Modern Engineering Thermodynamics

Problems 315
55.* Determine the maximum possible hot side exit temperature in a
vortex tube using air when the cold side temperature is 0.00°C,
the hot side mass flow fraction is 50.0%, the inlet pressure is
0.800 MPa, and both exits are at atmospheric pressure. Assume
constant, specific heat, ideal gas behavior and neglect any
changes in kinetic and potential energy.
56. Determine the heat transfer rate and the entropy production rate
for the steady state, steady flow of water flowing through a
straight horizontal 1.0 in. inside diameter pipe 10.0 ft long at a
rate of 5.00 ft 3 /min. The Darcy-Weisbach friction factor for this
flow is 0.0320. The flow is isothermal at 75.0°F, and the mass
density of the water is 62.26 lbm/ft 3 .
57.* Determine the entropy production rate and heat transfer in a
newly designed valve with 1.10 kg/s of an incompressible
hydraulic oil flowing through it. The minor loss coefficient of the
valve is 26.3, and the inlet and outlet oil temperatures are 53.0
and 49.0°C, respectively. The surface temperature of the valve is
constant at 46.0°C. The flow velocity through the valve is constant
at 3.00 m/s, and the specific heat of the oil is 1.13 kJ/(kg · K).
58.* A valve in an air handling system has a mass flow rate of
2.90 kg/min and a minor loss coefficient of 11.56. The valve inlet
temperature is 20.0°C and the valve is insulated. The inlet and
exit pressures of the valve are 1.66 and 0.300 MPa, respectively.
The air velocity through the valve is 2.70 m/s. Assuming air to
be an ideal gas with constant specific heats, determine the exit
temperature and entropy production rate of the valve.
59. Air enters a sudden contraction in a pipe at a rate of 0.300 lbm/s.
The contraction coefficient is 0.470, the exit temperature is 156°F
and the exit diameter is 2.00 in. The pipe is insulated. The
pressure across the contraction drops from 185 to 50.0 psia.
Determine the entrance temperature and the entropy production
rate of the contraction. Assume the air to be an ideal gas with
constant specific heats.
60.* Liquid mercury enters an insulated sudden expansion with a
velocity of 1.15 m/s at 20.0°C. The inlet and exit areas are
1.00 × 10 –3 and 0.00 × 10 –2 m 2 , respectively. Assuming the
flow is incompressible and adiabatic, determine the exit
temperature and the entropy production rate of the expansion.
The specific heat and density of the mercury are 0.1394 kJ/(kg · K)
and 13,579 kg/m 3 .
61.* It is required to dissipate 3.30 kJ/s of water flow energy in a
spillway with a hydraulic jump. An amount of 1000. kg/s of water
enters at 15.0°C and a depth of 0.500 m. The water passes through
the jump fast enough to be considered adiabatic. Determine
a. The required hydraulic jump depth, y 2 – y 1 (see Table 9.4).
b. The exit water temperature.
c. The entropy production rate of the hydraulic jump.
62.* Suppose 0.730 kg/s of oil at 1.20 MPa and 20.0°C enters a
system at 8.00 m/s and exits the system at 0.800 MPa, 40.0°C,
and 4.00 m/s. The exit is 4.00 m below the inlet. The specific
weight of the oil is 6000. N/m 3 , which is a constant throughout
thesystem.Thespecificheatoftheoilisconstantat1.21kJ/(kg· K).
Determine (a) the heat transfer rate and (b) the entropy production
rate of this system if its boundary temperature is maintained
constant at 40.0°C.
63. Determine the entropy production rate for the isothermal steady
laminar flow of a constant specific heat, incompressible power law
non-Newtonian fluid in a horizontal circular tube of radius R whose
velocity profile is given by
h
i
V = V m 1 − ðx/RÞ ðn+1Þ/n
where V m is the maximum (i.e., centerline) velocity, x is the radial
coordinate measured from the tube’s centerline, and n is the power
law exponent (a positive constant). Sketch a plot of _S p vs. n and
determine the value of n that minimizes _S p .
64. The steady laminar flow of a constant specific heat,
incompressible Newtonian fluid through a horizontal circular
tube of radius R has a velocity profile given by
V = V m 1 − ðx/RÞ 2
where V m is the maximum (i.e., centerline) velocity and x is the
radial coordinate measured from the tube’s centerline. If there is
a uniform heat flux _q s at the tube wall, then the temperature
profile within the fluid (neglecting axial conduction) is given by
T = T o + ð _q s R/k t Þ α − ðx/RÞ 2 + 0:25ðx/RÞ 4
(a)
where T o and α are constants, and k t is the thermal conductivity
of the fluid.
a. Combine the heat transfer and viscous work entropy
production rates per unit volume to show that the total
entropy production rate per unit volume for this system is
given by
σ = 4μV2 m x2
R 4 T
+ 1
_q s
2
2x/R − ðx/RÞ3
(b)
k t T
b. Comment on the integration of σ over the system volume.
Remember that the T in Eq. (b) is given by Eq. (a), and in a
polar cylindrical coordinate system, dV = 2πLx dx. Carry out
the integration analytically, if you can.
c. What factors in Eq. (b) can be manipulated to minimize _S P ?
65. The velocity and temperature profiles established in the free
convection of a fluid contained between two flat parallel vertical
walls maintained at different isothermal temperatures T 1 and
T 2 are
and
V = ρβgb 2 ðT 2 − T 1 Þðx/bÞ½ðx/bÞ 2 − 1Š/12μ
T = ðT 1 + T 2 Þ/2 − ðT 2 − T 1 Þðx/2bÞ
where x is measured from the centerline between the plates, β is
the coefficient of volume expansion, ρ is the density, and 2b is
the distance between the plates (Figure 9.33). Determine a
formula for the entropy production rate per unit depth and
length of this flow due to viscous effects.
FIGURE 9.33
Problem 65.
T 1 T
V
T 2
x
b
b