The mechanical aspect of entropy-exergy relationship, together with the thermal aspect usually considered, leads to a formulation of physical exergy based on both useful work and useful heat that are the outcomes of available energy of a thermodynamic system with respect to a reservoir. This approach suggests that a mechanical entropy contribution can be defined, in addition to the already used thermal entropy contribution, with respect to work interaction due to pressure and volume variations. The mechanical entropy is related to energy transfer by means of work interaction and it is complementary to the thermal entropy that accounts energy transfer by means of heat interaction. Furthermore, the study proposes a definition of exergy based on Carnot cycle that is reconsidered in the case the inverse cycle is adopted and, as a consequence, the concept that work depends on pressure similarly as heat depends on temperature, is pointed out. Then, the logical sequence to get mechanical exergy expression to evaluate useful work withdrawn from available energy is demonstrated. Based on mechanical exergy expression, the mechanical entropy set forth is deduced in a general form valid for any process. Finally, the formulation of physical exergy is proposed that summarizes the contribution of either heat or work interactions and related thermal exergy as well as mechanical exergy that both result as the outcome from the available energy of the composite of the system interacting with a reservoir. This formulation contains an additional term that takes into account the volume and, consequently, the pressure that allow to evaluate exergy with respect to the reservoir characterized by constant pressure other than constant temperature. The basis and related conclusions of this paper are not in contrast with principles and theoretical framework of thermodynamics and highlight a more extended approach to exergy definitions already reported in literature that remain the reference ground of present analysis.
The online version of this article (doi:10.1186/2251-6832-3-4) contains supplementary material, which is available to authorized users.
Competing interests
The author declares that he has no competing interests.
Abkürzungen
A
System
Cp
Specific heat under constant pressure (J ⋅ kg−1 ⋅ K−1)
CV
Specific heat under constant volume (J ⋅ kg−1 ⋅ K−1)
E
Energy (J)
EXM
Mechanical exergy (J)
EXT
Thermal exergy (J)
HP
High pressure
HT
High temperature
LP
Low pressure
LT
Low temperature
P
Pressure (MPa)
Q
Total heat interaction over the whole Carnot cycle (J)
QHT
High temperature heat (J)
QLT
Low temperature heat (J)
R
Reservoir
Universal
gas constant (J ⋅ kgmol−1 ⋅ K−1)
SM
Mechanical entropy (J ⋅ kgmol−1 ⋅ K−1)
ST
Thermal entropy (J ⋅ kgmol−1 ⋅ K−1)
T
Absolute temperature (K)
U
Internal energy (J)
V
Volume (m3)
W
Total work interaction over the whole Carnot cycle (J)
WHP
High pressure work (J)
WLP
Low pressure work (J)
Carnot direct cycle efficiency
Carnot inverse cycle efficiency
ε
γ
Ω
Available energy
Background
The first purpose of the present paper is to highlight the mechanical aspect of entropy property, complementary to the thermal aspect, and the definitions already existing in classical thermodynamics literature. The second purpose, on the basis of entropy property structure and the thermal and mechanical aspects of entropy-exergy relationship, is to propose a formulation of physical exergy based on both useful work and useful heat. This formulation would be suitable to evaluate either the net useful work or the net useful heat of a system being both the outcome of available energy, considering the role of pressure other than temperature, in the evaluation of work and heat interactions in combination with a reservoir as defined in the study of Gyftopoulos and Beretta[1]. The formulation, already established in literature, is based on mutual stable equilibrium state between system and reservoir due to temperature equality only. The tentative of the paper is to adopt the equality of pressure as a further condition of mutual stable equilibrium and to consider the pressure, too, in the formulation of the physical exergy as well as in the formulation of entropy. Pressure is accounted by means of the mechanical entropy related to volume. This is based on the relationship existing between entropy and exergy, either for heat interaction (thermal exergy based on thermal entropy) or for work interaction (mechanical exergy based on mechanical entropy).
