EXERGY ANALYSIS
3.2 PHYSICAL EXERGY
When evaluated relative to the environment, the kinetic and potential en- ergies of a system are in principle fully convertible to work as the system is brought to rest relative to the environment, and so they correspond to the kinetic and potential exergies, respectively. Accordingly,
where V and denote velocity and elevation relative to coordinates in the environment, respectively. We may then write Equation as
e
+
Considering a system at rest relative to the environment = = 0), the physical exergy is the maximum theoretical useful work obtainable as the system passes from its initial state where the temperature is T and the pressure is p to the restricted dead state where the temperature is and the pres- sure is The chemical exergy is the maximum theoretical useful work ob- tainable as the system passes from the restricted dead state to the dead state where it is in complete equilibrium with the environment. (The use of the term chemical here does not necessarily imply a chemical reaction, however;
see Section 3.4.2 for an illustration.) In each instance heat transfer takes place with the environment only. Physical exergy is considered further i n the next section and chemical exergy is the subject of Section 3.4.
3.2 PHYSICAL EXERGY
The physical exergy of a closed system at a specified state is given by the expression
where V , and denote, respectively, the internal energy, volume, and entropy of the system at the specified state, and and are the values of the same properties when the system is at the restricted dead state.
3.2.1 Derivation
Equation 3.3 for the physical exergy can be derived by applying energy and entropy balances to the combined system shown in Figure 3.1, which consists
EXERGY ANALYSIS
.
\ \
\
System
Heat and work
Boundary of the I combined
system.
interactions
,
are allowed.interactions Total volume is between the system I
\ and the i
\ environment
\ Environment at
Figure 3.1 Combined system of closed system and environment.
of a closed system and the environment. The system is at rest relative to the environment. As the objective is to evaluate the maximum work that could be developed by the combined system, the boundary of the combined system allows only energy transfers by work across it, ensuring that the work devel- oped is not affected by heat transfers to or from the combined system. And although the volumes of the system and environment may vary, the boundary of the combined system is located so that the total volume remains constant.
This ensures that the work developed is useful: fully available for lifting a mass, say, and not expended in merely displacing the surroundings of the combined system.
An energy balance for the combined system reduces to
or
where is the work developed by the combined system, and is the internal energy change of the combined system: the sum of the internal energy changes of the closed system and the environment. The internal energy of the closed system initially is denoted by At the restricted dead state, the in- ternal energy of the system is denoted by Accordingly, can be ex- pressed as
3.2 PHYSICAL EXERGY
where denotes the internal energy change of the environment. Since and the composition of the environment remain fixed, is related to changes in the entropy and volume of the environment through Equation
As' - A V
Collecting the last three equations,
-
- As' -As the total volume of the combined system is constant, the change in volume of the environment is equal in magnitude but opposite in sign to the volume change of the closed system: = - V ) . The expression for work then becomes
This equation gives the work developed by the combined system as the closed system passes to the restricted dead state while interacting only with the environment. The maximum theoretical value for the work is determined using the entropy balance as follows: Since no heat transfer occurs across its boundary, the entropy balance for the combined system reduces to give
where accounts for entropy generation within the combined system as the closed system comes into equilibrium with the environment. The entropy change of the combined system, AS,, is the sum of the entropy changes for the closed system and environment, respectively,
- S) +
where and denote the entropy of the closed system at the given state and the restricted dead state, respectively. Combining the last two equations, solv- ing for and inserting the result into the expression for gives
The value of the underlined term is determined by two states of the closed system-the initial state and the restricted dead state-and is independent of the details of the process linking these states. However, the value of depends on the nature of the process as the closed system passes to the restricted dead state. In accordance with the second law, this term is positive when irreversibilities are present and vanishes in the limiting case where there
EXERGY ANALYSIS
are no irreversibilities; it cannot be negative. Hence, the maximum theoretical for the work of the combined system is obtained by setting to zero, leaving
By definition, the physical exergy, is this maximum value; and Equation 3.3 is obtained as the appropriate expression for calculating the physical ergy of a system. Various idealized devices can be invoked to visualize the development of work as a system passes from a specified state to the restricted dead state [2,
3.2.2 Discussion
The physical exergy can be expressed on a unit-of-mass or molar basis.
On a unit-of-mass basis, we have
= (u - - - - (3.4)
For the special case of an ideal gas with constant specific heat ratio k, Equation 3.4 can be expressed as
The derivation is left as an exercise, Equation 3.5 is shown graphically in Figure 3.2. That the physical exergy vanishes at the restricted dead state and is positive elsewhere is evidenced by the contours of this figure.
For a wide range of practical applications not involving chemical reaction, mixing, or separation of mixture components, knowledge of the physical, kinetic, and potential exergies at various states of a system suffices. An ex- plicit evaluation of the chemical exergy is not required because the chemical exergy value is the same at all states of interest and thus cancels when dif- ferences in exergy values between the states are calculated. This is observed in applications with closed systems and control volumes alike; for illustrations see References 1, 2. In such special applications knowledge of the chemical composition of the environment is not required. Only the pressure and temperature have to be specified. Furthermore, the exergy change between two states of a closed system is determined from Equations 3.1 and 3.3 as