Modern Engineering Thermodynamics

250 CHAPTER 8: Second Law Closed System Applications
8.1.1 Closed System Indirect Method
1. The entropy produced by a change of state is given by Eq. (7.75) as
Z Z
_q
1ðS P Þ 2
= ms ð 2 − s 1 Þ−
dA dt (8.1)
τ Σ T b act
2. The entropy production rate of a closed system is given by differentiating Eq. (8.1) with respect to time, or
Z
_q
_S P = _S − dA (8.2)
Σ T b act
where T b is the local system boundary temperature evaluated at the point where _q crosses the system boundary Σ.
8.1.2 Closed System Direct Method
1. The entropy produced by a change of state is given by
Z 2
Q
1ðS P Þ 2
=
1 T dT + dW
2
act
T
V
irr
Z Z
=
τ V
− _q T 2 dT
dx
+ σ W
dV dt (8.3)
2. The entropy production rate of a closed system is given by
Z
_S P = − _q
dT
+ σ
T 2
W dV (8.4)
dx
where σ W =(σ W ) viscouw +(σ W ) electrical +(σ W ) diffusion + ⋯ and T is the local temperature inside the system volume
evaluated at the point where the heat transfer and work irreversibilities occur.
Both the direct method and the indirect method give accurate answers for entropy production if applied correctly.
Which method you choose to solve a particular problem depends entirely on the type of information given to
you in the problem statement (usually only one of the two methods works for a given problem scenario).
In the examples presented in this chapter, we are concerned mainly with determining the amount of entropy
production for a process by calculating it from the closed system entropy balance (i.e., by using the indirect
method).
Last, before we begin the example problems, keep in mind the basic reason why the evaluation of entropy production
is important. The entropy production is a measure of the “losses” within the system, so the larger the
entropy production, the more inefficient the system is at carrying out its function. We continually look for ways
to minimize a system’s entropy production and thus improve its overall operating efficiency.
8.2 SYSTEMS UNDERGOING REVERSIBLE PROCESSES
In Chapter 7, we define a reversible process as any process for which the entropy production or entropy production
rate is zero. Thus, a system is said to be reversible if there are no losses due to friction, viscosity, heat transfer,
diffusion, or the like anywhere within the system. Consequently, the entropy production of a system
undergoing a reversible process is always zero, or
_S P = 1 ðS P Þ 2 = 0 for all reversible processes
and the closed system entropy balance (SB) given in Eq. (7.75) reduces to
Z Z
_q
S 2 − S 1 = ms ð 2 − s 1 Þ =
dA dt (8.5)
τ Σ T b rev
where _q is the heat flux, T b is the system boundary absolute temperature, A is the system boundary area, and τ is
the time required to change from state 1 to state 2. The closed system entropy rate balance (SRB) is obtained
from this equation by differentiating it with respect to time as 1
Z
dS
dt = _S = m_s =
Σ
_q
T b
dA (8.6)
rev
1 Remember that the system mass m is always constant in a closed system.