Modern Engineering Thermodynamics

240 CHAPTER 7: Second Law of Thermodynamics and Entropy Transport and Production Mechanisms
a vaporization process), Q rev > Q act . The irreversibilities involved in a phase change process arise largely from the
heat transfer required to produce the phase change and from the mechanical moving boundary work associated
with any volume change between the phases. If the real system is truly isothermal, then heat transfer irreversibilities
may be allocated to the system’s surroundings. As for the work mode irreversibilities, for a reversible
process, ðdWÞ rev
= −mp dv, and the differential energy balance then gives ðdQÞ rev
= mdu+ ð pdvÞ = mdh ð Þ.However,
for an actual irreversible process ðdWÞ act = f ðη W ÞðdWÞ rev ,wheref(η W ) is a function of the work transport
energy efficiency, given in Eqs. (4.71) and (4.72), immediately preceding Eq. (7.46). Then, ðdQÞ act =
m½du + f ðη W ÞpdvŠ ≠ mdh ð Þ. Consequently, if there is negligible change in system volume during the phase change
or the moving boundary work is carried out very efficiently, then the work mode irreversibilities also are insignificant.
Under these conditions, the actual phase change can be accurately approximated as a reversible
process.
7.17 ENTROPY BALANCE AND ENTROPY RATE BALANCE
EQUATIONS
The resulting entropy balance for a closed system of mass m is given by integrating Eq. (7.48) and substituting
in Eqs. (7.60), (7.62), (7.65), and (7.69) to produce
Z
τ
General closed system entropy balance ðSBÞ
Z
_q
dA dt + 1 ðS P Þ
Σ T 2 = ðS 2 − S 1 Þ system =½mðs 2 − s 1 ÞŠ system (7.75)
b act
For the simplified case of isothermal boundaries, this equation reduces to
where in each case
Isothermal boundary closed system entropy balance
1Q 2
+ 1 ðS P Þ
T 2
= ms ð 2 − s 1 Þ (7.76)
b act
Z Z
1ðS P Þ 2 =
τ V
− _q
dT
T 2 + σ W
dx
dV dt
In Chapter 9, Eq. (7.75) is expanded into the following open system general entropy rate balance equations.
General open system entropy rate balance ðSRBÞ
Z
_q
dA +∑ _ms − ∑ _ms + _S P = _S system (9.4)
Σ T b act inlet outlet
and, for the simplified case of isothermal boundaries, this equation reduces to
Isothermal boundary open system entropy rate balance
_Q
_ms −∑ _ms + _S p = _S system (9.5)
T b act
out
+∑
in
There are two effective ways for calculating the entropy production or the entropy production rate for any
process: the direct and the indirect methods.
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Direct method involves calculating the amount of entropy produced for a process from its defining
equations. For example, for closed systems, the direct method of calculating the entropy production rate is
Z
_S P = ðσ Q + σ W ÞdV
V
where
σ Q = _q
dT
T 2 dx
actual
and
σ W = σ viscous + σ electrical + ⋯