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Top1. Introduction
The term exergy was coined in 1956 by Zoran Rant (1904-1972), (Rant, 1956). Earlier terms for this concept have been available energy and availability, inter alia (Kline, 1999). For a bibliography on exergy-related work, see Wall (1997). For arguments supporting the further development and application of this concept, please see e. g., Rosen et al. (2008).
Exergy may be defined as the maximum amount of mechanical work that can be extracted from a source with given properties. The original concept relies on the existence of an environment that has constant intensive properties (temperature, pressure, chemical potentials, etc.), which are not affected by the processes taking place during the extraction. If a maximal amount of mechanical work is to be extracted, the processes involved need to be ideal in the sense that no entropy is generated, i.e., the processes are reversible.
The main development in this paper is based on the following universal principle (Grubbström, 1985):
“The exergy of a system of objects equals its total inner energy less the inner energy of a single body having the same total extensive properties as those of the system.”
The new basic results in this article are obtained from reinterpreting an object in disequilibrium as a system of infinitely many infinitesimal objects with heterogeneous intensive properties. These results are summarised in the single formula (21) in Section 4. By “thermodynamic disequilibrium” we thus mean that the object under consideration on a macroscopic level has a heterogeneous distribution of at least one intensive property (such as temperature or pressure). An object in equilibrium has all its intensive properties at constant levels throughout.
We summarise classical developments in Section 2, followed by formulae for the exergy of a set of finite objects in Section 3. Our new considerations are mainly developed in Section 4, in which a finite object in disequilibrium is interpreted as a continuous set of infinitely small objects, each in microscopic equilibrium, but with different intensive properties in the finite perspective. Section 5 offers three simple examples illustrating our findings, and a final section contains a summary and some ideas for further developments. The concept of microscopic equilibrium is elaborated in Section 4.
In our treatment, we make use of two abstract spaces, on the one hand the thermodynamic configuration space erected by one axis for each thermodynamic property considered, on the other, the geometric space in which the object is physically embedded, the latter space erected by axes representing spatial co-ordinates.
Top2. The Inner Energy Function And Classical Exergy Expression
Table 1. | U | Inner Energy Function [J] |
| S | Entropy [J/K] |
| V | Volume |
| N | Molar content [mole] |
| T | Absolute temperature [K] |
| p | Pressure [J/(volume unit)] |
| μ | Chemical potential [J/mole] |
 | Gradient of function f |
| n | Number of (extensive/intensive) properties of thermodynamic system, dimension of thermodynamic configuration space |
| x | Vector of extensive properties, n-dimensional |
| y | Vector of intensive properties, n-dimensional |
| m | Number of objects (sources),  |
 | Vector of extensive properties of object i, i = 1, 2, …m |
 | Vector of intensive properties of object i, i = 1, 2, …m |
| E | Exergy [J] |
| a* | Equilibrium value of property a, equilibrium value of intensive property in case of infinite environment |
| c | Specific heat capacity at constant volume [J/(mole·K)] |
| R | Universal gas constant [J/(mole·K)] |
 | Density (per unit of volume) of extensive property a,  |
 | Vector of densities of extensive properties |
 | Spatial coordinates |
 | Vector of spatial coordinates |