 presents the distribution of temperature and exergy losses in the waste-heat boiler used for the preheating of feed water and secondary superheating of steam.

Figure 4.3: Temperature distribution at simultaneous heating of two fluid streams.

80 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Figure 4.4: Scheme of the combined power plant with the secondary steam superheater shifted to the waste-heat boiler of the gas turbine.

Figure 4.5: Temperature distribution and exergy losses in the waste heat boiler used for the preheating of feed water and secondary superheating of steam.

PRACTICAL RULES FOR IMPROVING THERMODYNAMIC IMPERFECTION 81

7. The flow heat capacities of the streams exchanging heat should be similar. If great differences appear, consider the partitioning of the stream with a greater heat capacity between two or more heat exchangers heated (or cooled) with additional streams. The exergy losses should be uniformly distributed within the heat exchanger. Example 4.8. Let us consider a heat exchanger fed with hot combustion gases and heating a fluid which has a stream with a heat capacity smaller than that of the combustion gases (fig. 4.6). The required final temperature of the heated stream is near the initial temperature of the combustion gases. However, the drop of the temperature of combustion gases is smaller than the increase of the temperature of the heated medium. Therefore, the outlet temperature of the combustion gases is not sufficiently low, and their utilization degree is not satisfactory. This utilization degree can be improved by reducing the flow rate of combustion gases and using the remaining part for heating another medium. The mentioned reduction of the flow rate at 21.4% would decrease the outlet temperature of combustion gases down to 110qC. This would require, however, an increase of the heat transfer area in a proportion of 1.56.

Figure 4.6: Temperature distribution when heating a stream with a small heat capacity.

82 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

8. Minimize the use of intermediate heat transfer fluid while exchanging heat between two streams. The application of an intermediate fluid increases the total heat transfer area or increases the total exergy loss. It might be sometimes justified because of the local conditions or by the long distance transportation of heat. Example 4.9. Figure 4.7 presents the temperature distribution in a system of two heat exchangers heating the combustion air for the Cowper stove by means of combustion gases. The lack of place within the duct of combustion gases makes it impossible to apply an immediate heat transfer from the combustion gases to the heated air. Therefore, an intermediate heat carrier with a high volumetric heat capacity has been applied to transfer heat from the stream of combustion gases to the combustion air. In the considered case the total exergy loss remains unchanged after introducing an intermediate heat carrier. However, the same heat stream will be transferred twice, with a smaller temperature difference between the heating and the heated medium. Therefore, the required increase of the total heat transfer area will be considerable. Assuming the same heat transfer coefficient as by immediate heating, we obtain a mean temperature difference twice smaller. The required total heat transfer area will be four times larger. Usually the application of an intermediate heat carrier with a higher volumetric heat capacity ensures an increase of the heat transfer coefficient, but it cannot increase twice. If it increases, for example, 1.5 times, the total heat transfer area will increase 2.7 times in comparison with the system operating without any intermediate heat carrier.

Figure 4.7: Application of an intermediate heat carrier in the heat exchanger.

PRACTICAL RULES FOR IMPROVING THERMODYNAMIC IMPERFECTION 83

Example 4.10. In a vapour-compression heat pump an intermediate heat carrier is usually used between the bottom heat source and the evaporator [83] (fig. 4.8). This leads to a lowering of the evaporation temperature, increasing the compression work and some increase of useful heat. The value of COP becomes lower. The quantity Tb in fig. 4.7 represents the temperature of the bottom heat source, Tib the mean temperature of the intermediate heat fluid of this cycle is represented by the area 1-2-3-5-1. After introducing the intermediate heat carrier the cycle passes through the points 1c-2c-3-4c-1c, and the internal driving work per unit of the working fluid is represented by the area 1c-2c-3-5c-1c. The heat absorbed from the bottom heat source changes only slightly. Let us assume the temperature of the bottom heat source (Tb = 8qC), the mean temperature difference between the bottom source and the intermediate heat carrier (Tic–Tb = 3K), between the evaporating fluid and the intermediate heat carrier (Tib–Tev = 3K), the working fluid F22, the condensation temperature (Tcn = 45qC), the internal adiabatic efficiency of the compressor 0.75. The internal COP (without taking into account the mechanical efficiency of the compressor and the electromechanical efficiency of the electric motor) amounts to 5.25 with the consider ed intermediate heat carrier and 6.15 without this heat carrier.

Figure 4.8: Influence of an intermediate heat carrier between the bottom heat source and the evaporator of a heat pump.

84 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

9. Exergy losses due to the hydraulic friction or irreversible heat transfer are the greater, the lower the temperature in the process is. Minimize these losses especially below the ambient temperature. The influence of hydraulic friction has been discussed in section 1.3. (fig. 1.2) presents the results of hydraulic friction during the adiabatic flow of a fluid stream. Below the ambient temperature, in a chilling stream, the absorption of friction heat reduces the amount of heat removed from the process and thereby requires an increase of the refrigerant flow rate and driving energy in the refrigerator. Synergistic negative effects appear. 10. Minimize the throttling of steam or other gases. Throttling causes a destruction of exergy. The throttling exergy loss increases as the volume becomes greater. Usually it does not have any economic justification, but sometimes is unavoidable. Example 4.11. Figure 4.9 presents the entropy increase due to the throttling of working fluid in the vapour-compression Linde-cycle. It results in a decrease Gqb of the amount of heat extracted from the bottom source. Thus, it decreases the useful effect of a refrigerator or heat pump. However, the efforts to eliminate this exergy loss have not given any positive results. For example, the realization of the non-throttling periodically operating Granryd-cycle introduces new exergy losses which can be greater than the initial throttling loss [79].

Figure 4.9: Deleterious impact of the throtting in the Linde-cycle.

PRACTICAL RULES FOR IMPROVING THERMODYNAMIC IMPERFECTION 85

11. Fans or compressors should be located in the coolest place of the process. The driving power of compressors and fans is proportional to the volume, and hence, decreases if the temperature drops. Figure 4.10 presents the adiabatic enthalpy increase of the compressed gas or vapour. The higher the initial temperature, the greater the required compression work is. 12. Eliminate leaks in pipelines, valves and combustion chambers. Losses of compressed gases, losses of hot combustion gases or penetration of excessive air into combustion gases ducts or combustion chambers leads to a great decrease in the efficiency of the total system. 13. Remember that in systems driven with chemical, nuclear or mechanical energy the rejection of heat to the environment in condensers of turbines, refrigerators, etc. is a reflection of the irreversibilities within the plant. Decrease of the heat losses from condensers can be attained only by a reduction of irreversibilities within the plant. The rejection of heat in condensers cannot be eliminated by the recompression of vapour (for example, in evaporator systems).

Figure 4.10: Adiabatic compression of a gas.

86 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Example 4.11. Figure 4.11 presents a multistage evaporation plant. The first stage is fed with steam (for example, from a heat-and-power plant). The steam generated in this stage flows to the second stage and evokes an evaporation of the liquid in this stage. A similar task is performed by the steam generated in this stage. The steam from the last stage flows to the condenser. Figure 4.12 presents a single-stage evaporator equipped with a compressor of the steam generated inside the evaporator. The compressed steam delivers heat to the evaporation process. Apparently this installation does not need any condenser. However, it is driven with electricity, usually delivered from a condensation steam power plant. Hence, the condensation has not been eliminated, it is only moved to another place. 14. Avoid the compression of steam which has been expanded previously. In this case the exergy loss appears two times. Figure 4.13 presents an adiabatic irreversible expansion of the steam and the subsequent adiabatic irreversible compression. The work of compression is greater than that of expansion. However, the recompression of the expanded steam has sometimes an economic justification. Example 4.13. Let us consider the transportation of a counter-pressure steam from the heat-and-power plant to the industrial consumer, which demands a large stream of 0.4MPa steam and a small stream of 0.8MPa steam. It is possible to produce two kinds of counter-pressure steam in the considered heat-and-power plant, but it would be necessary to build two pipelines from the heat-and-power plant to the industrial consumer. The investment expenditures can be reduced by building only one pipeline for the 0.4MPa steam and to apply the recompression of the necessary part of it within the plant of the consumer.

Figure 4.11: Multi-stage evaporator.

PRACTICAL RULES FOR IMPROVING THERMODYNAMIC IMPERFECTION 87

Figure 4.12: Single-stage evaporator with a vapour compression. 15. If you can reduce some exergy loss, try not to increase another exergy loss appearing parallel. Example 4.14. In heating systems equipped with a simple boiler, large exergy losses appear inside the heated rooms due to the irreversible heat transfer between the heating medium and the heated room. The introduction of a low-temperature heating system (for example, a floor heating system) can reduce considerably the exergy losses in this room, but increases the exergy losses in the boiler. Example 4.15. The increase of the combustion temperature, for example, by means of the preheating of combustion air, decreases the exergy losses connected with the combustion, but increases the irreversibility of heat transfer between the combustion gases and the heated agent.

Figure 4.13: Recompression of a previously expanded stream.

88 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

16. Avoid the elongation of the chain of thermodynamic processes. Every new process introduces new exergy losses and additional investment expenditures. Example 4.16. The proposed schemes of the humid air turbine (HAT) usually contain an aftercooler of the compressed air installed after the compressor (for example [14]). It makes it possible to decrease the temperature of the outlet gases discharged to the environment. However, the compressed air flowing from the aftercooler is again heated in the regenerative preheater fed with the working fluid from the turbine. Hence, the aftercooler elongates the chain of processes, increases the investment expenditures and introduces additional exergy losses. Figure 4.14 presents a scheme of HAT with multi-point water injection, blade cooling and aftercooler of the compressed air. Szczygiel [90] compared the scheme with an aftercooler of the total stream of the compressed air (fig. 4.14) with a scheme with a smaller aftercooler applied only for cooling the stream of cooling air (fig. 4.15). The obtained results are presented in table 4.1. The best efficiency values appear for the scheme with an aftercooler applied only for decreasing the temperature of the cooling air.

Figure 4.14: Scheme of the humid air turbine with the aftercooler of the compressed air CL, CH – low pressure and high pressure stage of the compressor, T1}T4 — stages of the turbine, C1 — intercooler and preheater of the injection water, C2 — external intercooler, C3 — aftercooler, CC — combustion chamber, H1}H3 — humidifiers of the compressed air, D1, D2 — distributors of the cooling air, R, R1 — regenerative preheater of air, W1,W2 — streams of the injection water, 1}12 — streams of the working fluid.

PRACTICAL RULES FOR IMPROVING THERMODYNAMIC IMPERFECTION 89

Figure 4.15: Scheme of the humid air turbine with an aftercooler of the compressed air applied only for cooling the stream of cooling air.

17. Try to introduce cogeneration processes producing simultaneously two or more useful effects. A cogeneration process (for example, the cogeneration of heat and electricity) denotes a considerable shortening of the chain of processes and therefore ensures a great reduction of exergy losses within the entire system. 18. Consider the influence of the proposed changes in energy management on the exergy losses in other links of the system. For example, the electrical heating of rooms results in an increase of the exergy losses in the power plant. Remember also, that the separate optimization of some link of energy systems is usually not exact because of the mutual influence of particular processes taking place within the system. Table 4.1: Influence of the aftercooler on the efficiency of the HAT power plant. Temperature after the turbine qC

without aftercooler

with aftercooler

aftercooler applied only to the cooling air

1200

47.67

45.35

47.88

1300

48.85

47.15

49.35

1400

49.82

48.57

50.69

Energy efficiency, %

90 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

19. Remember that the cost of exergy increases along the chain of processes. For example, the cost of the fuel exergy in a power plant is smaller than the production cost of the steam exergy, and the latter is smaller than the production cost of electricity. 20. Try to reduce the exergy losses in places where they are the greatest, and in places where they are most expensive. Some of the 2nd Law inefficiencies cannot be avoided, others can. Concentrate on those which can. For example, the losses of electricity in an electrical grid are very undesirable. Leites, Sama and Lior [33] published 12 directives resulting from the 2nd law and pertaining to the chemical technology. Many of them are similar to the above cited rules (for example to the rules 2, 3, 5, 7), however, some of them may be especially important for chemical technological processes. For example, the directive 4 is worth citing: “If the reaction is exothermic, it should begin at the heightened temperature. An endothermic reaction should begin at a lowered temperature. If the reaction runs with an increase of volume, begin it at a heightened pressure. The reaction running with decreasing volume should begin at a lowered pressure. The cited rule is only apparently not conform with the law of le Chatelier”. The conditions of the chemical reaction should not be favorable for the reaction rate because every increase of this quantity denotes an increase of the rate of exergy loss. Leites advises to begin the selection of the scheme of a technological process from a quasistatic course, running through the equilibrium states, without any exergy losses. So we can find indications how to select the real scheme burdened with possibly minimum exergy losses.

Exercises 4.1. Calculate the minimum attainable increase of the heat transfer area due to the introduction of the intermediate heat carrier. Assume that the heat transfer coefficient between the intermediate heat carrier and the walls of the heat exchangers is infinitely large. The resistance of heat conduction in the walls of heat exchangers may be neglected. The heat transfer coefficients on the side of the main heat carriers and their heat capacities are the same. 4.2. In a simple steam power plant the exergy efficiency values of its main parts are: boiler 0.45, turboset 0.82, transformation and transmission system 0.9. Assume that the flow rate of the chemical exergy of fuel amounts to 1000MW. Calculate how the increase of exergy loss amounting to 10MW influences the amount of electricity delivered to the consumer, if it appears, (1) in the boiler, (2) in the transformation and transmission system.

Chapter 5 Depletion of non-renewable natural exergy resources; thermo-ecological cost 5.1. Definition of the thermo-ecological cost The inevitable depletion of non-renewable natural resources is very dangerous for the future existence of mankind. Exergy can be accepted as a common quality measure of all natural resources. Therefore, the cumulative exergy consumption of non-renewable natural resources, termed ecological cost, has been proposed by Szargut as a measure of their depletion [62, 65, 76, 77]. Recently the term thermoecological cost has been introduced [80]. The set of eqns (3.6) can be used for the calculation of thermo-ecological costs but the quantity Ej should comprise only nonrenewable resources. Additionally, the deleterious ecological impact of waste products should be taken into account. The thermo-ecological cost can be calculated in a global or regional scope. In the second case the influence of the interregional exchange of goods should be taken into consideration.

5.2. Evaluation of the thermo-ecological cost of the waste products The waste products of every process usually exert a deleterious impact on the environment. Three kinds of this impact can be named: the corrosion of buildings, machines and equipment, the reduction of agricultural and forest production and the damage in human health. The corrosion is most important All the mentioned negative effects should be compensated by an additional consumption of useful products substituting the destroyed useful products or preventing possible damage [76]. The determination of the coefficients expressing the additional consumption of useful products is very difficult. The assumption that ecological losses are proportional to the exergy of waste products [2], is not justified. For example, CO has a relatively great exergy, but does not cause any corrosion of useful products. Therefore, Szargut [77] proposed to evaluate the deleterious impact of waste products by means of their monetary index of harmfulness. The additional exergy

92 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

consumption of non-renewable natural resources due to the emission of waste products (for example in kJ/kg) can be expressed as: ȟk

Bı k DCP  ¦ Pk ı k

(5.1)

k

where

B = annual consumption of the non-renewable exergy from own sources of the considered region Vk = monetary index of harmfulness of the kth waste product DCP = domestic consumption product. Pk = annual emission of the kth waste product in the country.

The domestic consumption product (DCP) expresses the monetary value of all useful final products used in the consumption sector, except those used in production processes. Equation (5.1) takes into account the reduction of DCP due to the emission of waste products (the last term in the denominator). The values of the monetary index of harmfulness have been assessed by many authors, and the obtained results differ considerably [20, 93, 107]. Usually gaseous waste products have been analyzed: sulfur oxides, nitric oxides, CO, benzo-a-pyrone. It is very difficult to determine the considered index for CO2 because its emission changes the composition and parameters of the natural environment, evoking the greenhouse effect. It leads to climatic changes and to a rise of the sea level. So far the evaluation of the monetary index of the harmfulness of CO2 is conventional and very low.

5.3. Balance equations After introducing the coefficients [k the set of balance equations determining the value of the specific thermo-ecological cost takes the form: ȡ j  ¦ ( f ij  aij )ȡi i

¦b  ¦ p sj

s

k

kj

ȟ k  ¦ arj ȡ r

(5.2)

r

where Uj,Ui = specific thermo-ecological cost of the major product of the jth and ith process bsj = exergy consumption of the sth non-renewable natural resource, per unit of the jth product pkj = emission of the kth waste product per unit of the jth product Ur = specific thermo-ecological cost of the imported rth semi-finished product. The thermo-ecological cost of the production of machines and installations may be taken into account by means of the consumption coefficient aij from eqn (3.9).

DEPLETION OF NON-RENEWABLE NATURAL EXERGY RESOURCES 93

The set of eqns (5.2) can be formulated separately for a group of production processes having weak connections with the remaining part of the network of production processes.

