Modern Engineering Thermodynamics

228 CHAPTER 7: Second Law of Thermodynamics and Entropy Transport and Production Mechanisms
then
ðdS T Þ rev = d Q
Q T
rev
(7.45)
which, when integrated, produces Eq. (7.43) again.
In an irreversible process, the total production of entropy is always positive, by virtue of the second law of thermodynamics.
Therefore, Eq. (7.25) can be viewed as an entropy balance equation for a closed system undergoing
irreversible heat and work transports of energy. Then,
dS = dS T + dS P
= dQ
+ dW rev − dW act
T
act
T
where dS P > 0 by the second law of thermodynamics. To reconcile the difference between the actual work and the
reversible work terms in this equation, we use the conceptofaworktransportenergyefficiency,η W ,whichwas
introduced in Chapter 4. For a work-absorbing system such as a pump, we have
Work-absorbing system
and, for a work-producing system such as an engine, we have
W act = W rev /η W (4.71)
Work-producing system
W act = η W W rev (4.72)
Let us define an irreversible work component, W irr , which is always a positive number, as
and
W irr = ð1 − 1/η W ÞW rev , for a work-absorbing system
W irr = ð1 − η W ÞW rev , for a work-producing system (7.46)
Note that η W is a positive number between zero and unity (0 < η W < 1) and that W rev is negative for a workabsorbing
system and positive for a work-producing system. Consequently, Eq. (7.46) produces positive values
for W irr for both work-absorbing and work-producing systems. Then, we can write for either a work-absorbing or
a work-producing system that
and the entropy balance equation becomes 9
dS = dS T + dS P =
W rev − W act = W irr
dQ T
act
+ dW T
irr
(7.47)
We now wish to identify the individual heat and work components of the entropy transport and production
terms. For a closed system, we may decompose the transport and production terms as follows:
and
so that
dS = ðdS T
dS T = ðdS T
dS P = ðdS P
Þ Q
+ ðdS T
Þ Q + ðdS T Þ W
Þ Q + ðdS P Þ W
Þ W
+ ðdS P Þ Q
+ ðdS P Þ W
(7.48)
Substituting Eq. (7.44) into (7.47) gives
dS = d Q
+ Q
T
act
T 2 dT + dW
act
T
irr
(7.49)
9 Note that W irr is always positive for both work-absorbing and work-producing systems, so its sign is correct here.