In order to simplify this analysis, the following assumptions have been posited: the system is considered as ‘simple’ according to the terminology reported in literature[1]; the system consists in a perfect and single-phase homogeneous gas; there are no phase changes or chemical and nuclear reaction mechanisms inside the system; the existence of an external reference system that behaves as a reservoir is assumed[1]; the analysis is focused on stable equilibrium states; the kinetic and potential energy of the system as a whole are neglected. On the basis of these assumptions, chemical exergy will be out of the domain of present study and reference will be made to physical exergy only.
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Methods
The methods adopted in the present analysis are based on the dualism deduced by the existence of the thermal and mechanical aspects characterizing the concept of energy and by the correlations existing between them; this dualism is developed in order to identify the two components of entropy and exergy properties correlated to heat interaction and work interaction exchanged by the system with a thermo-mechanical reservoir. In addition, the symmetry of the implications of temperature with respect to thermal energy and of pressure with respect to mechanical energy constitutes the paradigm for achieving the definition of the extended physical exergy that depends on both temperature and pressure.
Thermal and mechanical contribution of entropy property
As reported in literature, entropy property is related to the second law and the thermal aspect of thermodynamic systems behavior and properties. An investigation on the mechanical aspect can also be accomplished with an analysis of thermodynamic processes considered as ideal without thermal and mechanical irreversibilities. In particular, the isothermal reversible process is evaluated here in which the internal energy of a system with no chemical and nuclear reactions and constant amount of constituents is formulated by the classical expression of thermodynamic potential known as Euler relation:
(1)
which implies that the variation of internal energy alongside a reversible isothermal process is null as follows for a homogeneous single-phase system without chemical reactions:
(2)
where δQ and δW both represent positive energy received into the system by convention for reference transfer direction. The mechanical term d(PV) is equal to zero, in fact:
(3)
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being PdV = − VdP alongside an isothermal process. The mechanical term d(PV) is equal to zero also because, having assumed the state equation as valid, the term PV is constant, being T as constant by definition alongside the isothermal process. The following result is obtained:
(5)
that is valid if and only if entropy remains constant but, on the other side, it should change due to heat interaction occurring alongside the isothermal process. Thus, entropy should be constant and should change at the same time. This is an apparent inconsistency that can be resolved if S is intended to be the total entropy. In fact, it is here posited that total entropy is the result of contribution of entropy due to heat interaction, related to temperature, and entropy due to work interaction, related to pressure.
Another consideration addresses to the adiabatic reversible process, namely isoentropic, that is accomplished at constant thermal entropy, while temperature and pressure change according to the following formula:
(5a)
or
(5b)
Thus, thermal entropy variation related to heat interaction is null depending on a compensation effect due to the combination of either temperature and pressure variation.
The above considerations in the special case of isothermal reversible process suggest that both internal energy and entropy remain constant while a heat-to-work conversion occurs isothermally due to equal quantities of heat and work interactions between the system and the reservoir. Nevertheless, the heat interaction implies a transfer of entropy; therefore, a transfer of entropy under ‘thermal’ form requires an entropy conversion into ‘mechanical’ form in order to render the balance of entropy (total) equal to zero as required by the expression (Equation 4). Therefore, it can be posited that this mechanical form of entropy is associated to work interaction by means of pressure and volume variations. Mechanical entropy is complementary to and does not contradict the classical expression of entropy calculated alongside an isothermal reversible process that corresponds to the thermal entropy associated to heat interaction only.
This analysis suggests that the entropy (total) can be considered as constituted by two components, namely ‘thermal entropy’ that is constant in adiabatic reversible (isoentropic) processes where work interaction only occurs, and a ‘mechanical entropy’ that is constant in isovolumic reversible processes where heat interaction only occurs. In addition, it can also be posited that entropy, appearing in the expression of internal energy, represents the thermal component of entropy (or thermal entropy) that, thus, results consistent with the properties requested to comply with the above behavior. The relationship between entropy and exergy represents the basis for assuming and proving that thermal and mechanical components set forth for entropy can be retrieved in exergy property as well.