5.4. Influence of the interregional exchange The thermo-ecological costs of particular products differ in various regions because of differences in the applied technologies. The influence of the interregional exchange should be taken into account when determining the indices of thermo-ecological costs in the considered region. In eqn (5.2) the thermo-ecological costs of imported goods can be replaced by the equivalent thermo-ecological costs of exported goods [68]. The financial means for the import are acquired by the export. Hence, the mentioned equivalence can be determined only by means of classical economic indices. It can be assumed that the thermo-ecological costs of imported goods related to a unit of monetary value is the same as that burdening the exported goods. The index of the thermo-ecological costs of exported goods per monetary unit can be expressed as follows:

ȡm

¦S ȡ . ¦S D i

i

i

i

i

(5.3)

i

imported product is proportional to its monetary value Dr:

ȡr

ȡ m Dr

Dr

¦S ȡ . ¦S D i

i

i

i

i

(5.4)

i

Hence, the specific thermo-ecological costs of the rth imported product depends on unknown values Ui of the thermo-ecological costs of the exported domestic products. The component of eqn (5.2) representing semi-finished products can be expressed as follows:

¦a

rj

r

ȡr

ȡ m ¦ arj Dr r

1 ¦ Si Dr i

¦ȡ S ¦a i

i

i

rj

r

Dr

¦ȡ d i

i

ij

(5.5)

94 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

where:

dij

Si

¦a D ¦S D rj

r

i

i

r

(5.6)

.

i

All the quantities appearing on the right-hand side of eqn (5.6) are known. Hence, the auxiliary coefficients dij are known, too, and can be introduced into the set of eqns (5.2). The final form of eqn (5.2) is ȡ j  ¦ ( f ij  aij  dij )ȡi i

¦b  ¦ p sj

s

ȗ

kj k

(5.7)

k

The thermo-ecological cost (global or regional) can be smaller than the exergy when the considered product is burdened mainly with the consumption of the renewable primary exergy (for example, agricultural products burdened with the consumption of the renewable exergy of solar radiation or the electricity produced from renewable resources). The regional thermo-ecological cost can be smaller than the exergy also in the case when the considered region (for example, Switzerland) exports mainly high-tech products fabricated with a small consumption of domestic non-renewable exergy and imports energy carriers and semi-finished products requiring a great consumption of non-renewable exergy. Hence, interregional exchange can exert an essential influence on the regional thermo-ecological costs. The number of unknown quantities Ui appearing in the set of eqns (5.7) is equal to the number of equations. However, the application of the set of eqns (5.7) is not possible, if we want to perform the calculations in steps, as mentioned above because in every one of eqns (5.7) the thermo-ecological cost of all the exported goods does appear. Therefore, an iterative method has been proposed by Stanek [54]. The exported goods should be divided into some groups of similar products. For every group the thermo-ecological cost should be calculated by means of the sequence method applying an iterative procedure. The first approximation can be based upon the assumption that the thermo-ecological cost of the exported goods Um per unit of their monetary value equals the mean thermo-ecological cost per monetary unit of the domestic consumption product: ȡr

Dr ȡ m

Dr

B DCP

(5.8)

The values Ur resulting from eqn (5.8) should be used in the sequence calculation method of the particular groups of the exported goods. The obtained results should be introduced into eqns (5.3) and (5.4) to correct the values of Ur. These values can be used to correct the values of the mean products taken into account in the balance eqns (5.2). These corrected values can again be used in the sequence calculation of the thermo-ecological cost of exported goods.

DEPLETION OF NON-RENEWABLE NATURAL EXERGY RESOURCES 95

5.5. Calculation of the thermo-ecological cost In order to calculate the domestic thermo-ecological cost, its values for the deleterious waste products should be determined. According to eqn (5.1) the first step of the calculations is the determination of the annual consumption of non-renewable exergy. The second step comprises the application of the set of balance eqns (5.7) to the main strongly connected products. In the third step the sequence method can be applied for weakly connected products. Example 5.1. The annual exergy consumption of domestic non-renewable natural resources in Poland contains the exergy Bhc of hard coal, Bbc of brown coal, Bng of own natural gas, Bcr of own crude oil, BS of sulfur and BCu of copper ore. According to Stanek [53] in 1997 the mentioned values were: B 9

Bhc  Bbc  Bng  Bcr  BS  BCu

9

3608 u 10  638 u 10  140 u 109  13 u 109  33 u 109  15 u 109

4445 u 109 MJ/a

The domestic consumption product was 143.1×109 $/year. The thermoecological cost per unit of DCP was 31.1 MJ/$. Example 5.2. The annual emission of the main waste products (SO2, NOx and dust) amounted in Poland in 1997 to: PSO2

2181 u 103 ,

PNOx

1141 u 103 ,

Pd

1130 u 103 t/a.

The following monetary indices of harmfulness have been accepted: ıSO2

1500,

ı NOx

1500,

ıp

310 $/ t.

The indices of thermo-ecological cost of waste products, resulting from eqn (5.1) are: ȟ SO2

45,

ȟ NOx

45,

ȟp

9.3 MJ/kg.

Example 5.3. Determining the specific thermo-ecological cost of principal products burdened with a large consumption of non-renewable exergy, a system containing the following plants has been taken into account [53]: coal mines, natural-gas wells, a thermal power plant, a coking plant, an iron-ore sinter plant and a blast-furnace plant. The system consumes imported iron ore and natural gas. The major products of the considered system are as follows: 1 - hard coal used in power plants, 2 - hard coal used in the coke production and injected into the blast furnace, 3 - natural gas (domestic and imported),

96 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

4 - electricity, 5 - coke, 6 - iron ore sinter, 7 - pig iron, 8 - imported iron ore. The fabrication of the mentioned products is connected, but slightly, with other branches of industry. The consumption of steel in the mining of coal has been substituted by the consumption of pig iron. It has been assumed that coke-oven gas substitutes (with the substitution efficiency equal to one) the imported natural gas (which closes the balance of demand), the basic production of blast-furnace gas substitutes (with an efficiency of 0.7) the imported natural gas, and the peak production of this gas substitutes the hard coal used for the production of electricity. The consumption of the energy carrier for the transportation of the considered products has been taken into account. The influence of the injection of various auxiliary fuels (replacing partially coke) has been examined (coke-oven gas, natural gas and pulverized coal). The first step of approximation is presented in the considered example. The thermo-ecological cost of imported materials has been estimated by means of eqn (5.8). From this equation it results that Um = 34.4 MJ/$. The inaccuracy of this estimation does not influence much the calculated values of the thermo-ecological cost of domestic products. The thermo-ecological cost of imported iron ore (1.032 MJ/kg) results from its cost (30 $/t). The following balance equations have been formulated for the calculation of the thermo-ecological cost: Extraction and delivery of coal. The balance equations have the same form for both kinds of coal. Only the specific exergy is different: (1  a11 )ȡ1  Ȥ 3 a31ȡ3  a41ȡ 4  a71ȡ 7

b1  (1  Ȥ 3 )a31ȡ3r  ¦ pk 1ȟ k , k

(1  a22 )ȡ 2  Ȥ 3 a32 ȡ3  a42 ȡ 4  a72 ȡ 7

b2  (1  Ȥ 3 )a32 ȡ3r  ¦ pk 2 ȟ k , k

where F3 = fraction of domestic natural gas in the total consumption U3r = specific thermo-ecological cost of the imported natural gas. The following coefficients have been introduced: a11 a31

a71 p2 p

0.000041 kg/kg, (ȡ3 MJ kmol),

a32

a41 p1 p

0.0058 kg/kg, (ȡ1 ,ȡ 2 MJ kg)

a22 a42 a72 p1SO2 b1

0.175 MJ/MJ, (ȡ 4 MJ MJ), 0.004 kg/kg, (ȡ 2 MJ kg), p2SO2

21.8, b2

p1NOx

p2NOx

30.25 MJ/kg.

0.0001 kg/kg,

(5.9)

DEPLETION OF NON-RENEWABLE NATURAL EXERGY RESOURCES 97

Extraction and delivery of domestic natural gas: ȡ3  a43ȡ 4

where:

a43

(5.10)

b3

4.17 MJ/kmol,

802 MJ/kmol

b3

Production and delivery of electricity:

¦p

(1  a44 )ȡ 4  a14 ȡ1

4k

ȟk

(5.11)

k

where: a44 p4SO2

0.00218,

0,

a14

p4NOx

0.156 kg/MJ,

0.00101,

p4 p

0.00347 kg/MJ.

Production of coke:

¦p

ȡ5  a25ȡ 2  a45ȡ 4

5k

ȟ k  (a35  f35 )ȡ3r

(5.12)

k

with: a25 f 35

1.6 kg/kg,

a35

ak 5 [( MH L ) k ( MH L )5 ]v3 k

0.00458 kmol ng/kg,

f k 5 [( MH L ) k ( MH L )5 ]Q 3 k

0.0112 kmol ng/kg,

p5SO2

0.000473,

0.000129,

p5NOx

p5 p

a45

0.0958 kg/kg,

0.000613 kg/kg.

where ak5, fk5 = coefficient of the consumption and by-production of coke-oven gas (MHL)k,(MHL)3 = lower heating value of coke-oven and natural gas, MJ/kmol Q3–k = energy efficiency of the replacement of natural gas by cokeoven gas. Production of iron ore sinter: ȡ6  a46 ȡ 4  a56 ȡ5

¦p

6k

ȟ k  a86 ȡ8 r  a36 ȡ3r

(5.13)

k

with the values: a86

0.827 kg/kg,

a36

ak 6 [( MH L )k ( MH L )5 ]v3 k  aw6 [( MH L ) w ( MH L )3 ]v3 w

0.0002247 kmol/kg, p6SO2

0.003,

a46

0.1073 MJ/kg,

ȡ8 r

6.6 MJ/kg,

p6NO x

0.0005,

p6 p

a56

0.0612 kg/kg,

0.006 kg/kg.

98 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

where aw6, Q3–w = coefficient of the consumption of blast-furnace gas and the energy efficiency of the replacement of natural gas by blast-furnace gas (MHL)w = lower heating value of blast furnace gas, MJ/kmol. Production of pig iron (together with the production of blast, steam for blast humidification and oxygen). The by-production of electricity appears in the gas turbine (expansion of blast-furnace gas) and in the counter-pressure steam turbine delivering steam for blast humidification. The injected natural gas has been partitioned into the domestic and imported part [by means of the coefficient F3 from eqn (5.9)]: ȡ 7  (a17  f17 )ȡ1  a27 ȡ 2  Ȥ 3 a37 ȡ3  (a47  f 47 )ȡ 4  a57 ȡ5  a67 ȡ 6

¦p

7k

ȟ k  [(1  Ȥ 3 )a37  a37c  f 37c ]ȡ3 r

(5.14)

k

The specific consumption of the first kind of coal (used for the production of electricity): a17

D (1  ȗ D )(a1D  O 2 D a1t  X D a1 X )

(5.15)

where

D = consumption of dry blast per pig iron unit ]D = coefficient of blast losses in Cowper stoves O2D, XD = consumption of technical oxygen and humidifying steam per unit of dry blast a1D, a1t, a1X = specific consumption of coal (first kind) for blast compression, for the production of oxygen and for the production of humidifying steam. The coefficient of the by-production of coal (first kind) results from the byproduction of the peak part of blast-furnace gas (used in the boiler house): f17

(1  Į w ) f w7

( MH L ) w Ȟ1 w H L1

(5.16)

where fw7 = coefficient of the by-production of blast furnace gas, kmol/kg pig iron HL1 = lower heating value of coal, MJ/kg Dw = fraction of the basic part of the produced blast-furnace gas Q1–w = energy efficiency of the replacement of coal by peak blast-furnace gas. The basic part of blast-furnace gas replaces imported natural gas: f 37c

Į w f w7

( MH L ) w Q3 w ( MH L )3

(5.17)

DEPLETION OF NON-RENEWABLE NATURAL EXERGY RESOURCES 99

where Q3–w = energy efficiency of the replacement of natural gas by blast-furnace gas. The coefficient a27 of the consumption of the second kind of coal results from its injection into the blast furnace. The consumption of coke-oven gas injected into the blast furnace is expressed in units of the replaced imported natural gas:

a37c

ak 7

( MH L ) k Ȟ 3 k ( MH L )3

(5.18)

where ak7 = coefficient of the consumption of the injected coke-oven gas. The by-production of electricity results from the expansion of blast-furnace gas ftr7 and from cogeneration with the production of steam for humidification of the blast: f 47

f tr 7  D(1  ȗ D ) X D eX

(5.19)

where eX = by-production of electricity per unit of back-pressure steam. The coefficients of consumption ai7 and by-production fi7 depend on the operational parameters applied in the blast-furnace process: ai 7

f ( F , TD , O 2D , X D , pG ),

fi 7

F( F , TD , O 2D , X D , pG )

(5.20)

where TD = temperature of the preheated blast PG = pressure at the top of the blast furnace. Relations (5.20) have been determined by means of the balances of elements and energy [53]. The mean values of the emission indices of waste products have been assumed: p7SO2

0.182,

p7 NO x

0.042,

p7 d

0.367 kg/Mg pig iron.

The calculation results are presented in table 5.1. They should be corrected after an iterative determination of the thermo-ecological cost of the imported natural gas. The mutual connections between the considered pro-cesses result in a dependence of the values of the thermo-ecological cost of the remaining products (except pig iron) on the parameters of the blast-furnace process. However, this influence is very small. The thermo-ecological cost of the pig iron has been calculated with the following operational parameters: thermal parameters of the blast TD = 1100oC, O2D = 24%, pressure of the top gas pG = 0.3MPa, auxiliary fuel (pulverized coal) consumption 3GJ/(Mg pig iron).

100 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table 5.1: Exemplary values of the thermo-ecological cost. Material or energy

Thermo-ecological cost 1.12 HL 1.11 HL 815 MJ/kmol 3.13 MJ/MJ 49.6 MJ/kg 4.5 MJ/kg 32.2 MJ/kg

Coal for power plants Coal for coking plants Natural gas (domestic) Electricity from coal Coke Iron ore sinter Pig iron (liquid)

Example 5.4. In order to determine the values of the domestic thermoecological cost of imported fuels (natural gas and crude oil), the thermo-ecological cost of exported goods should be calculated. This problem has been solved by Stanek [54] using an iterative procedure. Table 5.2 contains after [54] data concerning the structure of the Polish domestic products exported in 1997. Table 5.3 presents the results of iterative calculations. In table 5.3 the thermo-ecological cost of exported machinery and devices is expressed in MJ/$. Table 5.2: Structure of the export of domestic products from Poland in 1997 [54].

Branch number Name and unit

Amount per year 106 kg/year $/year

1

Coke, kg

3234

2

Coal, kg

29 500

3,4

Steel blocks, kg

5

Monetary Thermo-ecological value per cost per year monetary unit, 106 $/year MJ/$ 273

542.6

1124.5

711.8

1124

243.3

195.6

Sulfur, kg

1127.4

48.5

571.9

6

Copper, kg

361.6

346

383.5

7

Cement, kg

2827

113.8

148.1

8

Steel products, kg

2935

9

Aluminium, kg

111.9

10

Machines and devices, $

1268.6

11 12

1128

146.1

302.7

92.5

1268.6

13.9

Agricultural prod. (meat), kg 444.8

643

17.5

Agricult. prod. (vegetal), kg

647.3

15.5

979.2

DEPLETION OF NON-RENEWABLE NATURAL EXERGY RESOURCES 101

Table 5.3: Iterative calculation of the thermo-ecological cost of exported goods, MJ/kg (No 10* in MJ/$) [54]. Iteration Branch

0

1

2

3

1

49.61

45.85

45.77

45.77

2

27.10

27.13

27.13

27.13

3

39.34

44.54

44.64

44.64

4

35.18

38.95

39.07

39.02

5

24.59

24.60

24.60

24.60

6

369.5

366.97

366.92

366.92

7

6.95

5.96

5.96

5.96

8

49.42

56.10

56.23

56.23

9

249.36

249.62

249.63

249.63

10*

13.00

13.09

13.92

13.92

11

13.92

25.26

25.48

25.48

12

5.59

10.19

10.27

10.27

In the last column of table 5.2 the values of the thermo-ecological cost of the exported products are cited. Most advantageous, from the point of view of the depletion of domestic natural resources is the export of machinery and food. Most disadvantageous is the export of raw materials and fuels, particularly coal. The results of iterative calculations of the thermo-ecological cost of imported fuels are presented in table 5.4 [54]. Table 5.4: Thermo-ecological cost of the imported fuels [54]. Thermo-ecological cost per monetary unit MJ/$

Imported crude oil MJ/kg

Imported natural gas MJ/kmol

0

31.13

6.85

95.89

1

218.1

47.9

671.7

2

221.7

48.7

682.8

3

221.8

48.8

683.0

4

221.8

48.8

683.0

Iterative step

102 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

The domestic thermo-ecological cost of imported fuels depends on the structure of the export. This dependence has been presented in table 5.5 after Stanek [54]. The smaller the ecological cost of the exported product per monetary unit, the smaller is the thermo-ecological cost of imported fuels.

