The Carnot cycle efficiency is an application of the Second Law of Thermodynamics, establishing the upward bound for the amount of useful work a heat engine can produce from heat. With the maximum conversion efficiency established, it is possible to consider the high-temperature reservoir itself as a source of work. By itself, the reservoir at high temperature, TH, in Fig. 3.6(a) could be seen as having the potential to do work if it could transfer heat to a heat engine (the type in Fig. 3.4), as in Fig. 3.6(b). The potential to do work is embodied in a property called exergy. Exergy is the term used to quantify the work potential of a system from its initial state to the dead state, which is the state of the environment, usually at standard temperature and pressure (STP, 25°C and 1 atm). (The dead state could be at any reference state.)
Besides the work potential of temperature, exergy also applies to pressure because the pressure could be relieved through a turbine, which would convert the pressure to shaft work. Temperature and pressure together define a state, which has its associated properties, so properties such as internal energy and
enthalpy also have a work potential. In this section, the exergy of enthalpy will be derived. It applies to thermal and mechanical reactions (ignoring chemical and nuclear reactions).
The First Law is written in its most general form for a control volume using the sign convention of positive for inputs and negative for outputs. E represents the total energy.
in UEout dEsystem
For the steady-flow device at state T and P shown in Fig. 3.7, its work potential is the amount of useful work that it can perform as it changes to the dead state, To and Po. Because this is an open system with mass flow, the property that accounts for both the internal energy and the flow work of the mass is enthalpy, H. As the system changes to the dead state, it rejects heat and does work, and both are outputs and written with minus signs:
Because exergy is the maximum work potential of a system, the process must be reversible to achieve the maximum work, with no losses caused by irreversibilities, such as heat transfer through a finite temperature difference from T to To. But if a reversible heat engine were used to bridge the temperature difference, using T as the heat source and To as the heat sink, the heat transfer would become reversible, and additional useful work would be performed. (For the derivation of exergy, the heat source is considered to be a high-temperature reservoir able to maintain its temperature as it transfers heat to another system. Derivations based on equations instead of physical representations also make the same consideration.) Therefore, the heat could be supplied to a reversible heat engine that operates on the Carnot cycle. The heat supplied to the heat engine, 8Qin,Calnot, is converted to work with a Carnot cycle efficiency— see Eq. (3.24). The relationship between entropy and heat from Eq. (3.6) allows for substitution in the second term.
= SQi in,Carnot Todl^m,Carnot
SQm,Carnot = SWCainot + Tod Sin,Cainot
But the equation should be written in terms of the system, sys, instead of the Carnot cycle heat engine. Because the heat leaving the system, Qout,sys, is the heat entering the heat engine, Qin,Calnot, and following the sign convention of "in" being positive and "out" being negative, the signs are opposite:
Likewise, the entropy change accompanying the heat transfer between the system and the heat engine follows the same sign convention. An assumption has been made that the heat transfer between the system and the engine occurs isothermally, which allows the use of the relationship dS = 8Q/T. The system loses entropy because it loses heat (-Q), and the engine gains entropy because it gains heat. Both processes occur at the same temperature, T.
j c _ A c dSout,sys dSin,Carnot
The work, 8W, that the steady-state device can do is shaft work, 8Wshaft.
Substituting into the original expression (-8Q - 8W = dH)
- (8 Wcarnot - T0dS) - 8Wshaft = dH 8Wshaft + 8Wcamot = - dH + T0dS
Integrating from the initial state to the final, dead state:
The total useful work potential in Eq. (3.25) is called the exergy, and for a steady-flow device it is called the "exergy of enthalpy," xh:
The exergy change from state 1 to state 2 (for a control volume with no change in kinetic or potential energies) is shown in Eq. (3.27).
The change in exergy represents the exergy destroyed, meaning that work potential has been consumed during the change of state.
Just as entropy can only increase for irreversible processes, as shown in Eq. (3.12), exergy can only decrease for irreversible processes. Next, the Decrease in Exergy principle will be derived for the device in Fig. 3.8.
Energy and entropy balances are written for the isolated system in Fig. 3.8, and E represents the total energy (internal, kinetic, potential, etc.) of the system. The following derivation is for an isolated system and is based on both First and Second Laws.
The energy balance for the system of Fig. 3.8 is reduced to H1 = H2 because it is isolated.
pin pout ~
Heat, work, mass internal, flow, kinetic, potential
For the entropy balance, based on Eq. (3.13), there is no mass flow in or out across the extended boundary.
Multiplying the entropy equation (Sgen) by To and making it equal to the energy balance gives Eq. (3.28).
The exergy change was derived earlier in Eq. (3.27) for a control volume, and it is written with molar properties in Eq. (3.29).
Subtracting Eq. (3.29) from Eq. (3.28) gives Eq. (3.30).
From the Increase in Entropy Principle presented in Eq. (3.12), Sgen > 0, so the exergy change of an isolated system is negative. This is also called the Decrease in Exergy Principle. Although the derivation was based on a control volume enclosed by an extended boundary, this principle is universally applicable because all systems can be made isolated systems by an extended boundary as in Fig. 3.8.
The decrease in exergy is "destroyed" exergy, Xdestroyed, which is then a positive quantity because of the "negative" connotation of the term "destroyed."
Xdestroyed = ToSgen (3.32)
Exergy destroyed is identical to irreversibility, I, which is the difference between the reversible work and the actual work.
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