Modern Engineering Thermodynamics

322 CHAPTER 10: Availability Analysis
10.4 MAXIMUM REVERSIBLE WORK
Any mathematical point function can be interpreted as the potential
of a conservative vector. Since all thermodynamic properties (both
extensive and intensive) are point functions, they must also represent
such potentials. Some of these vector potential relations are well
known. For example, Fourier’s lawrelatestheconduction heat transfer
rate per unit area vector to its potential, the local temperature T, as
!
q
! = − kt ∇ ðTÞ, where k t is the thermal conductivity.
Since the potential of a conservative force is equal to the reversible
work done on or by a system, perhaps it would be useful to know the
potential for the maximum possible reversible work that a system could
produce from any given state. Using a combination of the energy and
entropy balance, we can compute the reversible work produced or
absorbed by a closed system with isothermal boundaries at temperature
T b , as the system changes from state 1 to state 2, as
ð 1
W 2 Þ rev
= E 1 − E 2 + ð 1
Q 2 Þ rev
= E 1 − E 2 − T b ðS 1 − S 2 Þ
where E = U + mV 2 /2g c + mgZ/g c . To determine the maximum possible
reversible work we must define a minimal energy reference state,
which we call the ground state of the system. 4 We denote quantities
System with total
energy E and total
entropy S
Reversible
engine
Ground state at
temperature T 0 with
total energy E 0 and
total entropy S 0
System boundary
temperature is the
same as the ground
state temperature,
T b = T 0
W max rev
FIGURE 10.2
The maximum reversible work a system can
produce. Note that the system boundary must be
at the same temperature as the ground state to
produce the maximum reversible work.
in the ground state with a zero subscript (e.g., p 0 and T 0 denote the pressure and temperature of the ground
state). Then, for an arbitrary starting state, the maximum reversible work that a closed system can perform is
ðWÞ maximum
reversible
= E − E 0 − T 0 ðS − S 0 Þ (10.1)
where the system boundary is now assumed to be at the ground state temperature T 0 (see Figure 10.2). Note that
the change in entropy in Eq. (10.1) is solely due to a heat transfer by the system and not as a result of irreversibilities
(either internal or external) that may occur during the process of bringing the system to the ground state.
Now, what constitutes a suitable ground state? If we are to have a minimum system energy in the ground state,
then clearly the kinetic and potential energies of this state should be zero, so we set V 0 = Z 0 = 0. Beyond this,
the remaining properties of the ground state can be arbitrarily chosen.
10.5 LOCAL ENVIRONMENT
If the maximum reversible work is to be a useful concept in
engineering analysis, a practical ground state must be chosen. It
is easy to see that the most convenient ground state for any
given system is its local environment. Recallthatthetermsurroundings
refers to everything outside the system boundaries,
consequently the surroundings must include the local environment.
We therefore define the local environment and the ground
state as in the display boxes. Since the ground state and the
local environment are essentially identical, we denote all the
properties of the ground state and local environment with a
zero subscript (see Figure 10.3).
System boundary
System at E,∀, p, T, and S
Local environment = Ground state
at E 0 , ∀ 0 , p 0 , T 0 , and S 0
FIGURE 10.3
The surroundings, local environment, and the ground
state.
WHAT IS THE LOCAL ENVIRONMENT?
The local environment is a portion of the total surroundings in contact with the system boundaries. It must be large
enough for all of its intensive properties to be constant, and it must be insensitive to state changes of the system.
4 Some authors call this the dead state. However, the term ground state is more easily understood, since it is conceptually similar to the
zero energy ground level reference state commonly used in gravitational potential energy analysis.