5.6. Sustainability index The ratio of the thermo-ecological cost to the exergy characterizes the fraction of the non-renewable exergy in the fabrication process of the considered useful product. Usually its value is greater than 1. For some products fabricated with a large participation of the renewable exergy we can obtain values smaller than 1 (for example, for forest or agricultural products, for electricity from wind- or water power stations). The development of human economy should result in a reduction of the discussed index. Its value smaller than 1 is desirable as a condition of sustainable development. Therefore, the name sustainability index has been proposed for the thermo-ecological cost to exergy ratio [5]: rB

ȡ b

(5.21)

In table 5.6 some exemplary values of the sustainability index have been presented. The indices for industrial products result from the calculations of Stanek [54]. To compare the industrial products with the agricultural ones, two values have been determined according to Learch [32]. When considering the production of corn, the thermo-ecological cost is smaller than the exergy of the product since this product is generated mainly from the exergy of solar radiation. However, further Table 5.5: Influence of the structure of export on the thermo-ecological cost of imported natural gas [54].

Variant of structure

Thermo-ecological Thermo-ecological cost per monetary cost of natural gas unit of export, MJ/$ MJ/kmol

Ratio U/b

Structure of export similar to that of production

31.1

95.9

0.12

Structure of export similar to that of production

31.1

95.9

0.12

Structure of export corresponding with table 5.2.

221.8

683.1

0.85

Export based on coal

711.8

2192

2.7

Export based on machinery

13.9

42.8

0.05

Export based on agricultural productss

16.6

51.1

0.06

DEPLETION OF NON-RENEWABLE NATURAL EXERGY RESOURCES 103

Table 5.6: Sustainability index of some industrial and agricultural products. Material or energy kind

Thermo-ecological cost

Sustainability index

Coal for power plants

22.6 MJ/kg

1.037

Coal for coke production

31.1 MJ/kg

1.028

Natural gas (domestic)

815 MJ/kmol

1.016

Natural gas (imported)

683 MJ/kmol

0.85

Crude oil (imported)

48.8 MJ/kg

1.07

Electricity

3.13 MJ/MJ

3.13

Coke

49.6 MJ/kg

1.554

Pig iron*

32.2 MJ/kg

3.68

Corn

4.2 MJ/kg

0.35

Flour

22 MJ/kg

1.75

Meat

26 MJ/kg

2.50

(*) See table 5.1. processing uses mainly non-renewable exergy, and just after the second step (the production of flour) the thermo-ecological cost becomes greater than the exergy of the product. The value for meat [54] results from mean statistical data concerning the energy consumption in the meat industry.

Exercises 5.1. Calculate the sustainability index characterizing the production of heat by means of the heat pump considered in exercises 2.15 and 3.2. 5.2. Calculate the sustainability index characterizing the production of heat by means of a compressor heat pump extracting the bottom heat from the ground and compare this heat pump with that absorbing the bottom heat from the atmospheric air. The mean temperature of the ground is 8qC. The mean air temperature during the heating season is 2qC. The working fluid is freon R22. Its evaporation temperature is 6qC (for heat absorption from the ground) and 0qC (heat absorption from the atmospheric air). The condensation temperature is 45qC. The internal isentropic efficiency of the compressor is 0.8, the electromechanical efficiency of a complex compressor-electrical motor is 0.75. The heat of mechanical and electrical losses does not flow to the heated room. The tempe-rature of the heated room is 20qC. The influence of the consumption of materials used for the construction of the heat pump may be neglected. The

104 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

cumulative efficiency of electricity delivery is 0.31. Electricity is produced by means of non-renewable fuels. 5.3. Calculate the thermo-ecological cost and sustainability index characterizing the production of heat by means of a solar collector. Assume that the mean yearly thermal power of the collector is 100 W/m2. The temperature of the preheated heat carrier is 40qC and of the return carrier 25qC. The consumption of materials per 1 m2 of the collector area is: copper tubes and plate 27 kg/m2, steel 20 kg/m2, glass 10 kg/m2. The recovery coefficient of metallic scrap, eqn (3.9) is: copper 0.5, steel 0.3. The thermo-ecological cost of glass is 33.4 MJ/kg. The amount of glycol filling the system is 20 kg/m2, its specific chemical exergy amounts to 19450 kJ/kg, and CExE = 0.72. The life time of the collector is 15 years. The annual mean ambient temperature is 8qC. 5.4. Calculate the thermo-ecological cost and sustainability index characterizing the production of electricity in a wind power plant. The annual utilization time of the nominal power is 2000 h/year, the steel consumption per 1 MW of nominal power is 240 t/MW, the life time is 15 years, the utilization efficiency of the steel scrap (after the wear of the installation) is 0.35. The efficiency of electricity transformation and transmission 0.9. 5.5. Compare the thermo-ecological cost of electricity produced in Poland in a power plant fed with hard coal and imported natural gas. The energy efficiency of the power plant is 0.38 for hard coal and 0.37 for natural gas. The cumulative exergy efficiency of coal delivery 0.93. The efficiency of the transformation and transmission of electricity is the same in both cases and amounts to 0.9. According to table 5.4 the thermo-ecological cost of imported natural gas is 683 MJ/kmol. Its lower heating value is 778 MJ/kmol. The thermo-ecological cost of the production equipment may be neglected.

Chapter 6 Economic applications of exergy 6.1. Exergo-economics Exergy losses appearing in particular links of the analyzed system are not all of the same importance. In the course of thermodynamic processes the specific economic value of exergy increases. For example, in a steam power plant the specific economic value of fuel exergy is smaller than that of live steam, and the latter one is smaller than the specific economic value of the produced electricity. Hence, exergy losses are more harmful, the farther the analyzed link is located in the chain of partial processes. The cited statement stimulated many authors to propose economic applications of exergy called thermoeconomics, exergo-economics or exergonomics [18, 28, 29, 94–96, 98]. However, exergy is a thermo-dynamic concept, not an economic one [61]. An interesting variant of exergo-economics elaborated by Tsatsaronis and his coworkers [95] aims at the determination of the costs of exergy losses appearing in particular links of the considered system. The comparison of the calculated costs can indicate the links which should be improved. The sum of the costs of exergy losses can be used as an objective function when optimizing the design and operational parameters (which should be performed using the iterative method). Two methods of optimization of production processes can be applied: the method of the minimization of production costs or the method of minimization of the cost of exergy losses. The first method is usually applied. A scientifically correct determination method of the cost of exergy losses is difficult, it should be based on the comparison of costs of a real process and a reversible process. Tsatsaronis does not apply the mentioned comparison. He uses the equations of exergy balance and of costs balance. He differentiates the components representing the ‘fuel’ and those representing the useful product. For example, when analyzing an adiabatic steam turbine the exergy of fuel is expressed by means of the difference of the exergy B1 of the delivered steam and exergy of

106 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

the outlet steam B2. The performed mechanical work W is the useful product. The cost of the produced work Cw results from the balance of costs: B1c1  B2 c2  R

Cw

(a)

where c1, c2 = specific cost of exergy of the components of ‘fuel’, R = component of costs resulting from the investment expenditures. The cited author maintains that from the costs balance results the equality c1

(b)

c2

However, the equality (b) is purely conventional, because the outlet “fuel component” differs in parameters and quality from the inlet “fuel component”. The exergy balance of the real and reversible process has a form: B2  W  įB, B1s

B1

B2 s  W

(c)

where W = the performed work GB = sum of exergy losses in the considered process (together with external losses if they exist). The index s indicates the quantities appearing in the reversible process. The balance of costs of a comparative reversible process may be expressed in the form: Cws

B1s c1  B2 s c2 s  Rs

(d)

Assuming: c2

c2 s

c1 , R

Rs

(e)

and taking into account the difference of exergy balances (c) we obtain after subtracting eqns (a) and (d) the economic loss Cirr due to the irreversibility of the considered process: Cirr

Cw  Cws

įBc1

(6.1)

Equation (6.1) results from the convention (e). However, despite the formulated critical remarks, the method of Tsatsaronis is worth consideration and application, because it makes it possible to evaluate approximately and compare the economic onerosity of exergy losses appearing in particular links of the considered system. Other variants of thermo-economy assume that the specific cost of exergy is the same for all the useful products of a cogeneration process. This assumption is used at the partitioning of production costs between the useful products appearing simultaneously in the same point of the considered system. The cited

ECONOMIC APPLICATIONS OF EXERGY 107

assumption is also purely conventional. It does not take into account the influence of the co-generation process on the economic effects of the total system of energy and technological processes of the state. The partitioning of the production costs between the useful products appearing simultaneously in the same point of the system can be made by means of purely economic tools, that is using the principle of avoided expenditures [81]. In every cogeneration process a major product can be distinguished. Its demand determines the location of the plant and the production rate. The additionally fabricated product can be called a by-product if it substitutes a major product of another specialized process. The production cost of the by-product can be determined according to the production costs avoided in the substituted specialized process. Thus, the principle of avoided expenditures takes into account the influence of the co-generation process on the total energo-technological system. The principle of avoided expenditures can also be used to determine the consumption of the driving exergy burdening the fabrication of the major product of a cogeneration (combined) process: b fm

1 Gm

§ bb · ¨© B f  Gb Ș ¸¹ Bs

(6.2)

where bfm = specific consumption of the driving exergy burdening the major product Gm,Gb = amount of the major product and by-product Bf = overall consumption of driving exergy bb = specific exergy of the by-product KBs = exergy efficiency of the avoided process of separate fabrication of the by product. Some authors propose a minimization of the entropy generation in the considered link of the system, as a method of determining the optimum operational and construction parameters [4]. However, entropy is generated not only in the considered link of the system, but also in the preceding links producing semi-finished products and energy carriers supplied to the considered link [78]. When the considered process produces also waste products rejected to the environment (external exergy losses), the entropy generation connected with the considered process appears also in the environment. Hence, the minimization of entropy generation does not ensure any minimization of cumulative exergy losses connected with the considered process. It is additionally worth stressing that even the minimization of cumulative exergy losses is not always expedient. If the considered process is supplied with renewable energy carriers, the objective function for the optimization should not contain the exergy losses (entropy generation) connected with the utilization of renewable exergy. For example, when optimizing the solar collector it would not be reasonable to take into account the entropy generation connected with the absorption of solar

108 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

radiation because the destruction of exergy of solar radiation is not connected with any economic or environmental losses.

6.2. Optimization of the thermo-ecological cost The thermo-ecological cost may be accepted as an objective function when optimizing the operational and construction parameters of some installation. The objective function should contain the operational component comprising the thermo-ecological cost of raw materials, semi-finished products and energy carriers supplied to the investigated process and the investment component comprising the thermoecological cost of the applied machines and installations. According to the proof presented in chapter 3.2 the thermo-ecological cost of human work should not be introduced into the objective function. A considerable reduction of the depletion of non-renewable resources may be attained by the secondary utilization of the materials remaining after the wear and dismantling of the installation [15, 16]. This effect can be expressed by means of the recovery factor um taking into account the fraction zs of the initial amount of the mth material which can be used as a secondary raw material and the ratio of the specific thermo-ecological cost Us of the substituted raw material to the initial thermo-ecological cost of the mth material: um

zs

ȡs ȡm

(6.3)

When the investigated process produces one useful major product and some useful by-products, and the required production capacity is given, the objective function expressing the depletion of non-renewable exergy, and related to 1 year has the form taking into account the principle of avoided expenditures and the thermo-ecological cost of the products replaced by the useful by-products of the investigated process [86]: PA

§ IJn ¨ ¨© 1ª  « IJ «¬

¦ G ȡ  ¦ P ȟ  ¦ G ȡ s j

j

j

k

u

¦ G ȡ 1  u ¦ m m

m

·

u i iu ¸

k k

m

r

¸¹

(6.4)

º Gr ȡ r » »¼

where G j ,ȡ j = nominal flow rate and specific thermo-ecological cost of the jth raw material, semi-finished product or energy carrier supplied to the production process Pk , ȟ k = nominal flow rate and index of the specific thermo-ecological cost of the kth deleterious waste product rejected to the environment

ECONOMIC APPLICATIONS OF EXERGY 109

G u = nominal production rate of the useful uth by-product

UiSiu = specific thermo-ecological cost of the ith major product replaced by the uth by-product and the replacement ratio in units of the ith replaced product per unit of the uth by-product Wn = annual operation time with nominal capacity W = nominal life time of the installation, years um = expected recovery factor of the mth material, eqn (6.2) Gm,Um = consumption and specific thermo-ecological cost of the mth material or energy carrier used for the construction of the installation Gr,Ur = expected consumption and specific thermo-ecological cost of the rth material or energy carrier used in repairs. Example 6.1. The optimization of the pipe diameter of a heat exchanger [86]. A tubular heat exchanger is considered. The consumption of heating fluid (for example, steam), its temperature, the flow rate of heated water per one pipe, the initial and final temperature of the heated water are given. The internal diameter of the pipe is to be optimized. The decrease in this diameter increases the flow velocity of heated water, and so decreases the heat transfer resistance (hence, the heat transfer area), but increases the flow resistance (hence, the consumption of electricity driving the water pump). The resistance of the heat conduction in the pipe wall can be neglected. A constant heat transfer resistance on the external surface of the considered pipe can be assumed. According to eqn (6.4) the objective function can be formulated as follows:

PA

§ įp · 1 IJ n ¨ G s ȡ s  Vw ȡel ¸  ¬ª ʌLDtȖ st ȡ st 1  ust  ȡ P ¼º ȘP © ¹ IJ

(6.5)

where G s ,ȡ s = nominal flow rate and specific thermo-ecological cost of the heating steam Vw , įp = volumetric flow rate and pressure loss of the heated water KP = efficiency of the complex: water pump with driving electric motor Uel = specific thermo-ecological cost of electricity L,D,t = length, internal diameter and wall thickness of the considered pipe Jst,Ust = mass density and specific thermo-ecological cost of the steel pipe UP = thermo-ecological cost of the pump. The thermo-ecological cost of the heating steam is constant and can be omitted in the objective function. A constant value of the thermo-ecological cost of the

110 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

pump may also be accepted. Introducing additionally the known heating power Q one can obtain: PA

IJ nVw

 įp 1 Qt ȡel  Ȗ st ȡ st (1  ust ) ȘP IJ Į'Tm

(6.6)

where 'Tm = the mean temperature difference between the heating and heated fluid D = convective heat-transfer coefficient on the internal surface of the considered pipe. The pressure loss of the heated water can be expressed as follows: įp

Ȝf

L w2 Ȗw D 2

Ȝ f LȖ w

8Vw2 ʌ 2 D5

Ȝf

8V 2 Q Ȗ w 3 w6 Į'Tm ʌ D

(6.7)

where Of = coefficient of hydraulic friction D = internal diameter of the considered pipe w = flow velocity of water Jw = mass density of water. Additionally, the following expressions are introduced: D

4Vw Ȗ w , Ȝf ʌ Re Șw

0.046 Re 0.2 , Į

Ȝw 0.023Re0.8 Pr 0.4 D

(6.8)

where KwOw = dynamic viscosity coefficient and thermal conductivity of water. The following formula expresses the operational component of the annual thermo-ecological cost, where the quantity Re represents the decision variable: PAop

IJ n ȡel 0.154 Ș5w Re 4 Q 0.4 Ș P 'Tm Ȝ w Pr Ȗ 4wVw2

(6.9)

In order to develop a similar formula for the investment component of the thermo-ecological cost, a linear dependence between the thickness of the pipe wall and its internal diameter has been assumed: t

t0  aD

t0  a

4Vw Ȗ w ʌ Re Șw

(6.10)

The investment component has the form PAinv

Ȗ st ȡ st (1  ust ) 55.4Vw Ȗ w Q 0.4 IJ 'Tm Ȝ w Pr Șw Re1.8

§ 4Vw Ȗ w · ¨ t0  a ʌȘ Re ¸ © ¹ w

(6.11)

The form of eqns (6.9) and (6.11) indicates that the result of optimization does not depend on the value of the expression Q ( 'Tm Ȝ w Pr 0.4 ) . The introduction of these quantities into the calculations makes it possible to determine only the annual thermo-ecological cost of the considered device.

ECONOMIC APPLICATIONS OF EXERGY 111

The following data have been accepted for exemplary calculations: Wn = 4200 h/year, W = 10 years, Uel = 3.15, KP = 0.7, Ust = 60 MJ/kg, ust = 0.35, Jst = 7800 kg/m3, Vw = 0.0005 m3/s, Jw = 1000 kg/m3, Ow = 0.68 W/(m K), Pr = 1.7, Kw = 2.8u10–4 N˜s/m2, Q = 30 kW, 'Tm = 40 K, t0 = 0.001 m, a = 0.075. After introducing these data eqns (6.9) and (6.11) take the form: PAop 6.45 u1023 Re4 GJ/a, PAinv

5.97 u108 § 170.5· ¨ 0.001 ¸ GJ/a Re ¹ Re1.8 ©

(6.12)

where the quantity 0.001 + 170.5/Re expresses the thickness of the pipe wall. The calculation results are presented in fig. 6.1. The optimum internal diameter of the pipe results from the optimum value of Re, according to eqn (6.10): Reopt 80000 Dopt

0.0284 m

The optimum flow velocity of the heated water is 1.27 m/s.