Thermal aspect of entropy-exergy relationship
The definition of exergy here, considered for the scope of the present analysis, is the one based on heat and work interactions; in particular, the exergy formulated as the maximum net useful work obtained from the available energy on the basis of the thermodynamic efficiency of the Carnot direct cycle operating between the variable temperature T of system A and the constant temperature TR of reservoir R, considered as the external reference system, is discussed in this section:
(6)
where δWREVCONVER is the net amount of work resulting as the balance of a direct cycle that converts the available heat at a temperature T into work by means of a cyclic machinery in combination with the thermal reservoir at constant temperature TR; δWREVTRANSF is the net amount of available energy transferred by means of a work interaction in a cyclic machinery, resulting from system volume variation (cyclic machinery is an ideal device that returns to its initial state at the end of whatsoever process it accomplishes). For sake of generality, heat and work interactions are considered occurring either successively or simultaneously, and both result from generalized available energy of a simple system as defined in the study of Gyftopoulos and Beretta[1]. Therefore, in differential terms:
(7)
in which δQHT represents the infinitesimal heat interaction along the process at temperature T different from the temperature TR of the reservoir; δWHP is the infinitesimal work interaction at high pressure P, alongside the process, different with respect to the reservoir pressure PR. The formula (Equation 7) corresponds to the already known classical definition of physical exergy[2, 3]. This expression is used to define the exergy that is identified by the superscript T (that stands for ‘thermal’) according to the definition reported in the literature[2‐4] as above pointed out. In finite terms, considering that, δWHP = − PdV:
(8)
where W10 is the maximum net useful work output from the generalized available energy[1] as the result of interaction between the system and the reservoir; Q10HTis the heat interaction from higher isothermal curve at T to lower isothermal curve at TR (as a particular case, heat interaction can occur alongside an isovolumic process); and W10HP is the work interaction from higher isothermal curve at T to lower isothermal curve at TR. The sum of Q10HT and W10HP can also be expressed as:
(9)
that represents the equivalence with the amount of heat interaction only in the isovolumic process between two different temperatures. Thus, thermal exergy can also be associated to a sequence of isovolumic-isothermal process connecting the generic state 1 with a different generic state 0 of the system A. The integration function in the expression of thermal exergy (Equation 8) represents, in infinitesimal terms, the definition of entropy as per Clausius formulation or, as here proposed, the thermal component of entropy property identified by the superscript thermal, and therefore:
(10)
as the result of combination of the system and the reservoir at constant temperature TR and constant pressure PR. This formulation is also consistent with the formulation presented by Gyftopoulos and Beretta[1], deduced from the definition of generalized available energy with respect to an external reference system at constant temperature TR and constant pressure PR that behaves as a reservoir.
Mechanical aspect of entropy-exergy relationship
The correlation between thermal entropy and thermal exergy clarified so far suggests to assume the mechanical aspect of entropy-exergy relationship to attempt the evaluation of the concept of exergy related to work and pressure, based on the existence of the mechanical component of entropy previously assumed. This assumption derives from the fact that equality of pressure between the system and the reservoir is a further condition for stable equilibrium state of the composite system-reservoir other than the equality of temperature.
Now that we have defined exergy formulated by the direct cycle as thermal exergy which highlights the role of temperature in heat-to-work conversion, we may now search a definition of mechanical exergy that is expressed by the inverse cycle with the intent of evaluating the role of pressure in the opposite process, that is, in a work-to-heat conversion. Also for the mechanical aspect, the general formulation of exergy, in infinitesimal terms, derives from the relationship built around the Carnot cycle and its expression of thermodynamic efficiency.
The Carnot cycle used in the formulation of thermal exergy is defined in literature as a symmetric cycle consisting of four processes, each pair of which is of the same type (isodiabatic) as represented in Figure1.
Figure 1
Symmetry property of Carnot cycle.P, pressure; QHT, high temperature heat; WHP, high pressure work ; QLT, low temperature heat; WLP, low pressure work; V, volume; T, absolute temperature; S, entropy; 0 C, point 0 of Carnot cycle; 1 C, point 1 of Carnot cycle.
×
If the working system is a perfect gas as assumed, then the alternating polytropic (isoentropic) processes provide the following property to the cycle 0-1-1 C-0 C in Figure1:
(11)
From which, it may be inferred that, since the ends of the processes are proportional, the work interaction between the system and the external environment (reservoir) is the same both along the isoentropic compression from 0 to 1 and in isoentropic expansion from 1 C to 0 C. The amount of work interaction
(12)
depends on, and therefore, W is equal for the two different adiabatic reversible (isoentropic) processes.
