The Generalized Exergy Equation: A Rigorous Development and Detailed Presentation Suitable for Presentation to Advanced Undergraduates and Beginning Graduate Students

Sheldon Jeter

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Conference: 2022 ASEE Annual Conference & Exposition
Location: Minneapolis, MN
Publication Date: August 23, 2022
Start Date: June 26, 2022
End Date: June 29, 2022
Conference Session: Mechanical Engineering: Poster Session
Page Count: 16
DOI: 10.18260/1-2--41533
Permanent URL: https://peer.asee.org/41533
Download Count: 234

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Sheldon Jeter Georgia Institute of Technology

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Sheldon M. Jeter has mechanical engineering degrees from Clemson, the University of Florida, and Georgia Tech. He has been on the academic faculty at Georgia Tech since 1979 and will retire in August 2022. He has written over 250 refereed journal articles and conference papers and numerous research reports and other articles. He has supervised 16 Ph. D. graduates and numerous other research students. His research interests are thermodynamics, experimental engineering, heat and mass transfer, solar energy, and energy systems including concentrating solar power and other solar issues, building energy systems, and HVAC issues in health care facilities.

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Abstract

The Generalized Exergy Equation: A Rigorous Development and Detailed Presentation Suitable for Presentation to Advanced Undergraduates and Beginning Graduate Students Exergy is well known as the potential for a system to produce useful work in interaction with its surroundings, which are assumed to remain in equilibrium in this context. The generalized form of the exergy equation applies when the engineering system is open to the surrounding medium so that the matter in the system can merge with and react with and ultimately become identical with the constituents of the medium. The medium in its minimal form could be just the surrounding gaseous atmosphere. This gaseous atmosphere can conceivably be as simple as a hypothetical single component such as “dry air”, but more likely the simplest worthwhile and interesting medium is “moist air” (water vapor and dry air) in HAVC and similar applications or a slightly more complicated mixture of dry gases (nitrogen, oxygen, argon, carbon dioxide) and water vapor in combustion applications. For desalination systems, the medium must include the liquid saline ocean in equilibrium with the gaseous atmosphere. If more complex chemical processing applications are considered, an inert version of the crust of the earth must be included in the medium. The latter rather complex medium must contain the stable chemical versions of the ores or other inert feedstocks of interest that are in equilibrium with the gaseous atmosphere. A straightforward, if somewhat necessarily complicated, derivation of the unrestricted version of the control volume (CV) exergy equation will be detailed in the proposed presentation. In literature or practice, this equation is typically referred as the “Generalized Exergy Equation” (GEE) or an equivalent name, so this convenient and descriptive terminology is adopted herein. The nature of the applicable or assumed dead state is critical in the development and formulation of an exergy equation. For any form of the Restricted Exergy Equation, the system is kept separated from its surrounding, with the pertinent dead state being the Restricted Dead State (RDS) at ambient temperature and pressure. A commonly encountered special case of the RDS is the situation in which the composition of the matter, usually a fluid, is unchanged between the prevailing state and the dead state. This RDS can be called the Fixed (or Frozen) Dead State (FDS). In contrast, for the GEE addressed herein, the ultimate or General Dead State (GDS) must be considered. A typical textbook or heuristic presentation of the GEE involves the expected combination of the energy and exergy equations along with a somewhat heuristic (or at least challenging) and superficially plausible introduction of the FDS properties. While the result is useful, there are logical and pedagogical deficiencies in this approach, specifically (1) it could be erroneously inferred from some presentations that the mole numbers of the components of the system are fixed, (2) the expected forms of the system and stream exergies do not appear naturally unless the FDS is specifically introduced as an intermediate state, (3) the role of the FDS in distinguishing and quantifying the Chemical Exergy is not demonstrated, and (4) an equivalent form of the total exergy often seen in the literature and practice of chemically oriented disciplines is not encountered. The alternative straightforward and rigorous development generates an equation governing the generalized exergy by introducing equalities involving the chemical potentials of the components of the system that prevail in the medium. Note of course, that some components of the system, such as fuels, will not exist as independently defined components of the medium. In these cases, consideration of chemical equilibrium will lead to unambiguous formulation of the chemical potentials for such species. At this point a fully accurate and useful form of the GEE will have been generated. Nevertheless, this equation can be reformulated to better serve mechanical engineering applications and expectation by introducing the FDS properties both for the matter in the CV and for matter at the inlets and exits. The revised formulation splits the total exergy into the chemical exergy and the more familiar thermomechanical energy. The result is a highly adaptable and informative GEE, and the process generating this equation should be a significant teaching tool and learning experience that emphasizes the breath and rigorous nature of thermodynamics. Graduate and advanced senior students in mechanical or related engineering interested in advanced energy conversion and chemical process systems should be aware of the GEE in addition to the two forms of the REE. For background, such students should be well versed in energy and entropy analysis and mass conservation. In addition, some introduction to applied physical chemistry should be provided or required. The required chemistry is mostly just familiar stoichiometry; however, a working knowledge of chemical equilibrium and an appreciation of the chemical potential is also necessary. From a pedagogical view, the development of the GEE is a challenging but rewarding exercise in the integration of most of the principles of basic thermodynamics and elementary chemical thermodynamics. Knowledge and understanding of the GEE will be helpful in the thermodynamic analysis and, perhaps more importantly, the technoeconomic analysis of combustion and reaction systems, desalination and chemical process systems and even some advanced HVAC systems among others.