Figure 6.1: Determination of the optimum internal diameter of the pipe of a water preheater.

112 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Example 6.2. Optimization of a shell-and-tube water preheater [86]. A tubular water heater with a once-through flow of water has been considered. The stream of water and its initial and final temperature are given. Saturated steam with a given temperature is the heating medium. When analyzing this heat exchanger two decision variables should be optimized, for example, the number of tubes (or water stream per one tube) and the internal diameter of the tubes (or the internal Reynolds number). The following data have been accepted for exemplary calculations: Q = 600 kW,

Vwt = 0.005 m3/s,

ust = 0.5,

W = 15 years,

t0 = 0.0015 m.

The remaining data are the same as in the Example 6.1. The investment component of the objective function should be supplemented in comparison with (6.11) because the consumption of steel should take into account not only the tubes but also the remaining parts of the heat exchanger. Additionally, the consumption of electricity for the mechanical working and welding of the parts of the heat exchanger should be introduced into the objective function. The constant part of the mentioned components of the objective function may be omitted. It can be assumed that the diameter of the shell is proportional to the diameter of the tubes and to the square root of the number of tubes. The mass of the shell is additionally proportional to the length of the tubes. Hence, the changing part of the mass of the remaining parts of the heat exchanger (except the tubes) may be presented as follows: Gstc

țD n (1 ȤL)

(6.13)

where N,F = coefficients resulting from the design of heat exchangers. Similarly, the changing part of the consumption of electricity for the mechanical working and welding of the parts of the heat exchanger should be divided into two components, one proportional to the total perimeter of the crosssection of the tubes, (hence, to the product nD), and the second one to the diameter of the shell (hence, to the product D n ): Eel

ȝDn  ȞD n

§ Ȟ · Dn ¨ ȝ  ¸ © n¹

(6.14)

where P,Q = coefficients of proportionality. The following coefficients of proportionality have been accepted: ț 700 kg/m Ȝ = 0.15 l/m, ȝ = 0.08 GJ/m, Ȟ 0.3GJ/m.

The form of eqns (6.13), (6.14) and the values of the coefficients of proportionality depend on the type of the heat exchanger. The pressure loss of the heated water should take into account the flow resistance inside the pipes and the local resistance at the inlet and outlet of the

ECONOMIC APPLICATIONS OF EXERGY 113

tubes. It has been assumed that the sum of local resistances equals the dynamic pressure of the water stream inside the pipes: 1 § L · Ȗww2 ¨ Ȝ f 1¸ © D ¹ 2

įp

(6.15)

where Of = coefficient of hydraulic friction L, D = length and internal diameter of the considered pipe w = flow velocity of water The following formula expresses the operational component of the annual thermo-ecological cost, where the quantity Re represents the decision variable: PAop

WnUel 2 4 ª 0.154Kw5 Kw4 º Q n Re « 0.308  » 0.4  2 4 KP VwtJ w3 ¼» ¬« 'TmOw Pr Vwt J w

(6.16)

The investment part has a more complicated form:

PA inv

 J J § Ust 1  ust ª 55.4QV 4aVwt J w · wt w st t  « ¸ 0.4 1.8 ¨ 0 W SKw n Re ¹ «¬ 'TmOwKw Pr Re © 

·º 4NVwt J w § 13.85OQ » ¨1  0.4 0.8 ¸ SKw n Re © 'TmOw Pr Re ¹ »¼ 

(6.17)

Uel 4Vwt J w § Q · P W SKwRe ¨© n ¸¹

After introducing the given data, eqns (6.16) and (6.17) take the form: PAop 3.868 u1023 n2Re4 GJ/a

PAinv

2.754 u1011 § 1706· 3.18 u104 § 3.7 u104 · 0.0015 ¨ ¸ ¨1 ¸ n Re ¹ n Re1.8 © n Re © n Re0.8 ¹

(6.18)

(6.19)

48 § 30 · GJ  ¨8  ¸ Re © n¹ a

Calculations may be performed for some values of the number of tubes. For every assumed value of n, the local optimum value of Re can be determined. After that the optimum values of two decision variables may be selected. The calculation results are presented in table 6.1. The optimum values can be accepted as follows: nopt = 25,

Reopt = 48 000, Dopt = 0.019 m, Lopt = 1.78 m, wopt = 0.711 m/s, Dopt = 5686 W/(m2 K), Vw = 0.0002 m3/s.

114 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table 6.1: Optimization of a shell-and-tube water preheater. n

Re

PA op GJ/year

PA inv GJ/year

PA op + PA inv GJ/year

22

40 000 50 000 60 000

0.0479 0.1170 0.2426

0.6476 0.4739 0.3717

0.6955 0.5909 0.6143

24

40 000 48 000 50 000

0.0570 0.1183 0.1392

0.4271 0.3199 0.2949

0.4841 0.4332 0.4341

25

40 000 48 000 50 000

0.0619 0.1283 0.1511

0.4017 0.3024 0.2830

0.4707 0.4307 0.4341

26

40 000 50 000 60 000

0.0669 0.1634 0.3389

0.3918 0.2725 0.2059

0.4587 0.4359 0.5448

27

40 000 50 000 60 000

0.0722 0.1762 0.3654

0.3766 0.2629 0.1992

0.4488 0.4391 0.5646

6.3. Optimization of the exergetic cost According to the opinion of Sciubba [49, 50] the optimization of the operational and construction parameters should be performed by means of a quantity containing not only the cumulative consumption of exergy of all the consumed materials and energy carriers, but also the cumulative consumption of exergy burdening the human work. In the present book the quantity introduced by Sciubba has been named exergetic cost. The mean exergetic cost of human work should result from the ratio of the total consumption of non-renewable exergy in the considered country to the total amount of work-hours. The total exergy cost of the human work should be equal to the total consumption of non-renewable exergy. If the optimization comprises only the exergy cost of human work (immediate or embodied), the results should be similar to that of classical economic optimization because in both cases the main task of the optimization would be the economy of human work. The optimization of the thermo-ecological cost (determined without any thermo-ecological cost of human work and demonstrated in the previous paragraph) ensures the economy of non-renewable exergy resources. In this case the total exergetic cost of all the consumed materials should also equal the total consumption of non-renewable exergy. Hence, the optimization of the exergetic cost leads to a result representing some compromise between the economy of human work and the economy of non-renewable exergy resources.

ECONOMIC APPLICATIONS OF EXERGY 115

6.4. Correction of the economic optimization Sometimes the economic analysis does not take into account the influence of the parameters of an energy carrier on its quality. Such an inaccuracy can be corrected by means of the exergy analysis. For example, the heat lost from the pipeline of live steam in a steam power plant has a better quality, than the mean quality of heat delivered inside the boiler. The economic value of the lost heat is higher than its mean value. It can be approximately evaluated by means of exergy: k km

T  T0 Tm T Tm  T0

(6.20)

where km,k = mean specific cost of the heat delivered inside the boiler and local specific cost of the heat lost from the pipeline Tm,T = mean temperature during the absorption of heat inside the boiler, and local temperature in the pipeline. After this correction the calculated optimal thickness of the thermal insulation of the pipeline increases.

6.5. Influence of the thermodynamic imperfection on the investment cost The improvement of the exergy efficiency of some thermal installation requires an increase of investment expenditures. However, the higher the exergy efficiency of the selected solution, the larger additional investment expenditures are required to improve the installation. The investment cost becomes infinitely large, if the required exergy efficiency amounts to 100%. Therefore, after Szargut and Maczek [63] an approximate equation can be formulated expressing the dependence between the exergy efficiency KB and the rated exergy capacity B of single-purpose installations: J

 § ȘB · J0  Bj 0¨ © 1 ȘB ¸¹

m

(6.21)

where J0 = constant component of the investment cost j0 = factor of investment cost increment per unit of exergy capacity m = exponent related to the type of installation. The annual operational cost of the installation can be expressed as follows: §k · K BIJn ¨ B  K¸  K0 © ȘB ¹

where Wn = annual time of operation with a rated capacity kB = unit cost of driving exergy

(6.22)

116 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

ki = unit cost of service, maintenance, and renovation K0 = constant component of the annual operational cost. The total annual cost of the production of the required annual useful exergy effect is KA İJ  K

(6.23)

where H denotes the annual capital recovery factor (factor of fixed costs). The exergy efficiency is a sole decision variable in eqn (6.22). The optimum value of this efficiency, ensuring a minimum annual production cost, is expressed as follows: ȘBopt

1 1 (mL)1/(m1)

(6.24)

where: L

İj0 kB IJn

(6.25)

is the dimensionless number of thermoeconomic similarity. The presented equations do not take into account the technical limitations that can arise in the design of the installation.

6.6. Evaluation of the natural mineral capital and the freshwater resources of the Earth 6.6.1. Natural mineral capital of the Earth Valero and his co-workers have undertaken an attempt to evaluate the mineral capital of the Earth in the area of the production of the most important metals [99, 100]. They considered 42 elements, mainly metals. The mineral capital has been defined as the economy of the consumption of primary exergy which can be attained thanks to the extraction and utilization of natural resources whose useful components are more concentrated than in the reference environment or appear in the form of chemical compounds with positive chemical exergy (for example in the form of sulfides). Hence, the quantity determining the mineral capital contains two components, the concentration component and the chemical component. The concentration component of the mineral capital results from the difference of the cumulative consumption of the primary exergy which is necessary to extract the useful component from the reference environment and from the mined mineral. In both cases the final concentration of the useful component should equal the value required for further metallurgical treatment. The approximate evaluation is based upon the assumption of ideal solutions.

ECONOMIC APPLICATIONS OF EXERGY 117

The equations formulated in the present book differ from those of Valero et al. A scheme of the separation of the mined material is presented in fig. 6.2. To obtain raw material prepared for metallurgical treatment, containing 1 mol of the useful component with its molar concentration yr, the amount of the mineral extracted from the mine should be 1/y1 (where y1 is the molar fraction of the useful component in the mined mineral), and the obtained amount of the raw material will be 1/yr. The minimum work necessary to perform the separation presented in fig. 6.2 (assuming that the considered materials have the properties of an ideal solution) may be expressed as follows: ª y 1 yr 1 yr § 1 1 · 1 º  ¨  ¸ ln W1 RT0 «ln r  ln » yr 1 y1 © y1 yr ¹ 1 y1 »¼ «¬ y1

(6.26)

The minimum work W2 necessary to separate the metallurgical raw material from the reference environment (with the molar fraction y0 of the useful component) should be calculated under the assumption that this separation does not influence the composition of the environment (thanks to the considerably larger mass of the environment): W2

§ y 1 yr 1 yr · RT0 ¨ ln r  ln yr 1 y0 ¸¹ © y0

(6.27)

The difference W2–W1 expresses the theoretical economy of exergy attainable thanks to the use of natural resources which have a larger concentration of the useful component in comparison to the reference environment. The real economy, characterizing the natural mineral capital, should additionally take into account the cumulative exergy efficiency Ș s of the separation process: BNCi

ni ns

(W2 W1)

ni ns

ª y 1 y0 1 º 1  ln(1 y0 )  ln(1 y1)» RT0 «ln 1  1 y y y y 1 r 1 ¬ 0 ¼

(6.28)

where ni denotes the amount of the mol of the ith component in natural resources of this component.

Figure 6.2: Scheme of the separation of the mined mineral u — useful component, w — waste component.

118 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

The second term in square brackets of eqn (6.28) has usually a very small value. Example 6.4. In the case of producing iron, the compound Fe2O3 has been assumed as a useful component of natural resources. The minimum economic exploitable concentration has been assumed as y1 = 0.5. The molar fraction in the reference environment is y0 = 0.0068. The concentration required for metallurgical treatment is yr = 0.7. According to Valero et al [99] the world reserves of Fe2O3 in iron ores are 74u109 ton (4.63u1011 kmol). The cumulative exergy efficiency of separation is ns 0.03 (assuming the immediate efficiency 0.1 and cumulative efficiency of driving exergy production 0.3). From eqn (6.28) the concentration part of the mineral capital of iron-rich minerals is BNCFe2O3

6.95 u 109 GJ

5500 Mtoe.

Valero et al [99] calculated the value 10,164 Mtoe. They estimated the total concentration component of the natural capital to be 22,000 Mtoe (about 15% of the proven oil reserves). According to Valero et al [100] the concentration component of the annual consumption of the mineral capital is highest for iron ores (51.5%, 140 Mtoe/year), followed by phosphorus (11.6%), copper (8.3%), sodium (5.7%), zirconium (5.8%), aluminium (5.6%) and magnesium (5.1%). Natural resources applied for the production of metals sometimes have a high positive chemical exergy. Most important are sulfides. The consumption of exergy for the production of metal from sulfide can be smaller than in the case of production from oxides. Valero et al estimated the average attainable economy of exergy: about 80 GJ/ton metal. The natural mineral capital of the reserves of sulfides resulting from their chemical composition is about 1250 Mtoe, considerably less than the concentration component. 6.6.2. Resources of fresh water Valero et al [101] determined also the resources of the exergy of fresh water. This evaluation bases upon the statement that the chemical and potential exergies of seawater equal zero. Two components of the resources of fresh water have been considered: the stationary resources contained in nature (in subterranean reservoirs, lakes, rivers and ice sheets), and the renewable resources generated by the evaporation of liquid water from the ocean and continents surface. The evaluated stationary resources of fresh water (about 2.5% of total resources) are 35u1018 kg (about 69% in the ice sheet and 30% in subterranean reservoirs). According to the example 2.8 (p. 46) the total rate of evaporated water is 15u109 kg/s. Valero cites a slightly different value 18u109 kg/s. The evaporated water returns to the Earth’s surface in form of rain and snowfalls. However, only about 23% of them reach the surface of continents. Hence, the generation rate of renewable fresh water resources is about 3.45u109 kg/s. If the stationary

ECONOMIC APPLICATIONS OF EXERGY 119

resources had to replace the natural generation of fresh water, they would be exhausted in about 300 years. The minimum desalination exergy (minimum consumption of exergy for the separation of fresh water from the seawater) results from eqn (2.12a). According to the data from table 2.2, the average molarity of salt in seawater is 0.474 mol/kg H2O. Hence, the molar fraction of salt is 0.00854. From (2.12a) at T0 = 298K it results that bdes = 1.18 kJ/kg H2O. From the cited rate of precipitations reaching the surface of continents results the rate of the renewable chemical exergy of fresh water: 4.1 TW. This is near the rate of the potential energy of all rivers taken together (cited in example 2.8). Valero [101] cites the values of practical consumption of exergy in desalination processes. From comparison with the above cited chemical exergy result the following values of exergy efficiency of desalination processes: multistage evaporative installation inverse osmosis heat and power plant with evaporator

0.39% 2.7% 0.59–0.72%.

Exercise 6.1. Calculate the optimum thickness of a thermal insulation of the external wall of a house from the point of view of the economy of non-renewable exergy resources. The insulation layer consists of polystyrene foam with the thermal conductivity 0.04 W/(m K) and density 20 kg/m3. The initial heat transfer coefficient is 1.3 W/(m2 K). The internal temperature is 20qC. The mean ambient temperature in the heating season is 2qC, the duration of the heating season 5400 h/year. The thermo-ecological cost of heat is 1.32 J/Jth. The thermoecological cost of polystyrene production is 165 MJ/kg. The lifetime of the insulation layer is 20 years.

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Chapter 7 Application of exergy for determining the pro-ecological tax 7.1. Necessity of a new tax The income tax imposed in all countries represents some kind of penalty for the positive effects of human activity (effective work, invention etc.). Only VAT (the value added tax) burdens the consumption, but the height of VAT is established arbitrarily by the state administration. Therefore, some authors proposed to introduce taxes imposed for the negative effects of human activity, like the deleterious impact on the natural environment. Slesser [51] proposed to introduce a new environmentally sensitive concept in taxation. The new tax, named unitax, should replace VAT. According to this concept the unitax should be imposed on primary energy sources where they enter the economy. This denotes a considerable increase in the price of primary and transformed energy. Slesser does not use exergy. His concept does not give any possibility of taking into account the harmful effects of the environmental pollution because this pollution does not appear at the level of gathering the primary energy. Wall [105], Gong and Wall [23] proposed to apply the use of exergy of nonrenewable resources together with the exergy of waste products released to the environment as a base for an internationally governed tax. The income from this tax should support the research and other activities improving the exergy efficiency and the use exergy. Hence, Wall proposed an exergy-based tax as an auxiliary mean, supplementing the actual taxes. He has not formulated any equations determining the cumulative depletion of non-renewable natural exergy resources. Repetto, Dover et al [46] propose that the tax should rather be imposed for negative effects, like pollution, waste generation, congestion. The shift of taxes from the positive to negative effects of human activity by the introduction of a pollution charge should ensure not only ecological but also general economic profits because it should enhance a more rational use of natural resources and a better protection of the environment. According to Repetto et al [46], the shift of 1 US$ from the positive zone to the negative one can bring an economic profit of as much

122 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

as 0.45–0.8 US$. Hence, these authors proposed to substitute entirely the existing tax by a new pro-ecological pollution tax. Von Weizsäcker et al [102] write that the pollution tax could permit to attain a doubled well-being, at a twice lowered consumption of resources. Tsatsaronis [96] proposes to introduce a worldwide taxation agreement imposed on the use of electricity and fossil fuels to enhance the use of renewable resources. Environmental damages are not the sole and most important negative result of human activity. The depletion of non-renewable natural resources can be even more dangerous for the future of humankind [62]. Wall [103] proposed to apply exergy as a general measure of the quality of natural resources. Szargut [62, 80] introduced the concept of thermo-ecological cost expressing the cumulative consumption of non-renewable primary exergy, appearing in all the links of the energo-technological system as a result of the fabrication of some considered final product. Szargut [62] stressed the necessity to minimize the thermo-ecological cost. An important tool of this minimization can be a pro-ecological tax substituting the actual taxes. It should be proportional to the cumulative depletion of nonrenewable natural resources [85]. In the first step, the existing VAT could be substituted by an objective, non-arbitrary pro-ecological tax. In the second step also the personal income tax might be substituted.