As a result, the balance of the work interaction between the system and the reservoir along the isoentropic processes alone is null, and the balance contribution over the whole cycle is due exclusively to the isothermal processes where heat and work is exchanged simultaneously in directly proportional and equal amounts. This property enables to express the thermodynamic efficiency of the Carnot cycle of a closed nonbulk-flow system both in terms of heat and work interactions. The efficiency can be expressed either in terms of heat only or in terms of work only due to the equality of heat and work interactions alongside the isothermal process:
(13)
With regard to the inverse cycle, if the roles of used heat QHT and utilized total work W are replaced by used work WHP and utilized total heat Q, the following expression applies (that does not correspond to the coefficient of performance reported in literature):
(14)
As may be noted, since there is a linear proportionality between the two expressions due to the identity of two of the terms, it may be inferred that the meaning of used heat QHT in the direct cycle corresponds to the meaning of used work WHP in the inverse cycle. The meaning of the utilized total work W in the direct cycle corresponds to the meaning of the utilized total heat Q in the inverse cycle as well. The conclusion is that the efficiency of Carnot cycle, that depends on isothermal process only, does not change if inverse cycle is compared to direct cycle. In fact:
(15)
Now that we have defined exergy formulated by the direct cycle as thermal exergy which highlights the role of temperature in heat-to-work conversion, we may now define as mechanical exergy the exergy that may be expressed by means of the inverse cycle. In this case, the role of pressure in the opposite conversion, that is from work into heat, is due to the pressure level of work interaction alongside the higher temperature (and higher pressure) isothermal process of inverse Carnot cycle. As a consequence of the concept of interconvertibility (Gaggioli et al.[5‐7]): ‘useful work is not better than useful heat, and the available energy results in maximum net useful heat or, equivalently, maximum net useful work or the combination of both’. Thus, the definition of mechanical exergy that represents, in this case, the maximum net useful heat obtained from the available energy in infinitesimal terms may be expressed by the following relationship:
(16)
where δQREVCONVER is the net amount of heat resulting as the balance of an inverse cycle that converts available work at a pressure P into heat by means of a cyclic machinery in combination with the mechanical reservoir at constant pressure PR; δQREVTRANSF is the net amount of available energy transferred by means of heat interaction in a cyclic machinery resulting from system thermal entropy variation (cyclic machinery is an ideal device that returns to its initial state at the end of whatsoever process it accomplishes). For sake of generality, work and heat interactions are considered occurring either successively or simultaneously, and both result from generalized available energy of a simple system as defined by Gyftopoulos and Beretta[1]. Therefore, in differential terms:
(17)
in which δWHP represents the infinitesimal work interaction along the process at the pressure P different from the pressure PR of reservoir. It is noteworthy that the role of pressure here corresponds to the role of temperature with respect to heat in thermal exergy.
The formulation of thermal exergy is now reversed to define the mechanical exergy, identified by the superscript M (that stands for mechanical) that is not referred to potential and kinetic energy; in fact, in this case, ‘potential exergy’ and ‘kinetic exergy’ terms and definitions have already been adopted as components of exergy, namely kinetic exergy and potential exergy[2‐4]. After replacing work with heat, mechanical exergy is formulated as follows:
(18)
The expression in finite terms is:
(19)
where Q10 is the maximum net useful heat output from the generalized available energy[1] as the result of interaction between the system and the reservoir; W10HP is the work interaction from higher isothermal curve at T, and in correspondence of the point at P, to lower isothermal curve at TR (as a particular case, work interaction can occur alongside an adiabatic reversible process); the sum of Q10HT and W10HP can also be expressed as:
(20)
that represents the equivalence with the amount of work interaction only in the isoentropic process between two different temperatures. Thus, mechanical exergy can also be associated to a sequence of isoentropic-isothermal process connecting the generic state 1 with stable equilibrium state 0 of the composite of system and reservoir.
