Modern Engineering Thermodynamics

316 CHAPTER 9: Second Law Open System Applications
+Z
Air
(101.3 kPa)
1
Dam
3
H = 21.49 m
Air
turbine
W T
Exhaust
water
pipe
4
80.77 m
Air entrained
in downflow
of water
2
Compressed air
Seperation
tank
Air bubbles rise to surface
FIGURE 9.34
Problem 66.
66.* Figure 9.34 is a schematic of a novel hydraulic air compressor
built in Michigan in 1906 at the Victoria Copper Mining
Company and operated until 1921. This plant used the power
of falling water from the Ontonagon River to produce highpressure
air with no moving mechanical parts. The falling
water compressed entrained air bubbles, which were then
separated from the water in a large underground separation
tank. The efficiency of this system is defined to be rate of
energy removed
η =
Rate of energy removed from the compressed air
Net hydraulic power of the water
= _m aðh 2 − h 3 Þ a
_m w ðgH/g c Þ
where the a subscript refers to the air and the w subscript
refers to the water. Using the data given in Table 9.5,
determine
a. The separation tank temperature, assuming the air and water
are in constant thermal equilibrium in an adiabatic
downflow.
b. The air turbine output power.
c. The entropy production rate for the compressor, assuming it is
completely adiabatic.
In your analysis, assume air to be a constant specific heat ideal gas
and water to be an incompressible liquid. Neglect any changes in
kinetic energy of the air and the water. Note that the air turbine
discharges to the atmosphere, so p 3 = p atm =101.3kPa.
67.* A bottle of beer is emptied adiabatically at a rate of 0.100 kg/s.
What is the rate of change of entropy of the contents of the beer
bottle if the properties of the beer are
h = 53:0 kJ/kg T = 10:0°C
p = 0:1013 MPa s = 1:000 kJ/ðkg ⋅ KÞ
and the entropy production rate is 0.0100 kJ/(s · K).
Table 9.5 Data for the Victoria Hydraulic Air
Compressor
η = 57:4%
p 2a = 882:6 kPa
_m a = 5:430 kg/s
_m w = 6119:1 kg/s
T 1a = T 1w = 20:0°C
H = 21:49 m
Source: Data from W. Rice, “Performance of Hydraulic Gas Compressors.”
ASME Journal of Fluids Engineering (December 1976), pp. 645–653.
68. Helium at 70.0°F enters a 3.00 ft 3 rigid tank that is filled to a
final pressure of 2200. psia. Assuming the tank is initially
evacuated and that the helium behaves as an ideal gas with
constant specific heats, determine the amount of entropy
produced when the tank is filled (a) adiabatically and
(b) isothermally at 70.0°F.
69. Plot the isothermal to adiabatic entropy production ratio for
filling an initially evacuated container with equal amounts of an
incompressible liquid vs. the dimensionless ratio pv/(cT) asthis
ratio ranges from 0.00 to 10.0.
70.* A0.0370m 3 hydraulic cylinder is filled with 30.0 kg of oil (an
incompressible liquid) from a supply at 20.0°C and 50.0 MPa.
The specific heat of the oil is 1.83 kJ/(kg · K). Determine
a. The entropy produced if the cylinder is filled isothermally at
20.0°C.
b. The final temperature and the entropy produced if the
cylinder is filled adiabatically.
71. Equation (9.47) shows that filling a rigid container adiabatically
with an incompressible liquid produces less entropy than filling
it isothermally. However, an adiabatic filling process produces a
temperature rise in the vessel (see Eq. (6.35)) such that T final =
T 2 > T 1 .
a. Develop a formula for the additional entropy produced
when the rigid container adiabatically filled with an
incompressible liquid is cooled back to the initial
temperature by submerging it into a large isothermal bath at
temperature T1.
b. Compare the total entropy produced by the adiabatic filling
plus the cooling process described in part a with that produced
during an isothermal filling process at temperature T 1 .
Design problems
The following are elementary open-ended design problems. The
objective is to carry out a preliminary thermal design as indicated.
A detailed design with working drawings is not expected unless
otherwise specified. These problems do not have specific answers, so
each student’s design is unique.
72. A liquid cavitates when the local pressure drops below the
saturation pressure and the liquid begins to vaporize, or boil.