7.2. Structure of the pro-ecological tax In many countries a tax burdening the consumption of goods and services, called VAT is applied. However, the height of this tax is arbitrary; it depends on the decision of the state administration. The proposed pro-ecological tax could be very similar to VAT, but should be proportional to the thermo-ecological cost (cumulative consumption of non-renewable primary exergy), and should substitute the VAT-tax being in force. The personal income tax could also be eliminated after introducing the pro-ecological tax, according to the principle that the tax should be imposed only for negative results of human activity. The index of proportionality between the pro-ecological tax and the thermoecological cost should depend on the total consumption of non-renewable exergy in the considered country. This quantity may be determined by means of statistical data. For example, after Stanek [53] the total consumption of renewable exergy in 1995 amounted in Poland to 4122 PJ. This value takes into account hard coal (3314.7 PJ), lignite (579.2 PJ), domestic natural gas (144.6 PJ), copper ore (15.2 PJ), sulfur (46.2 PJ) and domestic crude oil (22.1 PJ). The mentioned coefficient of proportionality should be expressed in monetary units per unit of exergy. The value of the coefficient may depend on the state policy. It should take into account the demand for financial means for investments, social needs, education, military needs, police, administration, etc. The drop of the coefficient would denote an approval of the increase of consumption.

APPLICATION OF EXERGY FOR DETERMINING THE PRO-COLOGICAL TAX 123

The explained principle needs some correction, if the pro-ecological tax should take into account also the harmful impact of waste products. To avoid a double counting, the coefficient of proportionality x might be lowered. The thermo-ecological costs of waste products should be added to the total consumption of the non-renewable exergy: x

¦Ȥ B¦Pȟ k

(7.1) k

k

where x = coefficient for the calculation of the pro-ecological tax ¦ Ȥ = the sum of all the tax values acquired by the state administration B = annual domestic consumption of non-renewable exergy Pk,[k = see eqn (5.1). The coefficient x should be corrected every year according to the rate of inflation, and to changes in the denominator of eqn (7.1). Changes of state expenditures should also be taken into account. Example 7.1. In 1997 the sum of the values of VAT amounted in Poland to 11.25u109 US$/year, and the sum of personal income tax to 9.10u109 US$/year. The value of the quantity in the denominator of eqn (7.1) is B  ¦ Pk ȟ k

4455 u 109  174 u 109

4629 u 109

MJ/a

k

From eqn (7.1) it results that x = 0.0044 US$/MJ. Example 7.2. According to Stanek [53], in 1997 the thermo-ecological cost of electricity in Poland was 3.13 MJ/MJ. The pro-ecological tax per unit of electricity would be 0.0141 US$/MJ= 0.0495 US$/kWh. The thermo-ecological cost of the domestic natural gas was 815 MJ/kmol. The ecological tax imposed on the domestic natural gas would be 3.59 US$/kmol = 0.160 US$/mn3 (m3 in normal conditions). Equation (7.1) can be explained by means of a simplified example. Assume that hard coal is the only non-renewable resource consumed by two plants only. The first of them burdens the environment with an amount Ps of deleterious waste products, the specific thermo-ecological cost of which is [s. The second plant does not produce any harmful waste products. Assume that the consumption of semi-finished products and energy carriers in coal mines can be neglected. According to eqn (7.1) the proportionality coefficient determining the pro-ecological tax can be expressed as: x

Ȥ1  Ȥ 2 Bc  Ps ȟ s

(7.2)

124 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

where F1, F2 = tax burdening the first and second plant, Bc = exergy of the consumed coal. The tax burdening the first and second consumer is Ȥ1

( Bc1  Ps ȟ s ) x, Ȥ 2

Bc 2 x

(7.3)

The sum of the thus determined values of the tax satisfies eqn (7.2). A double burdening of ecological losses with the tax has been avoided. The structure of the pro-ecological tax may be similar to that of VAT. The sale price of every product should be burdened with a tax resulting from the values burdening the input components used in the last production link and with a tax resulting from the thermo-ecological cost of the waste products of the last link. Every intermediate producer should receive a refund of the tax after he has sold his product. So the total value of the tax would be paid by the final consumer. The calculation of the thermo-ecological cost would not be necessary because the subsequent values of the tax would be calculated by the producers to receive the refund of the tax. The calculation of the refunded tax may be based on a balance equation similar to eqn (5.2) determining the thermo-ecological cost: Ȥ j  ¦ f uj Ȥ u u

¦a Ȥ ¦a ij

i

i

r

rj

§ · Ȥ r  ¨ ¦ bsj  ¦ pkj ȟ k ¸ x © s ¹ k

(7.4)

where aij, arj = consumption coefficient of the ith domestic and rth imported semi-finished product or energy carrier per unit of the jth major product together with the by-products, fuj = coefficient of the by-production of the uth product per unit of the jth major product, bsj = coefficient of the immediate exergy consumption of the sth nonrenewable resource per unit of the jth major product, together with the by-products, Fi, Fu = pro-ecological tax burdening the jth major product and uth byproduct, Fi, Fr = pro-ecological tax burdening the consumed domestic and imported semi-finished products and energy carriers, x = pro-ecological tax burdening the consumption of a unit of primary non-renewable exergy, eqn (7.1), Pkj = amount of the kth aggressive component of the waste products rejected to the environment per unit of the jth major product. Equation (7.4) does not contain only the tax burdening the jth major product but also the tax burdening the by-products. The product belongs to the by-products if it replaces the major product of another specialized process. Therefore, the pro-

APPLICATION OF EXERGY FOR DETERMINING THE PRO-COLOGICAL TAX 125

ecological tax burdening the by-product may be calculated by means of the replacement ratio and the tax burdening the ith replaced major product, eqn (3.7): Ȥu

siu Ȥ i ,

f ui

f ij siu

(7.5)

where the replacement ratio siu expresses the number of units of the replaced product per unit of the uth by-product. The replacement ratio should be determined by a qualified regulation office. The correctness of the calculation results depends on the proper treatment of the mines extracting non-renewable materials from natural deposits. The exergy of these materials should be carefully determined. Any mine (of coal, crude oil, natural gas, mineral raw materials) should pay the tax immediately after the extraction, and receive the refund together with the tax burdening the bought materials and the energy carriers. It might be most difficult to determine the tax burdening the waste products. Even if some partially conventional values of the monetary coefficients of harmfulness have been accepted and eqn (7.1) has been applied, additionally all the production installations should be equipped with monitoring instruments determining the deleterious emissions. The correction resulting from deleterious emissions is not very large, and therefore only the biggest production plants should have monitoring equipment. For small producers the component of tax taking into account deleterious emissions might be assumed conventionally (depending, for example, on the kind and consumption of the used fuel and on the kind of the applied technology).

7.3. Pro-ecological tax resulting from the use of machines and installations The pro-ecological tax burdening the purchase of machines and installations results from the procedure of determining this tax. This tax should be added to the sale price and reimbursed to the producer of the machines and installations. The purchaser using these machines and installations in his production processes should burden the price of his products with a respective fraction of the mentioned tax. Calculating this fraction, the degree of the utilization of machines and installations in the production process and their durability should be taken into account. The fraction of the tax burdening the ith machine or installation per unit of the jth product can be calculated as follows: aij

mij s jGj

(7.6)

where mij = number of ith machines or installations to be used in the planned period of the fabrication of the jth product sj = planned period of production of the jth product in years Gj = annual production of the jth product

126 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

The application of eqn (7.6) has an important disadvantage. The producer buying the production machine or installation pays the discussed tax immediately at the moment of purchase because it is contained in the price of the machine or installation. So the producer grants some kind of loan to the state, and the reimbursement will last some years in the form of an additional component of the price of his products. Therefore, a discount calculus should be applied. It can be done by means of the correction of the thermo-ecological cost of the production machine, or more simply by the correction of the coefficient aij in eqn (7.6). The discounted coefficient aijd, calculated without taking into account the rate of inflation, may be expressed as follows: aijd

mij

r

Gj

1 1 s (1  r ) j

aij

sjr 1 1 s (1  r ) j

(7.7)

where r = annual rate of the discount. The component of the pro-ecological tax added to the selling price of the jth product to refund the tax Fi paid when purchasing the production installation, calculated without inflation may be expressed as follows: Ȥ id

Ȥ i aijd

(7.8)

The discounted coefficient is higher than the not-discounted. When the value mij is not an integer, one of the machines is not worn after the finished period of production, the producer cannot get back the total initially paid tax included into the selling price of the product, but can sell the partially worn machine and so get back the remaining part of the initial tax. Example 7.3. Let us assume r = 0.08, s = 15 years. The ratio of the discounted and not discounted coefficient is aijd aij

1.752.

The value resulting from eqn (7.8) should be actualized every year according to the rate of inflation: Ȥ in

(1  in 1 )Ȥ i , n 1

(7.9)

where: n = order number of the year of operation of the production installation, in–1 = annual rate of inflation in the preceding year.

7.4. Burdening of imports and exports The pro-ecological tax burdening the exported products might be refunded to the exporter similarly as in the case of other intermediate producers and should not be added to the export price. On the other hand, it should be taken into account

APPLICATION OF EXERGY FOR DETERMINING THE PRO-COLOGICAL TAX 127

that exported products burden the domestic economy with some depletion of the resources of non-renewable exergy, but the financial means obtained from this export make it possible to import some necessary foreign products. Therefore, the imported products might be burdened with such a pro-ecological tax which can compensate the tax refunded to the exporters. The calculation can be performed by means of the monetary value of the export assuming that the monetary unit of import should be burdened with a pro-ecological tax equal to that determined for the monetary unit of export. The following formula resulting from this principle determines the pro-ecological tax burdening the imported products: Ȥi

Di

¦Ȥ

e

E

(7.10)

where Fi = pro-ecological tax burdening the unit of the ith imported product Di = monetary value of the unit of the ith imported product ¦ Ȥ e = sum of the values of pro-ecological tax, calculated for the exported products and refunded to the exporters E = monetary value of the annual export. Example 7.4. After Stanek [54] in 1997 in Poland the mean thermoecological cost per unit of the monetary value of the exported goods amounted to 220.3 MJ/US$. Thus, the pro-ecological tax per unit of the price of imported goods would be 0.97 US$/US$. The price of imported natural gas was 0.1 US$/ m3n . The pro-ecological tax burdening the imported gas would be 0.0976 US$/ m3n , hence, less than domestic natural gas. When the monetary value of the import is greater than the value of export, from eqn (7.4) results the sum of the values of the pro-ecological greater than the sum refunded to the exporters. Such an effect can enhance export and lead to a reduction of the import.

for the tax the

7.5. Discussion The introduction of the pro-ecological tax would not mean any increase in the total sum of taxes. Only the distribution of taxes would be changed. The introduction of the discussed taxes would stimulate the limitation of the use of products burdened with a large consumption of non-renewable resources and enhance the introduction of new technologies based upon renewable resources. Therefore, it would stimulate new investments and decrease unemployment. The proposed tax could reduce the consumption of less necessary products. It is worth stressing that actually the economic conjuncture is realized by an enhancement of the consumption. Such an economic policy is short-sighted because it leads to an acceleration of the depletion of scarce non-renewable natural resources of the Earth.

128 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Exercises 7.1. Calculate the pro-ecological tax which should be added to the selling price of electricity produced in a wind power plant. Assume the thermoecological cost of the equipment of the wind power plant 12000 MJ/kW. The annual time of utilization of the nominal power is 2000 h/year, the life time 15 years, the transformation and transmission efficiency of electricity 0.9. The annual rate of discount is 0.08, the rate of pro-ecological tax per unit of the thermo-ecological cost is 0.0044 US$/MJ. Assume that the financial indices relate to the first year of operation. The consumption of materials for the maintenance may be neglected. 7.2. Calculate the pro-ecological tax which should be added to the selling price of heat produced in a water boiler fed with hard coal. The energy efficiency of the boiler is 0.85. The thermo-ecological cost of the boiler may be neglected. Take the rate of the pro-ecological tax per unit of the thermo-ecological cost from exercise 7.1. The thermo-ecological cost of coal per unit of its heating value take from table 5.1. 7.3. Calculate the pro-ecological tax which should be added to the selling price of heat produced in a compressor heat pump with the energy index COP = 3.5. The thermo-ecological cost of electricity take from table 5.1. The thermoecological cost of the heat pump installation per unit of thermal power is 15 GJ/kW. The life time of the installation is 20 years, the annual time of utilization of the nominal power is 3500 h, the annual rate of discount is 0.08. Take the rate of the pro-ecological tax per unit of the thermo-ecological cost from exercise 7.1. 7.4. Calculate the total reduction of the pro-ecological tax attained thanks to the application of an additional layer of the thermal insulation to the external wall of a house. The thickness of the additional insulation is 0.2 m. Take the parameters characterizing the additional insulation, the conditions of heating and the values of the specific thermo-ecological cost from Exercise 6.1., and the rate of the specific pro-ecological tax from Exercise 7.1. The calculation result should include the total life time of the insulation, relate to 1 m2 of the area of the wall and be discounted to the first year after introducing the new insulation layer.

Appendix 1 Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 Al s 26.9815 930.69 795.7 Al4C3 s 143.959 694.51 4217.4 s 133.3405 467.18 352.2 AlCl3 101.9612 185.69 15.0 Al2O3 s, D corundum Al2O3·H2O s, boermite 119.9765 128.35 9.4 s, gibbsite 156.0072 24.13 24.1 Al2O3·3H2O s 150.155 3313.81 2705.3 Al2S3 s 342.148 596.8 344.3 Al2(SO4)3 s, andalusite 162.046 28.03 9.2 Al2SiO5 s, kyanite 162.046 25.94 12.9 Al2SiO5 s, sillimanite 162.046 0 15.3 Al2SiO5 s, kaolinite 258.1615 68.25 12.0 Al2SiO5·(OH)4 mullite 426.0536 630.11 63.2 3Al2O3·2SiO2 Ba BaCO3 BaCl2 BaO BaO2 Ba(OH)2 BaS BaSO4

s, II s, II s s s s s s, barite

137.34 197.35 208.25 153.34 169.34 171.36 169.4 233.4

C C CCl4

s, graphite s, diamond l

12.01115 12.01115 153.823

747.77 75.18 48.69 194.15 113.38 45.93 1012.88 0 393.509 395.406 578.95

775.1 53.7 88.7 252.0 196.7 160.3 929.0 30.7 410.26 413.16 473.1

130 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa (continued) Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 C2N2 g, cyanogen 52.0357 1096.14 1118.9 CH4 g, methane 16.04303 802.3 831.6 g, ethane 30.0701 1427.8 1495.8 C2H6 C3H8 g, propane 44.0972 2045.4 2154.0 g, n-butane 58.1243 2658.4 2805.8 C4H10 g, n-pentane 72.1514 3274.4 3463.3 C5H12 C2H4 g, ethylene 28.0542 1323.1 1361.1 C3H6 g, propylene 42.0813 1927.7 2003.9 C2H2 g, acetylene 26.0382 1255.6 1265.8 C6H6 g, benzene 78.1147 3171.6 3303.6 C6H6 l, benzene 78.1147 3137.7 3298.5 C7H8 l, methylbenzene 92.1418 3736.4 3931.0 C8H10 l, ethylbenzene 106.1689 4347.7 4587.9 C10H8 s, naphtalene 128.1753 4984.2 5255.0 C14H10 s, anthracene 178.2358 6850.9 7218.1 CH2O2 l, formic acid 46.0259 213.0 291.7 C2H6O l, ethylalcohol 46.0695 1235.9 1357.7 C2H4O2 l, acetic acid 60.0529 786.6 908.0 C3H6O l, acethone 58.0807 1659.6 1798.5 C6H6O s, phenol 94.1141 2925.9 3128.5 C2H2O4 s, oxalic acid 90.0358 202.7 368.7 CH4ON2 s, urea 60.0558 544.7 689.0 CO g 28.0105 282.984 275.1 CO2 g 44.0095 0 19.87 CS2 l 76.139 2934.09 1694.7 Ca s, II CaC2 s s, aragonite CaCO3 CaCO3·MgCO3 s, dolomite CaCl2 s s CaFe2O4 Ca2Fe2O4 s Ca2Mg5Si8O22(OH)2 s, tremolite s Ca(NO3)2 CaO s CaO·Al2O3 s CaO·2Al2O3 s 3CaO·Al2O3 s