If the state equation applicable to ideal gas is used with the proper substitutions, the mechanical exergy can be written as follows:
(21)
The integrand function depending on the integration operator symbol ∫ is formally similar to the integrand function that represents the definition of entropy property as per Clausius formulation and, in particular, thermal entropy according to the definition proposed here. On the basis of this formal analogy applied to work interaction, it is now possible to define a ‘mechanical entropy’ by means of the expression:
(22)
where the factor would represent the integrating factor of the infinitesimal work δWHP that makes the integration function an exact differential function. In fact, going back to the expression of mechanical exergy (Equation 21) set forth and considering that δWHP = − PdV it is now possible to write:
(23)
that relates to the work interaction with environmental system (mechanical reservoir); therefore:
(24)
where the formal analogy with the equation of thermal exergy can be noted. In order to complete the proposed analogy, if reference is made to the integrating function of Equation 23, the thermal entropy according to Clausius formulation is as follows:
(25)
whereas the mechanical entropy is formulated as:
(26)
Thus, being dSM an exact differential function, then SM is a state property and can be adopted as the formal definition of mechanical entropy. It depends on the volume that is a state property, and therefore, SM behaves as a state property as well. Furthermore, since volume is additive, it can be proved that mechanical entropy is additive as well. With regard to the dimensional analysis, since logarithmic function is dimensionless, then the dimension of mechanical entropy is related to and is (J·kg−1·K−1) that is identical to thermal entropy dimension.
The relationship between mechanical exergy and volume (and the pressure as a consequence) is the reason why the equality of pressure can be assumed as a further condition of mutual stable equilibrium of the composite system-reservoir that can be considered in the definition of entropy property. The physical meaning of mechanical exergy can be ascribed to the combination of pressure level that characterizes the internal energy of the system and the pressure level of work interaction at that pressure level of the system itself.
Physical exergy related to thermo-mechanical reservoir
The definition of thermal entropy and mechanical entropy, deduced and formulated from thermal exergy and mechanical exergy respectively, is adopted here to extend the concept of physical exergy to the work interaction other than heat interaction between the system and the reservoir as defined by Gyftopoulos and Beretta[1]. By virtue of the concepts of equivalence and interconvertibility conceived by Gaggioli et al.[5‐7], the available energy of a system results in two outcomes:
· Available work or maximum net useful work that can be evaluated as thermal exergy
· Available heat or maximum net useful heat that can be evaluated as mechanical exergy.
The extension of physical exergy proposed here is, therefore, implicated with thermal exergy underpinned by the efficiency of Carnot direct cycle and the high temperature heat together with the mechanical exergy underpinned by the efficiency of Carnot inverse cycle and the high pressure work, whereas the environment is adopted as reservoir. A more specific definition of reference external system that would be a thermo-mechanical reservoir should behave as a thermal reservoir at constant temperature and variable pressure and as mechanical reservoir at constant pressure and variable temperature. This extension includes the effect of pressure in work interaction that results in different heat amounts depending on different pressure level of the same work amount. In other terms, the concept is that the same amount of available work can be used at different pressure of the system (at different thermodynamic potential at constant temperature and variable pressure), with respect to the constant pressure of the mechanical reservoir, to be converted into heat at different temperatures. Therefore, the useful work resulting from the available energy is further evaluated in terms of second law process to calculate the amount of heat that it is capable to produce. Thus, heat is converted into work and work is converted into heat or thermal exergy is converted into mechanical exergy, and vice versa, mechanical exergy is converted into thermal exergy. With different wording, ‘extended physical exergy’ can be translated into ‘exergy of exergy’ that makes becoming work interaction equivalent to and interconvertible with heat interaction and vice versa.
Before arriving to the formulation of the extended physical exergy, the internal energy as expressed by the Gibbs relation:
(27)
can be reformulated in different terms introducing the concepts of thermal entropy and mechanical entropy:
(28)
that can be transformed using the state equation in the following form:
(29)
where dSTOTAL = dST − dSM, and in finite terms
(30)
that, associated to the temperature, takes into account either heat or work interactions that contribute to the variations of internal energy. The expression (Equation 28) can be also written as:
(31)
and in finite terms, considering that U is a state property that depends on two independent variables:
(32)
If the term ΔSTOTAL is expressed by means of Equation 30 also using Equation 5, the total entropy, resulting from the addition of thermal and mechanical components of entropy, resolves the apparent inconsistency that thermal entropy would be constant in an isothermal reversible process that requires heat interaction by means of thermal entropy exchange as pointed out in relation to Equation 4.
