The maximum amount of useful work can be conducted by a system, if the system is reversibly brought into equilibrium with its environment. Based on technical and etymological considerations, Rant [114] proposed the term exergy for the maximum amount of useful work in 1956, which was soon widely accepted and is commonly used these days. Several definitions, detailed introductions, and derivations
to the concept of exergy can be found in [7, Chapter 3.3; 11, Chapter 3; 134, Chapter 11]. The basic approach is described in the following.
Neglecting nuclear, magnetic, electrical, and surface tension effects [11, Chapter 3], the equilibrium with the environment is characterized in terms of physical, chemical, kinetic, and potential equilibrium. If the environment itself is not in equilibrium, contradictions to the second law of thermodynamics arise [7, Chapter 3.3]. Consequently, an exergy analysis requires the definition of an equilibrium refer- ence environment.
Corresponding to the equilibrium conditions considered, the exergyEsysof a system is composed of the four parts physical exergy Ephy, chemical exergy Ech, kinetic exergy Ekin, and potential exergy Epot [11, Chapter 3]
Esys= Ephy+ Ech+ Ekin+ Epot. (4.57)
Physical exergy is almost always considered in exergy analyses [7, Chapter 3.3], describing the departure from the thermodynamic equilibrium with the reference environment. For closed systems the physical exergy becomes
Ephy= M
(en − enref) + pref(ρ−1− ρref−1) − Tref(s − sref) , (4.58)
with mass-specific internal energy, mass-specific entropy, density and pressure denoted byen, s, ρ, and
p. The subscript ref indicates quantities of the system being in thermodynamic equilibrium with the
reference environment. For open systems the physical exergy is defined as
Ephy= M (h − href) − Tref(s − sref),
(4.59) with the mass-specific enthalpy denoted byh. Only if chemical reactions and mixing of different species are part of the process to analyze, chemical exergy varies and has to be included in an exergy analysis. As these effects are not relevant for the PHES systems considered in this work, the underlying theory is not presented here. An introduction to chemical exergies and the related concept of fuel exergies is given in [7, Chapter 3.3; 11, Chapter 3]. Rarely, the parts kinetic and potential exergy are considered in an exergy analysis [7, Chapter 3.3; 97], which are equal to the respective energiesEn
Ekin= Enkin, (4.60)
Epot= Enpot. (4.61)
Compared to the physical exergy, kinetic and potential exergies can be neglected in the analysis of PHES systems. Consequently the exergy of a system considered in this work is composed solely of physical exergy
Esys= Ephy. (4.62)
Recalling the definition of exergy, it becomes clear that useful work supplied to or discharged by a system is pure exergy. Forms of work relevant in the context of PHES systems are electrical and mechanical work
Eel= Enel, (4.63)
Emech= Enmech. (4.64)
dEin,1 dEin,2 dEin, jmax ... dEout,1 dEout,2 dEout,kmax ... dEsys+ dED
Figure 4.16: Sample system illustrating an exergy balance. The system Exergy Esysdepends on the exergies entering the system Ein, j, the exergies leaving the system Eout,k, and the exergy destruction within the system ED.
Although electrical work is the thermodynamically correct term, the terminology electrical energy is al- most exclusively used in technology and science. To avoid confusions, the commonly used term electrical energy is used throughout this work. The maximum amount of useful work that can be conducted by a heatQ does not only depend on the amount of heat, but also on the temperature of the heat TQand the temperature of the reference environmentTref
EQ= 1 − TTref Q Q. (4.65)
Exergy balances are essential parts of an exergy analysis. Considering the sample system of Fig- ure 4.16, the corresponding exergy balance in differential form becomes
dEsys+ dED= jmax X j=1 dEin, j− kmax X k=1 dEout,k. (4.66)
The system ExergyEsysis increased by the exergies entering the systemEin, jand decreased by the exergies leaving the system Eout,k. The operation of real, non-ideal systems is accompanied by an irrecoverable exergy destruction ED. The smaller the exergy destruction, the closer comes the system to an ideal, thermodynamically reversible operation. In the limiting case of thermodynamically reversible system operation, the exergy destruction vanishes. Negative exergy destructionsED< 0 are thermodynamically
impossible. The exergies leaving the system could either be fueling downstream systems or they could be released to the reference environment being considered as exergy lossesEL.
The thermodynamic performance of a system cannot be evaluated solely based on exergy destruction. On the one hand, exergy destruction scales with system size. On the other hand, exergy destruction does not capture exergy losses to the environment. These effects are accounted for by the exergetic efficiency, which is defined as the ratio of product exergyEP to fuel exergyEF
ηex= dEP
dEF = 1 −
dED+ dEL
dEF . (4.67)
Exergetic efficiencies evaluate the operation of a system with respect to the thermodynamic optimum. Consequently they scale between zero and one and are therefore superior to energetic efficiencies.
In contrast to the exergy destruction, which is unambiguously defined by the balance Equation (4.66), exergetic efficiencies depend on the definition of product and fuel exergy. Equation (4.66) can be ex-
pressed in terms of product and fuel exergies, which in turn are functions of the system exergy and the exergies entering and leaving the system
dED= dEF− dEP− dEL
dED= dEFdEin, j, dEout,k, dEsys − dEPdEin, j, dEout,k, dEsys − dELdEout,k . (4.68)
For often analyzed systems and system components, common product and fuel definitions can be found in literature [11, Chapter 3; 141]. However, for complex and seldom analyzed systems or components, universally valid definitions are not available. For clarity, all exergetic efficiencies provided in this work are accompanied by the corresponding product and fuel definitions.
Being based on identical product and fuel definitions, comparisons between exergetic efficiencies of similar components (e.g. compressors) are valid. In contrast, comparisons between exergetic efficiencies of dissimilar components (e.g. compressor and heat exchanger) are not meaningful [11, Chapter 3], because the maximum possible exergetic efficiencies depend on the component-specific state of the art.