40.08 64.1 100.09 184.411 110.99 215.77 271.85 812.41 164.0898 56.08 158.04 260.00 270.2

813.57 1541.18 0 0 178.21 161.07 321.00 425.49 –124.90 178.44 351.66 541.71 716.72

712.4 1468.3 1.0 15.1 87.9 104.0 194.7 78.9 –81.4 110.2 122.7 138.4 381.4

APPENDIX 131

Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa (continued) Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 12CaO·7Al2O3 s 1386.68 3415.71 1542.0 CaO·Al2O3·SiO2 s, anortite 222.038 273.92 65.6 Ca(OH)2 s 74.09 69.04 53.7 310.18 0 19.4 Ca3(PO4)2 s, D CaS s 72.14 1056.57 844.6 s, anhydrite 136.14 104.88 8.2 CaSO4 CaSO4·½H2O 145.15 83.16 12.1 s, D CaSO4·2H2O s, gypsum 172.17 0 8.6 CaSiO3 s, volastonite 116.16 90.24 23.6 Ca2SiO4 172.24 232.28 95.7 s, E s 282.32 424.94 219.8 Ca3SiO5 Cd Cd CdCO3 CdCl2 CdO Cd(OH)2 CdS CdSO4 CdSO4·H2O

s, D s, Q s s s s s s s

Cl2 Cl

g g

Cr Cr3C2 Cr7C3 CrCl2 CrCl3 Cr2O3 Cu CuCO3 CuCl CuCl2 CuFe2O4 CuO Cu2O

112.4 112.4 172.41 183.31 128.4 146.41 144.46 208.46 226.48

357.1 356.51 0 126.04 98.95 38,26 920.6 149.24 84.79

293.8 293.2 40.6 73.4 67.3 59.5 746.9 88.6 80.6

70.906 35.453

160.44 201.9

123.6 87.1

s s s s s s

51.996 180.01 400.005 122.902 158.355 151.99

569.86 2415.85 5007.63 361.91 281.05 0

584.7 2493.0 5156.7 352.2 301.9 117.2

s s s s s s s

63.54 123.55 98.99 134.45 239.23 79.54 143.08

201.59 0 144.57 151.95 60.62 44.27 234.56

134.2 31.5 76.2 82.1 36.1 6.5 124.4

132 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa (continued) Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 Cu(OH)2 s 97.55 –6.37 15.3 CuS s 95.00 873.87 690.3 Cu2S s 159.14 1049.1 791.8 CuSO4 s 159.60 155.65 89.8 Cu2SO4 s 223.14 377.15 253.6 D2 D2O D2O

g g l

Fe Fe3C FeCO3 FeCl2 FeCl3 FeCr2O4 Fe0.947O FeO Fe2O3 Fe3O4 Fe(OH)3 FeS FeS2 FeSO4 FeSi FeSiO3 Fe2SiO4 FeTiO3

s, D s, D cementite s siderite s s s s, wustite s s, hematite s, magnetite s s s, pyrite s s s s, fyalite s

H2 H HCl HDO HDO HNO3 H2O H2O H3PO4 H2S H2SO4

g g g g l l g l s g l

4.02946 20.02886 20.02886 55.847 179.552 115.856 126.753 162.206 223.837 68.8865 71.846 159.692 231.539 106.869 87.911 119.075 151.909 83.933 131.931 203.778 151.75 2.01594 1.00797 36.461 19.0213 19.0213 63.0129 18.01534 18.01534 98.0013 34.08 98.077

249.199 0 –45.401 412.12 1654.97 65.06 230.77 253.29 107.1 124.01 140.16 0 117.98 –48.14 1037.54 1684.72 209.11 1249.42 118.07 255.3 118.9 241.818 338.874 108.82 0.21 –44.38 –53.19 0 –44.012 –76.26 946.61 153.25

263.8 31.2 22.3 374.3 1553.9 123.8 195.5 228.1 207.0 111.3 124.9 12.4 116.3 37.5 883.5 1426.6 170.9 1155.5 159.9 232.3 129.6 236.09 331.3 84.5 18.8 10.0 43.5 9.5 0.9 89.6 812.0 163.4

APPENDIX 133

Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa (continued) Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 K s 39.102 356.63 366.6 KAlSi3O8 s, adulare 278.337 66.26 7.4 K2CO3 s 138.213 –43.58 85.1 KCl s 75.555 0 19.6 KClO4 s 138.553 6.67 136.0 K2Cr2O7 s 294.184 –190.4 34.3 KNO3 s 101.1069 –135.90 –19.4 K2O s 94.203 350.04 413.1 KOH s 56.109 52.72 107.6 K2S s 110.268 1024.4 943.0 K2SO3 s 158.266 300.47 302.6 K2SO4 s 174.266 4.62 35.0 K2SiO3 s 154.288 75.9 138.2 Mg MgAl2O4 MgCO3 MgCl2 MgFeO4 MgO Mg(OH)2 Mg(NO3)2 Mg3(PO4)2 MgS MgSO4 MgSiO3 Mg2SiO4 Mg3Si2O5(OH)4 Mg3Si4O10(OH)2 Mg2TiO4

s s, spinel s s s s s s s s s s s s, chrysilite s, talc s

24.312 142.273 84.321 95.218 200.004 40.311 58.327 148.3218 262.879 56.376 120.374 100.396 140.708 277.134 379.298 160.52

725.71 274.17 23.43 244.65 121.53 124.38 42.73 –64.34 76.59 1105.11 166.22 87.73 188.35 117.06 140.26 231.48

626.1 45.3 30.2 158.2 68,1 59.1 33.2 49.7 78.1 893.9 73.0 14.8 59.8 38.8 14.8 119.2

Mn Mn3C MnCO3 MnCl2 MnFe2O4 MnO MnO2 Mn2O3 Mn3O4

s, D s s s s s s s s

54.9381 520.03 176.82545 1958.2 114.9475 19.42 124.844 199.18 230.63 118.36 70.9375 134.81 86.0369 0 157.8744 81.09 228.8119 172.26

487.7 1878.5 87.2 170.8 122.6 124.8 26.5 100.2 187.8

134 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa (continued) Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 Mn(OH)2 s, amorphous 88.9528 66.47 112.7 MnS s, green 87.002 1031.23 878.9 MnSO4 s 151.000 180.2 147.8 MnSiO3 s 131.022 110.08 108.0 N2 N2 atmospheric NH3 NH4Cl NH4NO3 (NH4)2SO4 NO NO2 N2O N2O4 N2O5

g g g s s s g g g g g

28.0134 28.1541 17.0305 53.491 80.04348 132.138 30.0061 46.0055 44.0128 92.011 108.0104

0 0 316.62 249.43 118.08 511.84 90.25 33.18 82.05 9.163 11.3

0.72 0.69 337.9 331.3 294.8 660.6 88.9 55.6 106.9 106.5 125.7

Na NaAlO2 NaAlSi2O6·H2O NaAlSi3O8 Na2CO3 NaCl NaHCO3 NaNO3 Na2O NaOH Na2S Na2SO3 Na2SO4 Na2SiO3 Na2Si2O5 Na4SiO4

s s s, analcime s, low albite s s s s s s s s s s s s

22.9898 81.9701 220.055 262.2245 105.9891 58.443 84.0071 84.9947 61.979 39.9972 78.044 126.042 142.041 122.064 182.149 184.043

330.9 128.4 35.41 72.75 –75.62 0 –101.94 –135.62 243.82 23.79 1014.84 297.63 0 11.31 13.28 151.45

336.6 67.2 20.3 21.9 41.5 14.3 21.6 –22.7 296.2 74.9 921.4 287.5 21.4 66.4 68.2 256.9

Ni Ni3C NiCO3 NiCl2 NiO Ni(OH)2

s s s s s s

58.71 188.14 118.72 129.62 74.71 92.72

239.74 1180.09 –49.93 94.85 0 –48.13

232.7 1142.9 36.4 97.2 23.0 25.5

APPENDIX 135

Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa (continued) Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 NiS s 909.77 883.15 762.8 Ni3S2 s 240.26 1967.14 1720.2 NiSO4 s 154,77 92.25 90.4 –266.75 53.6 NiSO4·6H2O s, D, tetragonal, 262.86 green O2 O O3

g g g

31.9988 15.9994 47.9982

0 249.17 142.67

3.97 233.7 168.1

P P P4O10

s, D, white s, red, triclinic s, hexagonal

30.9738 30.9738 283.8892

840.06 822.49 376.21

861.4 849.2 767.7

Pb PbCO3 PbCl2 PbO PbO PbO2 Pb3O4 Pb(OH)2 PbS PbSO4 PbSiO3 Pb2SiO4

s s s s, yellow s, red s s s s s s s

207.2 257.2 278.1 223.19 223.19 239.19 685.57 241.2 239.25 303.25 283.27 506.46

305.64 0 106.67 88.32 86.65 28.24 198.53 32.48 930.64 111.12 70.88 159.07

232.8 23.5 42.3 46.9 45.9 19.4 72.2 20.6 743.7 37.2 31.5 75.8

S SO2 SO3

s, rhombic g g

725.42 428.59 329.7

609.6 313.4 249.1

Si SiC SiCl4 SiO2 SiO2 SiO2 SiS2

s s, D, hexagonal l s, D, quartz s, D, cristobalite s, amorphous s

28.086 40.097 169.898 60.085 60.085 60.085 92.214

910.94 1241.69 544.81 0 1.46 7.45 2149.23

854.9 1204.9 482.2 2.2 3.1 8.2 1866.6

Sn Sn

s, I, white s, II, gray

118.69 118.69

580.74 578.65

551.9 552.0

32.064 64.0628 80.0622

136 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table I: Enthalpy of devaluation and standard chemical exergy of pure substances, Tn = 298.15 K, pn = 101.325 kPa (continued) Standard Molecular Enthalpy of chemical mass devaluation exergy Substance State M Dn, kJ/mol bch n, kJ/mol 1 2 3 4 5 SnCl2 s 189.6 416.08 393.5 SnO s 134.69 294.97 297.0 SnO2 s 150.69 0 36.2 SnS s 150.75 1205.74 1063.2 SnS2 s 182.82 1863.8 1611.7 Ti TiC TiO TiO2 Ti2O3 Ti3O5 TiS2

s s s s, rutile s s s

47.90 59.91 63.9 79.90 143.80 223.7 112.03

944.75 1154.16 425.14 0 368.66 375.1 2060.45

907.2 1137 419.2 21.7 386.1 414.1 1876.2

U UCl3 UCl4 UCl5 UO2 UO3 U3O8

s s s s s s s

238.03 344.39 379.84 415.3 270.03 286.03 842.085

1230.1 577.35 499.39 536.93 145.19 0 115.49

1196.6 556.0 491.1 519.5 168.9 49.8 236.2

W WC WO2 WO3 WS2

s s s s s

183.85 195.86 215.85 231.85 249.98

842.87 1195.84 253.18 0 2084.51

827.5 1199.5 297.5 69.3 1796.6

Zn ZnCO3 ZnCl2 ZnFe2O4 ZnO Zn(OH)2 ZnS ZnSO4 Zn2SiO4

s s s s s s, E s, sphalerite s s

65.37 125.38 136.28 241.06 81.37 99.38 97.43 161.43 222.82

419.27 0 583.93 74.08 70.99 1918 938.71 161.87 112.74

339.2 23.9 93.4 32.2 22.9 25.7 747.6 82.3 18.1

APPENDIX 137

Table II: Group contribution for the enthalpy of devaluation and standard chemical exergy of organic compounds* Gases Liquids b qch b qch Dq Dq No Group kJ/mol kJ/mol kJ/mol kJ/mol 1 1

2 | C  |

3

4

5

6

398.57

462.77

403.54

462.64

509.77

557.40

485.75

545.27

3

|  CH | |  CH 2

614.91

654.51

607.38

651.46

4

—CH3

713.47

747.97

715.35

752.03

5

| = C

440.53

513.35

443.16

473.02

6

| = CH

551.86

576.31

535.08

569.95

7

=CH2

660.26

678.74

680.26

675.68

8

=C=

543.04

554.23

539.28

559.21

9

{ C

510.20

519.38

494.34

515.27

10

{ CH

625.37

630.28

623.86

634.34

413.34

451.01

379.03

425.11

2

522.78

561.37

468.76

543.05

13

|  C  (ring) | |  CH (ring) | |  CH 2 (ring)

629.05

662.29

614.16

653.63

14

| = C  (ring)

442.71

466.41

–

–

11

12

138 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table II: Group contribution for the enthalpy of devaluation and standard chemical exergy of organic compounds (continued) Gases

D No

q

Liquids q

bch

D

q

q

bch

Group

kJ/mol

kJ/mol

kJ/mol

kJ/mol

2

3

4

5

6

559.18

576.65

542.95

568.28 86.52

1 15

| = CH (ring)

16

—O—

111.59

89.11

131.17

17

=O

246.86

245.09

91.46

18

—O— (ring)

117.42

97.12

126.27

106.64

19

§ | · —OH ¨ to  C ¸ ¨© | ¸¹

68.42

165.48

137.54

80.08

20

§ | · —OH ¨ to  CH¸ ¨© | ¸¹

63.90

66.78

85.11

52.59

21

| · § —OH ¨ to  CH 2 ¸ ¨© ¸¹

56.66

42.89

84.82

51.34

22

—OH (to CH3)

77.00

25.52

76.87

33.97

23

§ · | —OH ¨ to  CH ring¸ ¨© ¸¹ |

65.16

46.78

70.47

58.16

66.12

52.01

81.64

47.57

24

—OH (attached to aromatic)

–

25

| C = O

262.38

293.87

231.58

281.36

26

| H  C= O

388.64

412.68

356.72

400.21

27

| O =C O 

65.69

108.30

35.90

101.15

APPENDIX 139

Table II: Group contribution for the enthalpy of devaluation and standard chemical exergy of organic compounds (continued) Gases Liquids

D No 1 28

Group 2 | | O = C  O  C= O

q

kJ/mol 3

q

D

kJ/mol 4

kJ/mol 5

296.94

382.66

q

q

bch

244.81

bch kJ/mol 6 362.70

29

| O = C  OH

–

168.04

30

 O  CH = O

207.01

250.09

31

| O = C (ring) |

–

305.66

382.87

415.07

379.60

410.21

97.03

142.05

64.60

131.09

–

155.11 183.96

–

– 277.76

| H C= O

32

33

(attached to aromatic) | N 

34

|  NH

181.49

213.38

137.18

195.56

35

—NH2

258.43

290.20

235.43

284.39

36

=N—

56.82

72.98

103.43

37

{N

24.18

23.06

0

29.97

38

|  NH (ring)

186.19

209.24

151.20

199.37

237.80

240.16

216.84

269.24

153.58

196.27

39

40

—NH2 (attached to aromatic) |  NH (attached

to aromatic)

–

–

–

140 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table II: Group contribution for the enthalpy of devaluation and standard chemical exergy of organic compounds (continued) Gases

D No 1 41

Group 2 |  N  (attached

q

kJ/mol 3

Liquids q

D

kJ/mol 4

kJ/mol 5

77.07

134.06

q

q

bch

–

bch kJ/mol 6 –

to aromatic) 42

—NO2

42.30

1.45

43

—O—NO

19.66

18.89

44

—O—NO2

89.91

45

—N=C—

46

58.32

12.16

–

–

89.77

121.71

23.88

585.26

592.73

551.97

584.03

N { C

516.93

527.50

510.53

522.14

47 48

—S— —SH

761.07 862.06

636.88 724.36

741.74 848.67

642.32 732.26

49

=SO

692.04

553.78

696.47

566.88

50

| SO 2

439.87

373.78

414.68

–

51

—S— (ring)

764.08

633.45

755.42

687.25

52

—F

15.00

16.25

8.85

34.70

53

—Cl

46.08

26.24

42.89

32.01

54

—F (attached to aromatic)

15.08

45.16

8.85

34.70

55

—Cl (attached to aromatic)

46.08

26.24

42.89

32.01

Ŷ56

Ortho (1,2)

4.35

7.66

0.0

0.0

Ŷ57

Meta (1,3)

1.76

3.47

0.0

0.0

Ŷ58

Para (1,4)

1.26

4.60

0.0

0.0

Ŷ59

1,2,3 position

12.89

20.08

0.0

0.0

Ŷ60

1,2,4 position

9.05

13.60

0.0

0.0

Ŷ61

1,3,5 position

6.28

14.27

0.0

0.0

Ŷ62

1,2,3,4 position

14.06

26.94

0.0

0.0

APPENDIX 141

Table II: Group contribution for the enthalpy of devaluation and standard chemical exergy of organic compounds (continued) Gases

D No 1

*

q

Liquids q

bch

D

q

q

bch

Group

kJ/mol

kJ/mol

kJ/mol

kJ/mol

2

3

4

5

6

Ŷ63

1,2,3,5 position

12.80

23.93

0.0

0.0

Ŷ64

1,2,4,5 position

12.38

24.27

0.0

0.0

Ŷ65

1,2,3,4,5 position

17.99

35.36

0.0

0.0

Ŷ66

1,2,3,4,5,6 position

19.66

62.34

0.0

0.0

x67

3-atom saturated ring

62.30

49.04

83.68

83.68

x68

4-atom saturated ring

50.84

43.76

87.82

82.30

x69

5-atom saturated ring

50.38

45.52

0.0

0.0

x70

6-atom saturated ring

83.05

61.25

28.79

0.0

x71

7-atom saturated ring

73.81

46.32

–

–

x72

8-atom saturated ring

72.59

33.47

–

–

x73

Pentene ring

50.38

45.52

0.0

0.0

x74

Hexene ring

83.05

61.25

28.79

0.0

Recalculated with permission from the data of Fan and Shieh [17] to the reference level adopted in the present book. Ŷ Branching in aromatic. x Ring correction.

