Modern Engineering Thermodynamics

8.3 Systems Undergoing Irreversible Processes 269
then
ðσ W
Þ vis
= μ T
dV 2 ω 2 R 4
=
1 μ
dx
ðR 2 2 − R2 1Þ 2 T
Equation (7.69) gives the entropy production rate for viscous effects as
_S P
W
vis
Z
=
ðσ W
V
Þ vis
dV
R 2 2
2
x + 1 2
For the differential volume element dV, we use the volume of an annulus of thickness dx, ordV = 2πLx dx (Figure 8.15).
Annular differential
volume
d = 2πLxdx
A
x
L
x + dx
FIGURE 8.15
Example 8.13, annular differential volume.
Putting these expressions for ðσ W Þ vis and dV into Eq. (7.70) and carrying out the integration gives
ðS P Þ Wvis
= 2πLω2 R 4 1 μ
ðR 2 2 − R2 1Þ 2 T 2R2 2 ln R 2
+ R4 2
R 1
2R 2 1
− R2 1
2
where μ = 0.700 N · s/m 2 , L = 0.100 m, and
ω = 1000: rev
min
2π rad
rev
R 1 = 0.0500 m, R 2 = 0.0510 m, and T = 20.0°C = 293.15 K.
Substituting these values into the preceding formula gives
1 min
60 s
_S P = 2:08 W/K
W
vis
= 104:7 rad/s
In this example, we have a reasonably high entropy production rate. This is due to the very small gap between the cylinders
and the high viscosity of the engine oil. Check for yourself to see that _S P
W-vis ! 0asR 2 becomes much larger than R 1 ,or
as μ → 0. Conversely, check to see that _S P
W-vis ! ∞ as R 1 ! R 2 or as μ → ∞. Some of these elements are explored in the
following exercises.
Exercises
23. Determine the entropy production rate _S P in Example 8.13 if the oil is changed from SAE-40 motor oil with a viscosity of
μ =0.700N·s/m 2 to SAE-10 motor oil with μ =0.150N·s/m 2 at 20.0°C. Keep the values of all the other variables the
same as they are in Example 8.13. Answer: _S P = 0:455 W=K:
W-vis
24. Recalculate the entropy production rate _S P in Example 8.13 when the bearing gap is reduced from 1.00 mm to 0.500 mm
by making R 1 = 0.0500 m and R 2 = 0.0505 m. Keep the values of all the other variables the same as they are in
Example 8.13. Answer: _S P = 4:13 W/K:
25. Determine the entropy production rate _S P in Example 8.13 when the shaft rotation is increased to 8000. rpm.
Keep the values of all the other variables the same as they are in Example 8.13. Plot the entropy
(Continued )