In fact, a consequence deduced from the definition of total entropy is that Euler equation now results in finite terms:
(33)
that is verified being Equation 30 valid; in fact, in the particular case of a perfect single-phase homogeneous gas:
(34)
and
(35)
(36)
The typical thermodynamic processes are analyzed more in detail to prove that the above expression of total entropy is valid in general for all processes.
and confirms that ΔU = 0 for an isothermal reversible process since ΔSTOTAL = 0 as required to resolve the inconsistency of condition (Equation 4) discussed in ‘Thermal and mechanical contribution of entropy property’ section.Isovolumic process, following the same procedure:
Adiabatic reversible (isoentropic) process, considering the validity of Equation 36, the following applies:
(44)
(45)
(46)
that demonstrate the existence of the relationship between pressure that changes with volume and total entropy also, in case the thermal entropy, is null due to the absence of heat interaction. Thus, the procedure adopted so far to explicit the mechanical entropy leads to clarify the apparent inconsistency highlighted in Equation 4 of ‘Thermal and mechanical contribution of entropy property’ section and underlines the role of pressure in the formulation of mechanical entropy and mechanical exergy.
Another conclusion that arises from the above procedure is that the total entropy is, in all above cases, reduced to an expression that depends on temperature only. This result complies with the fact that internal energy, under the assumption of system constituted by a perfect single-phase homogeneous gas, depends on and is characterized by the temperature only.
On the basis of the relationship between total entropy and internal energy (Equation 32), if the reservoir behaves as a mechanical reservoir at constant pressure other than as a thermal reservoir at constant temperature, the internal energy balance of system and reservoir, adopting the symbology in[1], is:
(47)
whereΔUWSYSTEM + ΔUR,W = mechanical exergy converted into thermal exergy andΔUQSYSTEM + ΔUR,Q = thermal exergy converted into mechanical exergy
(48)
or, in other terms,
(49)
where QR is the minimum heat interaction with the thermal reservoir and WR is the minimum work interaction with the mechanical reservoir. The symbol EXPHYSICAL(or XPHYSICAL), instead of W and Q, is proposed for identifying physical exergy to generalize its meaning that here addresses either to useful work or useful heat.
Going back to the Figure1, it is to be noted that QAR = WAR alongside the isothermal process; however, TR = const and PR ≠ const. Therefore, WAR at decreasing pressure constitutes an amount of exergy that is lost since it is released isothermally to the reservoir while exchanging heat acquired to lead the system A in the condition of stable equilibrium with the reservoir. This isothermal process, as recognized in ‘Thermal and mechanical contribution of entropy property’ section, makes energy conversion and entropy conversion accounted in the expression of extended physical exergy:
(50)
where the term − TR · ΔSTOTAL,Rrepresents the contribution to entropy conversion (symbol ‘CONVER’) only occurring inside the reservoir and the terms + PR · ΔVR − TR · ΔST,R represent the contribution transferred (symbol ‘TRANSF’) from the system to the reservoir. It is noteworthy that entropy conversion is inherent to energy conversion, as demonstrated in ‘Thermal and mechanical contribution of entropy property’ section, and that entropy conversion requires the additional term that contributes to exergy balance expressed in the above extended formulation (Equation 50) that, therefore, considers the effect of both energy and entropy conversion processes.
Therefore, it can be deduced, substituting the expressions (Equations 10 and 24), that:
(51)
using the expression of mechanical entropy:
(52)
that can be considered as the formulation of the extended physical exergy expressed by means of the contribution of thermal entropy and mechanical entropy. Using the state equation, the above can be also written as:
(53)
The termorof the above expressions can be defined as an ‘entropic-mechanical’ term that takes into account the entropy conversion that occurs alongside the isothermal reversible process as pointed out in the previous sections. The entropy conversion balance takes into account the mechanical entropy other than thermal entropy that is included in the above entropical mechanical term of the extended formulation.