142 EXERGY METHOD: TECHNICAL AND ECOLOGICAL APPLICATIONS

Table III: Standard chemical exergy of species dissolved in ideal aqueous solutions (molarity 1 mol/kg H2O),Tn = 298.15 K, pn = 101.325 kPa Standard chemical exergy Chemical formula 1 AgCl CaCl2 CuCl CuCl2 FeCl2 H2CO3 HCl HF HNO3 H3PO4 H2S H2SO4 KCl KNO3 KOH K2SO4 LiCl MgCl2 NH3 NH4OH Na2CO3 NaCl NaHCO3 NaNO3 NaOH Na2SO4 O2 PbCl2 ZnCl2 ZnSO4

Dissociated

Undissociated q ch

b kJ/mol 2 59.4 – – – – 29.57 48.9 56.6 20.9 80.6 817.9 – 15.0 –19.8 46.9 – 58.1 – 327.4 328.8 – 5.5 21.4 29.4 37.7 – 20.3 – – 51.8

q

bch , kJ/mol

Form 3 Ag+, Cl– Ca2+, 2Cl– Cu+, Cl– Cu2+, 2Cl– Fe2+, 2Cl– 2H+, CO32– H+, HCO3– H+, Cl– H +, F – H+, NO3– 3H+, PO43– 2H+, HPO42– – 2H+, SO42– K+, Cl– K+, NO3– K+ OH– 2K+, SP42– Li+, Cl– Mg2+, 2Cl– – NH4+, OH– 2Na+, CO32– Na+, Cl– Na+, HCO3– + Na , NO3– Na+, OH– 2Na+, SO42– – Pb2+, 2Cl– Zn2+, 2Cl– Zn2+, SO42–

4 77.9 19.7 114.7 61.2 158.7 124.6 65.6 48.5 74.6 13.0 204.4 134.1 – 108.6 14.6 20.7 46.9 41.3 29.7 40.4 – 322.1 37.7 5.1 21.4 30.2 37.7 22.2 – 69.5 53.2 80.2

Solutions of exercises 1.1. The maximum energy efficiency of the prime mover, eqn (1.2) is KEmax = 0.667. The exergy efficiency, eqn (1.8), KB = 0.582. 1.2. The exergy efficiency of the heat pump, eqn (1.9) KB = 1.025 > 1. The information given in the prospectus is not credible. 1.3. The top and bottom temperature values in the considered Carnot cycle are Tmax = 880 K, Tmin = 320 K. The energy efficiency of the considered prime mover is 0.636. The feeding heat flow rate Q f = 10/0.636 = 15.7 MW. The flow rate of heat rejected to the environment Q 0 = 15.710 = 5.7 MW. The rates of exergy losses result from eqn (1.17). For the feeding heat delivery į B f = 15.7 u

900  880 u 300 = 0.12 MW 900 ˜ 880

Similarly for the rejection of the waste heat į B 0 = 0.36 MW. 1.4. The considered mixing process does not influence the enthalpy of CO2 or atmospheric air nor the composition of air. Only the entropy of CO2 increases due to its expansion. From eqn (1.22) it results: įB =

R 100 000 = 111 kJ/kg T0 ln 33 M

1.5. Heat delivered from the environment q0 = ( 273  T1 ) u 2.14 + 333.4 + ( T0  273 ) u 4.18 = 438.4 kJ/kg

The entropy increase of the ice and liquid water: 's

2.14 u ln

273 333.4 293   4.14 u ln 263 273 273

The exergy loss: įb

§ q · T0 ¨ 's  0 ¸ T © 0 ¹

29.52 kJ

1.597 kJ/(kg K)

144 SOLUTIONS OF EXERCISES

2.1. The temperature-dependent component of exergy results from eqn (2.5a): bphT = 26 882298 u 42.916 = 14 093 J/mol From eqn (2.5b) it results that: § T · 1100· § b phT ¦yicpi ¨ T T0  RT0 ln ¸ 33.525 u ¨ 1000 298 298uln ¸ 13840J/mol. © 298 ¹ © ¹ T 0 i

The error of the approximate value amounts to 1.8%. The pressure-dependent component of exergy results from eqn (2.6a) bph p

8.3143 u 298 u ln 4.2

3556 J/mol

2.2. The specific chemical exergy of the considered combustion gases results from eqn (2.26a): bch

0.718 u 0.69  0.087 u 19.87  0.175 u 9.5  0.02 u 3.97  8.3143 u 103 u 298 u (0.718 u ln 0.718  0.087 u ln 0.087  0.175 u ln 0.175  0.02 u ln 0.02) 1.901 kJ/mol.

The ratio of the chemical exergy to the temperature-dependent component of the physical exergy amounts to 0.135. In combustion processes the chemical exergy of combustion gases represents one of the components of external exergy losses. The possibilities of the utilization of this component are very small. 2.3. Specific enthalpy and entropy of the steam under consideration h = 3483.3 kJ/kg,

s = 6.592 kJ/(kg K)

At ambient temperature and pressure water appears in a liquid state h0 = 83.9 kJ/kg,

s0 = 0.296 kJ/(kg K)

The specific physical exergy amounts to bph = 3483.3–83.9–293 (6.592–0.296) = 1554.7 kJ/kg. The physical exergy may be determined also by means of the h,s diagram of steam as a difference between the initial enthalpy and the enthalpy h2s after isentropic expansion to the ambient isotherm: bph = 3483.3–1931.3 = 1552 kJ/kg. 2.4. Different reference states for enthalpy and entropy are assumed in various tables and diagrams. From the diagram in which the reference parameters hr = 0.5 MJ/kg, sr = 2.0 kJ/(kg K) for liquid ammonia at 0qC we obtain the specific enthalpy and entropy of liquid ammonia at 15qC h = 432.8 k/kg,

s = 1.742 kJ/(kg K).

SOLUTIONS OF EXERCISES 145

For gaseous ammonia in standard state h0 = 1859 k/kg,

s0 = 7.608 kJ/(kg K).

The specific physical exergy: bph = 432.8–1859–298u(1.742–7.608) = 321.9 kJ/kg. 2.5. Standard chemical exergy of SO2, table I bch = 313.4 kJ/mol. From eqn (2.10) bch = –RT0 ln y0 313 400 = 8.3143u298uln y0, y0 = 1.16u10–55. 2.6. From the exergy balance of the standard reversible reaction of formation, eqn (2.8) bch = 2191.6 + 2u729.1 + 854.9 + 2u3.97 = 129.4 kJ/mol. 2.7. The compound B(OH)3 in aqueous solution has been accepted as a reference species for B. The standard free energy of formation of the reference species amounts to 968.8 kJ/mol. eqn (2.15) with the values j = 1, z = 0, b ch H2 = 236.1 kJ/mol, b ch O 2 = 3.97 kJ/mol, m = 3.4u10–4, J = 1 gives: 968.81.5u(236.1 + 3.97)8.3143u10–3u298uln (3.4u10–3) = 628.5 kJ/mol. 2.8. From the exergy balance of the considered standard reversible reaction, eqn (2.7) it results that 'G q

b ch CaSO 4  b ch SO 2  0.5 b ch O 2  b ch CaO 24.9 313.4  0.5 u 3.97 110.2  417.4 kJ/mol.

2.9. The free energy decrease (maximum work) of the considered reaction results from the exergy balance. The chemical exergies of N2, O2 and CO2 contained in the standard environment, equal zero: w max

q q b ch CaCO 3  b ch Ca( NO 3) 2 16.3 1.4

19.1 kJ/mol 17.7 kJ/kg CaCO3.

The calculated value is much smaller than for technical fuels. The considered reaction should run spontaneously in nature, but is kinetically blocked. 2.10. The ratio E according to the formula (2.18a): ȕ 1.043  0.1896 u

4 8 1  0.0617 u  0.0428 u 55 55 55

1.0672

146 SOLUTIONS OF EXERCISES

According to eqn (2.22): bch = (24880 + 2442u0.1)u1.0672 + 9755u0.02 + 50u0.10 + 31.6u0.2 = 26 812 + 195 + 5 + 6 = 27 019 kJ/kg The influence of water and ash content is very small. The ratio of the chemical exergy to the lower heating value bch/CL = 1.086. 2.11. The contribution of chemical groups for the standard chemical exergy, taken from table II is CH=545.27 kJ/mol F=34.70 Cl=32.01 u 2 bchq =643.99 kJ/mol

2.12. The number of moles of the particular elements per 1 kg of steel: nFe

17.548, nC

0.5, nMn

0.1274, nS

0.0936, nP

0.0323, nSi

0.1068 mol

The number of moles of chemical compounds: nMn3C

1/ 3 nMn

nC  nMn 3C

0.0425, nFe3C

0.4575 mol.

The amount of moles of free Fe: nFe  3nFe3C

nFe cc

16.1755 mol.

Weighted sum of the chemical exergy values of the components:

¦nb i

q ch i

7020 kJ/mol

i

Exergy decrease due to the formation of solution (assuming an ideal solution): nT0 R ¦ y i ln y i

0.564 kJ/kg

i

The mixing component may be neglected. 2.13. The exergy efficiency of the boiler results from eqn (2.33). Assuming a constant heat capacity of liquid water we obtain the mean thermodynamic temperature of the heated water, eqn (2.34): Tm

60 383 ln 323

352 K

SOLUTIONS OF EXERCISES 147

From eqn (2.33):

KB

0.86 § 263 · u ¨ 1 ¸ © 1.09 352 ¹

0.199 .

2.14. The values determined in Exercise 2.3 may be used. The specific enthalpy and entropy of feed water are: hw = 1038 kJ/kg, sw = 2.701 kJ/(kg K). The mean thermodynamic temperature of the working fluid is Tm

3483.3  1038 6.592 2.701

628.5 K

The exergy efficiency of the boiler amounts to 0.469. 2.15. Per 1 J of driving heat the amount of the produced useful heat is 1.6 J, and 0.6 J of bottom heat is absorbed from the partially utilized geothermal water. The mean thermodynamic temperature values, eqn (2.34), and the values of the Carnot-factor, eqn (1.5), are: heating water 391.6 K and 0.3156; bottom water 307.4 K and 0.1282; useful water 331.6 K and 0.1918. The exergy efficiency of the heat pump:

KB

1.6 u 0.1918 0.3156  0.6 u 0.1282

0.782.

The calculated value is high, but it does not take into account the exergy losses burdening the production of the driving heat. 3.1. The exergy efficiency of the considered heat pump, eqn (1.9), amounts to 0.255. The cumulative exergy efficiency is 0.0738. 3.2. Per 1 J of driving heat the cumulative consumption of fuel exergy (see table 3.1) is * b ch 1.06/ 0.88 1.205 J

The cumulative exergy efficiency of the considered heat pump is *

ȘB

1.6 u 0.1918 1.205  0.6 u 0.1282

0.239

It is considerably smaller than the immediate exergy efficiency. 3.3. The cumulative exergy efficiency of the production of steel tubes, eqn (3.10), is 0.12. 3.4. Per 1 J of the chemical energy (or 1.09 J of exergy) of coal the amount of produced electricity and useful heat is 0.208 and 0.692 J. The ratio of the exergy increase to the enthalpy increase of network water amounts to 0.253. Hence, the exergy increase of the network water is 0.175 J. The cumulative consumption of the exergy of coal (per 1 J of chemical energy) is 1.121 J (see CExE of coal production, table 3.1). The avoided cumulative consumption of coal exergy due to the production of electricity is 0.208/0.29 = 0.717. Hence, the cumulative consumption

148 SOLUTIONS OF EXERCISES

of coal exergy burdening the heat production is 1.1210.717 = 0.404. The cumulative exergy efficiency of heat production is 0.175/0.404 = 0.433. 3.5. The CExC value for methanol: b

*

3308 6082 425  283 554 u 1.06 45348 u1.04  u   0.29 0.40 425 0.785 0.98

65 360 MJ/t.

The CExE value: 22410 65360

*

ȘB

0.343

3.6. The partial exergy losses burdening the production of particular energy carriers and raw materials, per 1 t of methanol and in % of CExC are electricity

įb1

§ 1 · 3308 u ¨ 1 © 0.29 ¸¹

heat

įb2

6082 u

coke

įb 3

natural gas

įb 4

production of methanol

įb5

8099 MJ/t 17.9%

425  283 § 1 · u¨  1 1355 MJ/t 2.1% © 0.4 ¸¹ 425 § 1 · 554 u 1.06 u ¨  1 161 MJ/t 0.25% © 0.785 ¸¹ § 1 · 45348 u 1.04 u ¨  1 962 MJ/t 1.5%. © 0.98 ¸¹

1  0.343  0.179  0.021  0.002  0.015 0.44

44%

3.7. The CExC value: *

b

2076 19 393 u 1.09 48 649 u 1.06 0.0077 u 7040    0.29 0.972 0.785 0.19 0.003 u 8200  = 95 035 MJ/t. 0.161

The CExE value: *

ȘB

281 95 035

0.003

0.3%.

4.1. After introducing the intermediate heat carrier two heat exchangers will be necessary to preheat the air. The heat flow rate in both heat exchangers will equal that in a single heat exchanger, the heat transfer coefficient will be twice higher and the mean temperature difference twice lower. Hence, the total necessary heat transfer area will be twice larger.

SOLUTIONS OF EXERCISES 149

4.2. In initial conditions the electric power delivered to the consumers amounts to: B el

E el

1000 u 0.45 u 0.82 u 0.9

332.1 MW

In the boiler the rate of the exergy increase of the working fluid is 450 MW. After increasing the exergy loss in the boiler the rate of exergy increase of the working fluid will be 440 MW, and the electric power delivered to the consumers will decrease to: Belc

440 u 0.82 u 0.9

324.7 MW .

If the additional exergy loss appears in the transformation and transmission system, the electric power delivered to the consumers will be 322.1 MW. Hence, the increase of the exergy loss is more disadvantageous in the final link of the system than in the initial link. 5.1. The thermo-ecological cost should not take into account the consumption of renewable exergy. Hence, the thermo-ecological cost per exergy unit of useful heat (the sustainability index) is ȡ b

1.205 1.6 u 0.1918

3.93

The thermo-ecological cost per energy unit of useful heat is 0.753. 5.2. The specific enthalpy of the working fluid before its compression is h1 = 404.7 or 402.8 kJ/kg (first value for the heat extraction from the ground). The specific enthalpy after compression is h2 = 441.8 or 445.8 kJ/kg. The specific enthalpy after condensation h3 = 255.1 kJ/kg. The values of COP are 3.77 or 3.32. The ratio of the exergy increase of the heated room to the amount of heat is 0.061. The cumulative exergy efficiency of the heating system is 0.0718 or 0.0630. Hence, the index of sustainability is 13.9 or 15.8. The utilization of the renewable source of bottom heat slightly improves the sustainability index of the heating system, but its value remains unfavorable. 5.3. The thermo-ecological cost of the energy unit of heat should take into account its annual production and the annual cumulative consumption of exergy burdening the production of metals and glycol: ȡ q

27 u 150.7 u 0.5  20 u 58.7.1 u 0.7  10 u 33.4  20 u 19.45/0.72 15 u 100 u 106 u 8760 u 3600 0.0789 J J th

The exergy increase of the heat carrier is 'b q

1

281 313 u ln 15 298

0.0800.

150 SOLUTIONS OF EXERCISES

The sustainability index is ȡ 'b

0.986.

The obtained result is considerably smaller than in exercise 5.1, and fulfills the condition of sustainability. 5.4. The thermo-ecological cost of the production of electricity should take into account the mean annual consumption of steel, its recovery factor after wear and the annual amount of the delivered electricity ȡ b

240 u 58 700 u 0.65 0.9 u 15 u 2000 u 3600

0.0943.

