Modern Engineering Thermodynamics

246 CHAPTER 7: Second Law of Thermodynamics and Entropy Transport and Production Mechanisms
37. Determine the work mode (a) entropy transport rate and
(b) entropy production rate as 100. hp is continuously dissipated
in a mechanical brake operating isothermally at 300.°F.
38.* Determine the work mode entropy production as a 300. kg steel
block slides 10.0 m down a 60.0° incline. The coefficient of
friction between the block and the incline is 0.100, and the bulk
mean temperature of the sliding surface is 50.0°C.
39. A mechanical gearbox at a uniform temperature of 160.°F
receives 150. hp at the input shaft and transmits 145 hp out the
output shaft. Determine its work mode (a) entropy transport
rate and (b) entropy production rate.
40.* Determine the work mode entropy production as 0.500 m 3
of air is compressed adiabatically from 200. kPa, 20.0°C to
0.100 m 3 in a piston-cylinder apparatus with a mechanical
efficiency of 85.0%. Assume constant specific heat ideal gas
behavior.
41. The velocity profile for the steady laminar flow of an
incompressible Newtonian fluid in a horizontal circular pipe
(Figure 7.27) is
V = V max ½1 − ðx/RÞ 2 Š
where V max is the centerline (x = 0) velocity, R is the pipe radius,
and x is the radial coordinate measured from the centerline.
a. Determine the position in this flow where the entropy
production rate per unit volume is a minimum.
b. Determine the position in this flow where the entropy
production rate per unit volume is a maximum.
c. Comment on how you can minimize the total entropy
production rate for this flow.
C L
FIGURE 7.27
Problem 41.
42. The viscous work entropy production rate per unit volume in
three-dimensional Cartesian coordinates is
( "
ðσ W Þ vis
= μ
2 ∂V x
T 3 ∂x − ∂V 2
y
+ ∂V y
∂y ∂y − ∂V #
2
z
+ ∂V z
∂z ∂z − ∂V 2
x
∂x
+ ∂V x
∂y + ∂V 2
y
+ ∂V y
∂x
R
x
∂z + ∂V 2
z
+ ∂V z
∂y
V m
)
∂x + ∂V 2
x
∂z
Show that this can be written as
(
ðσ W Þ vis = μ − 2
∂V x
T 3 ∂x + ∂V y
∂y + ∂V 2
z
∂z
"
∂V 2
#
+ 2 x
∂V 2 y
+ + ∂V 2
z
∂x ∂y ∂z
+ ∂V x
∂y + ∂V 2
y
+ ∂V y
∂x ∂z + ∂V )
2
z
+ ∂V z
∂y ∂x + ∂V 2
x
∂z
43. Show that the three-dimensional Cartesian coordinate viscous
work entropy production rate per unit volume given in Problem
42 reduces to the following for two-dimensional incompressible
flow:
ðσ W Þ vis = μ
∂V x
T ∂y + ∂V 2
y
∂x
Hint: For incompressible fluids, ∂V x
∂x + ∂V y
∂y + ∂V
z
∂z = 0:
44.* Determine the entropy production rate of a 10.0 × 10 3 Ω
electrical resistor that draws a constant 10.0 mA of current. The
temperature of the resistor is constant throughout its volume at
35.0°C.
Computer Problems
The following computer programming assignments are designed to
be carried out on any personal computer using a spreadsheet or
equation solver. They are meant to be exercises in the manipulation
of some of the basic formulae of this chapter. They may be used as
part of a weekly homework assignment.
45. Develop a program that allows you to input any temperature
(i.e., value plus unit symbol) in either the relative or absolute
Engineering English or SI units system at the keyboard and
converts this input into all the following temperatures and
outputs them to the screen:
°C, °F, R, and K.
46. Develop a program that computes the change in specific internal
energy, enthalpy, and entropy for an incompressible material.
Input the initial and final temperatures and pressures, the
specific heat, and the specific volume or density of the material.
Output u 1 − u 2 , h 2 − h 1 , and s 2 − s 1 to the screen along with
their proper units. Allow the choice of working in either the
Engineering English or the SI units system.
47. Develop a program that computes the change in specific internal
energy, enthalpy, and entropy for a constant specific heat ideal
gas. Input the initial and final temperatures and pressures or
specific volumes (allow the user the choice of which to input),
the specific heats and gas constant. Output u 2 − u 1 , h 2 − h 1 , and
s 2 − s 1 to the screen along with their proper units. Allow the
choice of working in either the Engineering English or the
SI units system.
48. Repeat Problem 47 except allow the user to choose a gas from a
menu. Have all the specific heats and gas constants for the gases
resident in your program.
49. Develop a program that outputs the heat production rate ð_S P Þ Q
of entropy due to steady state, one-dimensional thermal
conduction. Utilize Fourier’s law of conduction and input the
appropriate temperatures, thermal conductivity, cross-sectional
area, and length in the proper units. Allow the choice of
working in either the Engineering English or the SI units
system.
50. Develop a program that outputs the work mode entropy
production rate _S P due to the viscous dissipation in the
w
steady one-dimensional flow of a Newtonian fluid in a circular
pipe with the velocity profile given in Problem 41. Input the
fluid’s viscosity, density, and mass flow rate and the appropriate
pipe dimensions in proper units. Allow the choice of working in
either the Engineering English or the SI units system.