Being an additive property, it can also be written as:
(54a)
(54b)
where Equation 54a is the component related to the variation of internal energy due to heat interaction only (in particular, alongside an isovolumic process), namely the thermal exergy; Equation 54b is the component related to the variation of internal energy due to work interaction only (in particular, alongside an isoentropic process), namely the mechanical exergy.
In general, being the internal energy additive, it follows that both contributions to internal energy due to heat and work interaction can occur successively or simultaneously along any process. The term (Equation 54a) of extended physical exergy represents the thermal exergy calculated alongside an isovolumic-isothermal process; on the other side, the term (Equation 54b) represents the mechanical exergy calculated alongside an isoentropic-isothermal process.
Results and discussion
For a more clear understanding of the implications deriving from the extended formulation, reference can be made to an adiabatic reversible process as a particular case. This process is characterized by null variation of thermal entropy (due to absence of heat interaction) and a non-null variation of mechanical entropy (due to work interaction occurring). As a consequence of the extended formulation, if the adiabatic reversible process is evaluated in terms of exergy, the available energy (in the form of pressure mechanical energy withdrawn from the system) is accounted in terms of its capability to be converted (not transferred) into maximum net useful heat, and therefore, the exergy calculation implies a lower amount if compared with the classical concept that identifies exergy exclusively with work interaction as it is. From a different perspective, the entropic-mechanical addendum of the extended formulation behaves as a reduction term that takes into account the amount of work interaction undergoing the (reversible) entropy conversion occurring alongside the isothermal process related to the thermodynamic conditions of the reservoir that renders this work not more useful to be converted back into heat interaction. In a broader view, the application of the extended exergy can be suitable in exergy method and thermoeconomic analysis of multi-conversion complex systems realized to perform direct and inverse cycle processes with work and heat flows productions for either civil and industrial engineering applications.
Conclusions
If reference is made to the Equations 54a and 54b, the physical exergy can be expressed by the sum of the two components namely thermal exergy and mechanical exergy:
(55)
that do not depend on a particular process adopted for its definition; thus, it can be considered as a general formulation valid for whatever process, reversible or irreversible, connecting two different thermodynamic states.
One of the outcomes of the present proposal is that the extended physical exergy takes into account the equality of pressure, other than equality of temperature, as a further condition of mutual stable equilibrium between system and reservoir. Looking forward to the implications of this additional condition and the generalization to ‘all systems (large and small) in all states (in particular, nonequilibrium)’ in the sense formalized and worded by Gyftopoulos and Beretta, the concept of extended physical exergy would require the reference to a mechanical reservoir in addition to the thermal reservoir[1, 8, 9].
To generalize the results of the present proposal, the formulation of entropy reported[1, 8, 9]:
(56)
should remain valid if the concept of mechanical reservoir is introduced, and the equality of pressure between the system and the mechanical reservoir is considered as a further condition for mutual stable equilibrium between system and reservoir. The additive property of entropy would lead to assume that:
(57)
where the mechanical component of entropy would be defined with reference to a mechanical reservoir at constant pressure.
Finally, the additivity of entropy components
(58)
should be proved to complete the formulation of total entropy, taking into account the general definitions proposed for thermal entropy and mechanical entropy.
Looking forward to the possible future research studies, the logical further extension would be the assumption of equality of total potentials as an additional condition of mutual stable equilibrium of the composite system-reservoir. Thus, the set of all conditions of mutual stable equilibrium would lead to a more complete formulation of exergy with the contribution of chemical and nuclear exergy related to a ‘thermo-chemical-nuclear-mechanical’ reservoir that implies the definition of chemical and nuclear entropy related to the chemical and nuclear architecture of any system in any state, in particular, nonequilibrium.
Author's information
Pierfrancesco Palazzo graduated with a masters degree in Mechanical Engineering from the Department of Mechanical and Aeronautical Engineering, University of Roma 1 ‘La Sapienza’ in Italy. He has been studying about basic thermodynamics, energy processes, exergy method and thermoeconomic analysis and the related applications, and participated to international congresses. At present, he is an engineering manager in Technip, a world leader in project management, engineering and construction for the energy industry operating in the onshore, offshore and subsea segments.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The author declares that he has no competing interests.