The calculated value expresses simultaneously the sustainability index. The obtained result is very advantageous. Not all the used materials have been taken into account. For example the additional consumption of copper would increase the obtained result. 5.5. The energy efficiency of the power plant is related to the lower heating value. Therefore, the ratio of the exergy to the lower heating value appears in the calculation of the coal-fed plant and the ratio of the specific thermo-ecological cost to the lower heating value is used when calculating the gas-fed plant. The thermoecological cost of electricity when produced by means of coal is ȡc

1.09 0.38 u 0.93 u 0.9

3.43

Electricity produced by means of gas has the thermo-ecological cost ȡg

683 0.37 u 778 u 0.9

2.64

In the considered case the specific thermo-ecological cost equals the sustainability factor. From the point of view of the economy of domestic non-renewable exergy resources the application of imported gas is more advantageous, when compared with coal. 6.1. The annual thermo-ecological cost of heating and thermal insulating per unit of the wall area is PA

'Tm IJ R ȡ h įȡ p Ȗ + 1 į Wl  k O

A C 1+ b į

where: k = initial heat transfer coefficient of the wall O, J, G, = thermal conductivity, mass density and thickness of the thermal insulation 'Tm = mean temperature difference between the heated room and the environment in the heating season

SOLUTIONS OF EXERCISES 151

WR = annual duration of the heating season Uh, Up = specific thermo-ecological cost of useful heat and thermal insulation Wl = life time of the thermal insulation layer, A

'Tm kIJ R ȡ h , b

Ȗȡp

k , C Ȝ

IJl

The optimum insulation thickness is į opt

1 § Ab ·  1¸ b ¨© C ¹

Introducing the given data we obtain: A = 600.5 MJ/(m2 a),

b = 32.5 1/m,

C = 165 MJ/(m3 a),

Gopt = 0.304 m.

7.1. According to eqn (7.7) and Example 7.3 the discounted rate of the proecological tax is 0.0044u1.752 = 0.0077 US$/MJ. The pro-ecological tax which should be added to the selling price of electricity is 12 000 u 0.0077 0.9 u 15 u 2000

0.0034 US$/kWh

The determined value is considerably smaller than in Example 7.2. 7.2. The energy efficiency of the boiler represents the amount of useful heat per unit of the heating value of coal. Hence, the pro-ecological tax which should be added to the selling price of a heat unit is 1.12 u 4.4 0.85

5.8 US$ GJ

7.3. The value of COP represents the amount of useful heat per unit of driving electricity. Hence, the pro-ecological tax which should be added to the selling price of a heat unit to reimburse the tax paid for the purchase of electricity is 3.13 u 4.4 3.5

3.9 US$ GJ .

The total amount of heat produced during the life time per 1 kW of thermal power is 3500u20u3600u10–6 = 252 GJ The corrected rate of the tax resulting from eqns (7.6) and (7.7): 0.0044 u 20 u 0.08 1  1.0820

0.00898 US$ MJ

8.98 US$ GJ

152 SOLUTIONS OF EXERCISES

The tax component which should reimburse the initially paid tax burdening the purchase of installation: 15 u 8.98 252

0.53 US$ GJ

The total tax is 3.9+0.53 = 4.43 US$/GJ. 7.4. The reduction of the annual thermo-ecological cost of heat per 1 m2 of the wall:

P  Pc

§ · 1 ¸ ¨ 'T ¨ k  IJ ȡ m 1 į¸ R h  ¸ ¨ © k Ȝ¹ 520.9 MJ m 2 a

1 · § 18 u ¨1.3  ¸ u 5.4 u 3.6 u 1.32 © 5.79 ¹



The thermo-ecological cost of thermal insulation: Up = 0.2u20u165 = 660 MJ/m2 The total discounted value of the reduction of the pro-ecological tax (net present value NPV) is § 1 · 0.0044 u ¨ ȡ p  ȡ h ¦ t ¸ t 1  r ¹ ©

ª 0.0044 « ȡ p  ȡ h «¬ 19.6 $ m 2

1§ 1 ·º ¨1  ¸» r © 1  r IJ ¹ » ¼

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Index Absorption of thermal radiation, 14 Activity coefficient, 27 Adiabatic: combustion, 13 compression, 88 expansion, 86 Aftercooler, 89 Ahrendts J., 22 Air, atmospheric reference species, 24 Anergy, 9 Anthropogenic exergy losses, 16 Atanasova, 52 Availability, ix Avoided expenditures, 60, 107 Back-pressure turbine, 72 Baehr H.D., 9 Balance: of cumulative exergy consumption, 59 of thermo-ecological cost, 91 of energy, 5, 43, 46 of exergy, 4, 42, 46, 49, 51 Black body emission, 14 Blast furnace, 63, 99 Boiler: efficiency, 44 exergy losses, 46 Boundary of system, 4 Bošnjakovi F., ix Brodyanskyi W.M., ix By-product, 63, 98, 108 Calcium carbide production, 69 Carnot M.L.S., 1

machine, 7 Chemical exergy, 21, 39 Coal, chemical exergy of, 35 Coefficient: of by-production, 60 of consumption, 60, 64 of performance (COP), 52 of sensibility, 8 Cogeneration, 68, 89 Coke: chemical exergy of, 35 production, 67 Combined (complex) processes, 8 Combustion, 13, 88 Compression: heat pump, 5, 48, 72, 103 refrigerator, 48 Compressor, 42, 87 Condenser, 10, 86 Cost of: exergy, 90, 105 exergy losses, 105 Counter-current process, 78 Cumulative: exergy consumption, 57, 64 exergy efficiency, 60, 73 exergy loss, 63 Depletion of non-renewable resources, 91 Devaluation: enthalpy of, 34 reaction of, 37 Diffusion, 12

162 Index

Discounted coefficient, 126 Dissipative structure, 16 Dissociation, 13 Domestic consumption product, 92, 95 Ecological cost, see: thermo-ecological cost Economic optimization, 115 Efficiency: electro-mechanical, 46 exergy, 6, 60 of fuel delivery, 61 of transformation and transmission, 61 Emission, 14 Energy: balance, 5 efficiency, 7, 42, 44, 97, 104 Enthalpy: chemical (chemical energy), 34 of devaluation, 34 physical, 20 Entropy: flux of radiation, 14 generation, 3, 15, 107 Environment: standard concentration of reference species, 25 temperature, 2 Esterification, 76 Evaluation of natural resources, 91 Evaporation plant, 86 Evaporator, 86 Exergetic cost, 114 Exergonomics, 105 Exergy: balance, 4, 42, 46, 49, 51 calculation, 19 chemical, 21 of fresh water, 119 of fuels, 35 of nitrates, 27 of solutions, 37 components, 19 cumulative consumption of, 57

definition, 1 economic application of, 105 efficiency, 6, 52, 61 losses, 3, 76, 89 cumulative, 63 external, 3, 42 internal, 3, 43 natural, 16 partial, 63, 72 nuclear, 41 of radiation, 38 physical, 19 types, 19 Export of domestic products, 127 Free Gibbs energy, 24 Fresh water, 118 Friction: 20 heat, 20 Fuel exergy, 35 Gaggioli, 75 Gas turbine, 42, 47, 53, 77, 79, Gibbs J.W., ix Gibbs free energy, 24 Gouy-Stodola law, 3 Gouy G., ix Grassmann P., 4 Green-house effect, 17 Group contribution method, 36, 137 Heat: exchanger, 76, 109, 112 pump, 5, 48, 52, 128 source, 1, 4 transfer, irreversibility of, 10 Heat-and-power plant, 8, 64, 71, 86 Heating value, 35 Heat-and-power plant, 8 Human work, 58 Humid air turbine, 88 Humidity of air, 25 Imported goods, 101, 127 thermo-ecological cost of, 101 Improvement of processes, 75

Index 163

Input-output equations, 59, 94, 124 Intermediate heat carrier, 83 Investment: cost, 113, 115 efficiency, 61 Ionic reference species, 26 strength, 28 Irreversibility: of adiabatic expansion, 9 of combustion, 13 of radiation emission and absorption, 16 of heat transfer, 10 of mixing, 12 Keenan J.H., ix Kinetic exergy, 19 Leites, 90 Limestone, 45 Linde cycle, 84 Lior N., 25, 90 Maximum work, 2 Mixing, 77 Morris D.R., ix, 27 Natural: exergy losses, 16 losses of utilizable exergy, 39 gas, 95, 100, 101 mineral capital, 118 non-renewable resources, 91, 94 Net coefficient of consumption, 64 Net present value, NPV, 152 Nitrates, chemical exergy, 27 Number of thermoeconomic similarity, 116 Operational cost, 110, 113 Optimization: economic, 116 objective function, 108 of thermo-ecological cost, 108 of exergetic cost, 114 Oxygen, 12, 22, 28, 62, 76, 98

Partial exergy losses, 63, 65, 72 Personal tax, 122 Petela R., ix, 39 Pig iron, 66, 98 Power plant, 1, 16, 41, 47, 61, 78 Preheating of combustion reactants, 14 Pressure loss, 109, 112 Prime mover, 5, 38 Principle of avoided expenditures, 60, 63, 107 Product: major, 60 semi-finished, 57 Pro-ecological tax, 121 Radiation: exergy of, 38 relict, 41 solar, 39 Rant Z., ix, 2, 9 Reaction of formation, 24 Recirculation, 77 Reference: level, 22 reaction, 22 species, 22 dissolved in seawater, 27 ionic, 25, 28 gaseous, 24 solid, 26 standard concentration of, 25 Refrigerator, 48 Repetto R., 121 Replacement ratio, 60, 63, 109 Reynolds number (Re), 110 Riekert L., 2 Sama D., 3, 75, 90 Sciubba E., iii, 115 Seawater: molarity of elements, 38 reference species, 26 salinity, 25

164 Index

Second law of thermodynamics, ix, 2 Sequence analysis, 68 Slesser M., 121 Solar: collector, 104 radiation, 39 Solution, chemical exergy of, 36, 142 Standard chemical exergy: data, 39 definition, 23 group contribution, 35 Stanek W., iii, 100, 102, 122, 123, 127 Steam: boiler, 44 power plant, 48, 87 turbine, 45, 66, 72, 85 Stefan-Boltzmann constant, 14 Stodola A., ix Styrylska T., 35 Substitution ratio = replacement ratio Sulfuric acid plant, 52 Sustainability index, 102 System boundary, 5, 19

Temperature: exergy scale, 11 mean thermodynamic, 45 Thermo-ecological cost 92, 108 of exported goods, 94, 102 of imported fuels, 101 of waste products, 95 Thermodynamic imperfection, ix, 57 Thermoeconomics, 105 Throttling, 12, 84 Tsatsaronis G., 105, 122 Unitax, 121 Valero A., 116, 118 Value added tax, VAT, 121 Wall G., 41, 121 Waste heat boiler, 85 Waste products, 3, 53, 91, 95, 99 aggressive components, 96 index of harmfulness, 91 Water: boiler, 128 preheater, 111 resources, 118 Weizsäcker von, 122 Wind power plant, 104, 128

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Modelling and Simulation of Turbulent Heat Transfer Editors: B. SUNDÉN, Lund Institute of Technology, Sweden and M. FAGHRI, University of Rhode Island, USA Providing invaluable information for both graduate researchers and R & D engineers in industry and consultancy, this book focuses on the modeling and simulation of fluid flow and thermal transport phenomena in turbulent convective flows. Its overall objective is to present state-of-the-art knowledge in order to predict turbulent heat transfer processes in fundamental and idealized flows as well as in engineering applications. The chapters, which are invited contributions from some of the most prominent scientists in this field, cover a wide range of topics and follow a unified outline and presentation to aid accessibility. Series: Developments in Heat Transfer, Vol 16

WIT eLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visitors to the WIT eLibrary can freely browse and search abstracts of all papers in the collection before progressing to download their full text. Visit the WIT eLibrary at http://www.witpress.com

ISBN: 1-85312-956-9 2005 £124.00/US$198.00/€186.00

360pp

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Condensation Heat Transfer Enhancement V.G. RIFERT, Thermodistillation, Kiev, Ukraine and H.F. SMIRNOV, State Academy of Refrigeration, Odessa, Ukraine Professors V.G. Rifert and H.F. Smirnov have carried out research on heat transfer enhancement by condensation and boiling for over 30 years and have published more than 200 papers on the topic. In this book they provide research results from the former USSR to which there has previously been little access and also describe different theoretical models of the condensation process. Partial Contents: Theoretical Principles of Heat Transfer at Film Condensation; Condensation on Horizontal Low-Finned Tubes – Theoretical Models; Experimental Study of Condensation Heat Transfer on Finned Tubes; Condensation on Vertical Profiled Surfaces and Tubes; Heat Transfer Enhancement at Film Condensation Inside Tubes; Condensation in the Electric Field; Hydrodynamics and Heat Transfer at Film Condensation of Rotating Surfaces. Series: Developments in Heat Transfer, Vol 10 ISBN: 1-85312-538-5 2004 £144.00/US$230.00/€216.00

392pp

Heat and Fluid Flow in Microscale and Nanoscale Structures Editors: M. FAGHRI, University of Rhode Island, USA and B. SUNDÉN, Lund Institute of Technology, Sweden Presenting state-of-the-art knowledge in heat transfer and fluid flow in micro- and nanoscale structures, this book provides invaluable information for both graduate researchers and R&D engineers in industry and consultancy. All of the chapters are invited contributions from some of the most prominent scientists in this active area of interdisciplinary research and follow a unified outline and presentation to aid accessibility. Contents: Miniature and Microscale Energy Systems; Nanostructures for Thermoelectric Energy; Heat Transport in Superlattices and Nanowires; Thermomechanical Formation and Thermal Detection of Polymer Nanostructures; Two-Phase Flow Microstructures in Thin Geometries – Multi-Field Modeling; Radiative Energy Transport at the Spatial and Temporal Micro/Nanoscales; Direct Simulation Monte Carlo of Gaseous Flow and Heat Transfer in a Microchannel; DSMC Modeling of Near-Interface Transport in Liquid-Vapor Phase-Change Processes with Multiple Microscale Effects; Molecular Dynamics Simulation of Nanoscale Heat and Fluid Flow. Series: Developments in Heat Transfer, Vol 13

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ISBN: 1-85312-893-7 2004 £117.00/US$187.00/€175.50

392pp

Transport Phenomena in Fuel Cells

Thermal Analysis of Welds

Editors: B. SUNDÉN, Lund Institute of Technology, Sweden and M. FAGHRI, University of Rhode Island, USA

N.T. NGUYEN, ETRS Pty Ltd., HRL Services, Mulgrave, Australia

This is the first book to provide a comprehensive analysis of transport phenomena in fuel cells, covering fundamental aspects of their function, operation and practical consequences. It will contribute to the understanding of such processes in Solid Oxide Fuel Cells (SOFC), Proton Exchange Membrane Fuel Cells (PEMFC) and Direct Methanol Fuel Cells (DMFC). Written by eminent scientists and research engineers, the chapters focus on various mathematical models and simulations of transport phenomena in multiphase flows. Relevant experimental data is also featured. A detailed summary of state-of-the-art knowledge and future needs, the text will be of interest to graduate students and researchers working on the development of fuel cells within academia and industry. Series: Developments in Heat Transfer, Vol 19 ISBN: 1-85312-840-6 2005 £142.00/US$227.00/€213.00

432pp

This book is written for postgraduate students, and welding, mechanical design and research engineers who deal with problems such as residual stresses and distortions of welds, design of welded structures, micro-structure modelling of welds and optimisation of welding sequences and procedures. The subject is approached in a simplified way, focusing on heat conduction and stepby-step derivation of the analytical solutions using fundamental calculus. Solutions are given in closed form and are ready for use by means of simple integrals, while applications are demonstrated through easily accessible case studies. Special features include new analytical solutions developed by the author for various 3D heat sources and a CD-ROM containing Visual Basic programs combined into the WHEATSIM (weld heat simulation) package. Series: Developments in Heat Transfer, Vol 14 ISBN: 1-85312-951-8 2004 352pp+CD-ROM £124.00/US$198.00/€186.00

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Plate Heat Exchangers Design, Applications and Performance L. WANG and B. SUNDÉN, Lund Institute of Technology, Sweden and R.M. MANGLIK, University of Cincinnati, USA During recent years research into PHEs has increased considerably and there is now an urgent need for a state-of-the-art review of knowledge on this topic. Fulfilling this need, this book will enable graduate students, researchers, and R&D engineers in industry to achieve a better understanding of transport processes. Partial Contents: Construction and Operation; Applications; Materials and Manufacturing; Multi-Pass Flow Arrangements; Thermal-Hydraulic Performance in Single-Phase Flows; Fouling; Extended Design and Operation Issues. Series: Developments in Heat Transfer, Vol 11 ISBN: 1-85312-737-X 2005 apx 350pp apx £125.00/US$193.00/€187.50

Advanced Computational Methods in Heat Transfer VIII Editors: B. SUNDÉN, Lund Institute of Technology, Sweden, C.A. BREBBIA, Wessex Institute of Technology, UK and A.C. MENDES, University of Beira Interior, Portugal Heat Transfer topics are commonly of a very complex nature and may involve several different mechanisms. Advances in numerical solution methods of nonlinear partial differential equations and access to high-speed, efficient computers have led to dramatic progress in recent years. There is still a need, however, to develop further innovative approaches for the solution of a variety of problems of current interest. Containing papers from the Eighth International Conference on Advanced Computational Methods in Heat Transfer, this book contains discussion of advanced topics, new approaches and applications of advanced computational methods. Series: Computational Studies, Vol 5 ISBN: 1-85312-705-1 2004 £186.00/US$298.00/€279.00

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536pp