Citation
Format

Jeter, S. (2022, August), The Generalized Exergy Equation: A Rigorous Development and Detailed Presentation Suitable for Presentation to Advanced Undergraduates and Beginning Graduate Students Paper presented at 2022 ASEE Annual Conference & Exposition, Minneapolis, MN. 10.18260/1-2--41533

TY  - CPAPER
AB  - The Generalized Exergy Equation: A Rigorous Development and Detailed Presentation
Suitable for Presentation to Advanced Undergraduates and Beginning Graduate Students
Exergy is well known as the potential for a system to produce useful work in interaction with its surroundings, which are assumed to remain in equilibrium in this context. The generalized form of the exergy equation applies when the engineering system is open to the surrounding medium so that the matter in the system can merge with and react with and ultimately become identical with the constituents of the medium. The medium in its minimal form could be just the surrounding gaseous atmosphere. This gaseous atmosphere can conceivably be as simple as a hypothetical single component such as “dry air”, but more likely the simplest worthwhile and interesting medium is “moist air” (water vapor and dry air) in HAVC and similar applications or a slightly more complicated mixture of dry gases (nitrogen, oxygen, argon, carbon dioxide) and water vapor in combustion applications. For desalination systems, the medium must include the liquid saline ocean in equilibrium with the gaseous atmosphere. If more complex chemical processing applications are considered, an inert version of the crust of the earth must be included in the medium. The latter rather complex medium must contain the stable chemical versions of the ores or other inert feedstocks of interest that are in equilibrium with the gaseous atmosphere. A straightforward, if somewhat necessarily complicated, derivation of the unrestricted version of the control volume (CV) exergy equation will be detailed in the proposed presentation. In literature or practice, this equation is typically referred as the “Generalized Exergy Equation” (GEE) or an equivalent name, so this convenient and descriptive terminology is adopted herein.
The nature of the applicable or assumed dead state is critical in the development and formulation of an exergy equation. For any form of the Restricted Exergy Equation, the system is kept separated from its surrounding, with the pertinent dead state being the Restricted Dead State (RDS) at ambient temperature and pressure. A commonly encountered special case of the RDS is the situation in which the composition of the matter, usually a fluid, is unchanged between the prevailing state and the dead state. This RDS can be called the Fixed (or Frozen) Dead State (FDS). In contrast, for the GEE addressed herein, the ultimate or General Dead State (GDS) must be considered. A typical textbook or heuristic presentation of the GEE involves the expected combination of the energy and exergy equations along with a somewhat heuristic (or at least challenging) and superficially plausible introduction of the FDS properties. While the result is useful, there are logical and pedagogical deficiencies in this approach, specifically (1) it could be erroneously inferred from some presentations that the mole numbers of the components of the system are fixed, (2) the expected forms of the system and stream exergies do not appear naturally unless the FDS is specifically introduced as an intermediate state, (3) the role of the FDS in distinguishing and quantifying the Chemical Exergy is not demonstrated, and (4) an equivalent form of the total exergy often seen in the literature and practice of chemically oriented disciplines is not encountered. 
The alternative straightforward and rigorous development generates an equation governing the generalized exergy by introducing equalities involving the chemical potentials of the components of the system that prevail in the medium. Note of course, that some components of the system, such as fuels, will not exist as independently defined components of the medium. In these cases, consideration of chemical equilibrium will lead to unambiguous formulation of the chemical potentials for such species. At this point a fully accurate and useful form of the GEE will have been generated. Nevertheless, this equation can be reformulated to better serve mechanical engineering applications and expectation by introducing the FDS properties both for the matter in the CV and for matter at the inlets and exits. The revised formulation splits the total exergy into the chemical exergy and the more familiar thermomechanical energy. The result is a highly adaptable and informative GEE, and the process generating this equation should be a significant teaching tool and learning experience that emphasizes the breath and rigorous nature of thermodynamics.
Graduate and advanced senior students in mechanical or related engineering interested in advanced energy conversion and chemical process systems should be aware of the GEE in addition to the two forms of the REE. For background, such students should be well versed in energy and entropy analysis and mass conservation. In addition, some introduction to applied physical chemistry should be provided or required. The required chemistry is mostly just familiar stoichiometry; however, a working knowledge of chemical equilibrium and an appreciation of the chemical potential is also necessary. From a pedagogical view, the development of the GEE is a challenging but rewarding exercise in the integration of most of the principles of basic thermodynamics and elementary chemical thermodynamics. Knowledge and understanding of the GEE will be helpful in the thermodynamic analysis and, perhaps more importantly, the technoeconomic analysis of combustion and reaction systems, desalination and chemical process systems and even some advanced HVAC systems among others.

AU  - Sheldon Jeter
CY  - Minneapolis, MN
DA  - 2022/08/23
PB  - ASEE Conferences
TI  - The Generalized Exergy Equation: A Rigorous Development and Detailed Presentation Suitable for Presentation to Advanced Undergraduates and Beginning Graduate Students
UR  - https://peer.asee.org/41533
DO  - 10.18260/1-2--41533
ER  -