Modern Engineering Thermodynamics

7.15 Work Mode Production of Entropy 235
7.15 WORK MODE PRODUCTION OF ENTROPY
From Eq. (7.53), we have ðdS P Þ W
= dW/T ,whereT is the local absolute temperature inside the system
irr
boundary evaluated at the point where the work irreversibility (e.g., friction) occurs. Then.
Z
ðS dW
P Þ W
=
(7.68)
T
and, if the system is isothermal throughout at temperature T as it changes from state 1 to state 2, then
Isothermal work mode entropy production:
ðS P Þ W
T =constant
=
irr
W irr
1 T 2
The corresponding entropy production rate equation is Eq. (7.58),
_S P W = d Z
dt
dW
T
irr
When the system local internal temperature T in Eq. (7.58) is independent of time, this equation simplifies to
(7.68a)
(7.58)
Steady state work mode entropy production rate:
_S P
W = _W
(7.58a)
T
irr
Therefore, only the work mode energy dissipated within the system contributes to the entropy production of the
system. This dissipated energy has been defined to be the difference between the reversible work (as given by
the work mode formula of Chapter 4) and the actual work (for which we have no specific formula except the
empirically based efficiency Eq. (7.46)). If one has experimentally measured the work efficiency, then W irr can
be found from Eq. (7.46). However, to evaluate ðS P Þ W
or _S P from Eq. (7.68) or (7.58), we need to know the
W
mathematical functional relation between W irr and the local absolute temperature T inside the system at all the
points where the irreversibility occurs, in order to carry out the integration.
EXAMPLE 7.10
Determine the work mode entropy production when a measured 42.0 × 10 3 ft·lbf of work are used to compress 1.00 lbm of
air (an ideal gas) from 14.7 psia to 50.0 psia isothermally at 70.0°F in a closed system.
Solution
First, draw a sketch of the system (Figure 7.20).
Air
m = 1.00 lbm
p 1 = 14.7 psia
T 1 = 70.0°F
State 1
Isothermal process
1 W 2 = 42.0 × 103 ft lbf
State 2
Air
m = 1.00 lbm
p 2 = 50.0 psia
T 2 = T 1 = 70.0°F
(S P ) W = ?
FIGURE 7.20
Example 7.10.
The unknown is the work mode entropy production for this system. Here, W act = 42:0 × 10 3 ft.lbf. For the isothermal compression
of an ideal gas, Eq. (4.28) gives
Z 2
Z 2 dV
W rev = pdV = p 1 V 1
1
1 V
= p 1 V 1 ln V 2 /V 1 = p 1 V 1 lnðp 1 /p 2 Þ
(Continued )