Modern Engineering Thermodynamics

7.15 Work Mode Production of Entropy 237
For the one-dimensional flow of a Newtonian fluid with viscosity μ, velocity distribution V = V(x), and local
internal absolute temperature T, we have
ðσ W Þ vis = μ
dV 2
T dx
Then, we can write
Entropy production due to fluid viscosity:
Z Z
μ
ðS P Þ
dV 2dt
W =
dV (7.71)
vis τ V T dx
and
Entropy production rate due to fluid viscosity:
Z
μ
dV 2dV
_S P W =
(7.72)
vis V T dx
To carry out these integrations, we need to know in advance the velocity distribution function V = V(x) and how
μ and V depend on the local internal absolute temperature T.
EXAMPLE 7.11
Determine the entropy production rate per unit volume due to the flow of a lubricating oil (μ = 0.10 N·s/m 2 ) at 30.°C ina
sliding bearing. Assume the lubricant is isothermal throughout its volume.
Solution
First, draw a sketch of the system (Figure 7.21).
Air
Oil
Bearing runner
1.00 m/s
x
V = x ×10 3 (in m/s when x is in meters)
FIGURE 7.21
Example 7.11.
The unknown is the entropy production rate per unit volume due to the flow of the lubricating oil. In this case, dV/dx =
1000 s −1 = constant, and since μ, T, and dV/dx are constant here, Eq. (7.72) can be integrated to give
and
_S P
W
vis
= μ
T
dV
dx
2V
_S P W-vis
σ W-vis = = μ
dV 2 ð0:10 N.s/m 2 Þ
=
V T dx ð30: + 273:15 KÞ 1000 s−1 2
N
= 330
m 2 . s.K = 0:33 kJ/ m3 .s.K
Exercises
19. Determine the entropy production rate per unit volume in Example 7.11 when bearing grease (μ = 30. N · s/m 2 ) is used
instead of lubricating oil. With the heavier lubricant, the bearing temperature is now 50.°C. Answer: σ W-vis =
93 kJ=ðm 3 .s.KÞ.
20. Determine the work mode entropy production due to viscosity in a system where the fluid has a constant velocity.
Answer: (S P ) W-vis = 0.
21. If a fluid flowing in a pipe has a constant entropy production rate per unit volume due to viscosity of 0.100 Btu/(ft 3·s·R)
and the volume of the fluid in the pipe is 0.475 ft 3 , determine the work mode production rate of entropy for this system.
Answer: (_S P ) W-vis = 0.0475 Btu/s·R.