Modern Engineering Thermodynamics

276 CHAPTER 8: Second Law Closed System Applications
production rate of each light source and comment on which is
the most efficient.
35. Rework Example 8.11 by setting T b = T s . Using the formula for
T s given in the example, show that
dQ
T b
= −
hA dt
1 + T ∞ e αt /ðT 1 − T ∞ Þ
where α = hA/(mc v ). Integrate this from t=0.00 to 5.00 s and
combine it with the m(s 2 – s 1 )resultfromtheexampletoget
1(S P ) 2 under this condition. What is the significance of this result?
36. Integrate Eq. (7.52) to determine the entropy production and
use Eq. (7.77) to find the total entropy change for an aergonic
closed system in which the temperature increases from T 1 =
70.0°F toT 2 = 200.°F for the cases where the heat transfer varies
with the system absolute temperature according to the relations
a. Q = K 1 T (convection).
b. Q=K 2 T 4 (radiation), where K 1 = 3.00 Btu/R and K 2 = 3.00 ×
10 –4 Btu/R 4 . The system boundary is maintained isothermal
at 212°F.
37. Determine the entropy production rate due to conduction heat
transfer inside a system having a volume of 3.00 ft 3 and a
thermal conductivity of 105 Btu/(h·ft·R) that contains the
following temperature profile:
T = 300: × ½expðx/3:00ÞŠ
where T is in R and x is in feet.
38. Using Eq. (7.64), show that the entropy production rate per unit
volume due to heat transfer (σ Q ) is a constant if the temperature
distribution due to conduction heat transfer is given by
T = C 1 ½expðC 2 xÞŠ
where C 1 and C 2 are constants. (Hint: Use Fourier’s law of heat
conduction to eliminate the _q term.)
39. The temperature distribution due to conduction heat transfer
inside a flat plate with an internal heat generation (Figure 8.21)
is given by
T = T 0 + ðT s − T 0 Þðx/LÞ 2
where T s is the surface (x = L) and T 0 is the centerline
(x = 0) temperature. Determine a formula for the entropy
production rate ð _S p Þ for this system.
40. Example 8.13 deals with the velocity profile in a liquid
contained in the gap between two concentric cylinders of radii
R 1 and R 2 > R l in which the inner cylinder is rotating with an
angular velocity ω and the outer cylinder is stationary. If this gap
is very small, the velocity profile can be approximated by the
linear relation
V = R 1 ωðR 2 − xÞ/ðR 2 − R 1 Þ
where R 2 >> (R 2 – R 1 )andx is measured radially outward
from the center of the inner cylinder. Using the data given in
Example 8.13, determine the entropy production rate using this
simpler velocity profile and compare your answer to that given
in Example 8.13 for the more complex nonlinear velocity
profile.
41. Example 8.13 deals with the entropy production rate in the flow
between concentric rotating cylinders in which the outer cylinder
was stationary and the inner cylinder rotates with a constant
angular velocity ω. If, instead, we allow both cylinders to rotate
in the same direction with constant angular velocities ω 2 at the
outer cylinder and ω 1 at the inner one, then the velocity profile
in the gap between the cylinders becomes
V = ω 2 R 2 2 − ω
1R1 2 x − ð ω2 − ω 1 Þ R 2 1 2 R2 /x / R
2
2
− R1
2
Find the expression for the entropy production rate due to
viscous effects in the fluid of viscosity μ contained between
these rotating cylinders of radii R l and R 2 > R 1 and length L.
Assume the fluid is maintained isothermal at temperature T.
42.* A dipstick heater is an electrical resistance heater plugged into a
regular 110. V ac outlet and inserted into the dipstick tube of an
automobile engine. Its purpose is to keep the engine oil warm
during the winter when the car is not in use, thus allowing the
engine to start more easily. Determine the entropy produced
during an 8.00 h period by a 100. W steady state dipstick heater
whose surface is isothermal at 90.0°C.
43.* The potential difference across the tungsten filament operating
at 2400.°C in a cathode ray vacuum tube is 25.0 × 10 3 V. The
filament is a small disk 2.00 × 10 –3 m in diameter and 1.00 ×
10 –4 m thick having a resistivity of 6.00 × 10 –4 Ω ·m. Assuming
all the voltage drop occurs uniformly across the thickness of the
disk, determine its entropy production rate.
44. A current of 100. A is passed through a 6.00 ft long stainless
steel wire 0.100 inch in diameter. The electrical resistivity of
the wire is 197 × 10 –5 Ω·in, and its thermal conductivity is
12.5 Btu/(h·ft·R). The outer surface temperature of the wire
is maintained constant at 300.°F and the temperature profile
inside the wire is given by
T = T w + ρ e J 2 e ðR2 − x 2 Þ/ð4k t Þ
FIGURE 8.21
Problem 39.
T s
T 0
T s
L
x
L
where T w is the wall temperature of the wire, R is its radius, and
x is measured radially out from the center of the wire.
Determine the total entropy production rate within the wire due
to the flow of electricity through it. Assume all the physical
properties are independent of temperature.
45. Determine the entropy produced when 3.00 lbm of carbon
dioxide at 70.0°F and 30.0 psia are adiabatically mixed with
7.00 lbm of carbon dioxide at 100.°F and 15.0 psia. The final
mixture pressure is 17.0 psia. Assume the carbon dioxide
behaves as a constant specific heat ideal gas.